Electrons
and
Chemical Bonding
Electrons and
Ch<
emicai boh
Harry B. Gray
Columbia University
1965
W. A. BENJAMIN, INC. New York Amsterdam
ELECTRONS AND CHEMICAL BONDING
Copyright © 1964 by W. A. Benjamin, Inc.
All rights reserved
Library of Congress Catalog Card Number 64'22275
Manufactured in the United States of America
The manuscript was put into production on January 16, 1964;
this volume was published on August 21, 1964; second
printing with corrections April 15, 1965
The publisher is -phased to acknowledge tlie assistance
of henore Stevens, who copyedited tlte manuscript, and
William Prokos, who produced the illustrations
and designed tlie dust jacket
W. A. BENJAMIN, INC.
New York, New York 10016
To my Students in Chemistry 10
Pref;
ace
"his book was developed from my lectures on chemical
_L bonding in Chemistry 10 at Columbia in the spring of 1962,
and is mainly intended for the undergraduate student in chemistry
who desires an introduction to the modern theories of chemical
bonding. The material is designed for a one-semester course in
bonding, but it may have greater use as a supplementary text in
the undergraduate chemistry curriculum.
The book starts with a discussion of atomic structure and
proceeds to the principal subject of chemical bonding. The
material in the first chapter is necessarily quite condensed and is
intended as a review. (For more details, the student is referred
to R. M. Hochstrasser, Behavior of Electrons in Atoms, Benjamin,
New York, 1964).
Each chapter in the bonding discussion is devoted to an impor-
tant family of molecules. Chapters II through VII take up, in
order, the principal molecular structures encountered as one pro-
ceeds from hydrogen through the second row of the periodic table.
Thus, this part of the book discusses bonding in diatomic, linear
triatomic, trigonal planar, tetrahedral, trigonal pyramidal, and
angular triatomic molecules. Chapters VIII and IX present an
introduction to modern ideas of bonding in organic molecules and
transition metal complexes. Throughout, our artist has used
small dots in drawing the boundary-surface pictures of orbitals.
The dots are intended only to give a pleasing three-dimensional
effect. Our drawings are not intended to be charge-cloud pictures.
Charge-cloud pictures attempt to show the electronic charge
density in an orbital as a function of the distance from the nucleus
by varying the "dot concentration."
vii
viii Preface
The discussion of atomic structure does not start with the
Schrodinger equation, but with the Bohr theory. I believe most
students appreciate the opportunity of learning the development
of atomic theory in this century and can make the transition
from orbits to orbitals without much difficulty. The student can
also calculate several important physical quantities from the
simple Bohr theory. At the end of the first chapter, there is a
discussion of atomic-term symbols in the Russell-Saunders
LSMjMs approximation.
In this book the molecular orbital theory is used to describe
bonding in molecules. Where appropriate, the general molecular
orbitals are compared with valence-bond and crystal-field descrip-
tions. I have written this book for students who have had no
training in group theory. Although symmetry principles are
used throughout in the molecular orbital treatment, the formal
group-theoretical methods are not employed, and only in Chapter
IX are group-theoretical symbols used. Professor Carl Ballhausen
and I are publishing an introductory Jecture-note volume on
molecular orbital theory, which was written at a slightly higher
level than the present book. The lecture notes emphasize the
application of group theory to electronic structural problems.
The present material includes problems integrated in the text;
most of these are accompanied by the worked-out solutions.
There are also a substantial number of problems and questions
at the end of each chapter.
It is a great pleasure to acknowledge the unfailing support
encouragement, and devotion of the seventy-seven fellows who
took the Columbia College course called Chemistry 10 in the
spring of 1962. I doubt if I shall ever have the privilege of working
with a finer group. The class notes, written by Stephen Steinig
and Robert Price, were of considerable help to me in preparing
the first draft.
I would like to thank Professors Ralph G. Pearson, John D.
Roberts, and Arlen Viste for reading the manuscript and offering
many helpful suggestions. Particularly I wish to thank one of my
students, James HaJper, who critically read the manuscript in
every draft. Finally 7 , a large vote of thanks goes to Diane Celeste,
Contents
Preface v
1 Electrons In Atoms 1
1-1 Introductory Remarks 1
1-2 Bohr Theory of the Hydrogen Atom (1.913) 1
1-3 The Spectrum of the Hydrogen Atom 5
1-4 The Need to Modify the Bohr Theory 9
1-5 Electron Waves 9
1-6 The Uncertainty Principle 11
1-7 The Wave Function 12
1-8 The Schrodiriger Wave Equation 13
1-9 The Normalization Constant 13
1-10 The Radial Part of the Wave Function 13
1-11 The Angular Part of the Wave Function 14
1-12 Orbitals 14
1-13 Electron Spin 17
1-14 The Theory of Many-Electron Atoms 20
1-15 Russell-Saunders Terms 22
1-16 Ionization Potentials 27
1-17 Electron Affinities 33
II Diatomic Molecules 36
2-1 Covalent Bonding 36
2-2 Molecular-Orbital Theory 38
xi
xii Contents
2-3 Bonding and Antibonding Molecular Orbitals 39
2-4 Molecular-Orbital Energy Levels 42
2-5 The Hydrogen Molecule 46
2-6 Bond Lengths of H 2 + and H 2 47
2-7 Bond Energies of H 2 + and H 2 47
2-8 Properties of H 2 + and H 2 in a Magnetic Field 48
2-9 Second-Row Homonuclear Diatomic Molecules 49
2-10 Other A 2 Molecules 58
2-11 Term Symbols for Linear Molecules 60
2-12 Heteronuclear Diatomic Molecules 62
2-13 Molecular-Orbital Energy-Level Scheme for LiH 67
2-14 Ground State of LiH 68
2-15 Dipole Moments 69
2-16 Electronegativity 69
2-17 Ionic Bonding 73
2-18 Simple Ionic Model for the Alkali Halides 75
2-19 General AB Molecules 78
III Linear Triatomic Molecules 87
3-1 BeH 2 87
3-2 Energy Levels for BeH 2 89
3-3 Valence-Bond Theory for BeH 2 93
3-4 Linear Triatomic Molecules with t Bonding 95
3-5 Bond Properties of C0 2 100
3-6 Ionic Triatomic Molecules: The Alkaline Earth
Halides 101
IV Trigonal -Planar Molecules 106
4-1 BF 3 106
4-2 a Molecular Orbitals 106
4-3 ir Molecular Orbitals 109
4-4 Energy Levels for BF 3 HI
4-5 Equivalence of <r x and a y Orbitals 112
4-6 Ground State of BF 3 114
4-7 Valence Bonds for BF 3 115
4-8 Other Trigonal-Planar Molecules 117
Contents xiil
V Tetrahedral Molecules 120
5-1 CH 4 120
5-2 Ground State of CH 4 122
5-3 The Tetrahedral Angle 122
5-4 Valence Bonds for CH 4 125
5-5 Other Tetrahedral Molecules 127
VI Trigonal-Pyramidal Molecules 129
6-1 NH» 129
6-2 Overlap in a x , a y , and <r 2 130
6-3 The Interelectronic Repulsions and H — N — H
Bond Angle in NH 3 135
6-4 Bond Angles of Other Trigonal-Pyramidal
Molecules 137
6-5 Ground State of NH 3 138
VII Angular Triatomic Molecules 141
7-1 H 2 141
7-2 Ground State of H 2 143
7-3 Angular Triatomic Molecules with tt Bonding:
NO, 148
7-4 <r Orbitals 148
7-5 7T Orbitals 148
7-6 Ground State of N0 2 152
VIII Bonding in Organic Molecules 155
8-1 Introduction 155
8-2 C 2 H 4 156
8-3 Energy Levels in G 2 H 4 159
8-4 Ground State of C 2 H 4 159
8-5 Bent-Bond Picture of C 2 H 4 160
8-6 Bond Properties of the C=C Group 162
8-7 The Value of (3 CC in C 2 H 4 164
8-8 H 2 CO 164
8-9 Ground State of H 2 CO 165
xiv Contents
8-10 the n — > -w* Transition Exhibited by the
Carbonyl Group 167
8-11 C 2 II 2 167
8-12 Ground State of G 2 H 2 168
8-13 CHgCN 168
8-14 C 6 H 6 170
8-15 Molecular-Orbital Energies in C 6 H 6 171
8-16 Ground State of C 6 H 6 173
8-17 Resonance Energy in C 6 il 6 173
IX Bonds Involving d Valence Orbitals 176
9-1 Introduction 176
9-2 The Octahedral Complex Ti(H 2 0) 6 3 + 176
9-3 Energy Levels in Ti(H 2 0) 6 3+ 179
9-4 Ground State of Ti(H 2 0) 6 3+ 181
9-5 The Electronic Spectrum of Ti(H 2 0) 6 3+ 183
9-6 Valence-Bond Theory for Ti(H 2 0) 6 3 + 184
9-7 Crystal-Field Theory for Ti(H 2 0) 6 3+ 186
9-8 Relationship of the General Molecular-Orbital
Treatment to the Valence-Bond and Crystal-
Field Theories 187
9-9 Types of ?r Bonding in Metal Complexes 188
9-10 Square-Planar Complexes 189
9-11 Tetrahedral Complexes 194
9-12 The Value of A 197
9-13 The Magnetic Properties of Complexes: Weak-
and Strong-Field Ligands 200
9-14 The Electronic Spectra of Octahedral Complexes 201
Suggested Heading 212
Appendix : Atomic Orbital Ionization Energies 217
Index 219
Physical Constants"
Planck's constant, h
Velocity of light, c = 2.997925 X 10 w cm sec" 1
Electron rest mass, m e = 9.1091 X 10" 2S g
Electronic charge, e = 4.80298 X 10" 10 esu (cm 3 ' 2 sec" 1 )
Bohr radius, « = 0.529167 A
Avogadro's number, N = 6.0247 X 10 23 mole" 1 (physical scale)
Conversion Factors
Energy
1 electron volt (eV) = 8066 cm" 1 = 23.069 kcal mole" 1
1 atomic unit (au) = 27.21 eV = 4.3592 X 10" 11 ergs
= 2.1947 X 10 6 cm" 1 = 627.71 kcal mole" 1
Length
1 Angstrom (A) = 10" 8 cm
"Values recommended by the National Bureau of Standards; see J. Chan.
Edac, 40, 642 (1963).
XV
I
Electrons in Atoms
1-1 INTRODUCTORY REMARKS
The main purpose of this book is the discussion of bonding in
several important classes of molecules. Before starting this dis-
cussion, we shall review briefly the pertinent details of atomic struc-
ture. Since in our opinion the modern theories of atomic structure
began with the ideas of Niels Bohr, we start with the Bohr theory
of the hydrogen atom.
1-2 BOHR THEORY OF THE HYDROGEN ATOM (1913)
Bohr pictured the electron in a hydrogen atom moving in a circular
orbit about the proton (see Fig. 1-1). Note that in Fig. 1-1, m e rep-
resents the mass of the electron, m n the mass of the nucleus, r the
radius of the circular orbit, and v the linear velocity of the electron.
For a stable orbit, the following condition must be met: the cen-
trifugal force exerted by the moving electron must equal the com-
bined forces of attraction between the nucleus and the electron :
centrifugal force = — — (1~1)
o r
There are two attractive forces tending to keep the electron in orbit :
the electric force of attraction between the proton and the electron,
Electrons and Chemical Bonding
Figure 1-1 Bohr's picture of the hydrogen, atom.
and the gravitational force of attraction. Of these, the electric force
greatly predominates and we may neglect the gravitational force:
electric force of attraction = -
(1-2)
Equating (1-1) and (1-2), we have the condition for a stable orbit,
which is
^ = ^ (1-3)
r r
We are now able to calculate the energy of an electron moving in
one of the Bohr orbits. The total energy is the sum of the kinetic
energy T and the potential energy V; thus
E= T+ V
where T is the energy due to motion
T = !«2 e V 2
(1-4)
(1-5)
Electrons iw Atoms 7
and V is the energy due to electric attraction.
XT A _g2
_ J ? dr= — (1 " 6)
Thus the total energy is
E = %m e v 2 — e 2 /r (1-7)
However, the condition for a stable orbit is
n e v 2 e 1 e 2 , .
= -z or m e v 2 = - (1-8)
Thus, substituting for nieV 1 in Eq. (1-7), we have
* = ¥-*=-¥ (1-9)
2r r ir
Now we need only specify the orbit radius r and we can calculate the
energy. According to Eq. (1-9), all energies are allowed from zero
(j = oo) to infinity (r = 0).
At this point Bohr made a novel assumption — that the angular
momentum of the system, equal to m e vr, can only have certain discrete values,
or quanta. The result is that only certain electron orbits are allowed.
According to the theory, the quantum unit of angular momentum is
h/2ir (h is a constant, named after Max Planck, which-is defined on
page 5). Thus, in mathematical terms, Bohr s assumption was
m e vr = n[ —J (1-10)
with n = 1, 2, 3 • • • (all integers to °°). Solving for v in Eq.
(1-10), we have
v = «(A) -L (1-11)
\2ir/ m e r
Substituting the value of v from Eq. (1-11) in the condition for a
stable orbit [Eq. (1-8)], we obtain
m l m e -r r
'hr i m e e i
Electrons and Chemical Bonding
¥ (1-13)
Equation (1-13) gives the radius of the allowable electron orbits for
the hydrogen atom in terms of the quantum number, n. The energy
associated with each allowable orbit may now be calculated by sub-
stituting the value of r from Eq. (1-13) in the energy expression
[Eq. (1-9)], giving
2tHW 4 n l£ ,
E ~~ ~^¥~ (1 ~ 14)
PROBLEMS
1-1. Calculate the radius of the first Bohr orbit.
Solution. The radius of the first Bohr orbit may be obtained directly
from Eq. (1-13)
n 2 h 2
Substituting n = 1 and the values of the constants, we obtain
(1) 2 ( 6.6238 X 10" 27 erg-sec) 2
4(3. 1416) 2 (9. 1072 X 1CT 28 g)(4.8022 X 1CT 10 abs esu) 2
= 0.529 X 10~ 8 cm = 0.529 A
The Bohr radius for n = 1 is designated a .
1-2 . Calculate the velocity of an electron in the first Bohr orbit
of the hydrogen atom.
Solution. From Eq. (1-11),
(h)l
v = n\ — I —
\1t!lm e r
Substituting » = 1 and r = ao = 0.529 X 10~ 8 cm, we obtain
_ (6.6238 X 10" 27 erg-sec)
_37l7__
1
X
(9.1072 X 10- 28 g)(0.529 X 10" 8 cm)
2.188 X 10 8 cm sec" 1
Electrons in Atoms 5
1-3 THE SPECTRUM OF THE HYDROGEN ATOM
The most stable state of an atom has the lowest energy and
this is called the ground state. From Eq. (1-14) it is clear that the
most stable electronic state of the hydrogen atom occurs when « = 1.
States that have n > 1 are less stable than the ground state and
understandably are called excited states. The electron in the hydrogen
atom may jump from the n = 1 level to another n level if the correct
amount of energy is supplied. If the energy supplied is light energy,
light is absorbed by the atom at the light frequency exactly equiva-
lent to the energy required to perform the quantum jump. On the
other hand, light is emitted if an electron falls back from a higher n
level to the ground-state (n = 1) level.
The light absorbed or emitted at certain characteristic frequencies
as a result of the electron changing orbits may be captured as a series
of lines on a photographic plate. The lines resulting from light ab-
sorption constitute an absorption spectrum, and the lines resulting from
emission constitute an emission spectrum. The frequency v of light
absorbed or emitted is related to energy E by the equation deduced
by Planck and Einstein,
E = hv (1-15)
where h is called Planck's constant and is equal to 6.625 X 10~ 27
erg-sec.
It was known a long time before the Bohr theory that the positions
of the emission lines in the spectrum of the hydrogen atom could be
described by a very simple equation
*n = Ru(~ - A) (H6)
where n and m are integers, and where Ru is a constant, called the
Rydberg constant after the man who first discovered the empirical cor-
relation.
This equation can be obtained directly from the Bohr theory as
follows: The transition energy (-Eh) of any electron jump in the
hydrogen atom is the energy difference between an initial state I and
a final state II. That is,
£h = £n - £1 OH)
6
I
electrons ai
or, from Eq. (1-14),
£11 =
\ ni 2 h 2 ,
27rW 4 / 1
1 )
h 2 V«i 2
«II 2 /
(1-18)
(1-19)
Replacing £ H with its equivalent frequency of light from Eq. (1-15),
we have
27T 2 »2 e g 4 / 1 1 \ , .
„ n = — — i (1-20;
/) d \«r «ir/
Equation (1-20) is equivalent to the experimental result, Eq. (1-16),
with »i = n, nn = m, and Rh = (2ir' l m e e f )/h' i . Using the value of
9.1085 X 10 -28 g for the rest mass of the electron, the Bohr-theory
value of the Rydberg constant is
2ttW_ 2(3- 1416) 2 (9. 1085 X 10" 28 )(4.8029 X lO' 10 ) 4
Kit rs —
> 3 (6.6252 X 10- 27 ) 3
= 3.2898 X 10 15 cycles/sec (1-21)
It is common practice to express Rh in wave numbers v rather than
in frequency. Wave numbers and frequency are related by the equa-
tion
v = cv (1-22)
where c is the velocity of light. Thus
v 3.2898 X 10 15 cycles/sec ,„.,-_ _ t ri n ^
Rh = —- , nmn < TT7^~1 = 1Q 9,737 cm 1 (1-23)
2.9979 X 10 10 cm/ sec
The accurately known experimental value of Rh is 109,677-581 cm -1 .
This remarkable agreement of theory and experiment was a great
triumph for the Bohr theory.
PROBLEMS
1-3. Calculate the ionization potential of the hydrogen atom.
Solution . The ionization potential (IP) of an atom or molecule is the
energy needed to completely remove an electron from the atom or
molecule in its ground state, forming a positive ion. For the hy-
drogen atom, the process is
Electrons in Atoms
H-*H+ + e
E = IP
We may start with Eq. (1-19),
27r 2 ff2 e e 4
P —
/I 1
hli P
\«i 2 nu"
For the ground state, m — 1; for the state in which the electron is
completely removed from the atom, »n = m . Thus,
Recall that
and therefore
2ttW
£ 2
P-
4it 2 m e J
1 2.it l m e e l
Tan V 1
Then
e 1 (4.8022 X lCr 10 abs esu) 2
IP = — = - — 7 — — -—- = 2.179 X 1CT U erg
2a 2(0.529 X KT 8 cm) s
Ionization potentials are usually expressed in electron volts. Since
1 erg = 6.2419 X 10 11 eV, we calculate
IP = 2.179 X 10- 11 erg = 13.60 eV
The experimental value of the IP of the hydrogen atom is 13-595 eV.
1-4. Calculate the third ionization potential of the lithium atom.
Solution. The lithium atom is composed of a nucleus of charge
+3(2 = 3) and three electrons. The first ionization potential I Pi
of an atom with more than one electron is the energy required to
remove one electron; for lithium,
Li ->■ Li+ E = IPi
The energy needed to remove an electron from the unipositive ion
Li + is defined as the second ionization potential IP2 of lithium,
Li+ -> Li 2 + E = IP,
and the third ionization potential IP 3 of lithium is therefore the
energy required to remove the one remaining electron in Li 2+ .
Electrons and Chemical Bonding
The problem of one electron moving around a nucleus of charge
+3 (or +.Z) is very similar to the hydrogen atom problem. Since
the attractive force is -Ze 2 /V 2 , the condition for a stable orbit is
meV 2 Ze 1
r r 1
Carrying this condition through as in the hydrogen atom case and
again making the quantum assumption
(h
\2ir,
we find
n 2 h 2
4ir 2 m e Ze 2
and
_ 27r 2 tfz e ZV
n 2 h 2
Thus Eq. (1-19) gives, for the general case of nuclear charge Z,
„ l^meZVf 1 1
h 2 \ki 2 nit
or simply E = Z 2 E H - For lithium, Z = 3 and IP 3 = (3) 2 (2.179 X
10" 11 erg) = 1.961 X 10~ 10 erg = 122.4 eV.
1-5 . The Lyman series of emission spectral lines arises from tran-
sitions in which the excited electron falls back into the n = 1 level.
Calculate the quantum number n of the initial state for the Lyman
line that has v = 97,492.208 car 1 .
Solution. We use Eq. (1-20)
liflm^ ( 1 1
"h = — ,-z — I — ; ;
in which «n is the quantum number of the initial state for an
emission line, and m = 1 for the Lyman series. Using the experi-
mental value
Rh = --^ = 109, 677. 581 cm- 1
err
Electrons in Atoms
we have
97,492.208 = 109,677.5811 1 -I or «„ = 3
1-4 THE NEED TO MODIFY THE BOHR THEORY
The idea of an electron circling the nucleus in a well-defined orbit
— just as the moon circles the earth — was easy to grasp, and Bohr's
theory gained wide acceptance. Little by little, however, it was
realized that this simple theory was not the final answer. One diffi-
culty was the fact that an atom in a magnetic field has a more compli-
cated emission spectrum than the same atom in the absence of a
magnetic field. This phenomenon is known as the Zeeman effect and
is not explicable by the simple Bohr theory. However, the German
physicist Sommerfeld was able to temporarily rescue the simple
theory by suggesting elliptical orbits in addition to circular orbits
for the electron. The combined Bohr- Sommerfeld theory explained the
Zeeman effect very nicely.
More serious was the inability of even the Bohr-Sommerfeld
theory to account for the spectral details of the atoms that have sev-
eral electrons. But these were the 1920s and theoretical physics was
enjoying its greatest period. Soon the ideas of de Broglie, Schro-
dinger, and Heisenberg would put atomic theory on a sound founda-
tion.
1-5 ELECTRON WAVES
In 1924, the French physicist Louis de Broglie suggested that elec-
trons travel in waves, analogous to light waves. The smallest units
of light (light quanta) are called -photons. The mass of a photon is
given by the Einstein equation of mass-energy equivalence
E = mc 1 (1-24)
Recall from Eq. (1-15) that the energy and frequency of light are re-
lated by the expression
E = hv (1-25)
io Electrons and Chemical Bonding
Combining Eq. (1-24) and Eq. (1-25), we have
m = ~ (1-26)
The momentum p of a photon is
p = mv = ?»c (1-27)
Substituting the mass of a photon from Eq. (1-26), we have
t = "f d-28)
Since frequency v, wavelength X, and velocity v are related by the
expression
X = - (1-29)
find
X=^ (1-30)
Equation (1-30) gives the wavelength of the light waves or elec-
tron waves. For an electron traveling in a circular Bohr orbit, there
must be an integral number of wavelengths in order to have a stand-
ing wave (see Fig. 1-2), or
nk = 2ti" (1-31)
Substituting for X from Eq. (1-30), we have
« { — I = l-K'f
V
or
' h
. = rp = angular momentum (1-32)
27T/
Thus de Broglie ivaves can be used to explain Bohr's novel postulate
[Eq. (1-10)].
Electrons in Atoms
ii
Figure 1-2 A standing electron wave with n = 5.
1-6 THE UNCERTAINTY PRINCIPLE
In 1927, Davisson and Germer demonstrated that electrons are
diffracted by crystals in a manner similar to the diffraction of X rays.
These electron-diffraction experiments substantiated de Broglie's
suggestion that an electron has wave properties such as wavelength,
frequency, phase, and interference. In seemingly direct contradic-
tion, however, certain other experiments, particularly those of
J. J. Thomson, showed that an electron is a particle with mass,
energy, and momentum.
As an attempt at an explanation of the above situation, Bohr put
forward the principle of complementarity , in which he postulated that
12. Electrons and Chemical Bonding
an electron cannot exhibit both wave and particle properties simul-
taneously, but that these properties are in fact complementary de-
scriptions of the behavior of electrons.
A consequence of the apparently dual nature of an electron is the
uncertainty principle, developed by Werner Heisenberg. The essential
idea of the uncertainty principle is that it is impossible to specify at
any given moment both the position and the momentum of an elec-
tron. The lower limit of this uncertainty is Planck's constant
divided by 4ir. In equational form,
(A^)(Ax) > A (i_ 33 )
Here Ap x is the uncertainty in the momentum and Ax is the uncer-
tainty in the position. Thus, at any instant, the more accurately it
is possible to measure the momentum of an electron, the more un-
certain the exact position becomes. The uncertainty principle means
that we cannot think of an electron as traveling around from point
to point, with a certain momentum at each point. Rather we are
forced to consider the electron as having only a certain probability
of being found at each fixed point in space. We must also realize that
it is not possible to measure simultaneously, and to any desired accu-
racy, the physical quantities that would allow us to decide whether
the electron is a particle or a wave. We thus carry forth the idea that
the electron is both a particle and a wave.
1-7 THE WAVE FUNCTION
Since an electron has wave properties, it is described as a wave func-
tion, ip or &C x >y,Z), the latter meaning that ^ is a function of coordi-
nates x,y, and Z- The wave function can take on positive, negative,
or imaginary values. The probability of finding an electron in any
volume element in space is proportional to the square of the absolute
value of the wave function, integrated over that volume of space.
This is the physical significance of the wave function. Measurements
we make of electronic charge density, then, should be related to \ip\ 2 ,
not \j/. Expressed as an equation, we have
probability (x,y,z) ^ I^C^iOl 2 (l~34)
Electrons in Atoms 13
By way of further explanation, it should be noted that the prob-
ability of finding an electron in any volume element must be real and
positive, and \^/\ 2 always satisfies this requirement.
1-8 THE SCHRODINGER WAVE EQUATION
In 1926, the Austrian physicist Erwin Schrodinger presented the
equation relating the energy of a system to the wave motion. The
Schrodinger equation is commonly written in the form
3ty = E4> (1-35)
where 3C is an operator called the Hamiltonian operator (after the
English physicist Hamilton) and represents the general form of the
kinetic and potential energies of the system; E is the numerical value
of the energy for any particular \p. The wave functions that give
solutions to Eq. (1-35) are called eigenjunctions; the energies E that
result from the solutions are called eigenvalues.
The Schrodinger equation is a complicated differential equation
and is capable of exact solution only for very simple systems. Fortu-
nately, one of these systems is the hydrogen atom.
The solution of the Schrodinger equation for the hydrogen atom
yields wave functions of the general form
*„«», = [N] [K nl (r)] [<S K faA] (1-36)
We shall now attempt to explain the parts of Eq. (1-36).
1-9 THE NORMALIZATION CONSTANT
In Eq. (1-36), N is a normalisation constant, fixed so that
X" X" X" '*' 2 dx & dz=1 ( 1_37 )
That is, the probability of finding the electron somewhere in space
must be unity.
1-10 THE RADIAL PART OF THE WAVE FUNCTION
RniC r ~) is tne radial part of the wave function. The value of
jR n ;(f) | 2 gives the probability of finding the electron any distance r
i a Electrons and Chemical Bonding
from the nucleus. The two quantum numbers n and / are associated
with the radial part of the wave function: n is called the principal
quantum number and defines the mean radius for the electron; 4/ nlmi can
only be an eigenfunction for n = 1, 2, 3 . . . integers. / is the quantum
number which specifies the angular momentum of the electron; \j/ n i mi
can only be an eigenfunction for /= 0, 1, 2, 3, • • • to « — 1.
1-11 THE ANGULAR PART OF THE WAVE FUNCTION
^im, Qx/r,y/r, z/r) is the angular part of the wave function. The
quantum numbers / and mi are associated with the angular part of
the wave function, mi is called the magnetic quantum number and de-
fines the possible values for the ^-axis component of the angular
momentum of the electron in a magnetic field. ^ n im, can only be
an eigenfunction for mi = +/, / — 1, / — 2, . . . to — /.
1-12 ORBITALS
The hydrogen eigenfunctions yp n im l are commonly called orbitals.
The orbitals for the hydrogen atom are classified according to their
angular distribution, or / value. Each different / value is assigned a
letter:
/ = is an j- orbital.
/ = 1 is a p orbital.
/ = lis a. d orbital.
/ = 3 is an/ orbital.
The letters s, p, d, and / are taken from spectroscopic notation. For
/ = 4 or more, alphabetical order is followed, omitting only the
letter j. Thus, / = 4 is a g orbital, / = 5 is an h orbital, etc.
An orbital is completely specified in this shorthand notation by
adding the n and mi values. The n value goes in front of the letter for
the / value. The mi value is indicated as a subscript, the total short-
hand being nl mi . Now for mi ^ 0, the nl mi orbitals are imaginary
functions. It is usually more convenient to deal with an equivalent
set of real functions, which are linear combinations of these nl mi
functions. The shorthand for the real hydrogen orbitals is again nl;
the added subscript now gives the angular dependency. The com-
plete set of real orbitals for hydrogen through n — 3 is given in
Table 1-1.
Electrons in Atoms 15
Table 1-1
Important Orbitals for the Hydrogen Atom 8
Orbital quantum Orbital Angular function,'
numbers desig- Radial function , b ^ (x j> z_\
n I mi nation R n i(r) lm i\r'r'r)
1
Is
2
2s
2
1
(l) d
2/>*
2
1
*Pz
2
i
(-l) d
2Pv
2e
3s
1
iV^ 2 ^ 2 T7f
1 r , V5~ (x/r)
uE re ~ k ~tt?~
1 _ r V3 (z/r)
2VS re /2 ~^W~
2 V6 2 VF
g^=(27-18r + 2r^-V3 ^
3 1 (l) d 3p x -* {r 2_ 6r)e -r A
4 , , „ , - r , ^3 (x/r)
3 10 3^ _^_ (r2 _ 6r)e -V3 ^i
3 1 (-1,- 3 Py -^^ r)e -V, ^M
3 2 (2> d *,._,, 3^^-/3 MI^^J
•J 9 /,»<! <w 4 2 r/ V30 (B/r')
3 2 (1) 3rf « 8l7S0 re 27fT~
3 2 3d -^ % V5[(3^-^) A1]
3 2 ° 3d * 2 81730 r e 4 V?
3 2 ( 1} id yz 8lVS0 r e 2V2¥
3 2(2) 3d xy 8 iVM r 6 2VF
a Both the radial and the angular functions are normalized to one; r is in atomic
units (that is, in units of a Q ; see problem 1-1).
^To convert to a general radial function for a one- electron atom with any nuclear
charge Z, replace r by Zr and multiply each function by (Z) ^ '.
c Often expressed in the spherical coordinates and by replacing x with r sin 8 x
cos <p t y with r sin B sin 0, and z with r cos 0.
d It is not correct to assign m; values to the real functions #, ji, * 2 — y 1 , xz, yz,
and xy.
i6
Electrons and Chemical Bonding
It is common practice to make drawings of the hydrogen orbitals,
outlining the region within which there is a large probability for
finding the electron. Remember that the electronic density in an
orbital is related to the square of the absolute value of the wave function.
Keep this in mind when you encounter dual-purpose drawings of the
boundary surfaces of orbitals, which outline 90 per cent, say, of jt^l 2 ,
and also indicate the + and — signs on the lobes given by the angular
part of ip. The boundary-surface pictures are very useful and should
be memorized. The boundary surfaces for j-, p, d, and / orbitals are
given in Figs. 1-3, 1-4, 1-5, and 1-6, along with radial-distribution
graphs for the different orbitals.
/
/
W
■ orbital
R(r)
(b)
Figure 1-3 (a) Boundary surface of an s orbital, (b) Plots
of the radial function R(r) vs. r for Is, 2s, and 3s orbitals. The
2s radial function changes sign as r increases. Thus there is a
point where R(r) = for the 2s radial function. Such a zero
point is called a node. The 3s radial function has two nodes.
Electrons in Atoms
p y orbital
s
/
p x orbital
w
orbital
Figure 1-4 (o) Boundary surfaces of the p orbitals. (b) Plots
of the radial function R(r) vs. r for 2p and 3p orhitals. The 3p
orbital has one node, as indicated.
1-13 ELECTRON SPIN
The three quantum numbers n, I, and mi ate all associated with the
movement of the electron around the nucleus of the hydrogen atom.
In order to explain certain precise spectral observations, Goudsmit
i8
Electrons and Chemical Bonding
>.y
+
0)
m
MB
d Iy orbital
X,..
^ orbital
</„ orbital
<>a
. ~.
H0
. ,•'•*.■. *:■•'.*
■ ' *y
' +
\',T"'
rf„ ? orbital
:•**.-:/
*#;■:..
/ + """
<f . . orbital
(*)
R(r)
Figure 1-5 (a) Boundary surfaces of the d orbitals. (b) Plot
of R(r) vs. r for a 3d orbital.
Electrons in Atoms
19
<■-
i
>*
• 1
-J..M
- -('
! '*"
1
•
»!
? V+'
4: ,.■,(•? !
■^
•»;:
/I*.
**
-■a
:--K' J '"
-'it'
*w
w
W
Figure 1-6 (a) Boundary surfaces of the / orbitals. (b) Plot
of R(r) vs. r for a 4/ orbital.
zo Electrons and Chemical Bonding
and Uhienbeck (1925) introduced the idea of electron spin (this is
analogous to the earth spinning about its own axis while moving in
an orbit around the sun). The spin of an electron is quantized in
half-integer units, and two more quantum numbers, s and m s , are
added to our collection: s is called the spin quantum number and equals
\; m s is related to s in the same way that mi is related to / and equals
±i
1~14 THE THEORY OF MANY-ELECTRON ATOMS
It has not been possible to solve the Schrodinger equation exactly
for atoms with two or more electrons. Although the orbitals for a
many-electron atom are not quite the same as the hydrogen orbitals,
we do expect the number of orbitals and their angular dependencies
to be the same. Thus the hydrogen orbitals are used to describe the
electronic structure of an atom with more than one electron. The
procedure is simply to assign to each electron in the atom a set of the
four quantum numbers n, I, mi, and m s (j is always |), remembering
that no two electrons can have the same jour quantum numbers. This is a
statement of the Pauli principle.
What we actually do, then, is to fill up the hydrogen orbitals with
the proper number of electrons for the atom under consideration (the
aufbau, or building up, principle). One electron can be placed in each
orbital. Since an electron can have m s equal to +§ or — |, two elec-
trons may have the same orbital quantum numbers. The total num-
ber of electrons that the different orbital sets can accommodate is
given in Table 1-2.
The j-, p, d, f, etc., orbital sets usually are called subshells. The
group of subshells for any given n value is called a shell.
The ground-state electronic configuration of a many-electron atom
is of greatest interest. In order to determine the ground state of a
many-electron atom the orbital sets are filled up in order of increasing
energy until all the electrons have been accommodated. We know
from experimental observations that the order of increasing energy
of the orbital sets in many-electron neutral atoms is Ij- , Is, 1p, 3-f, 3p,
As, 3d, Ap, 5s, Ad, 5p, 6s, Af, 5d, 6p, Is, 5{^~- 6d. A diagram showing
the energies of the orbitals in neutral atoms is given in Fig. 1-7-
Electrons in Atoms
XI
Table 1-2
The s,p,d, and/ Orbital Sets
Type
of
orbital
Orbital
quantum numbers
Total number
Total of e lee irons
orbitals that can be
in set accommodated
s I = 0; m; = ° 1
p I = \;mi = 1,0,-1 3
d I = 2; m t = 2,1,0,-1,-2 5
/ I = 3; mi = 3,2,1,0-1 ,-2,-3 7
10
14
high energy
i,
low energy
« = 1
Figure 1—7 Relative energies of the orbitais in neutral atoms.
2.2.
Electrons and Chemical Bonding
1-15 RUSSELL-SAUNDERS TERMS
It is convenient to classify an atomic state in terms of total orbital
angular momentum L and total spin S (capital letters always are used
for systems of electrons; small letters are reserved for individual elec-
trons). This Russell-Saunders LSMlMs scheme will now be de-
scribed in detail.
For a system of n electrons, we define
Ml = mi t + mi 2 + »z 3 + • • ■ + m n (1-38)
Ms = m H + m H + »,,+ ••■ + m Sn (1-39)
We also have these relationships between L and Ml, S and Ms:
Ml = L, L - 1, L - 2, • • • , -L (1-40)
Ms = S, S - 1, S - 2, ■ ■ ■ , -S (1-41)
Let us take the lithium atom as an illustrative example. The
atomic number (the number of protons or electrons in the neutral atom)
of lithium is 3- Therefore the orbital electronic configuration of the
ground state is (1j) 2 (2j) 1 . The ground-state LSM L Ms term is found
as follows:
1. Find the possible values of Ml-
Ml — mi! + mi 2 + mi,
mi x = mi % = mi z = (all are s electrons)
Ml=
2. Find the possible values of L.
M L =
L=
3. Find the possible values of Ms-
Ms = m si + m s% + m sz
™n = +2, f» S2 = — |, m n = ±.\
Ms = +| or -\
4- Find the possible values of S.
M s = +| -i
2
f = 1
J — 2
Electrons in At
oms
z 3
A Russell-Saunders term is written in the shorthand notation 2S+1 L.
The superscript IS + 1 gives the number of different Ms values of any
state, often referred to as the spin multiplicity . As in the single-elec-
tron-orbital shorthand, letters are used for L. (L = is S; L = 1 is
P; L— 2 is D; L = 3 is F; etc.) For the lithium atom, the ground-
state term has L = and S = \ , designated 2 S. An excited electronic
configuration for lithium would be (1j) 2 (2|j) 1 . For this configura-
tion, we find M L = 1, 0, - 1(L = 1) and M s = ±|(^ = I)- There-
fore the term designation of this particular excited state is 2 P.
Admittedly the lithium atom is a very simple case. To find the
term designations of the ground state and excited states for more
complicated electronic structures, it helps to construct a chart of the
possible Ml and Ms values. This more general procedure may be
illustrated with the carbon atom. The carbon atom has six elec-
trons. Thus the orbital configuration of the ground state must be
(ls) 2 (2sy(lpy . It remains for us to find the correct ground-state
term.
First a chart is drawn as shown in Fig. 1-Sa, placing the possible
values of Ml in the left-hand column and the possible values of Ms
in the top row. We need consider only the electrons in incompletely
filled subshells. Filled shells or subshells may be ignored in con-
structing such a chart since they always give a contribution Ml =
0(L = 0) and Ms = 0(\f = 0). (Convince yourself of this before
proceeding.) For carbon the configuration (Ipf is important. Each
of the two p electrons has 1=1 and can therefore have mi = +1, 0,
or —1. Thus the values possible for Ml range from +2 to —2.
Each of the two p electrons can have m s = +| or — \. Thus the
values possible for Ms are 1, 0, and — 1.
The next step is to write down all the allowable combinations
(called microstates~) of mi and m s values for the two p electrons and to
place these microstates in their proper Ml, Ms boxes. The general
form for these microstates is
m sl m S2 • • • m Sn \ + stands for m s = +-|
mi,mi 2 ■ ■ ■ mij — stands for m s = — |
The microstate that fits in the M L = 2, Ms = 1 box is (1, 1). How-
ever, since for both the 2p electrons under consideration n = 2 and
M
Electrons and Chemical Bonding
1
-1
2
(1.1)
Pauli
(1.1)
i
+ +
(1.0)
(1,0) (1,0)
(1,0)
+ +
(1,-1)
(l-l) (1,-1)
(0 + ,0)
(1,-1)
-i
+ +
(-1.0)
(4,0) (-1,0)
(-1,0)
-2
(-1,-1)
(a)
V^;
1
-1
2
(1,1)'
1
+ +
(1,0)
(1,0) (1,0)
(1.0)
+ +
(1,-1)
! (0,0)
(i-i)
-1
(-1.0)
(~*m <-" 1, ° )
(-1,0)
-2
C-t-l)
Figure 1-8 (a) Ml, M s microstate chart for the (2p) 2 orbital
configuration. (6) Ml, M s microstate chart for the (2p) 2
orbital configuration; the 3 P term has been eliminated by
crossing out the six microstates in the Ms = 1 and M s = — 1
columns and, randomly; three microstates with Ml equal to
1, 0, and —1 in the M$ = column.
I = 1, this microstate is not allowable according to the Pauli prin-
ciple and is crossed out in Fig. l-8<z.
Proceeding to the M L = 1, Ms = 1 box, the microstate (1, 6)
fits and is allowable. The two electrons may both have mi— +1
Electrons in Atoms
x 5
and therefore Ml = 2 if their m„ values differ. Thus the microstate
(1, 1) is allowable and fits in the M L = 2, M s = box. This pro-
cedure is followed until the chart is completed.
From the completed chart the 2S+1 L terms may be written down.
Start at top left on the chart. There is a microstate with Ml = 1,
Ms = 1. This microstate may be considered the parent of a state
that has L = 1, S = 1, or 3 P. From Eqs. (1-40) and (1-41), we see
that a term with L = 1 and $ = 1 has all possible combinations of
Ml = 1,0,-1 and Ms = 1,0,-1. Therefore, a d P state must have,
in addition to the Ml = 1, Ms = 1 microstate, microstates with
Ml = 0, M s = 1; M L = -1, M s = 1; M L = 1, M s = 0; M L = 0,
M s = 0; Ml = -1, M s = 0; M L = 1, M s = -1; M L = 0, M s =
— 1; Ml = — 1, Ms = — 1. Thus a total of nine microstates are ac-
counted for by the 3 P term. Subtracting these nine microstates from
the chart, we are left with a new puzzle, as shown in Fig. l-8b.
Moving across the top row, there is a microstate with Ml = 2,
Ms = 0, which may be considered the parent of a state that has
L = 2, S = 0, or *D. The J D state also must have microstates Ml =
I, M s = 0;Mi= 0, M s = 0; Ml = -1,M S = 0; M L = -2,M S =
0. Subtracting these five combinations of the X T> state, we are left
with a single microstate in the Ml = 0, Ms = box. This micro-
state indicates that there is a term having L = 0, S = 0, or X S.
We now have the three terms, 3 P, : D, and V, which account for all
the allowable microstates arising from the (2f) 2 electronic configura-
tion. The ground-state term always has maximum spin multi-
plicity. This is Hund's first rule. Therefore, for the carbon atom,
the IP term is'the ground state.
The X D and l S terms are excited states having the (If) 2 orbital
electronic configuration. Hund' s second rule says that, when com-
paring two states of the same spin multiplicity, the state with the
higher value of L is usually more stable. This is the case with the
l D and X S terms for the carbon atom, since the 1 D state is more stable
than the 1 S state.
PROBLEMS
1-6. Work out the ground-state and excited-state terms for the
most stable orbital electronic configuration of the titanium atom.
Solution. The atomic number of titanium is 22. Thus the most
2.6
Electrons and Chemical Bonding
stable orbital electronic configuration is (l.0 2 (2.r) 2 (2£) e (3.r) 2 (3j) 6
(4j') 2 (3^) 2 . The only incompletely filled subshell is 3d.
Examine Table 1-3, the Ml, Ms chart for the (3/) 2 configuration.
The (2, 1) microstate is the parent of a 3 F term. The S F term
Table 1-3
Values of M L , M s for (3d) 2 Configuration
M L
M S
1
-1
4
(2,2)
3
(2,1)
(2,1X2,1)
(2,1)
2
+ +
(2,0)
(2,0X2,0X1,1)
(2,0)
1
+ + + +
(1,0)(2,-1)
(1,0X1,0)
(2,-l)(2,-l)
(1,0X2,-1)
+ + + +
(2,-2)(l,-l)
(2,-2)(2,-2)
(1,-1)
(1,-1X0,0)
(2,-2)(l,-l)
-1
+ + + +
(-l,0)(l,-2)
(-1, o)(- i,o)
(-2.1X-2.1)
X-l,0)(T,-2)
-2
+ +
(-2,0)
(-2, OX- 2, OX- 1,-1)
(-2,0)
-3
+ +
(-2,-1)
(-2, -IX- 2,-1)
(-2,-1)
-4
(-2,-2)
Electrons in Atoms 2.7
accounts for 21 microstates. Starting at the Ml = 1, Ms = 1 box,
there are two microstates. Thus there also must be a 3 P term. The
(2, 2) microstate is the parent of a l G term. The terms l D and 1 S
account for the remaining microstates in the Ms = column.
The ground-state term has maximum spin multiplicity and must
be either 3 F or 3 P. The 3 F state has the higher angular momentum
(L = 3) and is predicted to be the ground state. The 3 F term is the
experimentally observed ground state for the titanium atom. The
3 P state is the first excited state, with the 1 G, r D, and 1 S states more
unstable.
1-7. Using Table 1-4, work out the terms arising from the orbital
electronic configuration QidyQAtf) 1 , and designate the most stable
state.
Solution. The (3^) 1 (4^) 1 problem is slightly different from the (3<0 2
problem. Both electrons are d electrons with 1=2, but one has
+ +
« = 3 and one has n = 4. Thus, for example, the (2, 2) micro-
state does not violate the Pauli principle, since the n quantum num-
bers differ. The bookkeeping is simplified by adding a subscript
4 to the mi value for the Ad electron.
The terms deduced from the chart for the (j>dyQ\dy configuration
are S G, 3 F, 3 D, 3 P, S S, 1 G, 1 F, J D, l P, and ^. Following the spin-
multiplicity and angular-momentum rules, the 3 G state should be
most stable.
1-16 IONIZATION POTENTIALS
The ionization potential (abbreviated IP) of an atom is the mini-
mum energy required to completely remove an electron from the
atom. This process may be written
atom + IP(energy) — » unipositive ion + electron (1-42)
Further ionizations -are possible for all atoms but hydrogen. In
general, the ionization energy required to detach the first electron is
called IPi, and subsequent ionizations require IP 2 , IPs, IP4, etc.
Quite obviously, for any atom there are exactly as many IP's as
electrons.
The first ionization potentials for most of the atoms are given in
Table 1-5- For any atom, the IPi is always the smallest IP. This
is understandable since removal of a negatively charged particle
Table 1-4
Values of M L) M s for (3d) 1 (M) 1 Configuration
M L
M S
1
-1
4
+ +
(2,2 4 )
(2,2 4 )(2,2 4 )
(2,2 4 )
3
+ + + +
(2,1 4 )(2 4 ,1)
(2,1 4 )(2,1 4 )
(2 4! 1)(2 4 ,1)
(2,1 4 )(2 4 ,1)
2
+ + + +
(2,0 4 )(2 4 ,0)
(2,0 4 )(2,0 4 )(2 4 ,0)
(2,0 4 )(2 4 ,0)
+ +
(M 4 )
(2 4 ,0)(1,1 4 )(T,1 4 )
(lJ«)
+ + + +
U,0 4 )(1 4 ,0)
(1,0 4 )<1,0 4 )(1 4 ,0)
(T,o 4 )(I 4 ,o)
1
+ + + +
(2,_1 4 )(2 4> -1)
(T 4 ,0)(2,-l 4 )(2,-l 4 )
(2 4 ,_1)(2 4 ,-1)
(2,-l 4 )(2 4 ,-l)
+ + + +
(1,-1 4 )(1 4 ,-1)
(1,-1 4 )(!,-1 4 )(1 4 ,-I)
(T,-I 4 )(I 4) -T)
+ + + +
(2,-2 4 )(2 4 ,-2)
(l 4 ,-l)(2,-2 4 )(2,-2 4 )
(2,-I 4 )(2 4 ,-2)
+ +
(o,o 4 )
(2 4 ,-2)(2 4 ,_2)(0,0 4 )(0,0 4 )
(o,o 4 )
+ + + +
(-1,0 4 )(-1 4 ,0)
(-1,0 4 )(-T,0 4 )(-1 4 ,0)
(-7,o 4 )(-i 4 ,o)
-1
+ + + +
(-2,1J(-2 4 ,1)
(_T 4 ,0)(-2,T 4 )(-2,1 4 )
(-2 4 ,1)(-2 4 ,1)
(-2,1 4 )(-2 4 ,1)
+ + + +
(-2,0 4 )(-2 4 ,0)
(-2,0 4 )(-2,0 4 )(-2 4) 0)
(-2,0 4 )(-2 4 ,0)
-2
+ +
(-1,-1 4 )
(_2 4 ,0)(-l,-l 4 )
(-T.-1J
(-1,-T 4 )
-3
+ + + +
(_2,-l 4 )(-2 4 ,-l)
(_2,-7 4 )(-2,-l 4 )
(_2 4 ,-T)(-2 4 ,-l)
(_2,-l 4 )(-2 4 ,-l)
-4
+ +
(-2,-2 4 )
(-2,-2 4 )(-2,-2 4 )
(-2,-2 4 )
2,8
Electrons in Atoms
Z9
Table 1-5
The Electronic Configurations and Ionization Potentials of Atoms
Ground
Orbital electronic
state
z
Atom (A)
configura
Hon
term
IPi, eV &
1
H
Is
2 S
13.595
2
He
is 2
x s
24.580
3
Li
[He] 2 s
2 S
5.390
4
Be
[He] 2 s 2
l s
9.320
5
B
[He]2s 2 2/>
2 P
8.296
6
C
[He]2s 2 2p 2
3p
11.264
7
N
[He]2s 2 2p 3
4 S
14.54
8
O
[He]2s 2 2p 4
3p
13.614
9
F
[He]2s 2 2p 5
2 P
17.42
10
Ne
[He]2s 2 2/> 6
L s
21.559
11
Na
[Ne]3s
2 S
5.138
12
Mg
[Ne]3s 2
l s
7.644
13
Al
[Ne]3s 2 3i>
2 P
5.984
14
Si
[Ne]3s 2 3£ 2
3 P
8.149
15
P
[Ne]3s 2 3/> 3
4 S
11.0
16
S
[Ne]3s 2 3/> 4
3p
10.357
17
CI
[Ne]3s 2 3£ 5
2 P
13.01
18
Ar
[Ne]3s 2 3£ 6
's
15.755
19
K
[Ar]4s
2 S
4.339
20.
Ca
'[Ar]4s 2
x s
6.111
21
Sc
[Ar]4s 2 3rf
2 D
6.56
22
Ti
[Ar]4s 2 3d 2
3 F
6.83
23
V
[Ar]4s 2 3d 3
4 F
6.74
24
Cr
[Ar]4s3d 5
7 S
6.763
25
Mn
[Ar]4s 2 3d 5
e s
7.432
26
Fe
[Ar]4s 2 3d 6
S D
7.90
27
Co
[Ar]4s 2 3d 7
4 F
7.86
28
Ni
[Ar]4s 2 3d 8
3 F
7.633
29
Cu
[Ar]4s 3d 10
2 S
7.724
30
Zn
[Ar]4s 2 3d 10
*S
9.391
31
Ga
[Ar]4s 2 3d 10
4:p
2p
6.00
32
Ge
[Ar]4s 2 3d 10
iP 2
3 P
7.88
33
As
[Ar]4s 2 3d 10
4P 3
4 S
9.81
34
Se
[Ar]4s 2 3d lc
A P 4
3p
9.75
35
Br
[Ar]4s 2 3d 10
4/> 5
2 P
11.84
(continued)
3°
Electrons and Chemical Bonding
Table 1-5 (continued)
Ground
Orbital electronic
state
z
Atom (A)
configuration
term
IP j, eV
36
Kr
Ax]
4s 2 3d 10 4£ 6
*S
13.996
37
Rb
Kr
5s
2 S
4.176
38
Sr
Kr"
5 s 2
's
5.692
39
Y
Kr =
5s 2 4d
2 D
6.5
40
Zr
Kr
5s 2 4d 2
3 F
6.95
41
Nb
Kr"
5s Ad*
6 D
6.77
42
Mo
Kr"
5s 4d 5
7 S
7.10
43
Tc
Kr =
5s 2 4rf 5
6 S
7.28
44
Ru
Kr
5s 4e?
5 F
7.364
45
Rh
Kr
5s Ad 8
4 F
7.46
46
Pd
Kr
Ad 10
'S
8.33
47
Ag
Kr
5sAd X0
2 S
7.574
48
Cd
Kr
5 s 2 Ad 10
"S
8.991
49
In
Kr"
5s 2 Ad 10 5p
2 P
5.785
50
Sn
Kr =
5s 2 Ad 10 5P 2
3 P
7.342
51
Sb
Kr"
5s 2 Ad L0 5p 3
4 S
8.639
52
Te
Kr =
5s 2 4d 10 5£ 4
3p
9.01
53
I
Kr
5s 2 Ad 10 5p 5
2p
10.454
54
Xe
Kr
5s 2 Ad 10 5p 6
'S
12.127
55
Cs
Xe
6s
2 S
3.893
56
Ba
Xe
6 s 2
x s
5.210
57
La
Xe
6s 2 5rf
2 D
5.61
58
Ce
Xe
6s 2 Af 5d
3 H
6.91 b
59
Pr
Xe
6s 2 4/ 3
4 I
5.76 b
60
Nd
Xe
6s 2 4/ 4
5 I
6.31 b
61
Pm
Xe
6s 2 4/ 5
6 i/
62
Sm
Xe
6s 2 4/ 6
7 ^
5.6 b
63
Eu
Xe
6s 2 4/ 7
8 S
5.67 b
64
Gd
Xe
6s 2 4/ 7 5d
9 D
6.16 b
65
Tb
Xe
6s 2 4/S
e H
6.74 b
66
Dy
Xe
6s 2 4/ 10
5 I
6.82 b
67
Ho
Xe"
6s 2 4/ 11
4 I
68
Er
Xe
6s 2 4/ 12
3 H
6.08 c
69
Tm
Xe
6s 2 4/ 13
2 F
5.81 d
70
Yb
Xe
6s 2 4/ 14
X S
6.2 b
71
Lu
Xe
6s 2 4/ 14 5d
2 D
5.0 b
72
Hf
Xe
6s 2 4/ 14 5d 2
3 F
(continued)
Electrons in Atoms
31
1
able 1-5 (c
ontinu
ed)
Ground
Orbital electronic
state
z
Atom (A)
configuration
term
/Pi, eV
73
Ta
Xe]6s 2 4/ 14 5d 3
*F
7.88
74
W
Xe]6s 2 4/ 14 5d 4
5 D
7.98
75
Re
Xe]6s 2 4/ 14 5d 5
6 S
7.87
76
Os
~Xe]6s 2 4/ 14 5d 6
5 D
8.7
77
Ir
Xe]6s 2 4/ 14 5d 7
* F
9
78
Pt
;Xe]6s 2 4/ 14 5rf 9 "
%
S D
9.0
79
Au
Xe]6s 4/ 14 5d lc
2 S
9.22
80
Hg
[Xe]6s 2 4/ 14 5d 10
X S
10.43
81
Tl
r Xe]6s 2 4/ 14 5d 10
e,p
2p
6.106
82
Pb
rxe]6s 2 4/ 14 5rf 10
6p 2
3 P
7.415
83
Bi
Xe]6s 2 4/ 14 5d i0
6p 3
4 S
7.287
84
Po
Xe]6s 2 4/ 14 5d 10
6/>*
3 p
8.43
85
At
Xe]6s 2 4/ 14 5d 10
6p 5
2 P
86
Rn
Xe]6s 2 4/ 14 5d 10
5^ 6
x s
10.746
87
Fr
Rn]7 s
2 S
88
Ra
Rn]7 s 2
x s
5.277
89
Ac '
Rn]7s 2 6d
2 D
90
Th
Rn]7s 2 6d 2
3 F
6.95 e
91
Pa
Rn]7s 2 5/ 2 6d
4 K
92
U
Rn]7s 2 5/ 3 6d
5 L
6.1 e
93
Np
: Rn]7s 2 5/ 4 6d
6 L
94
Pu
Rn]7s 2 5/ 6
7 F
5.1 f
95
Am
Rn]7s 2 5/ 7
S S
6.0 g
96
Cm
"Rn]7s 2 5/ 7 6d
9 D
97
Bk
Rn]7s 2 5/ 9
e H
98
Cf
: Rn]7s 2 5/ 10
5 I
99
Es
"Rn]7s 2 5/ n
4 I
100
Fm
"Rn]7s 2 5/ 12
3 E
101
Md
Rn]7s 2 5/ 13
2 F
102
No
"Rn]7s 2 5/ 14
l S
103
Lw
;Rn]7s 2 5/ 14 6rf
2 D
a From C. E.Moore, "Atomic Energy Levels," NBS Circular 467, 1949,
1952, and 1958, except as indicated.
i>T. Moeller, The Chemistry of the Lanthanides , Reinhold, New York,
1963, p. 37.
C N. I. Ionov and M. A. Mitsev, Zhur. Eksptl. i Theoret. Fiz., 40, 741(1961).
d J. Blaise andB. Vetter, Compt. Rend., 256, 630 (1963).
e K. F. Zmbov, Bull. Boris Kidrich Inst. Nucl. Sci., 13, 17 (1962).
R.H.U.M. Dawton and K. L. Wilkinson, Atomic Energy Research Estab.
(Gt.Brit.), GR/E, 1906 (1956).
g M. Fred and F. S. Tompkins, J. Opt. Soc. Am., 47, 1076 (1957).
32. Electrons and Chemical Bonding
from a neutral atom is easier than its removal from a positively
charged ion.
In any column in the periodic table, the IP's decrease as the atomic
number increases. Let us examine, for example, the Li and Cs atoms.
Lithium, which has IPi = 5-390 eV, has the electronic configuration
[He]2j\ Cesium, with IPi = 3.893 eV, has the structure [Xe]6s.
The 2s electron in Li spends much more time near the nucleus than
the 6s electron does in Cs. This means that the net attraction be-
tween the electron and Z e tt, the shielded nuclear charge, is substan-
tially larger for the Li 2s electron than for the Cs 6s electron, a fact
that is illustrated in Fig. 1-9.
In any row in the periodic table, the IP's generally increase from
left to right, being smallest for the alkali metal atoms and largest for
the inert gas atoms. There are irregularities, however, since atoms
shielding due to Is 2 electrons in Li
w
shielding due to ls 2 2s 2 2p B 3s 2 3pHs 2 3d'Hp' i 5sHd 10 5p< i electrons
<*)
Figure 1-9 Ionization of an electron from (a) a lithium atom
and (b) a cesium atom.
Electrons in Atoms
33
with filled or half-filled subshells have larger IP's than might be
expected. For example, Be([He]2.r 2 ) has IPi = 9.320 eV and
BQ[He]lsnp 1 ') has IPi = 8.296 eV; N([He]2j»2f ) has IPi = 14.54 eV
and OQHe]2j 2 2^ 4 ) has IP, = 13.614 eV. The steady if slightly ir-
regular increase in IP's from Li (IPi =5.390 eV) to Ne (IPi = 21.559
eV) is due to the steady increase in Z e & observed between Li and Ne.
The electrons added from Li to Ne all enter Is and If orbitals and are
not able to completely shield each other from the increasing nuclear
charge.
The variation of the ionization potential of atoms with atomic
number is shown in Fig. 1-10.
1-17 ELECTRON AFFINITIES .
The electron affinity (abbreviated EA) of an atom is the energy
released (or needed, if the atom has a negative EA) when the atom
first transition series
gj second transition series
Xe
third transition series
lanthanides
(.(!
70
80
atomic number
Figure 1—10 Variation of atomic ionization potential with
atomic number.
M
Electrons and Chemical Bonding
adds an extra electron to give a negative ion. Thus we have the
equation
atom + electron — » uninegative ion + EA(energy) (1-43)
Table 1-6
Atomic Electron Affinities
Orbital electronic
Orbital electronic
Atom (A)
configuration
EA, eV
configuration of A'
H
Is
0.747 a
He
F
He]2s 2 2/> s
3.45 b
Ne
CI
"Ne]3s 2 3£ 5
3.61 b
Ar
Br
Ar|4s 2 3<2 10 4£ 5
3.36 b
Kr
I
: Kr]5s 2 4d 10 5Z> 5
3.06 b
Xe
: He]2s 2 2/) 4
1.47 c
[He]2s 2 2/> 5
S
: Ne]3s 2 3£ 4
2.07 d
[Ne]3s 2 3/> 5
Se
Ar]4s 2 3d 10 4/> 4
(1.7) e
[Ar]4s 2 3d 10 4£ 5
Te
Kr]5s 2 4d 10 5/> 4
(2.2) e
[Kr]5s 2 4d 10 5£ 5
N
He]2s 2 2£ 3
(-o.i) £
[He]2s 2 2/> 4
P
Ne]3s 2 3/> 3
(0.7) f
[NeJSs^ 4
As
Ar]4s 2 3d 10 4/> 3
(0.6) f
[Ar]4s 2 3d 10 4£ 4
C
He] 2 s 2 2 /> 2
1.25 g
[He]2s 2 2/> 3
Si
Ne]3s 2 3/> 2
(1.63) £
[NejSs'S/' 3
Ge
Ar]i4s 2 3d 10 4/> 2
(1.2) £
[Ar]4s 2 3rf 10 4/> 3
B
He] 2 s 2 2/>
(0.2) a
[Ee]2s 2 2p 2
Al
Ne]3s?3p
(0.6) a
[Ne]3s 2 3/> 2
Ga
Ar]4s 2 3d 10 4£
(0.18) f
[Ar]4s 2 3d 10 4/> 2
In
Kr]5s 2 4d 10 5£
(0.2) £
[Kr]5s 2 4d 10 5/> 2
Be
He] 2 s 2
(-0.6) a
[He] 2 s 2 2p
Mg
^Ne]3s 2
(-0.3) a
[Ne]3s 2 3/>
Li
He] 2 s
(0.54) a
[He] 2 s 2
Na
^Ne]3s
(0.74) a
[Ne]3s 2
Zn
Ar]4s 2 3d 10
(~0.9) £
[Ar]4s 2 3d 10 4£
Cd
: Kr]5s 2 4d 10
(~0.6) £
[Kr]5s 2 4d 10 5i&
a H. A. Skinner and H. O. Pritchard, Trans. Faraday Soc, 49, 1254 (1953).
b H. S. Berry and C. W. Riemann, J. Chem. Phys., 38, 1540 (1963).
C L. M. Branscomb, Nature, 182, 248 (1958).
d L. M. Branscomb and S. J. Smith, J. Chem. Phys., 25, 598 (1956).
e H. O. Pritchard, Chem. Revs., 52, 529 (1953).
£ A. P. Ginsburg and J. M. Miller, J. Inorg. Nucl. Chem., 7, 351 (1958).
g M. L. Seman and L. M. Branscomb, Phys. Rev., 125, 1602 (1962).
Electrons in At
oms
35
Unfortunately, as a result of certain experimental difficulties, very
few EA values are precisely known. A representative list is given in
Table 1-6.
The halogen atoms have relatively large EA's, since the resulting
halide ions have a stable filled-shell electronic configuration. Atoms
with filled subshells often have negative EA values. Good examples
are Be, Mg, and Zn.
It is interesting to note that the atoms in the nitrogen family, with
the electronic configuration s 2 p 3 (?S~), have very small EA's. Thus we
have additional evidence for the greater stability of a half-filled sub-
shell.
SUPPLEMENTARY PROBLEMS
1(a). Compare the velocity and radius of an electron in the fourth
Bohr orbit with the velocity and radius of an electron in the first
Bohr orbit; (b) Derive the expression, dependent only on the variable
n, for the velocity of an electron in a Bohr orbit.
2. Calculate the energy of an electron in the Bohr orbit with n = 3-
3. Calculate the second ionization potential of He.
4. Calculate the frequencies of the first three lines in the Lyman
series (the lowest-frequency lines).
5. The Balmer series in the spectrum of the hydrogen atom arises
from transitions from higher levels to n = 2. Find which of the
Balmer lines fall in the visible region of the spectrum (visible light
wavelengths are between 4000 and 7000 A).
6. Following the Pauli principle and Hund's first rule, give the
orbital configuration and the number of unpaired electrons in the
ground state for the following atoms: (a) N; (b) S; (c) Ca; (d) Fe;
(e) Br.
7. Find the terms for the following orbital configurations, and
in each case designate the term of lowest energy: (a) Is; (b) If",
(c) If is; (d) 2p3p; (e) 2pld; (f) 3d s ; (g) W; (h) 3d"; (i) 2sAf; (j) If;
(k) ?,d :i 4s.
8. Find the ground-state term for the following atoms: (a) Si;
(b) Mn; (c) Rb; (d) Ni.
Diatomic Molecules
2-1 COVALENT BONDING
A molecule is any stable combination of more than one atom.
The simplest neutral molecule is a combination of two hydro-
gen atoms, which we call the hydrogen molecule or H 2 . The H 2
molecule is homonuclear, since both atomic nuclei used in forming the
molecule are the same.
The forces that hold two hydrogen atoms together in the H 2
molecule are described collectively by the word bond. We know this
bond to be quite strong, since at ordinary temperatures hydrogen
exists in the form H 2 , not H atoms. Only at very high tempera-
tures is H 2 broken up into its H atom components. Let us try to
visualize the bonding in H 2 by allowing two hydrogen atoms to ap-
proach each other, as illustrated in Fig. 2-1. When the atoms are at
close range, two electrostatic forces become important: first, the
attraction between the nucleus H„ and the electron associated with
Isb, as well as that between the nucleus H& and the electron associated
with ls a ; and second, the repulsion between H B and H& as well as that
between ls a and ls b .
The attractive term is more important at large H a -Hb distances,
but the situation changes as the two atoms come closer together, the
importance of the H a -H 6 repulsion increasing as internuclear dis-
tances become very short. This state of affairs is described by an
36
Diatomic Molecules 37
.••; .; •y-.% ; : ; :-. .:;©.• ;•
.'©. : ;':;:v^;-:S;^ « ;7/!v-.v.\-v.-'-i#
Figure 2-1 Schematic drawing of two hydrogen atoms ap-
proaching each other.
energy curve such as that shown in Fig. 2-2. The energy of the
system falls until the H a -Hf, repulsion at very short ranges forces
the energy back up again. The minimum in the curve gives both
the most stable intemuclear separation in the Hz molecule and its gain
in stability over two isolated H atoms.
One of the early successful pictures of a chemical bond involving
electrons and nuclei resulted from the work of the American physical
chemist, G. N. Lewis. Lewis formulated the electron-pair bond, in
which the combining atoms tend to associate themselves with just
enough electrons to achieve an inert-gas electronic configuration.
The hydrogen molecule is, in the Lewis theory, held together by an
electron-pair bond (Fig. 2-3). Each hydrogen has the same partial
claim to the electron pair and thus achieves the stable 1j 2 helium
configuration. A bond in which the electrons are equally shared by
the participating nuclei is called a covalent bond.
The remainder of this book will be devoted to the modern ideas
of bonding in several important classes of molecules. The emphasis
will be on the molecular-orbital theory, with comparisons made
from time to time to the valence-bond theory. Of the many scientists
involved in the development of these theories, the names of R. S.
Mulliken (molecular-orbital theory) and Linus Pauling (valence-
bond theory) are particularly outstanding.
38
Electrons and Chemical Banding
high energy
ii
low energy
R =
"separated atoms"
increasing R
Figure 2-2 Energy of a system of two hydrogen atoms as
a function of internuclear separation.
2-2 MOLECULAR-ORBITAL THEORY
According to molecular-orbital theory, electrons in molecules are
in orbitals that may be associated with several nuclei. Molecular
orbitals in their simplest approximate form are considered to be linear
combinations of atomic orbitals. We assume that when an electron in a
molecule is near one particular nucleus, the molecular wave function
is approximately an atomic orbital centered at that nucleus. This
means that we can form molecular orbitals by simply adding and
subtracting appropriate atomic orbitals. The method is usually ab-
breviated LCAO-MO, which stands for linear combination of atomic
Diatomic Molecules
39
'."'• •;*•'• •.';'• ••:'.•*.•)''. vfit£
electron-pair bond
Figure 2—3 Electron-pair bond in the hydrogen molecule.
orbitals— molecular orbitals. We shall use the abbreviation MO in this
text for a molecular orbital.
Atomic orbitals that are in the proper stability range to be used in
bonding are called valence orbitals. The valence orbitals of an atom
are those that have accepted electrons since the last inert gas and, in
addition, any others in the stability range of the orbitals that will
be encountered before the next inert gas. For example, the valence
orbital of the hydrogen atom is Is. The 2s and If orbitals of hydro-
gen are too high in energy to be used in strong bonding.
2-3 BONDING AND ANTIBONDING MOLECULAR ORBITALS
Let us consider now the MO bonding scheme for the simplest
imaginable molecule, one with two protons and one electron. This
combination is H 2 + , the hydrogen molecule-ion. Each hydrogen in
40 Electrons and Chemical Bonding
^ overlap region
Figure 2-4 The overlap of two hydrogen Is orbitals in H2 + .
the molecule has a Is valence orbital, as shown in Fig. 2-4. Notice
that the two atomic orbitals overlap in the heavily shaded region
between the two nuclei. It is just this overlap region that is affected
by adding and subtracting atomic orbitals to construct molecular
orbitals.
There are two different ways in which we can linearly combine
two Is hydrogen atomic orbitals. The first is to add them together
(Fig. 2-5). It is easy to see from this figure that an electron in MO
I will spend most of its time in the overlap region between the nuclei
H a and Hj. This maximizes the attractive force between the elec-
tron and the two nuclei; therefore an electron in this MO is more
k h 4 W$M}&M&i
K + \ = MO I
Figure 2-5 Schematic drawing of the formation of the bond-
ing MO of H 2 + .
Diatomic Molecules 4 1
stable than in either isolated Is atomic orbital. We refer to such an
MO as bonding. Furthermore, this MO is symmetric for rotation
about a line joining the two H nuclei. That is, if we place an arrow
through the two nuclei, and then turn the arrow, the MO still looks
exactly the same (Fig. 2-6). We call an orbital with such cylindrical
symmetry a a molecular orbital} The a bonding MO will be abbre-
viated <j b .
The other linear combination is formed by subtraction of one of
the two hydrogen 1j orbitals from the other (Fig. 2-7). This type of
MO has a node in the region between the two nuclei. Thus an elec-
tron in MO II will never be found halfway between the two nuclei;
instead it will be mainly confined to space outside the overlap region.
An electron in MO II is less stable than in an isolated Ij- hydrogen
atomic orbital, and we therefore say that II is antibonding. The
antibonding MO also has cylindrical symmetry and thus is a anti-
bonding or a*.
> — llHillw : H ' : 8 w-Q-~
>
M:
no change after rotation
Figure 2-6 Rotation of the bonding MO of H 2 + about the
internuclear axis.
1 In fact, any molecular orbital that does not have a nodal plane containing the inter-
nuclear axis is a cr molecular orbital.
42- Electrons and Chemical Bonding
o
K - h = MO II
Figure 2-7 Schematic drawing of the formation of the anti-
bonding MO of H 2 + .
2-4 MOLECULAR-ORBITAL ENERGY LEVELS
The approximate wave functions for the a b and a* molecular
orbitals are:
K« b ~) = n*(:u + U) (2-1)
K<r*~) = N*(ls a - 1st) (2-2)
Equations (2-1) and (2-2) are simply the analytical expressions for
the molecular orbitals shown in Figs. 2-5 and 2-7, respectively.
The values of the constants N 6 and N* in Eqs. (2-1) and (2-2) are
fixed by the normalization condition,
/ M 2 dx dy it = f |^| 2 dr - 1 (2-3)
Let us proceed to evaluate N 6 . First we substitute ^(cr 5 ) in Eq. (2-3),
giving
StyQW dr=l = f[W(ls a + Ujl 2 dr
= Qwyif(i Sa y dr + so-s b y dr
+ 2/OOCU) *■] (2-4)
Provided the atomic orbitals ls a and ls b are already normalized,
S(lSaXlSa)dT = / (l^XU) * = 1 (2"5)
The integral involving both ls a and 1j- 6 is called the overlap integral
and is denoted by the letter S:
S = overlap integral = f(lsa)(ls h ~) dr (2-6)
Diatomic Molecules 43
Thus, Eq. (2-4) reduces to
(N 6 ) 2 [2+2^] = 1 (2-7)
and
In our approximate scheme we shall neglect the overlap integral
in determining the normalization constant. 1 Therefore, arbitrarily
picking the positive sign in Eq. (2-8), we have
N 5 = V| (2-9)
The value of N* is obtained in the same fashion, by substituting Eq.
(2-2) in Eq. (2-3) and solving for N*. The result is
N * = ± r2(r^) (2 ~ 10 >
or, with the S = approximation,
N* = V| (2-11)
The approximate molecular orbitals for H2 + are therefore
K^ = ^fls a + ls b ~) (2-12)
^*) = -Jp-Sa ~ ISb) (2-13)
The energies of these molecular orbitals are obtained from the
Schrodinger equation,
Sty = &\> (2-14)
Multiplying both sides of Eq. (2-14) by ty and then integrating, we
have
ftfty dr = Ef^ dr (2-15)
1 This approximation involves a fairly substantial error in the case of H 2 + . The
overlap of ls a and Is/, in H 2 + is 0.590. Thus we calculate N 6 = 0.560, as compared to
N b = 0.707 for the S — approximation. In most other cases, however, the overlaps
are smaller (usually between 0.2 and 0.3) and the approximation involves only a small
error.
44 Electrons ani Chemical Bonding
Since J"f 2 dr = 1, Eq. (2-15) reduces to
E = JiO&P dr (2-16)
Substituting Eq. (2-12) in Eq. (2-16), we have
E[K* b ~)] = / WV)M<K^)] dr = i/(l Jo + U)3C(1j.H- LO </r
= i/*(U>)3C(l-0 <*r + |/(1jjXU) </t
+ i/(i Ja )3c(i^) </ r + i/xuMu) ^ (2-17)
We shall not attempt to evaluate the various integrals in Eq. (2-17),
but instead shall replace them using the following shorthand:
q a = fO-Sa)3C(lsa) dr (2-18)
q b = SO-s^MClst) dr (2-19)
/3 = fClsa^SCQlsi) dr = fClsOSCO-Sa) dr (2-20)
In this case, since ls a and lj-f, are equivalent atomic orbitals,
4a = qp = q (2-21)
We shall call q a and fo coulomb integrals. The coulomb integral repre-
sents the energy required to remove an electron from the valence
orbital in question, in the field of the nuclei and other electrons in
the molecule. Thus it is sometimes referred to as a valence ionization
potential.
We shall call /3 the exchange integral in this text. In other sources,
however, you may find /3 referred to as a resonance or covalent integral.
We have seen that an electron in the a b molecular orbital spends most
of its time in the overlap region common to both nuclei. Thus the
electron is stabilized in this favorable position for nucleus ^-electron-
nucleus b attractions. The exchange integral fi simply represents
this added covalent-bonding stability.
Simplifying Eq. (2-17), we have finally
EWp*)] = 4 + P (2~22)
The energy of the <r* molecular orbital is found in the same manner,
substitution in Eq. (2-16) giving
£[#>*)] = i/Tl'a - lft)3C(lj. - U) dr = 4- j3 (2-23)
Diatomic Molecules
45
This result shows that the antibonding molecular orbital is less
stable than the bonding molecular orbital by an amount equal to
— 2/3. An electron in the <r* molecular orbital has only a small prob-
ability of being found in the energetically favored overlap region.
Instead it is confined to the extreme ends of the molecule, which are
positions of high energy relative to the middle of the molecule.
It is convenient to show the relative molecular-orbital energies in a
diagram. Such a diagram for H 2 + is shown in Fig. 2-8. The valence
orbitals of the combining atoms are represented in the outside col-
umns and are ordered in terms of their coulomb energy. The most stable
valence orbitals are placed lowest in the diagram. Since \s a and \sh
have the same coulomb energy, these levels are placed directly oppo-
site one another.
The molecular-orbital energies are indicated in the middle column.
The a b orbital is shown to be more stable than the combining 1j
valence orbitals, and the cr* orbital is shown to be correspondingly
less stable.
The electron in the ground state of H 2 + occupies the more stable
molecular orbital; that is,
gro
■und state of H 2 + = a b
H, orbital
molecular orbitals
H,, orbital
S2
S
,i—i
_ _ __' q + p
Figure 2-8 Relative molecular-orbital energies for H 2 4
46 Electrons and Chemical Bonding
PROBLEM
2-1 . Calculate the energies of the a b and a* orbitals for H 2 + , in-
cluding the overlap integral S. Show that u* is destabilized more
than a h is stabilized if the overlap is different from zero.
2-5 THE HYDROGEN MOLECULE
The orbital electronic structures of molecules with more than one
valence electron are built up by placing the valence electrons in the
most stable molecular orbitals appropriate for the valence orbitals of
the nuclei in the molecule. We have constructed the molecular orbi-
tals for the system of two protons and two Is atomic orbitals. This
set of orbitals is appropriate for H 2 +, H 2 , H 2 ~, etc. The hydrogen
molecule, H 2 , has two electrons that can be placed in the molecular
orbitals given in the energy-level diagram (Fig. 2-8). Both elec-
trons can be placed in the <j h level, provided they have different spin
(m s ~) quantum numbers (the Pauli principle). Thus we represent the
ground state of H 2
ground state of H 2 = (o- 5 ) 2 or [a b (m« = +|)][cr ! '(»2 s = — §)]
which in our shorthand is (o- 5 )(cr 6 ).
This picture of the bond in H 2 involving two electrons, each in a
o b orbital but with opposite spins, is analogous to the Lewis electron-
pair bond in H 2 (Fig. 2-3). It is convenient to carry along the idea
that a full bond between any two atoms involves two electrons.
Thus we define as a useful theoretical quantity the number of bonds
in a molecule as follows:
number of bonds =
(number of electrons in bonding MO's) —
(number of electrons in antibonding MO's)
_
(2-24)
One electron in an antibonding MO is considered to cancel out the
bonding stability imparted by one electron in a bonding MO. Using
this formula we see that H 2 + has half a <r bond and H 2 has one a bond .
Diatomic Molecules 47
2-6 BOND LENGTHS OF H 2 + AND H 2
A useful experimental quantity reflecting electronic structure is
bond length. The standard bond length for a bond between any two
atoms is the equilibrium internuclear separation} We shall express this
distance between nuclei in Angstrom units and refer to it as R. The
bond lengths of H 2 + and H 2 in the ground state are 1.06 and 0.74 A,
respectively, as shown in Fig. 2-9. Thus the H 2 molecule, with one
a bond, has a shorter R than does H 2 +, with only half a a bond. In
general, when molecules with nuclei of approximately the same
atomic number are compared, the bond length is shortest between
the two atoms with the largest number of bonds.
2-7 BOND ENERGIES OF H 2 + AND H 2
Another useful experimental quantity that reflects electronic struc-
ture is bond-dissociation energy. The standard bond-dissociation energy
H h)
■-bond
H — — — — H 1 o-bond
Figure 2-9 Comparison of H 2 + and H 2 .
1 To make matters more complicated for us, nuclei in molecules are always vibrating.
For example, the bond in H 2 , say, stretches and contracts as shown schematically
below:
H-H<-
contr acted
-» H— H < — » H H
equilibrium stretched
internuclear
separation
stretching
The equilibrium internuclear separation about which the nuclei vibrate is the standard
bond length.
48 Electrons and Chemical Bonding
for a bond between any two atoms is the energy required to break the
bond, giving isolated ground-state atoms; i.e.,
H 2 + bond-dissociation energy — >■ H + H (2-25)
We shall express bond energy in kcal/mole units, and refer to a par-
ticular bond energy as DE (atom 1-atom 2). The bond energies of
H 2 + and H 2 are 61.06 and 103-24 kcal/mole, respectively. We see
that H 2 , with one a bond, has a larger bond energy than H 2 +. This
is again a very general result, since bond energies in an analogous
series of molecules increase with an increasing number of bonds.
2-8 PROPERTIES OF H 2 + AND H 2 IN A MAGNETIC FIELD
Most substances can be classified as either paramagnetic or ,
netic according to their behavior in a magnetic field. A paramag-
netic substance is attracted into a magnetic field with a force that is
proportional to the product of the field strength and field gradient.
A diamagnetic substance, on the other hand, is repelled by a mag-
netic field.
In general, atoms and molecules with unpaired electrons (S 9^ 0)
are paramagnetic. Since electrons possess spin, an unpaired electron
creates a permanent magnetic moment. There is in many cases a further
contribution to the permanent magnetic moment as a result of the
movement of the electron in its orbital about the nucleus (or nuclei,
in the case of molecules). In addition to the permanent paramag-
netic moment, magnetic moments are induced in atoms and molecules
on the application of an external magnetic field. Such induced mo-
ments are opposite to the direction of the field; thus repulsion occurs.
The magnitude of this repulsion is a measure of the diamagnetism of
the atom or molecule in question.
The paramagnetism of atoms and small molecules that results from
unpaired electrons is larger than the induced diamagnetism; thus
these substances are attracted into a magnetic field. Atoms and
molecules with no unpaired electrons (S = 0), and therefore no
paramagnetism due to electron spin, are diamagnetic and are repelled
by a magnetic field.
The H 2 + ion, with one unpaired electron (S = §), is paramagnetic .
Diatomic Molecules
49
The Ha molecule, with its two electrons paired (i 1 = 0), is
magnetic.
2-9 SECOND-ROW HOMONUCLEAR DIATOMIC MOLECULES
Let us proceed now to the atoms in the second row of the periodic
table, namely, Li, Be, B, C, N, O, F, and Ne. These atoms have Is,
2p x , 2p y , and 2p z valence orbitals. We first need to specify a coordi-
nate system for the general homonuclear diatomic molecule A 2 , since
the 2p orbitals have directional properties. The 1 axis is customarily
assigned to be the unique molecular axis , as shown in Fig. 2-10. The
molecular orbitals are obtained by adding and subtracting those
atomic orbitals that overlap.
<r Orbitals
The 2s and 2p z orbitals combine to give a molecular orbitals, as
illustrated in Fig. 2-11. The normalized wave functions are:
*0. & ) = .^|(2K + ipj
(2-26)
(2-27)
(2-28)
Figure 2-10 Coordinate system for an A 2 molecule.
50 Electrons and Chemical Bonding
■ j/lV'Vv" -4»i :'v :,^;j ■': . .' . y j fe ■ /. 0; ;/;:!;
(a) overlap of 2s valence orbitals
,-M$m%
!■■■'■ Vj 1 ' 1 .' ' . :".':'- : .i-;' >^V! "t-'- ' . ' .-'.' ' -' ' ' '' : ': : -'^'"- '^-w X ''''•■'■.'':■•■ ' '■•'. V'
fij overlap of 2p, valence orbitals
Figure 2-11 (a) Overlap of two 2s valence orbitals in A 2 .
(6) Overlap of two 2p z valence orbitals in A 2 .
K*$ = ^f 2 P: ~ 2 hd (2-29)
Notice that the <r z molecular orbitals are symmetric for rotation
about the z axis.
ir Orbitals
The 2p x and 2f v orbitals are not symmetric for rotation about the z
axis. The two 2f x orbitals overlap to give the molecular orbital
shown in Fig. 2-12. This molecular orbital has a plus lobe on one
side of the z axis and a minus lobe on the other side. So if we rotate
the molecular orbital by 180°, it simply changes sign. Multiplica-
tion by —1 restores the original orbital. In other words, there is a
node in theyz plane as shown in Fig. 2-13. A molecular orbital of
this type is called a -w molecular orbital. It is clear that the two 2f v
orbitals can also overlap to give x molecular orbitals, which have a
node in the xz plane. There will be w bonding (tr 8 ) and ir antibond-
ing (71-*) molecular orbitals; the more stable T b orbitals will have a
Diatomic Molecules
5 1
— o
Figure 2-12 Overlap of two %p x orbitals in A 2 .
concentration of electron density between the two A nuclei, whereas
the less stable w* orbital will have a node between the two nuclei.
Boundary surfaces of the a and ir molecular orbitals for A 2 molecules
with 2s and 2f valence orbitals are shown in Fig. 2-14. The nor-
malized wave functions for the it MO's follow:
Kir*"') = ^0-fr a + 2-tm)
K*J0 = ^p-ha + 2pn)
(2-30) .
(2-31)
(2-32)
(2-33)
tfOv*) = ^/| 0-ha - 2AJ
The energy-level diagram for the molecular orbitals that accept the
valence electrons can now be estimated. We know that the 2s level
is considerably more stable than 2f in the atoms. The red line at
1.85 eV in the emission spectrum of lithium is due to an electron fall-
ing from the If to the more stable 2s orbital. In fluorine, the 2j--2^
energy difference is over 20 eV. Thus we place 2f above 2s in the
energy-level diagram. 1 Then the a*, a*, ir b , and t* orbitals are placed
with the bonding levels more stable than the antibonding levels in
1 See Appendix for neutral-atom orbital energies.
5*
Electrons an I Chemical Bonding
nW i i i p« ' *i ' i Uj»)W)<i !i <i) i >nii i i » i - 1 . 1 - 1 h)
e
• 180°
nodal plane
r(-l)
original orbital
Figure 2-13 Rotation of a 7r molecular orbital by 180° about
the internuclear axis.
any given combination. The possible energy-level diagrams are
shown in Fig. 2-15.
The relative positioning of the a} level is uncertain. When the
2s-2p energy difference is large, tr/ is probably more stable than
ir Xj y b , as shown in Fig. 2-lSa. We should emphasize here that it is a
good approximation to consider the <7 S molecular orbitals as com-
Diatomic Molecules
53
-— -Ai. r^n ■#:£**
-^t-t*;' f*r**-.?-:...i-
i
i
• i .•
+
;>
... 1
1
1
1 >&■
1 • '*!» ~
.';,v I 1 !
—.-■J
- ■_ sf
... '. ■ J
■y ,' ',■ •'<*.' .. ■
.-.; *r
t'^.-V;.
1
*%tsl
fe$ 1 1 •■%&
a,"
<r,"
*
j;
X X
;
i
i
.
.
t j.
r>
A
■/+X}[
2
<l: °
?,&f. '-,
p^ : ■
£
<'"...•••:•.
■:~;Sy
■-•'.'—•. ;*
«>..
mi
T K *
t„' and jr,* are equivalent to sf* 6 and t x *
Figure 2—14 Boundary surfaces of the <r and tt molecular
orbitals formed from s and p valence orbitals for a homonuclear
diatomic molecule.
posed of the two 2s atomic orbitals only if the 2s-2p energy differ-
ence is large. For small 2s-2f energy differences, we must consider
the two 2s and the two 2f s orbitals together in an LCAO-MO
scheme. The most stable MO would be the combination
Krt) =
=(2Lr„ + t2K + 2j, + t2*0
V2(l + T 2 )
where the coefficient t is less than unity and represents the amount of
2f included in the <rf MO.
A ia) orbitals
A, orbitals
a
m
i
-ooo— <
;^^boo-
■o — :
A.., orbitals
/i, orbitals
/SL, orbitals
54
-ooo^t',/
., i^ooo-
/ 1 — o
I *
I /
,b I <
2s
w
Figure 2-15 Molecular-orbital energy-level diagrams for a
homonu clear diatomic molecule (a) with no a s —a e interaction;
(6) with appreciable tr 5 -tr s interaction.
Diatomic Molecules
55
The stabilization of oJ> and <r s * resulting from such s-p hybridiza-
tion is accompanied by a corresponding destabilization of tf 2 6 and
<r s *, these latter orbitals acquiring some 2s character in the process.
This effect is shown schematically in Fig. 2-16.
The final result for any reasonable amount o(s-p mixing is that the
oi 6 orbital becomes less stable than t x /, as shown in Fig. 2,-13$, As
we shall see in the pages to follow, the experimental information now
available shows that the tr z b level is higher energy than the ir x ,y h level in
most, if not all, diatomic molecules.
In Fig. 2-15 the irj> and ir b levels are shown on the same line.
There is no difference in overlap in the ic x and tt v molecular orbitals
and thus they have the same energy, or, in the jargon of the profes-
sion, they are degenerate.
Using the molecular-orbital energy levels in Fig. 2-15, we shall
discuss the electronic configurations of the second-row Aa molecules.
Li 2
The lithium atom has one Is valence electron. In Li, the 2s-2p
energy difference is small and the <r s b MO of Li 2 undoubtedly has
considerable 2p character. The two valence electrons in Li 2 occupy
the a s b MO, giving the ground-state configuration (vf) 2 . Consistent
with the theory, experimental measurements show that the lithium
energy difference
is larger
Figure 2-16 Schematic drawing of the effect of Vs-a? interac-
tion on the energies of <r, 1 ', ov*, o^, and ff s *.
c6 Electrons and Chemical Bonding
molecule has no unpaired electrons. With two electrons in a bond-
ing MO, there is one net bond. The bond length of Li 2 is 2.67 A as
compared with 0.74 A for H 2 . The larger K for Li 2 is partially due to
the shielding of the two o-., 6 valence electrons by the electrons in the
inner Is orbitals. This shielding reduces the attractions of the nuclei
and the electrons in the a s b MO. The mutual repulsion of the two Is
electron pairs, an interaction not present in H 2 , is also partly respon-
sible for the large R of Li 2 . The bond energies of H 2 and Li 2 are 103
and 25 kcal/mole, respectively. The smaller bond energy of Li 2 is
again undoubtedly due to the presence of the two lj- electron pairs,
as discussed above.
Be 2
The beryllium atom has the valence electronic structure Is" 1 . The
electronic configuration of Be 2 would be (cr/) 2 ((r s *) 2 . This configura-
tion gives no net bonds [(2 — 2)/2 = 0] and thus is consistent with
the absence of Be 2 from the family of A 2 molecules.
B 2
Boron is 2s 2 2p l . The electronic configuration of B 2 depends on the
relative positioning of the o-/ and the -Kx,,} levels. Experimental
measurements indicate that the boron molecule has two unpaired elec-
trons in the ir Xty b level. Thus the electronic configuration' of B 2 is
(ffs 6 ) 2 7 s*) 2 O r :& ! OO r i/0> giving one net "" bond. The bond length of
B 2 is 1.59 A. The bond energy of B 2 is 69 kcal/mole.
C 2
Carbon is 2s 2 2p 2 . In carbon the c/ and ir x J' levels are so spaced
that both the (<r s 6 ) 2 (^*) 2 0O 4 and the (cr/Xtr/Xir^/Xo/) con-
figurations have approximately the same energy. The latest view is
that the configuration (tf s 6 ) 2 (o- s *) 2 (7r T]! /0 4 is the ground state (by less
than 0.1 eV). In this state there are no unpaired electrons and a total
of two ir bonds. This means that a z b must be considerably higher
energy than ir x , y b in C 2 , since the lowest state in the (cr^Xo-.,*) 2
(j t x,v b yQ J z h ~) configuration has two unpaired electrons. Electron
pairing requires energy (recall Hund's first rule). The two bonds
predicted for C 2 may be compared with the experimentally ob-
served bond energy of 150 kcal/mole and the bond length of 1.31 A.
Diatomic Molecules 57
N 2
Nitrogen is 2s 2 2p s . The electronic configuration of N 2 is (o-/) 2
(cr s *) 2 (7r Xi /) 4 (cr s 6 ) 2 , consistent with the observed diamagnetism of this
molecule. The nitrogen molecule has three net bonds (one a and
two x), the maximum for an A 2 molecule, thus accounting for its un-
usual stability, its extraordinarily large bond energy of 225 kcal/
mole, and its very short R of 1.10 A.
We wish to emphasise here that the highest filled orbital in N2 is erf,
which is contrary to the popular belief that Tv x , y h is the higher level. The
experimental evidence comes from a detailed analysis of the electronic
spectrum of N2, and from spectroscopic and magnetic experiments that
establish that the most stable state for N 2 + arises from the configuration
2
2
Oxygen is 2s 2 2p i . The electronic configuration of 2 is Qrf)
(v3*y(^z b y(Kx,y h y(jx*')(jy*'). The electrons in ir XiB * have the same
spin in the ground state, resulting in a prediction of two unpaired
electrons in 2 ; the oxygen molecule is paramagnetic to the extent of
two unpaired spins in agreement with theory. The explanation of
the paramagnetism of 2 gave added impetus to the use of the molec-
ular-orbital theory, since from the simple Lewis picture it is not at
all clear why 2 should have two unpaired electrons.
Two net bonds (one a, one t) are predicted for 2 . The bond
energy of 2 is 118 kcal/mole, and R = 1.21 A. The change in bond
length on changing the number of electrons in the ir x , y * level of the
O2 system is very instructive. The accurate bond length of O2 is
1.2074 A. When an electron is removed from tt x , v *, giving 2 +, the
bond length decreases to 1.1227 A. Formally, the number of bonds
has increased from 2 to 2|. When an electron is added to the ir XyV *
level of 2 , giving 2 ~, the bond length increases to 1.26 A; addition of
a second electron to give 2 2_ increases the bond length still further
to 1.49 A. This is in agreement with the prediction of 1| bonds for
cv
Fluorine is 2s 2 2p 5 . The electronic configuration of F 2 is (o" s 6 ) 2
(a s *yQr s !'y(jx,y h y(Tr x ,y*y, leaving no unpaired electrons and one net
c8 Electrons and Chemical Bonding
bond. This electronic structure is consistent with the diamagnetism
of F 2 , the 36-kcal/mole F — F bond energy, and the E. of 1.42 A.
Ne 2
Neon has a closed-shell electronic configuration 2s 2 2p 6 . The hypo-
thetical Ne2 would have the configuration (ff s 6 ) 2 (ff s *) 2 (<7/) 2 (7r Xi /) 4
( 7r ai,i/*) 4 ( cr z*) 2 an< i zero net bonds. To date there is no experimental
evidence for the existence of a stable neon molecule.
2-10 OTHER A 2 MOLECULES
With proper adjustment of the n quantum number of the valence
orbitals, the MO energy-level diagrams shown in Fig. 2-15 for
second-row A 2 molecules can be used to describe the electronic struc-
tures of A 2 molecules in general.
Na2, K 2 , Kbi, Cs 2
The alkali metal diatomic molecules all have the ground-state con-
figuration (iT s *) 2 , with one a bond. They are diamagnetic. The
bond lengths and bond energies of Li 2 , Na 2 , K 2 , Rb 2 , and Cs 2 are given
in Table 2-1., The bond lengths increase and the bond energies de-
Table 2-1
Bond Lengths and Bond Energies of Alkali Metal Molecules 3
Bond energy,
Molecule
Bond length,
,4
kcal/mole
Li 2
2.672
25
Na 2
3.078
17.3
K 2
3.923
11.8
Rb 2
10.8
Cs 2
10.4
a Data from T. L. Gottrell, The Strengths of Chemical Bonds, Butter -
worths, London, 1958, Table 11.5.1.
Diatomic Molecules 59
crease, regularly, from L12 to Cs2. These effects presumably are due
to the increased shielding of the cr s b electrons by inner-shell elec-
trons in going from Li 2 to Cs2.
Ch, Bt% I2
The ground-state electronic configuration of the halogen molecules
is (o- s 6 ) 2 (o'»*) 2 (o"/) 2 (^,/) 4 C^,!/*) 4 , indicating one net er bond. The
molecules are diamagnetic. Table 2-2 gives bond lengths and bond
energies for F 2 , Q2, Br 2 , and I 2 . The bond lengths increase predict-
ably from F 2 to I2, but the bond energies are irregular, increasing from
F 2 to CU and then decreasing from CI2 to I2. The fact that the bond
energy of CI2 is larger than that of F 2 is believed to be due to the
smaller repulsions of electron pairs in the t orbitals of Cl 2 . One ex-
planation which has been advanced is that the reduced repulsions
follow from the interaction of the empty chlorine 3d orbitals in the
tv MO system. As a result of such p T -d T interaction, the electron
pairs in CI2 have a greater chance to avoid each other. However, it
is not necessary to use the p T -d T explanation, since we know from
atomic spectra that the interelectronic repulsions in the 2fi orbitals of
F are considerably larger than the repulsions in the 3p orbitals of CI.
Table 2-2
Bond Lengths and Bond Energies of Halogen Molecules 3
Bond energy,
Molecule Bond length, A kcal/mole
F 2 1.418 36
Cl 2 1.988 57.07
Br 2 2.283 45.46
I 2 2.667 35.55
a Data from T. L. Cottrell, The Strengths of Chemical Bonds, Butter-
worths, London, 1958, Table 11.5.1.
Go Electrons and Chemical Bonding
Table 2-3
Quantum Number Assignments for Molecular Orbitals
in Linear Molecules
Molecular orbitals m; Atomic orbitals
a
5
S ) Pz> d Z 2
±1
Px > Py ) d xz j dy Z
±2
d X y, d x z_y2
2-11 TERM SYMBOLS FOR LINEAR MOLECULES
Electronic states of a linear molecule may be classified conveniently
in terms of angular momentum and spin, analogous to the Russell-
Saunders term-symbol scheme for atoms. The unique molecular axis
in linear molecules is labeled the Z axis. The combining atomic orbitals
in any given molecular orbital have the same mi value. Thus an mi
quantum number is assigned to each different type of MO, as indi-
cated in Table 2-3- The term designations are of the form
2S+1
\Mz
where S has the same significance as for atoms. The Mj,-state
abbreviations are given in Table 2-4.
We shall work two examples in order to illustrate the procedure.
Table 2-4
State Symbols Corresponding to M/, Values in
Linear-Molecule Electronic- State Classification
State M L
2
n ±1
A ±2
* ±3
Diatomic Molecules 61
EXAMPLE 2-1
The ground-state term of H 2 is found as follows.
1 . Find Ml-' The two electrons are placed in the <r b MO shown in
Fig. 2-8, giving the (o- 6 ) 2 configuration. This is the most stable
state of H 2 . The MO is <r type, so each electron has mi = 0.
Then
Ml = m h + m h = + =
and the state is 2.
2. Find Ms: Since both electrons have mi = 0, they must have
different m, values (the Pauli principle). Thus,
Ms = m, x + m,, = (+i) +■(-!) =
with Ms = 0, S = 0. The correct term symbol 1 is therefore *2.
From the result in the H 2 case, you may suspect that filled molec-
ular orbitals always give M L = and Ms = 0. Indeed this is so,
since in filled orbitals every positive mi value is matched with a
canceling negative mi value. The same is true for the m, values; they
come in +|, —\ pairs in filled orbitals. This information eliminates
considerable work in arriving at the term symbols for states of mole-
cules in which there are many electrons, since most of the electrons
are paired in different molecular orbitals.
EXAMPLE 2-2
Let us now find the ground-state term for 2 . The electronic
configuration of 2 is (<r s '') 2 (^ s *) 2 (^^) 2 (^„;') 4 CT*./) 2 - All the orbit-
als are filled and give Ml = up to ir x , y *. The two electrons in
ir* can be arranged as shown in Table 2-5.
There is a term with M L = +2, —2, and M s = QS = 0); the
term designation is : A. There is a term with Ml — and Ms =
+ 1,0, — l(S = 1); the term designation is 3 2. This leaves one
microstate unaccounted for, with Ml = and Ms = 0(S = 0);
thus there is a X S term.
The ground state must be either 'A, 3 2, or '2. According to
1 There are additional designations possible in certain linear molecules, depending
on the symmetry properties of the molecular wave function. For example, the complete
symbol for the ground state of H 2 is 1 2„ + . A discussion of the complete notation is
given in C. J. Ballhausen and H. B. Gray, Molecular Orbital Theory, Benjamin, New
York, 1964, Chap. 3.
6x
Electrons and Chemical Bonding
Table 2-5
Ml, Mg Values for Example 2-2
M L
M S
1
-1
2
UiffJ
1
i + + i
( 7T! 7T_! )
( ffi vr.j )
-1
-2
(jf-jW.!)
Hund's first rule the ground state has the highest spin multiplicity;
the ground state is therefore 3 S. As we discussed earlier, the 3 S
ground state predicted by the molecular-orbital theory is consistent
with the experimental results, since 2 is paramagnetic to the
extent of two unpaired electrons (i 1 = 1). Spectroscopic evidence
also confirms the 3 S ground state for O2.
In Table 2-6 are listed the ground-state terms and other pertinent
information for several homonuclear diatomic molecules.
2-12 HETERONUCLEAR DIATOMIC MOLECULES
Two different atoms are bonded together in a heteronuclear diatomic
molecule. A simple example for a discussion of bonding is lithium
hydride, LiH.
The valence orbitals of Li are Is, 2p x , 2p y , and 2p z . The valence
orbital of H is Is. Fig. 2-17 shows the overlap of the hydrogen 1j
orbital with the Is, lp x , lp y , and 2p z lithium orbitals. The first step
is to classify the valence orbitals as a or ir types. The lj- of H and
the 2s and 2p z of Li are <r valence orbitals. Thus, the lithium Is and
Diatomic Molecules
2p z orbitals can be combined with the Is orbital of hydrogen. The
2fz and 2p y orbitals of Li are it valence orbitals and do not interact
with the 0- type Is orbital of H. The overlap of 2p x (or 2p y ~) with Is
is zero, as shown in Fig. 2-17.)
We shall now discuss the <r-molecular-orbital system in some de-
tail. Since the 2s level of Li is more stable than the 2p level, it is a
good approximation to consider the a b molecular orbital as composed
mainly of the hydrogen It and the lithium 2s orbitals.
It is also important to note that the Is orbital of H is much more
stable than the 2s orbital of Li. We know that in the free atoms this
stability difference is large, since the first ionization potential of
Li (lt*2s — > Ij- 2 ) is 5-4 eV and the ionization potential of H is 13-6 eV.
As a consequence of the greater stability of the hydrogen Is orbital,
an electron in the a b molecular orbital spends most of its time in
the vicinity of the H nucleus.
-;i : . — ; ;,„ . v . ; h-— — z — <. . . -„; ■ ■ ■ ■ ■&• ■; • •■ :'■■• ■ ■ h—
Is
equal + and — give zero
U : '■\j.VC S ir-r^- 2 samc for 2 Pv> U
net overlap 2p x ,\s is zero
Figure 2-17 Overlap of the hydrogen Is orbital with the
lithium valence orbitals.
64 Electrons and Chemical Bonding
Table 2-6
Properties of Homonuclear Diatomic Molecules 3
Ground
Bond
Bond-dissociation
Molecule
state
length, A
energy, kcal/mole
Ag 2
2 E?
39
As 2
X E
91
Au 2
X E?
52
B 2
3 £
1.589
69
Bi 2
x £
39.2
Br 2
X £
2.283
45.46
c 2
1 £( 3 n) b
1.3117
150
Cd 2
*£?
2.1
Cl 2
X E
1.988
57.07
Cl 2 +
2 n
1.891
Cs 2
X £
10.4
Cu 2
X E
47
D 2
?„j
0.7416
F 2
X E
1.418
36
Ga 2
35
Ge 2
65
H 2
*E
0.7415
103.24
H 2 *
2 E
1.06
61.06
He 2 +
2 E
1.08
Hg 2
X E
3.2
I 2
X E
2.6666
35.55
K 2
X L
3.923
11.8
Li 2
X E
2.672
25
N 2
X E
1.0976
225.0
(continued)
Diatomic
Molecui
les
t
Table
2-6 (continued)
Ground
Bond
Bond -dissociation
Molecule
state
length, A
energy
, kcal/mole
N/
2 Z
1.116
Na 2
x s
3.078
17.3
o 2
3 S
1.20741
117.96
O/
2 n
1.1227
o 2 -
2 n?
1.26
or
X E?
1.49
p 2
*£
1.8943
116.0
Pb 2
23
Rb 2
X E
10.8
s 2
3 S
1.887
83
Sb 2
'£
69
Se 2
3 E
2.152
65
Si 2
2.252
75
Sn 2
46
Te 2
2.59
53
Zn 2
X E?
6
65
a Data from G. Herzberg, Spectra of Diatomic Molecules, Van Nostrand,
New York, 1950, Table 39; T. L. Cottrell, The Strengths of Chemical Bonds,
Butterworths, London, 1958, Table 11.5.1; L. E. Sutton (ed.), "Interatomic
Distances," SpecialPublicationNo.il, The Chemical Society, London, 1958.
b A short discussion of the ground state of C 2 can be found in J. W.
Linnett, Wave Mechanics and Valency, Methuen, London, 1960, p. 134.
The a h orbital is shown in Fig. 2-18. The analytical expression for
the (T i MO of LiH has the form
#>*) = COs + Q2p z + c ^ (2-34)
In this case, G > G > G and their numerical values are restricted by
the normalization condition [Eq. (2-3)].
66 Electrons and Chemical Bonding
<sSftJ,itT
a"
Figure 2-18 Boundary surface of the a bonding molecular
orbital of LiH.
Since both the Is and the 2pz lithium orbital s are used in the <r
molecular orbitals, there are two a* orbitals, one involving the Is
and one involving the 2p z , These u* orbitals are mainly localized
on the Li, as shown in Fig. 2-19. The approximate wave functions
are:
#>.*) = Q2s - Qls; Q > a (2-35)
lK>.*) = C&, - C 7 1j; C, > Q (2-36)
-Ii . . ' . -; ■1i: H >~—
,.':."•+.'•••.:■
Figure 2-19 Boundary surfaces of the a* and a* MO's of
LiH.
Diatomic Molecules 67
2-13 MOLECULAR-ORBITAL ENERGY-LEVEL SCHEME FOR L1H
Figure 2-20 shows the MO energy-level scheme for LiH. The
valence orbitals of Li are placed on the left side of the diagram, with
the 2p level above the 2s level. On the right side, the hydrogen lj-
level is shown. The Is level of H is placed below the 2s level of Li,
to agree with their known stability difference.
The a* and tr* MO's are placed in the center. The a b MO is more
stable than the hydrogen Is valence orbital, and the diagram clearly
shows that o- 6 is mainly composed of hydrogen Is, with smaller frac-
tions of lithium 2s and 2p z . The <r s * MO is less stable than the lith-
ium 2s valence orbital, and the diagram shows that <j s * is composed
of lithium 2s and hydrogen 1j , with a much greater fraction of lith-
Li orbitals LiH orbitals H orbital
/ — O— ,
/ , — Q— ,'
. Ip's ' 1 »
_ooo-f-/-^oo— w
I
is A %
■o— ; \
\ »
\
\
v.
» » \ h
! — O— '
figure 2—20 Relative orbital energies in LiH.
o
68 Electrons ani Chemical Bonding
ium Is. The a s * orbital is shown less stable than 2p z , and it clearly
has considerable 1p z character.
The 2px and lp y orbitals of Li are shown in the MO column as
7r-type MO's. They are virtually unchanged in energy from the Li
valence-orbital column, since H has no valence orbitals capable of
7r-type interaction.
2-14 GROUND STATE OF LiH
There are two electrons to place in the MO energy-level scheme for
LiH shown in Fig. 2-20. This total is arrived at by adding together
the one valence electron contributed by hydrogen (\s) and the one
valence electron contributed by lithium (2s~). Both electrons are ac-
commodated in the a b MO, giving a ground-state configuration
Since the electrons in the a b MO spend more time in the vicinity of
the H nucleus than of the Li nucleus, it follows that a separation of
charge is present in the ground state. That is, the Li has a partial
positive charge and the H has a partial negative charge, as shown
below:
Li 8 +H 5 -
A limiting situation would exist if both electrons spent all their time
around the H. The LiH molecule in that case would be made up of
a Li+ ion and a H~ ion; that is, § = 1. A molecule that can be formu-
lated successfully as composed of ions is described as an ionic molecule.
This situation is encountered in a diatomic molecule only if the
valence orbital of one atom is very much more stable than the valence
orbital of the other atom. The LiH molecule is probably not such an
extreme case, and thus we say that LiH has partial ionic character.
A calculation of the coefficients C\, C 2 , and C 3 would be required to
determine the extent of this partial ionic character. One such calcu-
lation (unfortunately beyond the level of our discussion here) gives
a charge distribution
Li 0.8+ H 0.8-
which means that LiH has 80 per cent ionic character.
Diatomic Molecules 69
2-15 DIPOLE MOMENTS
A heteronuclear diatomic molecule such as LiH possesses an electric
dipole moment caused by charge separation in the ground state. This
electric moment is equal to the product of the charge and the distance
of separation,
dipole moment = jx = ell (2-37)
Taking JR. in centimeters and e in electrostatic units, p. is obtained in
electrostatic units (csu). Since the unit of electronic charge is 4.8 X
10 -10 esu and bond distances are of the order of 10~ 8 cm (1 A), we see
that dipole moments are of the order of 10~ 18 esu. It is convenient to
express p. in Debye units (D), with 10~ 18 esu = 1 Debye. If, as a first
approximation, we consider the charges centered at each nucleus, K
in Eq. (2-37) is simply the equilibrium internuclear separation R in
the molecule.
Since it is possible to measure dipole moments, we have an experi-
mental method of estimating the partial ionic character of hetero-
nuclear diatomic molecules. The dipole moment of LiH is 5-9 Debye
units (5.9 D). ForR = 1.60 A (or 1.60 X 10- 8 cm), we calculate for
an ionic structure Li + H~ a dipole moment of 7.7 D. Thus the partial
charge from the dipole moment datum is estimated to be 5-9/7.7 =
0.77, representing a partial ionic character of 77 per cent. This
agrees with the theoretical value of 80 per cent given in the last
section.
Dipole moments for a number of diatomic molecules are given in
Table 2-7.
2-16 ELECTRONEGATIVITY
A particular valence orbital on one atom in a molecule which is
more stable than a particular valence orbital on the other atom in a
molecule is said to be more electronegative. A useful treatment of elec-
tronegativity was introduced by the American chemist Linus Pauling
in the early 1930s. Electronegativity may be broadly defined as the
ability of an atom in a molecule to attract electrons to itself. It must
be realized, however, that each different atomic orbital in a molecule
has a different electronegativity, and therefore atomic electronega-
70
Electrons and Chemical Bonding
Table 2-7
Dipole Moments of Some Diatomic Molecules 8
Molecule
Dipole moment, D
LiH
5.88
HF
1.82
HC1
1.07
HBr
0.79
HI
0.38
o 2
CO
0.12
NO
0.15
IC2
0.65
BrCl
0.57
FC1
0.88
FBr
1.29
KF
8.60
KI
9.24
a Data from A. L. McClellan, Tables of Experi-
mental Dipole Moments, Freeman, San Francisco,
1963.
tivities vary from situation to situation, depending on the valence
orbitals under consideration. Furthermore, the electronegativity, of
an atom in a molecule increases with increasing positive charge on
the atom.
The Pauling electronegativity value for any given atom is obtained
by comparing the bond-dissociation energies of certain molecules
containing that atom, in the following way. The bond-dissociation
energy (DE) of LiH is 58 kcal/mole. The DE's of Li 2 and H 2 are 25
and 103 kcal/mole, respectively. We know that the DE's of Li 2 and
Hi refer to the breaking of purely covaient bonds — that is, that the
two electrons in the a h levels are equally shared between the two
hydrogen and the two lithium atoms, respectively. If the two elec-
trons in the o- 6 MO of LiH were equally shared between Li and H, we
might expect to be able to calculate the DE of LiH from the geo-
metric mean; thus
Diatomic Molecules
71
DEl,-h = VDEh, X DEll (2-38)
This geometric mean is only 51 kcal/mole, 7 kcal/mole less than the
observed DE of LiH. It is a very general result that the DE of a mole-
cule AB is almost always greater than the geometric mean of the DE' s of A<i
and Bi. An example more striking than LiH is the system BF. The
DE's of B 2 , F 2 , and BF are 69, 36, and 195 kcal/mole, respectively.
The geometric mean gives
DEbf = V69 X 36 = 50 ^ 195 (2-39)
This "extra" bond energy in an AB molecule is presumably due to
the electrostatic attraction of A and B in partial ionic form,
Pauling calls the extra DE possessed by a molecule with partial ionic
character the ionic resonance energy or A. Thus we have the equation
A = DEab - VD^XD^ (2-40)
The electronegativity difference between the two atoms A and B is
then defined as
X A - X B = 0.208VA (2-41)
where Xa and X B are electronegativities of atoms A and B and the
factor 0.208 converts from kcal/mole to electron-volt units. The
square root of A is used because it gives a more nearly consistent set
of electronegativity values for the atoms. Since only differences are
obtained from the application of Eq. (2-41), one atomic electro-
negativity value must be arbitrarily agreed upon, and then all the
others are easily obtained. On the Pauling scale, the most electro-
negative atom, fluorine, is assigned an electronegativity (or EN) of
approximately 4. The most recent EN values, calculated using
the Pauling idea, are given in Table 2-8.
Another method of obtaining EN values was suggested by R. S.
Mulliken, an American physicist. Mulliken's suggestion is that
atomic electronegativity is the arithmetic mean of the ionization
potential and the electron affinity of an atom; i.e.,
EN = IP t E - (2-42)
Table 2-8
Atomic Electronegativities
I
II
III
II
II
II
II
II
II
II
I
n
III
IV
III
n
I
H
2.20
Li
0.98
Be
1.57
B
2.04
C
2.55
N
3.04
O
3.44
F
3.98
Na
0.93
Mg
1.31
Al
1.61
Si
1.90
P
2.19
S
2.58
CI
3.16
K
0.82
Ca
1.00
Sc
1.36
Ti
1.54
V
1.63
Cr
1.66
Mn
1.55
Fe
1.83
Co
1.88
Ni
1.91
Cu
1.90
Zn
1.65
Ga
1.81
Ge
2.01
As
2.18
Se
2.55
Br
2.96
Rb
0.82
Sr
0.95
y
1.22
Zr
1.33
Mo
2.16
Rh
2.28
Pd
2.20
Ag
1.93
Cd
1.69
In
1.78
Sn
1.96
Sb
2.05
I
2.66
Cs
0.79
Ba
0.89
La
1.10
W
2.36
Ir
2.20
Pt
2.28
Au
2.54
Hg
2.00
Tl
2.04
Pb
2.33
Bi
2.02
Ce
1.12
Pr
1.13
(HI)
Nd
1.14
(HI)
Sm
1.17
(HI)
Gd
1.20
(HI)
1.22
(HI)
Ho
1.23
(HI)
Er
1.24
Tm
1.25
(HI)
Lu
1.27
(III)
U
1.38
(HI)
Np
1.36
(m)
Pu
1.28
(HI)
From A.
the molecules
L. Allred, J. Inorg. Nucl. Chem., 17,
which were used in the calculations.
215 (1961); roman numerals give the oxidation state of the atom in
Diatomic Molecules
73
Equation (2-42) averages the ability of an atom to hold its own
valence electron and its ability to acquire an extra electron. Of
course the EN values obtained from Eq. (2-42) differ numerically
from the Pauling values, but if the Mulliken values are adjusted so
that fluorine has an EN of about 4, there is generally good agreement
between the two schemes. 1
2-17 IONIC BONDING
The extreme case of unequal sharing of a pair of electrons in an
MO is reached when one of the atoms has a vety high electronega-
tivity and the other has a very small ionization potential (thus a
small EN). In this case the electron originally belonging to the
atom with the small IP is effectively transferred to the atom with the
high EN,
M- + X-^M+ :X- (2-43)
The bonding in molecules in which there is an almost complete elec-
tron transfer is described as ionic. An example of such an ionic di-
atomic molecule is lithium fluoride, LiF. To a good approximation,
the bond in LiF is represented as Li + F _ . The energy required to
completely separate the ions in a diatomic ionic molecule (Fig. 2-21)
is given by the following expression:
potential energy = electrostatic energy + van der Waals energy
A' 1 . $r
A'* + B<-
Figure 2—2 1 Dissociation of an ionic molecule into ions.
1 However, note that the two scales are in different units.
74 Electrons and Chemical Bonding
The electrostatic energy is
where £1 and #2 are charges on atoms M and X and K is the inter-
nuclear separation.
There are two parts to the van der Waals energy. The most im-
portant at short range is the repulsion between electrons in the filled
orbitals of the interacting atoms. This electron-pair repulsion is illus-
trated in Fig. 2-22. We have previously mentioned the mutual re-
pulsion of filled inner orbitals, in comparing the bond energies of Li2
and H2.
The analytical expression commonly used to describe this inter-
action is
van der Waals repulsion = be" aR (2-45)
where b and a are constants in a given situation. Notice that this
repulsion term becomes very small at large R values.
The other part of the van der Waals energy is the attraction that
results when electrons in the occupied orbitals on the different atoms
correlate their movements in order to avoid each other as much as
possible. For example, as shown in Fig. 2-23, electrons in orbitals
on atoms M and X can correlate their movements so that an instan-
taneous-di-jiok-induced-dipole attraction results. This type of potential
Figure 2-22 Repulsion of electrons in filled orbitals. This
repulsion is very large when the filled orbitals overlap (recall
the Pauli principle).
Diatomic Molecules 75
Figure 2-23 Schematic drawing of the instantaneous -
dipole-induced-dipole interaction, which gives rise to a weak
attraction.
energy is known as the London energy, and is denned by the expres-
sion
London energy = — ^ (2-46)
where d is a constant for any particular case. The reciprocal R e type
of energy term falls off rapidly with increasing R, but not nearly so
rapidly as the be~ aR repulsion term. Thus the London energy is more
important than the repulsion at longer distances.
2-18 SIMPLE IONIC MODEL FOR THE ALKALI HALIDES
The total potential energy for an ionic alkali halide molecule is
given by the expression
B**.:=5£ +*-»-£ (2-47)
We need only know the values of the constants b, a, and d in order
to calculate potential energies from Eq. (2-47)- The exact values of
these constants for alkali metal ions and halide ions are not known.
However, the alkali metal ions and the halide ions have inert-gas
electronic configurations. For example, if LiF is formulated as an
ionic molecule, Li+ is isoelectronic with the inert gas He, and F - is
isoelectronic with the inert gas Ne. Thus the van der Waals inter-
action in Li+F~ may be considered approximately equal to the van
7 6
Electrons and Chemical Bonding
der Waals interaction in the inert-gas pair He-Ne. This inert-gas-
pair approximation is of course applicable to the other alkali halide
molecules as well.
The inert-gas-pair interactions can be measured and values for the
b, a, and d constants are available. These values are given in Table
2-9. Using Eq. (2-47), we are now able to calculate the bond energy
of LiF.
EXAMPLE
To calculate the bond energy of LiF, we first calculate the energy
needed for the process
LiF -> Li+ + F^
We shall calculate this energy in atomic units (au). The atomic
unit of distance is the Bohr radius, a a , or 0.529 A. The atomic unit
of charge is the electronic charge. The b, a, and d constants in
Table 2-9
van der Waals Energy Parameters 3
Interaction pair
a
6
d
He -He
2.10
6.55
2.39
He-Ne
2.27
33
4.65
He-Ar
2.01
47.9
15.5
He-Kr
1.85
26.1
21.85
He-Xe
1.83
42.4
33.95
Ne-Ne
2.44
167.1
9.09
Ne— Ar
2.18
242
30.6
Ne-Kr
2.0-2
132
42.5
Ne-Xe
2.00
214
66.1
Ar— Ar
1.92
350
103.0
Ar— Kr
1.76
191
143.7
Ar— Xe
1.74
310
222.1
Kr-Kr
1.61
104
200
Kr-Xe
1.58
169
310
Xe-Xe
1.55
274
480
All values are in atomic units. Data from E. A. Mason, J. Chem. Phys .
23, 49 (1955).
Diatomic Molecules
Table 2-9 are given in atomic units. Finally, 1 au of energy is equal
to 27.21 eV. The bond length of LiF is 1.52 A; this is equal to 1.52/
0.529 = 2.88 au. For Li+F - , q\ = f t = 1 au and e l = 1 au.
Thus, on substitution of the b, a, and d parameters for He-Ne, Eq.
(2-47) becomes
PE = -Z_L -j_ 33 e (-2.27)(2.88) 4.65
77
2.88 (2.88) 6
PE = -0.347 + 33(0.00144) - ^
571
PE = -0.308 au = -8.38 eV
Accordingly, the energy required to separate Li+ from F~ at a bond
distance of 2.88 au is 8.38 eV. This is called the coordinate-bond
energy. However, we want to calculate the standard bond-dissocia-
tion energy, which refers to the process
DE
LiF > Li + F
That is, we need to take an electron from F~ and transfer it to Li+:
T . v 8.38 eV -IP^Li) T . , „
LiF > Li + + F > Li + F
+ EA F
We see that the equation which allows us to calculate the DE of an
alkali halide is
DE = -PE - IPi + EA
Since IPi(Li) = 5.39 eV and EA F = 3.45 eV, we have finally
DElsf = 8.38 - 539 + 3.45 = 6.44 eV
The calculated 6.45 eV, or 149 kcal/mole, compares favorably with
the experimental DE of 137 kcal/mole.
Experimental bond energies and bond distances for the alkali halide
molecules are given in Table 2-10. The alkali halides provide the best
examples of ionic bonding, since, of all the atoms, the alkali metals
have the smallest IP's; of course the halogens help by having very-
high EN's. The most complete electron transfer would be expected
7 8
Electrons and Chemical Bonding
Table 2-10
Bond Properties of the Alkali Halides 3
Bond-dissociation
Molecule
Bond length, A
energy.
, kcal/mole
CsF
2.345
121
CsCl
2.906
101
CsBr
3.072
91
Csl
3.315
75
KF
2.139 b
118
KC1
2.667
101
KBr
2.821
91
KI
3.048
77
LiF
1.520 b
137
LiCl
2.029 b
115
LiBr '<■
2.170
101
Lil
2.392
81
NaF
1.846 b
107
NaCl
2.361
98
NaBr
2.502
88
Nal
2.712
71
RbF
2.242 b
119
RbCl
2.787
102
RbBr
2.945
90
Rbl
3.177
77
a Ground-state terms are *S . Data from T. L. Cottrell, The Strengths
of Chemical Bonds , Butterworths , London, 1958, Table 11.5.1
Estimated values; see L. Pauling, The Nature of the Chemical Bond,
Cornell Univ. Press, Ithaca, N.Y., 1960, p. 532.
in CsF and the least complete in Lil.
be of considerable importance.
In Lil, covalent bonding may
2-19 GENERAL AB MOLECULES
We shall now describe the bonding in a general diatomic molecule,
AB, in which B has a higher electronegativity than A, and both A
and B have s and p valence orbitals. The molecular-orbital energy
Diatomic Molecules
79
levels for AB are shown in Fig. 2-24. The s and p orbitals of B are
placed lower than the s and p orbitals of A, in agreement with the
electronegativity difference bet-ween A and B. The o- and tt bonding
and antibonding orbitals are formed for AB in the same manner as for
Aj, but with the coefficients of the valence orbitals larger for B in the
bonding orbitals and larger for A in the antibonding orbitals. This
means that the electrons in the bonding orbitals spend more time
near the more electronegative B. In the unstable antibonding orbit-
als, they spend more time near the less electronegative A. The
A orbitals
AB orbitals
B orbitals
/
tip ! „
-ooo=^ /
« w
Figure 2-24 Relative orbital energies in a general AB mole-
cule, with B more electronegative than A,
8o
Electrons and Chemical Bonding
.+
*&*+-
•zMi
^.-.■ t^
'■^-■Rvfe'';'-'
••.•I.V.'-' ' •-.>F' : I
; :*v^
■.'•"Ivp!
T-Srr
TTy orbitals are equivalent to Tj
Figure 2-25 Boundary surfaces of the MO'i of an AB mole-
cule, with B more electronegative than A.
boundary surfaces of the molecular orbitals for a general AB molecule
are given in Fig. 2-25- The following specific cases illustrate the use
of the bonding scheme shown in Fig. 2-24.
BN Q] Valence Electrons)
The ground-state electronic configuration for BN is (jyfyQr^y
(j r x,y b y(<r z b '). This gives a % state and a prediction of two bonds
Diatomic Molecules 81
(| c, f 7r). The BN molecule is thus electronically similar to C2. The
bond lengths of C2 and BN are 1.31 and 1.28 A, respectively. The
BN bond energy is only 92 kcal/mole, as compared to 150 kcal/mole
for C 2 .
50, CN, C0+ (9 F*&»« Electrons')
The BO, CN, and CO+ molecules all have the ground-state con-
figuration (Vs') 2 ^*) 2 ^,/) 4 ^/), and thus a 2 2 ground state. There
are 2| bonds predicted, which is \ more than for BN. The bond lengths
are all shorter than that of BN (or C 2 ), being 1.20 A for BO, 1.17 A
for CN, and 1.115 A for CO+. The bond energies are higher than
that for BN, being 185 kcal/mole for BO and 188 kcal/mole for CN.
CO, N0+, CN- flO Valence Electrons)
The CO, NO+, and CN~ molecules are isoelectronic with N2,
having a *2 ground state. The configuration (c s i> ) 2 (o' s *) 2 (7r Ii ,/) 4 ((r 2 6 ) 2
predicts one <r and two t bonds. The bond lengths of NO+, CO, and
CN" increase with increasing negative charge, being 1.062 A for
NO+, 1.128 A for CO, and 1.14 A for CN". Comparing molecules
having the same charge, the bond lengths of NO+, CO, and CN~ are
shorter than those of BO, CN, and CO+, as expected. The bond
energy of CO is 255-8 kcal/mole, which is even larger than the bond
energy of 225 kcal/mole for N2.
NO (11 Valence Electrons)
The electronic configuration of NO is (ff s 6 ) 2 (<r s *) 2 (7r :c , ! /') 4 (a-/') 2
(jx, v *), giving a V ground state. Since the eleventh electron goes
into a ir* orbital, the number of bonds is now 2-|, or | less than for
NO+. The bond length of NO is 1 . 15 A, longer than either the CO or
NO+ distances. The bond energy of NO is 162 kcal/mole, consider-
ably less than the CO value.
The bond properties of a number of representative heteronuclear
diatomic molecules are listed in Table 2-11.
8x
Electrons and Chemical Bonding
Table 2-11
Properties of Heteronuclear Diatomic Molecules 3
Ground
Bond
Bond dissociation
Molecule
state
length., A
energy, heal/mole
AlBr
ls
2.295
99
A1C1
X S
2.13
118
AiF
X E
1.65
158
A1H
X S
1.6482
67
All
J S
90
AlO
2 S
1.6176
138
AsN
*£
115
AsO
2 n
113
BBr
X S
1.88
97
BC1
X E
1.715
117
BF
X E
1.262
195
BH
*E
1.2325
70
BN
3 n
1.281
92
BO
2 £
1.2049
185
BaO
2 E
1.940
130
Be CI
2 £
1.7
69
BeF
2 S
1.3614
92
BeH
2 S
1.3431
53
BeO
X S
1.3308
124
Br CI
X S
2.138
52.1
BrF
X S
1.7555
55
BrH
X E
1.408
86.5
BrET
2 n
1.459
CF
2 n?
1.270
106
(continued)
Diatomic Molecules
83
Table 2-11 ~ (continued)
Ground
Bond
Bond dissociation
Molecule
state
length, A
energy, kcal/mole
CH
2 n
1.1198
80
CN
2 Z
1.1718
188
CN*
1.17 27
CN"
1.14
co
J E
1.1282
255.8
CO*
2 E
1.1151
CP
2 E
1.562
138
cs
X E
1.5349
166
CSe
2 S?
1.66
115
CaO
'Tj
1.822
100
C1F
X E
1.6281
60.5
CsH
'■L
2.494
42
GaCl
J E
2.208
115
GaF
J E
1.775
142
GeO
ls
1.650
157
HC1
j e
1.27 44
102.2
HC1 +
2 n
1.3153
HD
2 E
0.7413
HF
x e
0.9175
134
HI
X E
1.608
70.5
HS
2 n
1.3503
80
IBr
X E
41.90
IC1
X E
2.32070
49.63
IF
2 E
1.985
46
InBr
J E
2.5408
85
(continued)
84
Electrons and Chemical Bonding
Table 2-11 (continued)
Ground
Bond
Bond dissociation
Molecule
state
length, A
energy, kcal/mole
InCl
l s
2.4012
104
InF
x s
1.9847
125
InH
X E
1.8376
57
Inl
'Ti
2.86
65
KH
X S
2.244
43
LiH
l s
1.5953
58
MgO
X E
1.749
92
NH
3 E
1.038
85
NH +
2 n
1.084
NO
2 n
1.150
162
NO*
X E
1.0619
NP
X S
1.4910
NS
2 n
1.495
115
NS +
1.25
NaH
X S
1.8873
47
NaK
'E
14.3
NaRb
j e
13.1
OH
2 n
0.9706
101.5
OH +
3 S
1.0289
PH
3 E
1.4328
PN
*E
1.4910
138
PO
2 n
1.448
125
PbH
2 n
1.839
42
PbO
X E
1.922
94
PbS
*£
2.3948
75
(continued)
Diatomic Molecules
Table 2-11 (continued)
Ground
Bond
Bond dissociation
Molecule
state
length, A
energy, kcal/mole
RbH
*£
2.367
39
SO
3 S
1.4933
119
SbO
2 n
74
SiF
2 n
1.603
88
SiH
2 n
1.520
74
SiN
2 E
1.572
104
SiO
X E
1.509
185
SiS
X E
1.929
148
SnH
2 n
1.785
74
SnO
"E
1.838
132
SnS
^
2.06
110
SrO
J S
1.9 20
83
TIBr
X E
2.6181
78
T1C1
X E
2.4848
87
TIF
X E
2.0844
109
T1H
'£
1.870
46
Til
J E
2.8136
65
a Data from G. Herzberg, Spectra of Diatomic Molecules , Van Nostrand,
New York, 1950, Table 39; T. L. Cottrell, The Strengths of Chemical Bonds,
Butterworths, London, 1958, Table 11.5.1; L. E. Sutton (ed.), "Interatomic
Distances," SpecialPublicationNo.il, The Chemical Society, London, 1958.
SUPPLEMENTARY PROBLEMS
1. Find the ground-state term for (a) B2; (b) F 2 ; (c) C2; (d) S2.
2. Discuss the bond properties of N2, P2, As2, Sb2, and Bi 2 in terras
of their electronic structures.
3. Discuss the bond properties of Cl» and Cl2 + using molecular-
orbital theory.
86 Electrons and Chemical Bonding
4. Calculate the bond energies of (a) CsF; (b) CsBr; (c) Nal;
(d) KCl. Compare your results with the experimental bond energies
given in Table 2-10.
5. Work out the ground-state term for (a) BeF; (b) BeO. Calcu-
late the bond energy of BeO, assuming ionic bonding.
6. Discuss the bond properties of the interhalogen diatomic mole-
cules— C1F, BrCl, IC1, IBr, etc.
7. Discuss the bond properties of NO, PO, AsO, and SbO.
8. Formulate the bonding in the hydrogen halide molecules (HF,
HC1, HBr, and HI) in terms of MO theory. Discuss the bond prop-
erties of these molecules.
Linear Triatomic Molecules
3-1 BeH 2
"et us investigate the molecular orbitals of BeH 2 , a very simple
linear triatomic molecule. As in a diatomic molecule, we tag
the molecular axis the £ axis (the H-Be-H line), as shown in Fig.
3-1. Beryllium has 2s and 2p valence orbitals; hydrogen has a 1j
valence orbital. The molecular orbitals for BeH 2 are formed by using
H __
Figure 3—1 Coordinate system for BeH 2 .
88 Electrons and Chemical Bonding
^£fe^-
Figure 3-2 Overlap of the hydrogen Is orbitals with the
beryllium 2*.
the 2s and 2p B beryllium orbitals and the lj orbitals of H a and HV
The proper linear combinations for the bonding molecular orbitals
are obtained by writing the combinations of ls a and lj 6 that match
the algebraic signs on the lobes of the central-atom (Be) 2s and 2p s
orbitals, respectively. This procedure gives a bonding orbital which
concentrates electronic density between the nuclei. Since the 2s orbital does
not change sign over the boundary surface, the combination
(ls a + lsb) is appropriate (see Fig, 3-2). The 2p z orbital has a plus
lobe along +£ and a minus lobe along — Z- Thus the proper combi-
nation of H orbitals is (ls a — 1j&) (Fig. 3-3).
We have now described the two different <r J molecular orbitals,
——ft; , u . n ,, . ; .ij U. . t .. .,.: •—. .#,
h + 2 ft - h b
Figure 3-3 Overlap of the hydrogen Is orbitals with the
beryllium 2p„.
Linear Tnatomic Molecules 89
which can be written as the following molecular-orbital wave func-
tions:
W) = Cilf + QClSa + ISb) (3-1)
#>,*) = cat* + Q(is a - ij 6 ) (3-2)
The antibonding molecular orbitals corresponding to ^(o-., 6 ) and
i/'Or/) will have nodes between the Be and the two H nuclei. That
is, we shall combine the beryllium Is with — (\s a + 1j&) and the
beryllium 2p z with — (ls a — l-st). The two a* molecular orbitals are
therefore
lft>„*) = 0.2J- - c,(lj a + Is,,) (3-3)
and
lK>.*) = Q2h ~ Cs(1j« - lJi) (3-4)
In order to describe these a b and cr* orbitals in more detail, we must
find good numerical values for the coefficients of the Be and H valence
orbitals. Though there are reasonably good approximate methods
for doing this, all are beyond the level of this book. However, since
the beryllium Is and 2p z orbitals are much less stable than the hydro-
gen Is orbitals (H is more electronegative than Be), we can confi-
dently assume that the electrons in the bonding orbitals spend more
time around the H nuclei — that is, that 2C 2 2 > Cr and 2Q 2 > Ci.
In an antibonding orbital, an electron is forced to dwell mostly in
the vicinity of the Be nucleus — that is, Q 2 > 2C 6 2 and C 7 2 > 2Ci.
(For further explanation of the relationships between the coeffi-
cients, see Problem 3-1.)
The 2p x and lp v beryllium orbitals are not used in bonding, since
they are ir orbitals in a linear molecule and hydrogen has no ir
valence orbitals. These orbitals are therefore nonbonding in the BeH 2
molecule. The boundary surfaces of the BeH 2 molecular orbitals are
given in Fig. 3-4-
3-2 ENERGY LEVELS FOR BeH 2
The molecular-orbital energy-level scheme for BeFf 2 , shown in
Fig. 3-5, is constructed as follows: The valence orbitals of the cen-
tral atom are indicated on the left-hand side of the diagram, with
go
Electrons ani Chemical Bonding
**
^rf'-
^%0* '-
^rm
+
■+(„—, — r*ftc- ;." ■..,• .• .a t w*
~*w££
.+>..".*•■■ '
feJtfe^
|pf
*-«— «.-»— H,-
Figure 3-4 Boundary surfaces of the MO's of BeHa.
Linear Trialomic Molecules
Be orbitals BeH, orbitals
H orbitals
2p's
^000^"/-- a OO SL \\
bp
g
figure 3-5 Relative orbital energies in BeH 2 .
the more stable Is level below the 2p. The Ij- orbitals of the two
hydrogens are placed on the right-hand side of the diagram. The
positioning of the Ij- hydrogen orbitals lower than either Is or 2p
of beryllium is based on EN considerations, In the middle of the
diagram are the molecular orbitals — bonding, nonbonding, and anti-
bonding. As usual, bonding levels are more stable than their com-
bining atomic orbitals, and antibonding levels are correspondingly
less stable. The 2p x and 2p y nonbonding Be orbitals are not changed
in energy in our approximation scheme. Thus they are simply
moved out into the molecular-orbital column.
The ground state of BeH 2 is found by placing the valence electrons
cji Electrons and Chemical Bonding
in the most stable molecular orbitals shown in Fig. 3-5. There are
four valence electrons, two from beryllium (2j-) 2 and two from the
two hydrogen atoms. The ground-state electronic configuration is
therefore
Qr. h y(<r, b y = ^
PROBLEM
3-1 ■ Assume that the electronic charge density is distributed in
the a b molecular orbitals as follows:
<r s <v Be, 30 per cent; 2H, 70 per cent
<r s h : Be, 20 per cent; 2H, 80 per cent
Calculate the wave functions for aj> and ai\ as well as the final
charge distribution in the BeH 2 molecule.
Solution. Since the normalization condition is f\ip\ 2 dr = 1, we
have for aj>
/K». l )l' dr = C?f(2sy dr + C?f{ls a y dr + G?f(\s h J dr
+ IdGftlsXlSa) dr + 2C 1 C 2 y(20(U) dr
+ 2C 2 2 /(lv>(U) dr = 1
If the atomic orbitals Is, ls„, and ls b are separately normalized, we
have
y |^(a/0S 2 dr = d 2 + C 2 2 + C 2 2 + overlap terms = 1
Making the simplifying assumption that the overlap terms are zero,
we have finally
f\K°f)\* dr = d 2 + 2C 2 2 = 1
The probability for finding an electron in the <r} orbital if all space
is examined is of course 1 . The equation C-? + 2C 2 2 = 1 shows that
this total probability is divided, the term Ci 2 representing the prob-
ability for finding an electron in a- a b around Be, and the term 2C7 2 2 the
probability for finding an electron in <r s b around the H atoms. Since
the distribution of the electronic charge density is assumed to be 30
per cent for Be and 70 per cent for the H atoms in <r,, 6 , the probabili-
ties must be 0.30 for Be and 0.70 for the H atoms. Solving for the
coefficients C\ and C» in a}, we find
G 2 = 0.30 or Ci = 0.548
and
2C 2 2 = 0.70 or C 2 = 0.592
Linear Tnatomic Molecules en
Similarly, we have the equation Ci -\- ICi = 1 for <r/; again solving
for coefficients on the basis of our electronic-charge-density assump-
tions,
Ci = 0.20 or C 3 = 0.447
and
Id 2 = 0.80 or C 4 = 0.632
The calculated wave functions are therefore
K°f) = (0.548)2j- + 0.592(lx o + 1x0
and
<A<>* 6 ) = (0.447)2^ + 0.632(lx o - ls h ~)
The ground-state configuration of BeHo is (o- s i, ) 2 (o-/') 2 . The distribu-
tion of these four valence electrons over the Be and H atoms is calcu-
lated as follows:
Be <rj>: 2 electrons X C? = 2 X 0.30 = 0.60
o-/.- 2 electrons X C 3 2 = 2 X 0.20 = 0.40
total 1 electron
H„ = H 6 o-A- 2 electrons X C 2 2 = 2 X 0.35 = 0.70
a}: 2 electrons X C 4 2 = 2 X 0.40 = 0.80
total 1.5 electrons per H
The BeH 2 molecule without the four valence electrons is represented
H+— Be++— H+
Introducing the electrons as indicated above, we have the final
charge distribution
-0.5 + -0.5
H— Be— H
It is most important to note from these calculations that the elec-
tronic charge densities associated with the nuclei in a normalized molecular
orbital are given by the squares of the coefficients' of the atomic orbitals (J,n
the zero-overlap approximation).
3~3 VALENCE-BOND THEORY FOR BeH 2
The molecular-orbital description of BeH 2 has the four electrons
delocalized over all three atoms, in orbitals resembling the boundary-
94 Electrons and- Chemical Bonding
surface pictures shown in Fig. 3-4 (jT s b and tr, 5 ). We may, however,
cling to our belief in the localized two-electron bond and consider
that the four valence electrons in BeH s are in two equivalent bonding
orbitals. By mixing together the 2j and 2p 3 beryllium orbitals, we
form two equivalent sp hybrid orbitals, as shown in Fig. 5~6. These
two hybrid orbitals, sp a and spb, overlap nicely with lj- a and 1^, re-
spectively, and the bonding orbitals are (see Fig. 3-7):
h = CiJp a + C 2 ls a (3-5)
^2 = Cispi, + Qlsb (3-6)
The use of equivalent hybrid a orbitals for the central atom is es-
pecially helpful for picturing the a bonding in trigonal-planar and
tetrahedral molecules.
.•i-'W.i.
S -*• *-
y<! . ■ > i'..y,.'.'B^
m —
>r
vi? : -'; r J:i ;
w
&
'■*:;i
■'■> + '■'■
jp
■^X
&pr
W??
)
2p,
/
2s
x^
: ^+^:.
2 -^
— 7*^;-^fig-
£..-i-.'.'-'.'.'.'l'.'-. (
Figure 3-6 Formation of two sp hybrid orbitals.
Linear Triatomic Molecules
95
localized electron-pair bonds
Figure 3-7 Valence bonds for BeH 2 , using two equivalent sp
hybrid orbitals centered at the Be nucleus.
PROBLEM
3-2. Show that the general molecular-orbital description of BeH 2
is equivalent to the valence-bond description if, in Eqs. (3-1) and
(3-2), Ci = C% and C 2 = &. (From the MO wave functions, con-
struct the localized functions ^i and ^ 2 .)
3-4 LINEAR TRIATOMIC MOLECULES WITH W BONDING
The C0 2 molecule, in our standard coordinate system, is shown in
Fig. 3-8. This molecule is an example of a linear triatomic molecule
in which all three atoms have ns and np valence orbitals. The 2s and
1p z carbon orbitals are used for u bonding, along with the 2p z orbitals
on each oxygen. 1 The o- orbitals are the same as for BeH2, except
that now the end oxygen atoms use mainly the 2p s orbitals instead
of the Is valence orbitals used by the hydrogen atoms. The o- wave
functions are:
lK>.») = <£* + C0-K + 2 *0 0-7)
1 The oxygen valence orbitals are If and 2f , Thus a much, bettet, approximate a-
MO scheme would include both If and If, oxygen orbitals. For simplicity, however,
we shall only use the 2p, oxygen orbitals in forming the <r MO's.
9 6
Electrons and Chemical Bonding
(3-8)
(3-9)
(3-10)
*(*.*) = Qls - Q(1K + 2?. 6 )
#>.*) = Q2p z + Q(2p Za - ip^
*0,*) = Qlpz - C g (2pz a - 2p Zb ~)
The ir molecular orbitals are made up of the 2p x and 2p y valence
orbitals of the three atoms. Let us derive the w x orbitals for CC>2.
There are two different linear combinations of the oxygen 1p x
orbitals:
2p Xa + 2p Xh (3-11)
2p Xa - 2p Xb (3-12)
The combination (2p Xa + 2p Xb ) overlaps the carbon 2p x orbital as
shown in Fig. 3-9- Since x and y are equivalent, we have the follow-
ing tt'' and 7r* molecular orbitals:
tfjrJO = Q2p x + do(2^ o + 2p Xb ) (3-13)
KV) = Q2p y + Ci»(2p ya + 2p Vb ) (3-14)
lK>**) = Cnlp* ~ Cn(2p Xa + 2p Xb ~) (3-15)
tf<V s *) = C u 2& - C 12 (2^ o + 2^) (3-16)
x
O. — -s—
Figure 3-8 Coordinate system for COa.
Linear Triatomic Molecules
97
.•"•itf/".^-';*- •'■■
v--p-
■■^■:
-r-.O-
2 fc,
ii
., .- .+ _
I /
no net overlap
.+.
-C-
..). I
%'•:*
no net overlap
Figure 3-9 Overlap of the 2pz orbitals of the carbon atom
and the two oxygen atoms.
The combination 0-p x ,— 2p^) has zero overlap with the carbon 2p x
orbital (see Fig. 3-9), and is therefore nonbonding in the molecular-
orbital scheme. We have, then, the normalized wave functions
lK^) = yj&P** - 2 iO
(3-17)
and
Electrons and Gwmkal Bonding
1
f(x») = ^f 2 P»* ~ 1 Pv b ')
(3-18)
The boundary surfaces of the MO's for CO2 are shown in Fig. 3-10.
The MO energy-level scheme for C0 2 is given in Fig. 3-11. Notice
-o-
<r<
£;*::
: : *i
1
. -. : • '
^+\i
O
1
1
■
■
1
:$$l
,' 1
5^/:
—
.'- ~ • . ' •'.
! ..+
& . . ' . -
-". ' !
'". ',■■■*
v- ! aft ! r.~r.
■ .j.::,,J»
7*-"— "
*c-*
-ffl:;;<;\;:
- ,V si. ...gfrS.
■■ ',-...
-G;-
v^;v-,.qr^, ; -;;
%iv" 1 '
.V::. : .-1':Y.. :; .'
' 1 *•:
■ 1 '- ; ' j '
"> ! '
w [ "f
:;* ' 3
;:': ^A
«t
,:■:
■$<£?:.
«i*
*#i
+
•'*&•:.
-o-
-p-
<m?
tt„ 5 , ttj,', and TT a are equivalent to tJ', t«*, and t x
Figure 3-10 Boundary surfaces of the MO's of CO2.
Linear Triatomic Molecules
99
C orbitals
C0 2 orbitals
O orbitals
-OOO^f''/
/ i\ -co—
t >
\ \ —CO — i}
* I
V—o^
2/,
a
Figure 3-11 Relative orbital energies in C0 2 .
that the oxygen orbitals are more stable than the carbon orbitals.
There are 16 valence electrons (C is IMf; O is 2s 2 2f) to place in the
levels shown in the scheme. The ground, state of C0 2 is therefore
ICO
Electrons and Chemical Bonding
1
Figure 3-12 Valence-bond structures for CO2.
There are four electrons in a b orbitals and four electrons in tt* orbitals.
Thus we have two cr bonds and two ir bonds for CO2, in agreement
with the two valence-bond structures shown in Fig. 3-12.
3-5 BOND PROPERTIES OF C0 2
The C — O bond distance in carbon dioxide is 1.162 A, longer than
the C — O bond distance in carbon monoxide. These bond lengths
Linear Triatomic Molecules 101
are consistent with the double bond (C=0) bet-ween C and O in CO2
and the triple bond (C=0) in CO.
There are two types of bond energies for C0 2 . The bond-dissocia-
tion energy, which we discussed in Chapter II, refers to the breaking
of a specific bond. In C0 2 , the process
DE
O^C^O > CO + O (3-19)
represents the dissociation of one oxygen from carbon dioxide,
leaving carbon monoxide; this DE is 127 kcal/mole. However, the
average C — O bond energy in C0 2 is obtained by completely splitting
C0 2 into ground-state atoms, breaking both C — O bonds:
O^C^O > C + O + O (3-20)
The average C — O bond energy (BE) is then one-half the value of E
in Eq. (3-20). Obviously E is the sum of DE(C0 2 ) and DE(CO),
DE(C0 2 ) DE(CO)
O— C— O — — -^ C— O + O — 4 C + O + O (3-21)
E = DE(C0 2 ) + DE(CO) = 127 + 256 = 383 kcal/mole
and
I = BE(C0 2 ) ^ 192 kcal/mole (3-22)
We shall use the abbreviations BE and DE in the bond-energy tables
in this book.
The ground states, bond lengths, and bond energies for a number
of linear triatomic molecules are given in Table 3-1-
3-6 IONIC TRIATOMIC MOLECULES : THE ALKALINE EARTH HALIDES
Molecules composed of atoms of the alkaline earth elements (Be,
Mg, Ca, Sr, Ba) and halogen atoms are probably best described with
the ionic model, since the electronegativity differences between alka-
line earth and halogen atoms are large. Thus we picture the bonding
as X~ — M++ — X~. Let us illustrate bond-energy calculations for
molecules of this type, using CaCl 2 as an example.
IOZ
Electrons and Chemical Bonding
Table 3-1
Properties of Linear Triatomic Molecules 8
Molecule
Ground
state
Bond
Bond
length, A
Bond energies,
kcal/mole
BeBr a
X E
Br Be- Br
Be- Br
89(BE)
BeCl 2
'D
ClBe-Cl
Be- CI
1.74
147(DE)
109(BE)
Bel 2
X S
IBe-I
Be- 1
69(BE)
C0 2
'E
oc-o
c-o
1.162
127(DE)
192(BE)
COS
x s
oc-s
1.561
128(DE)
cs 2
1 s
sc-s
c-s
1.554
128(BE)
CSe 2
J E
C-Se
112(BE)
CaCl 2
X S
CICa-Cl
Ca-Cl
2.54
176(DE)
113(BE)
CdBr 2
X S
BrCd-Br
2.39
76(DE)
CdCl 2
X E
CICd-Cl
2.23
84(DE)
Cdl 2
X E
ICd-I
2.58
50(DE)
HCN
J S
HC-N
H-CN
1.153
1.066
207 (DE)
114(DE)
HgBr 2
X E
BrHg-Br
Hg-Br
2.43
72(DE)
44(BE)
HgBrI
J £
BrHg-I
64(DE)
HgCl 2
X E
CIHg-Cl
Hg-Cl
2.30
81(DE)
54(BE)
HgClBr
l E
BrHg- CI
77(DE)
HgClI
X S
IHg-Cl
CIHg-I
75(DE)
63(DE)
(continued)
Linear Triatomic Molecules
Table 3-1 (continued)
103
Ground Bond Bond energies,
Molecule state Bond length, A kcal/mole
HgF 2
X E
FHg-F
Hg-F
Hgl 2
x £
IHg-I
Hg-I
2.60
N<V
l £
N-O
1.10
MgCl 2
X S
ClMg- CI
Mg-Cl
2.18
SiS 2
X S
Si-S
ZnCl 2
x £
ClZn-Cl
2.12
Znl 2
X E
IZn-I
100
DE)
66
BE)
60
DE)
35
BE)
136
f DE)
99
BE)
70
(BE)
96
(DE)
53
(DE)
a Data from T. L. Cottrell, The Strengths of Chemical Bonds, Butter-
worths, Loudon, 1958, Table 11.5.1.
EXAMPLE
Our purpose is to calculate the average Ca — CI bond energy in
CaCh:
R
-Ca+
R
-Ck-
Cl„
For CaCl 2 (or any MX 2 ) there are two attractions, Ca ++ — Cl„~ and
Ca ++ — Cl 6 ~, each at a distance of R. In addition there is one repul-
sion, Cl a ~ — C1& - , at a distance of 2R. The sum of these electrostatic
terms is represented
2e 2 2e 2 , e * 3.5* 2
electrostatic energy = -— - — + — = —
The energy per bond is one-half — 3.5e 2 /-R, or — 1.15e i /R. The van
der Waals energy can be approximated again as an inert-gas-pair
interaction. In this case we have one Ar-Ar interaction for each
bond. The inert-gas-pair approximation of the van der Waals
energy is not expected to be as good for the MXj molecules as for the
104 Electrons and Chemical Bonding
MX molecules, however, owing to the small size of M ++ compared
to that of the isoelectronic inert gas atoms (see Fig. 3-13). Thus
the actual Ca ++ — CI - van der Waals repulsion energy is probably-
less than that calculated.
The final expression for the energy of each Ca ++ — CI - bond is
— 1.1 5e i d
PE = potential energy = — + be^ aR
The Ca — CI bond length in CaCl 2 is 2.54 A, or 4.82 au. On substi-
tuting the Ar-Ar parameters from Table 2-9, we have
PE = — — — + $$&*-"**?(*#> - 1
4.82 (4.82) 8
or
PE = -0.337 au = -9.17 eV
The 9.17 eV is one-half the energy required to dissociate CaCl 2 into
ions,
E'
CaCl a > Ca++ + CI- + CI- E' = -2PE
For the average bond energy BE, we have the process
CaCl 2 — ^Ca + Cl+Cl
E= E'+ 2EA(C1) - IPi(Ca) - IP 2 (Ca) and BE = -
2
With EA(C1) = 3.61 eV, IPi(Ca) = 6.11 eV, IP a (Ca) = 11.87 eV,
and E' = 18.34 eV, we obtain E = 7.58 eV or 175 kcal/mole and
Ar K+ Ca 2 +
Figure 3-13 Relative effective sizes of Ar, K + , and Ca a
Linear Triatomic Molecules 105
BE(Ca — CI) ^ 88 kcal/mole. This calculated value of 88 kcal/
mole may be compared with the experimental value of 113 kcal/
mole. We see that the ionic model for CaCU is not as good as the
ionic model for the alkali halides. This is evidence that the alka-
line earth halides have more "covalent character" than the alkali
halides. Thus, it is likely that there are important covalent-bond
contributions to the bond energy of CaCl 2 .
Experimental bond energies for a number of alkaline earth halides
are given in Table 3-1.
SUPPLEMENTARY PROBLEMS
1. Work out the ground-state term for the molecule N 3 .
2. Calculate the Be — CI bond energy in BeCU- The value of
IP 2 (Be) is 18.21 eV.
3. Discuss the bonding in C0 2 , CS 2 , and CSe 2 in terms of MO
theory. Compare the bond properties of these molecules.
Trigonal-Planar Molecules
4-1 BF 3
^oron trifluoride has a trigonal-planar structure, with all F — B — F
bond angles 1 120°. Boron has Is and 2p orbitals that bond with
the fluorine Is and 2p orbitals. A convenient coordinate system for a
discussion of bonding in BF 3 is shown in Fig. 4-1.
We need only one <r valence orbital from each fluorine. We shall
use in the discussion only the 2p orbital, since the molecular orbitals
derived are appropriate for any combination of 2s and 2p. However,
it is probable that the very stable fluorine 2s orbital is not appre-
ciably involved in the a bonding. The ionization potential of an
electron in the 2s orbital of fluorine is over 40 eV.
4-2 a MOLECULAR ORBITALS
The <r molecular orbitals are formed using the 2s, 2p x , and 2p y
boron orbitals, along with the 2p Za , 2p % , and 2p Zc orbitals of the
fluorine atoms. We must find the linear combinations of 2p z , 2p Zb ,
and 2p Zc that give maximum overlap with 2s, 2p x , and 2p y . The
1 Bond angle is a commonly used term, meaning the angle between "internuclear
lines."
106
Trigonal-Planar Molecules
107
Figure 4-1 Coordinate system for BF 3 .
boron Is orbital is shown in Fig. 4-2. The combination (2p Za +
2 P* h + tyO overlaps the Is orbital. Thus the molecular orbitals
derived from the boron 2s orbital are (using the shorthand &, =» 2p z ,
Zb = 2pz b , and $, = 2^ c ):
#t>£) = C{2s + QQZa + «, + &) (4-1)
iK>/D = G&f - C 4 (^ + S + &) (4-2)
^ The boron 2p s orbital is shown in Fig. 4-3. The combination
0& — &) matches the positive and negative lobes of 2p v . The molec-
ular orbitals from 2f v are ;
* QrS - C 6 2fo + C 6 fe - * c ) (4-3)
iKo"v*) = c-ityv ~ c ^xi - O (4-4)
The boron 2p x orbital is shown in Fig. 4-4. A combination
Qt* —Kb— %d correctly overlaps the lobes of 2p x . There is a minor
complication, however: the overlaps of z a , $, and z* with 2p x are not
the same. Specifically, z°. points directly at the positive lobe of 2p x ,
io8
Electrons and Chemical Bonding
*- v
+
Is + z^ z b + z c
Figure 4-2 Overlap of the boron 2 s orbital with the 2p,
orbitals of the fluorine atoms.
whereas Zb and z_ c are 60° displaced from a comparable overlap with
the negative lobe. In order to relate Za to & and Zc we must find the
fraction of 2p x that can be resolved along the Zb line. This fraction is
simply cos 60°, of ^. We deduce that the sum fy, + Zc gives the same
overlap with 2p x as & does alone. Then the proper combination is
(&a ~ %Zb ~ 2ZJd, and the a molecular orbitals from 2p x are:
$(?/) = Q2p x + CioOfc, - \Zb - k<=)
■K<r x *) = Culpx - Cn(Xa - ks — k c )
(4-5)
(4-6)
Trigonal-Planar Molecules
109
: ::*-': ; .V.
%?sv
rt^trf
\' : i:>;p;. : : :">;■'
£feH
-+-J'
no net o verlap between 2^ + £
X
2p s + ^ - s.
Figure 4-3 Overlap of the boron 2pj, orbital with the 2p 2
orbitals of the fluorine atoms.
4-3 H" MOLECULAR ORBITALS
The ir molecular orbitals are formed using the boron 2p„ orbital and
the 2py orbitals of the fluorine atoms. The combination (j a + yb +
jy c ) matches the 2p z orbital, as shown in Fig. 4-5- Thus the bonding
and antibonding x molecular orbitals are:
#>■.*) = 6*2* + C u (j a + y„+ jc) (4-7)
\ffafi = QsZf $ - C ie (j a +y b + juD (4-8)
no
Electrons and Chemical Bonding
X
■H?;
-*~y
2p i + z^~ z b - s c
Figure 4—4 Overlap of the boron 2p x orbital with the 2p s
orbitals of the fluorine atoms.
Since we started with three fluorine 2p y orbitals, there are two
more independent linear combinations of y a , J&> and y . One satis-
factory pair is (j a — j c ) and (j a — 2y b + j c ). As shown in Fig. 4-6,
these orbital combinations do not overlap the boron 2p z orbital.
Thus they are nonbonding in BF3, and we have
lK>i) = 7sG*» ~ yd
K^i) = tt^J" ~ 2 y> + yd
(4-9)
(4-10)
Trigonal-Planar Molecules
ft
t
in
t
t
rrf,
2 P, + y. + a + ft
Figure 4-5 Overlap of the boron 2p 2 orbital with the 2p v
orbitals of the fluorine atoms.
4-4 ENERGY LEVELS FOR BF3
The molecular-orbital energy-level scheme for BF 3 is shown in
Fig. 4-7. The fluorine valence orbitals are more stable than the
boron valence orbitals, and so electrons in bonding molecular orbit-
als spend more time in the domain of the fluorine nuclei. The <r x
and tx y molecular orbitals are degenerate in trigonal-planar molecules
such as BF3. Since this is by no means obvious from Eqs. (4-3),
(4-4), (4-5), and (4-6), we shall devote a short section to an expla-
nation.
112.
Electrons and Chemical Bonding
&?b /
^
+ >F,
overlaps have opposite
signs
therefore net
overlap = ._";•'. +■
■w
X
■+~y
y«-y*
— !— , *_ y
net overlap =
% - 2 n + y<
Figure 4-6 Two combinations of the fluorine 2p„ orbitals
that have zero overlap with the boron 2jp a orbital.
4-5 EQUIVALENCE OF <T X AND <T y ORBITALS
The total overlap of the normalized combination Vf(^ a — $$, —
$&) with If* will be called S(v^); the total overlap of (1/V2)
0& ~~ &) with %i w iH b e called i"(0- -A- rffrS* «r overlap, such as the
overlap between %_ a and 2p x (Fig. 4-8), will be called S($,,pJ). To
evaluate S(a^) and ifo) in terms of SQp^p^), we use the following
calculations:
$>*) = vf f(ip x x?* - £& - k£ ^
= Vlt^fa flO + I cos 60°^O„ p r ~) + | cos 60° JO,, #«•)]
= Vf(f) [J(^, j,)] = Vf JTjfc, j>0 (4-11)
%r) = -^fQ-PvXZb - &3*
= 4|[cos 30° J^ #,) + cos 30° %,, ^)]
= ^2 (^ + ^r) 1 ^" *^ = Vfi,(? " ^ (4_12)
Since the overlaps are the same in <r :E and a v , and since the com-
Trigonal-Planar Molecules
113
B orbitals
BF 3 orbitals
MQQ^^
F orbitals
- -- _- _^
U
1 1,
"I
' '.
> 1'
' 1'
.'i
/
1 > 1
1 1 1
V-OCF-'/
1 I
2 P>
<XX>
000 —
000 —
2s.
2fc
i(>-
Figure 4-7 Relative orbital energies in BF 3 .
billing boron and fluorine valence orbitals have the same initial
energies, it follows that tr x and a v are degenerate in trigonal-planar
molecules. However, it is worth pointing out that <r* and a y are not
necessarily degenerate if the bond angles deviate from 120°.
ii4
Electrons and Chemical Bonding
-*~y
overlap of z a with 2p^
Figure 4-8 Standard two-atom er overlap between p orbitals.
4-6 GROUND STATE OF BF3
There are 24 valence electrons in BF 3 [7 from each fluorine (Is^lf),
3 from the boron (l^lp)] . Placing these electrons in the most stable
molecular orbitals, we obtain a ground-state configuration:
(2 Ja X2 Jt ) 2 (^) 2 C^) 2 C^) 2 C0 2 (^) 2 ^i) 2 (T2) 2 (2^ a ) 2 (2^) 2 C2^ c ) 2
s=
There are six electrons in a* orbitals to give a total of three <r bonds
for BF 3 ; in addition, the two electrons the t$ orbital indicate one w
Trigonal-Planar Molecules 115
bond. The B — F bond length in BF 3 is 1.291 A; the B — F bond
energy is 154 kcal/mole.
4~7 VALENCE BONDS FOR BF 3
The valence-bond description of the ground state of BF3 is com-
parable to the molecular-orbital description. Three equivalent sp 2
hybrid orbitals are formed first by mixing together the Is, 2p x , and
2p y boron orbitals, as shown in Fig. 4-9. Each j^ 2 hybrid orbital has
one-third s and two-thirds p character. These three sp 2 orbitals are
then used to make three electron-pair a bonds with the lp z fluorine
orbitals. In addition, the 2p z boron orbital can be used to make a t
bond with any one of the three fluorine 2p y orbitals. Thus there are
three equivalent resonance structures for BF 3 , as shown in Fig. 4-10.
Notice that the three resonance structures move the electron-pair
7r bond around the "ring"; this is analogous to having two electrons
in the delocalized -ir z h molecular orbital.
PROBLEM
4-1. Construct the wave functions for the three equivalent sp 2
hybrid orbitals.
Solution. It is convenient to use the coordinate system shown in
Fig. 4-1, directing the three sp" 1 hybrid orbitals at atoms a, b, and c.
The s, p x , and p y orbitals are used to form the sp 2 orbitals. Each
hybrid orbital has one-third s character. Of the two p orbitals,
only the p x is used to bond with atom a (j) y has zero overlap with a).
Since each sp 2 orbital has two-thirds p character, the wave function
for sp a 2 is
K'pS) = V*j + Vip x
The remaining third of the p x orbital is divided equally between b
and c. Since the p y orbital has not been used as yet, and since it
overlaps equally well with b and c, we split it up between b and c
to complete the two-thirds p character in spt 2 and sp 2 . Choosing
the algebraic signs in the functions so that large and equal lobes
are directed at b and c, we have:
K'pfi = Vfr - Vip x + V%,
Kspfi = Vf, - Vlp x - V\p„
nS
"Electrons and Cnemkal Bonding
■ 2s
-*- y
2p,-
: 'mixing
*p;
-*-y
W
Figure 4-9 Formation of three sp 2 hybrid orfaitals.
Trigonal-Planar Molecules
117
OW*"a~>
Figure 4-10 Valence-bond structures for BF 3
The boundary surfaces of sp^, sp^, and sp? are shown in Fig. 4-9.
Bonding orbitals are combinations of the sp* orbitals and appro-
priate cr orbitals of atoms a, b, and c:
4-8 OTHER TKIGONAL-PLANAK. MOLECULES
Elements in the boron family are the central atoms in many tri-
gonal-planar molecules. Also, several important molecules and
complex ions containing oxygen have trigonal-planar structures,
among them S0 3 , NOr, and C0 3 2 -. Bond properties of a number of
trigonal-planar molecules are given in Table 4-1. The BH 3 mole-
cule, which is presumably trigonal planar, is more stable in a dimeric
form,
u8 Electrons and Chemical Bonding
Table 4-1
Properties of Trigonal- Planar Molecules 3
Bond
Bond energy (BE),
Molecule
Bond
length, A
kcal/mole
BF 3
B-F
1.291
154
BC1 3
B-Cl
1.74
109
BBr 3
B-Br
1.87
90
BH 3
B-H
93
B(CH3) 3
B-C
1.56
89
Al( 0^)3
Al-C
61
B(OR) 3 b
B-OR
1.38
128
S0 3
S-O
1.43
104
NO3-
N-O
1.22
co 3 2 -
C-O
1.29
B0 3 3 "
B-O
1.38
"Data from T. L. Cottrell, The Stre.
worths, London, 1958, Table 11.5.1.
b R = CH 3 or C 2 H 5 ; R = H, 1.36A.
s of Chemical Bonds, Butter-
BH 3 + BH 3 -» B 2 H 6
The bonding in diborane B2H6 is described in a number of other
sources. 1
The B(CH 3 ) 3 and A1(CH 3 ) 3 molecules have trigonal-planar parts.
C
B
ind
C
I
Al
c c c c
The structure around each carbon is tetrahedral, as will be described
in Chapter V.
1 See, for example, F. A. Cotton and G. Wilkinson, Advanced Inorganic Chemistry>
Wiley-Interscience, New York, 1962, pp. 200-203: W. N. Lipscomb, Boron Hydrides,
Benjamin, New York, 1963, Chap. 2; C. J. Ballhausen and H. B. Gray, Introductory
Notes on Molecular-Orbital Theory, Benjamin, New York, 1965, Chap. 7.
Trigonal-Planar Molecules
lie
SUPPLEMENTARY PROBLEMS
1 . In most cases it is convenient to have a normalized linear combi-
nation of orbitals to bond with a central atom. For example, the
combination appropriate for 2s in a trigonal-planar molecule is
0&» + Zb + Zc)- The normalized combination is —i=(Xa + Zj> + &).
V3
Normalize the combinations (%, — £„) and (& — §&, — §&).
2. Show that the molecular-orbital and valence-bond descriptions
of a bonding in a trigonal-planar molecule are equivalent, if, in Eqs.
(4 = 1), (4-3), and (4-5), Q = Q = Q and C = V3C2 = V2C 6 =
VfCio. In general, do you expect that C x = C 5 ? Q = C 9 ? V3C2 =
V2C<? VlQ = VfCio? Explain.
V
Tetrahedral Molecules
5-1 CH 4
The methane molecule, CH 4 , has a tetrahedral structure. This
structure is shown in Fig. 5-1. With the carbon in the center of
the cube, the hydrogens are then placed at opposite corners of the
cube, as defined by a regular tetrahedron. The origin of the rec-
tangular coordinate system is chosen at the center of the cube, with
the x,y, and z. axes perpendicular to the faces. All the carbon valence
orbitals, 2s, 2p x , 2p y , and 2p z , must be used to form an adequate set of
<T molecular orbitals.
The overlap of the four Is hydrogen orbitals with the carbon 2s
orbital is shown in Fig. 5-2. The linear combination (lj a + ls b +
ls c + Ijvj) is appropriate. The bonding and antibonding molecular
orbitals are:
#>,») = COs + C s (1jv, + ls„ +-' lsc + 1 J d ) (5-1)
i/<«r s *) = Q2s - Q(lsa + 1st + U + ls d ~) (5-2)
The overlap of the four Is orbitals with the carbon 2p t orbital is
shown in Fig. 5-3- Hydrogen orbitals Lr„ and lsb overlap the
plus lobe, and orbitals ls c and ls,i overlap the minus lobe. Thus the
proper combination is (ls a + lsb — ls c — Isd).
X2JO
Tttraheiral Molecules
in
*-?
Figure 5-1 Coordinate system for CH 4 .
The 2p s and 2p y carbon orbitals overlap the four hydrogen orbitals
in the same way as 2p z . This is shown in Fig. 5-4. The linear com-
binations are (ls a + 1j,j — ls b — ls c ) with 2p y , and (ls a + Lfj, —
1^6 — lsa) with lp x . The molecular orbitals are given below.
lK>,*) = CtZf* + ^CU + U - lSc ~ lSi) (5-3)
$>**) - &2& ~ a(lj- + U - 1*» - lJd) (5-4)
lK«V*) - C.2A, -I- CwCLr. + )t« - U - Ud (5-5)
^C%*3 — Cu2p y — Cih(1jo + Ijvj — If* — lj e ) (5-6)
tfj?V) = Caljp* + Cutis* + U» - U, - lug (5-7)
f (ffx*) = C 1( 2j» - C w (1j + 1a - ls b - 1st) (5-8)
112-
EUctrons and Chemical Bonding
y$-:
:H;
i
^^H,
-*-y
'■ • «,
:x •.••: i'
k + k + k,
Figure 5-2 Overlap of the carbon 2 s orbital with the Is
orbitals of the hydrogen atoms.
5-2 GROUND STATE OF CH 4
The molecular-orbital energy-level scheme for CH 4 is shown in
Fig. 5-5. The <j x , o- v , and <r z orbitals have the same overlap in a tetra-
hedral molecule afld are degenerate in energy. This is clear from the
overlaps shown in Figs. 5-3 and 5-4.
There are eight valence electrons in CH 4 because carbon is Is^lf 2
and each of the four hydrogens contributes a Ij- electron. Thus the
ground state is
There are four <r bonds in CH 4 . The average C — H bond energy is
99.3 kcal/mole. The C— H bond length in CH 4 is 1.093 A.
5 _ 3 THE TETBAHEDKAL ANGLE
The H— C— H bond angle in CH 4 is 1Q9°28'. We can calculate the
tetrahedral angle by simple trigonometry. First, we place the CH 4
Tetrahedral Molecules
i rj
'--'.-^r---:d-
U, I h. h - h d
Figure 5—3 Overlap of the carbon %p, orbital with the Is
orbitals of the hydrogen atoms.
K + h d - h t - h.
h a + h c - h b - h d
Figure 5-4 Overlap of the carbon %p x and 2p y orbitals with
the 1$ orbitals of the hydrogen atoms.
124
Electrons <mi Chemical Bonding
C orbitals
CH. orbitais
H orbitals
hi
1-1
a
moo-;
-ooo^
^-KXXP' .'
Figure 5-5 Relative orbital energies in CH4.
molecule in a unit cube, as shown in Fig. 5-6. The lengths of the
sides defining the H a — C — H<j angle d are obtained by using the
Pythagorean theorem. Thus we have the result
cos T
V3
3
or B= 109°28'
(5-9)
Tttmhedral Molecules
1x5
Figure 5-6 Unit-cube model for evaluating the tetrahedral
angle.
5-4 VALENCE BONDS FOR CH4
Four equivalent valence orbitals centered on carbon can be con-
structed by scrambling together the Is, lp x , lp y , and 1p, orbitals.
These equivalent orbitals are called .r^ 3 hybrids, and their construc-
tion is shown schematically in Fig. 5-7. Each sf hybrid orbital has
one-fourth j- character and three-fourths p character.
The four sp s orbitals are directed toward the corners of a regular
tetrahedron, and thus are ideally suited for forming four localized
bonding orbitals with the four hydrogen lj- orbitals. The valence-
bond structure for CH 4 is shown in Fig. 5-8.
PROBLEM
5-1. The normalised wave functions for the four equivalent jft 3
hybrid orbitals are listed below (coordinate system as shown in
Fig- 5-7):
Kspi 3 ') = is + VJ(-^ -p y + p,-)
ii£
Electrons and Chemical Bonding
sf? hybrid orbitafs
Figure 5-7 Formation of four sp s hybrid orbitals.
<sp/) = h + vfe$& - fy - p$
<&) = h + Vi(-^ + Pv - ;,)
Show how these orbitals are obtained by following the procedure
used to solve Problem 4-1.
*■}$*,•■'■-
■ ~a
/
■■Wit?**..
i
•;;:;..■.• :■. .-v
•• .:-.;-'-;'Vi;.j'.v.
1
1
1
♦>
i
i
j
1
1
1
:'4W
« .; +
I
1 •'•*•*.
:-;^0- -
f^V^-H
;«£B
';■■:■:'::■■■'
■>-'■:.::■>■■■■
"C '
1
1 ^ i'
'^^
vf.
'
l • '
*'
Figure 5-8 Valence-bond structure for CH 4 ,
Tetrahedral Molecul
es
J.1SJ
5~5 OTHER TETRAHEDEAL MOLECULES
Members of the carbon family (carbon, silicon, germanium, tin,
and lead) readily form four a bonds with four adjacent atoms. The
resulting molecules invariably have a tetrahedral structure around
Table 5-1
Properties of Tetrahedral Molecules 8
Bond
Bond energy (BE),
Molecule
Bond
length, A
kcal/mole
CH 4
Cttr-H
1.093
101(DE)
C-H
99.3
CF 4
C-F
1.36
116
CC1 4
C-Cl
1.761
78.2
CBr 4
CBr 3 -Br
1.942
<50(DE)
SiH
Si~H
1.480
76
SiF 4
Si-F
1.54
135
SiCl 4
Si- CI
2.02
91
SiBr 4
Si- Br
2.15
74
sn
Si-I
2.43
56
Si(CH 3 ) 4
Si-C
1.93
72
Si(C 2 H 5 ) 4
Si-C
60
GeCl 4
Ge~Cl
2.08
81
GeBr 4
Ge-Br
2.32
66
Ge\
Ge-I
2.48
51
SnCl 4
Sn-Cl
2.30
76
SnBr 4
Sn — Br
65
Sn(CH 3 ) 4
Sn-C
2.18
Sn(C 2 H 5 ) 4
Sn-C
54
Pb(CH 3 ) 4
Pb-C
2.30
Pb(C 2 H 5 ) 4
Pb-C
31
S<V
S-O
1.49
cio 4 -
Cl-O
1.44
NH 4 *
N-H
1.03
BH 4 "
B-H
1.22
BF 4 -
B-F
1.43
a Data from T. L. Cottrell, The Strengths of Chemical Bonds, Butter-
worths, London, 1958, Table 11.5.1.
n8 Electrons and Chemical Bonding
the central atom. The bonding in these molecules involves the use
of one s and three p valence orbitals by the central atom, and of an
appropriate valence orbital by each of the four surrounding atoms.
A number of important oxyanions have a tetrahedral structure,
among them S0 4 2_ and C10 4 _ . Properties of a representative group
of tetrahedral molecules are given in Table 5-1.
SUPPLEMENTARY PROBLEMS
1 . Describe the bonding in CF 4 in terms of molecular orbitals, and
construct a molecular-orbital energy-level diagram. Around which
nucleus or nuclei do the electrons spend more time in the a b orbitals?
Do you expect any partial ionic character in the C — F bonds? What
is the dipole moment of CF 4 ? Why?
2. Under what conditions are the molecular-orbital and valence-
bond descriptions of bonding in CH 4 the same? From Eqs. (5-1),
(5-3), (5-5), and (5-7), construct the valence-bond bonding func-
tions that are shown in Fig. 5-8.
3. What is the structure of BH 4 -? ofNH 4 +? Are the CH 4 orbitals
appropriate for these molecules? Discuss the partial ionic character
you might expect in the B — H, C — H, and N — H bonds. Make an
estimate of the coefficients in Eqs. (5-1) through (5-8) that might
be expected for the BFLr, CH 4 , and NH 4 + molecules.
VI
Trigonal-Pyramidal Molecules
6-1 NH 3
A familiar example of a trigonal-pyramidal molecule is ammonia,
NH 3 . The NH 3 molecule is shown in Fig. 6-1. The three hy-
drogens, which are bent out of the x,y plane, form the base of a tri-
gonal pyramid that has the nitrogen at the apex. Each N — H makes
an angle 6 with Z- In addition, N — H„ is lined up with the x axis,
and N — H 6 and N — H c make 30° angles with -\-y and —y, respec-
tively. Thus NH 3 is aligned the same way we aligned a trigonal-
planar molecule (Fig. 4-1), but with the three peripheral atoms bent
down.
Bonding in NH 3 involves the hydrogen lj- valence orbitals and the
nitrogen 2s and 2p valence orbitals. Let us ignore the 2s nitrogen
orbital for the moment, and consider only the 2p-ls bonding.
The overlap of the three hydrogen lj- orbitals with the nitrogen 2pz
orbital is shown in Fig. 6-2. The correct combination of Is orbitals
is (1j- + 1J& + l-O- The <j z molecular orbitals are:
lK>.*) = Q2p z + C 2 Oa + 1st + U) (6-1)
$>.*) = c ^h - C 4 (U, + 1j* + 1j„) (6-2)
The overlap of 2p y with \s\, and \s c is shown in Fig. 6-3- The cor-
rect combination is (lsb — ls c j- The a y molecular orbitals are:
119
Electrons and Chemical Bonding
¥-*
Figure 6-1 Coordinate system for NH,.
K°f) = capv + GsCU - IjO (6-3)
lK<%*) = C 7 2ft, - QCU - 1-jQ (6-4)
The overlap of 2p x with lj- a , l.r&, and ls c is shown in Fig. 6-4. Since
lii and 1j- c make an angle of 60° with — x, the overlap of \tb or lj- c
with 2p x is only one-half (cos 60° = £) that of ls a with 2f> x (see Sec-
tion 4-2). Thus the proper Is combination is (ls a — 1 1st — J 1j- c ).
The (Tz molecular orbitals are:
^C<rJ>') = Cap, + C 10 Q.s a - | In - | U) (6-5)
#>»*) = C u 2f x - Co(Lr„ - Hft - I U) (6-6)
6-2 OVERLAP IN ff^, (Tj,, AND CT 2
A calculation of the overlap in the cr x , <r u , and 07. molecular orbitals
is easily carried out. The direct overlap of a 2p with a Ij- valence
Trigonal-Pyramidal Molecules
13 r
liB
2 P, + K + ii s + u.
Figure 6-2 Overlap of the nitrogen 2p a orbital with the Is
orbitals of the hydrogen atoms.
orbital is shown in Fig, 6-5; this we shall denote as S(ls,2pe~). We
then proceed to express the molecular-orbital overlaps in terms of
SCffi") = S 2p* — 7=(l-fa + 1^6 + Lf<D dr
V3
= -^[cos 6 $(lj,2p,~) + cos $ S(ls, 2j>„) + cos S(ls,2p,*)]
V3
= V3 cos d S0-s,2p a ')
(6-7)
K"v) = fl-Pv — ^C 1 ^ - lJe) dr
V2
= — - [cos 30° sin ^1j,2j>„) + cos 30° sin 6 S(ls,2p g ~)]
V2
= Vf sin 9 i*(lj,2^)
(6-8)
x%t
Electrons and Chemical Bonding
y-*-
z?» + h - k
Figure 6-3 Overlap of the nitrogen 2p„ orbital with the Is
orbitals of the hydrogen atoms.
K*$ = Sty ^iO-Sa - § U - i U) dr
= V% [sin 6 S(\s,lp^ + cos 60° sin 8 SQ.s^p,') + cos 60° sin
= Vf sin 8 SCls,2p°)
It is important to note from Eqs. (6-7), (6-8), and (6-9) that c v
and u x are equivalent, and therefore their energies will be the same for
any value of 6. When B = 90°, of course, we obtain the correct over-
lap values for a trigonal-planar molecule (see Section 4-5) '
S(a s *) =
Let us now investigate the case for an H — N — H bond angle of 90°
(</, = 90° in Fig. 6-1). Calling the N— H a length unity, the other
pertinent distances given in Fig. 6-6 can be easily obtained by geom-
Trigonal-Pyramidal Molecules
133
v*
XX
Figure 6-4 Overlap of the nitrogen 2p x orbital with the Is
orbitals of the hydrogen atoms.
.+ .
^rJTN-
-xi-
S(h,2p s ]
Figure 6-5 Standard two-atom a overlap between an $ and a
p orbital.
i 3 4
Electrons and Chemical Bonding
d> = 90°
Figure 6-6 Relative distances in the NH S molecule for an
H— N— H bond angle of 90°.
«ry. We see that for <j> = 90°, cos = —^ and sin 9 = vf . Thus,
Eqs. (6-7), (6-8), and (6-9) reduce to
In other words, the <r x , %] and <r z molecular orbitals are the same for
<t> - 90°. This is no surprise, since the lp z , 2 h , and 2p, orbitals make
90 angies with each other, and for <f> = 90° the Is orbitals can be
aligned along the x,y s and z axes, as shown in Fig. 6-7. Each hydro-
Trigonal-Pyramidal Molecules
*35
m?Mi
Figure 6-7 Simple picture of the bonding in. NH S , using only
the nitrogen 1p orbitals.
gen overlaps one 2p orbital, as in Eq. (6-10)- The total overlap in
<r x , ff y , and <r z is smaller for any other angle.
6-3 THE INTERELECTRONIC REPULSIONS AND H N H
BOND ANGLE IN NH3
The actual H— N— H bond angle in NH 3 is 107°, or 17° larger than
the angle predicted for pure 2p-ls bonding. It is probable that the
mutual repulsions of the one nonbonding pair (called a lone pair) and
the three bonding pairs of electrons are responsible for the 17° angle
opening. The four electron pairs must therefore be so arranged as to
minimize these interelectronic repulsions. One way to get the three
136
Electrons and Chemical Bonding
bonding pairs farther apart is to involve the nitrogen 2s orbital in the
bonding. In Fig. 6-8 is shown the overlap of the hydrogen Is orbit-
als with the nitrogen Is. Notice that the combination appropriate
for Is (ls a + ls b + \s c ~) is the Is combination in ft [Eqs. (6-1) and
(6-2)]. Thus e s "mixes together" with 05,* and <s* to give three
molecular orbitals which we shall call a g b , a z , and &*■ This addition
of Is "character" to the N — H bonding increases the H — N — H
angle from 90° to 107°. You may think of the angle opening by in-
clusion of 2r in the following way: The best H — N — H angle for
"pure" 2p bonding is 90°. The best H — N — H angle for "pure" 2s
bonding is 120°, since the symmetrical trigonal planar structure al-
lows the best overlap arrangement for three hydrogen Is orbitals
with a 2j orbital. (The Is orbitals are as far from each other as pos-
sible and do not compete for overlap of the same portion of the 2s. )
Thus inclusion of 2s character in a "pure" 2^-bonding scheme in-
creases the H — N — H angle from 90°.
Oi
Hi:
2s + k, + u. + u
s
Figure 6-8 Overlap of the nitrogen 2s orbital with the Is
orbitals of the hydrogen atoms.
Trigonal-Pyramidal Molecuhs 137
The similar valence-bond idea, particularly appealing, is that the
bonding pairs and the lone pair are in four tetrahedral sp s orbitals.
This structure places the four electron pairs as far away from each
other as possible. The "tetrahedral" structure of NHs is shown in
Fig. 6-9. The slight deviation of the H — N — H bond angle from the
tetrahedral angle of 109° is considered a result of the nonequivalence
of the bonding and non-bonding pairs of electrons.
6-4 BOND ANGLES OF OTHER TRIGONAL-PYRAMIDAL MOLECULES
The H— P— H and H— As— H angles in PH-, and AsH 3 are 94° and
92°, respectively. This probably indicates a high degree of phos-
phorus and arsenic ^-orbital character in the three bonding orbitals.
We assume that the mutual repulsions of bonding pairs of electrons
are reduced in going from nitrogen to phosphorus to arsenic. This
is a reasonable assumption, since we know from atomic spectra that
the atomic interelectronic repulsions, in the valence $ orbitals, de-
crease in the order N > P > As. The trihalides of nitrogen, phos-
phorus, arsenic, antimony, and bismuth are trigonal pyramidal. The
bond angles are all in the 95 to 105° range, as given in Table 6-1.
:."P.\vX.
s 1
„ rv.f .-.-; - .
? \
t "
\\i :':'•'
■a/
1
f
I
1 • .
1
1
1
1
1
\.S
1
1
■
1
1
1
1 ,-■ .* -
• '.'.'•'<-.' :•%••'.•
I 'T
1 /.";;'
1
s ' ■•\'v'-**
i'-'"-*." '
1 y
'M-fi'l
t y
_ -if
lone pair '
Figure 6-9 Valence-bond structure for NH3, using sp 3
orbitals for nitrogen.
I -3 8 Electrons and Chemical Bonding
Table 6-1
Properties of Trigonal- Pyramidal Molecules 3
AB 3
B-A-
■B
AB bond
AB bond energy (BE),
molecule
angle, deg
length, A
kcal/mole
NH3
107
1.014
93.4
NF 3
103
1.37
65
NCI3
46
PH 3
94
1.42
77
PF 3
104
117
PCI3
100
2.04
78
PBr 3
100
2.20
63
PI3
2.47
44
AsH 3
92
1.52
59
AsF 3
102
1.71
111
AsCl 3
98
2.16
70
AsBr 3
2.33
58
As^
2.54
43
SbCl 3
104
2.48
67
BiCl 3
2.48
67
Bi(CH3) 3
2.30
31
"Data from T. L. Cottrell, The Strengths of Chemical Bonds, Butter-
worths, London, 1958, Table 11.5.1; L. E. Sutton (ed.), "Interatomic Dis-
tances," Special Publication No. 11, The Chemical Society, London, 1958.
6-5 GROUND STATE OF NH 3
The molecular-orbital energy-level scheme for NH 3 is shown in
Fig. 6-10. The a x and <j v orbitals are degenerate. The eight valence
electrons give a ground-state configuration of
(VOV^WXO 2 s = o
There are three a bonds. The N — H bond length is 1.014 A, and the
average N — H bond energy is 93-4 kcal/mole. The electrons in the
bonding orbitals spend more time around nitrogen than around the
hydrogens. This means that in the ground state the nitrogen has a
small negative charge and the hydrogens carry a small positive
charge. Thus there are three bond difoles, as shown in Fig. 6-11.
These three bond dipoles add vectorially to give NH 3 a net dipole
N orbitals
NH, orbitals
H orbitals
4
1 1
/ ,v-oo^O
' /I
/ f
- K
I 1 '
1 * r»
1 i<
i \ \
t/\ ^0&-' ;
<,2j
a
■o-
Figure 6-10 Relative orbital energies in NH3.
add bond
*1\
H*
1.46 D ^S»h
total dipole moment
H'
1
H
H
Figure 6-11 Contributions to the dipole moment of NHs.
J 39
140
Electrons and C\
nanicai
bonding
Table 6-2
Dipole Moments of Some Trigonal- Pyramidal Molecules 8
Molecule
Dipole moment, D
NH3
NF 3
PH 3
PF 3
PCI3
PBr 3
AsH 3
AsF 3
AsCl 3
AsBr 3
Aslj
SbCl 3
SbBr 3
SbL,
.47
.23
.55
.03
.79
.61
.15
.82
.99
.67
.97
.93
.48
.59
a Data from A. L. McClellan, Tables of Experimental Dipole
Moments, Freeman, San Francisco, 1963:
moment. The total dipole moment, 1.46 D, also includes a contri-
bution from the lone-pair electrons in a z , as indicated in Fig, 6-11.
Dipole moments for a number of trigonal-pyramidal molecules are
given in Table 6-2.
SUPPLEMENTARY PROBLEMS
1. Why is the dipole moment of NH 3 larger than the dipole mo-
ment of PH 3 ? Why is the dipole moment of PF 3 larger than that of
PC1 3 ?
2. What structure would you expect for CH 3 ~ and H 3 0+. Discuss
the bonding in these molecules.
Angular Triatomic Molecules
7-1 H 2
The most familiar angular triatomic molecule is water, H 2 0.
The H — O — H bond angle in the water molecule is known to be
105°- We can conveniently derive the molecular orbitals for H 2
by placing the oxygen atom at the origin of an xyx. coordinate system.
The two hydrogens are placed in the x& plane, as shown in Fig. 7-1.
Imagine starting with a linear H — O — H along the % axis and bend-
ing the two hydrogens toward the x axis, until the H — O — H angle 9
corresponds to the observed 105°. It is convenient to bend each
hydrogen the same amount from the z axis, so that the x axis bisects
8. We can go through this procedure for any angular triatomic mole-
cule, independent of the value of d. Thus the a molecular orbitals for
H 2 are a representative set.
The valence orbitals involved are Is and 2p for oxygen and lj- for
hydrogen. The overlaps of the 2p orbitals with the two hydrogen
.Is orbitals are shown in Fig. 7-2. From these overlaps, we can write
the following set of wave functions:
#>**) = Q2p x + QQls a + l Sb ) (7-1)
*Ox*) = C,2p x - Clls a + l Sb ) (7-2)
lK<r,») = Q2p, + Q(ls a - 1st) (7-3)
</<<r s *) = Q2p, - Q(ls a - U) (7-4)
141
142-
Electrons awl Chemical Bowling
*•$
Figure 7-1 Coordinate system for HjO.
The 2jj, oxygen orbital has no overlap with either Lr a or 1ft, and
thus it is nonbonding in our scheme. Notice that 2f y is available for
ir bonding, but hydrogens do not have ir valence orbitals.
The overlap of 2s with ls a and ls b is shown in Fig. 7-3. The com-
bination (1j- + 1st), which was used in the <r x orbitals, is correct for
Is. This means that a s mixes with <r x . The result is three molecular
orbitals — a bonding orbital, an orbital that is nearly nonbonding, and
an antibonding orbital. We shall call these orbitals <r t b , <r x , and ay*,
respectively.
The molecular-orbital energy-level scheme is shown in Fig. 7-4,
with the hydrogen lj- orbital placed above the oxygen 2s and 2p
valence orbitals. The <r} molecular orbital is seen to be more stable
than the <r x , owing to the interaction of <r x with <rh
Angular Triatomic Molecules
X
2p, + h, - h t
2p x ■+ fc, + h b
J 43
Figure 7-2 Overlap of the oxygen 2p g and 2p;c orbitals with
the 1* orbitals of the hydrogen atoms.
7-2 GROUND STATE OP H 2
The ground-state electronic configuration of H 2 0, with eight
valence electrons (two from the hydrogens, 2/ 2 2^ 4 or six from oxy-
gen), is therefore
i44
Electrons and Chemical Bonding
+ .■:::■;■
mm
£tfSfe^
2a + h e +- lr t
Figure 7-3 Overlap of the oxygen 2s orbital with the Is
orbitals of the hydrogen atoms.
WK^)WW
i= o
We note that all the electrons must be paired, and H2O is diamag-
netic. There are four electrons in <r h orbitals, giving two <r bonds.
We might expect the H — O — H bond angle to be 90° if only the
2p x and Ifn orbitals were used in a bonding. That is, a 6 of 90° makes
1p x and 1p z equivalent with respect to overlap with the H valence
orbitals. This is easy to see if we place the two hydrogens along the
x and z axes, as shown in Fig. 7-5- The possibility of the Is orbital
being involved in bonding is one explanation for the 15° deviation
of the H — O — H angle from 90°. To demonstrate the angle "open-
ing," it is convenient (as for NH 3 ) to place the eight valence elec-
trons into four rf hybrid orbitals, as shown in Fig. 7-6. The fact
that the H — O — H angle in water is less than 109° is, according to
this view, a result of the different repulsions of electron pairs in
bonding and nonbonding orbitals. The nonbonding pairs would
repel each other more strongly than the bonding pairs, consistent
with a 105° angle between the bonding pairs.
Awgwlflr Triatomic Molecules
orbitals
H 2 orbitals
r-O-
'ti — O i -
ii
'i >
1
l* .
HPOO%--i
145
H orbitals
Figure 7-4 Relative orbital energies in H 2 0.
The bond angle in H 2 S is 92°, much closer to the 90° expected for
pure p bonding. In H 2 S, it is probable that there is strong Jp-ls
bonding. This is consistent with the fact that the interelectronic
repulsions in 3p orbitals on sulfur are known to be less than the inter-
electronic repulsions in lp orbitals on oxygen.
The electrons in <r* orbitals in H 2 spend more time near the oxygen
than near the hydrogens, owing to the larger electronegativity of
i^6
Electrons and Chemical Bonding
*~y
Figure 7-5 Simple picture of the bonding in H 2 0, using only
the oxygen 2p orbitals.
:Bf-f-.-
_
fc :
: t '• . ■
•. - • . : ' • *
x- <
1
1 cS
i- : . •
-,.-wV
lone pairs
Figure 7-6 Valence-bond structure for H2O, using sp 3 or-
bitals for oxygen.
Angular Triatomic Molecules
147
Figure 7-7 Separation of charge in H2O in the ground state.
oxygen. As a result, the hydrogens carry a small positive charge in
the ground state of H2O, as shown in Fig. 7-7.
The H s O molecule has a dipole moment of 1. 844 D. The moment is
due to the charge separation described above as well as to lone pairs,
as shuwn in Fig. 7-8. Each H — O bond has a small bond di-pole mo-
$+ 5-
ment resulting from the charge separation H — O. Since the H2O mole-
cule is angular, these bond moments add together to give a result-
ant dipole moment.
Table 7-1 gives dipole moments of several angular triatomic mole-
cules.
bond dipoles
r-=0-
total dipole moment
lone pairs
Figure 7-8 Contributions to the dipole moment of H 2 0.
148 Electrons ani Chemical Bonding
Table 7-1
Dipole Moments of Some Angular Triatomic Molecules*
Molecule Dipole moment, D
H 2 1.844
H.^ 0.92
S0 2 1.633
N0 2 0.39
3 0.52
a Data from A. L. McClellan, Tables of Experimental Dipole
Moments, Freeman, San Francisco, 1963.
7~3 ANGULAR TRIATOMIC MOLECULES WITH W BONDING: N0 2
The N0 2 molecule is an example of an angular triatomic molecule
with both a and ■k bonding. We place the N of N0 2 at the origin of
an xyz coordinate system shown in Fig. 7-9. The oxygens are
situated in the x£ plane, bent away from the X. ax is- The O — N — O
angle is 8. We shall consider the nitrogen Is and 2p and the oxygen
2p orbitals in constructing the molecular orbitals.
7-4 a ORBITALS
The nitrogen Is, 2p x , and 2p z valence orbitals are used to form a
molecular orbitals with the 2p s and 2p Zb of the oxygens. The a
molecular orbitals are very similar to those we obtained for H 2 0.
In order of increasing energy, we have a s b , a}, a x , a z *, and <r x * (see
Fig. 7-4).
7-5 7T ORBITALS
The nitrogen 2p v orbital overlaps the 2p Va and 2p Vb on the oxygens,
as shown in Fig. 7-10. The bonding molecular orbital is obtained
by adding the three orbitals together:
tKV) = co-h + C ^y« + >) 0-5)
Angular Triatomic Molecules
149
-b — -y
Figure 7-9 Coordinate system for NO a .
The aniibonding orbital has a node between O a and N and between
0, andN:
<K*ti*) - c ^py — &(y* + yt)
(7-6)
The other combination of the 2p y orbitals of 0„ and O b is
(lp y<l — 2^ b ). This combination has zero net overlap with the
nitrogen 2p y , and is the nonbonding molecular orbital:
K*v) = TTzOo- - yb)
(7-7)
We shall also consider the 2p x orbitals of O a and Ob nonbonding in
N0 2 . An approximate energy-level scheme for the molecular orbitals
of NO2 is given in Fig. 7-11.
150
Electrons and Chemical Bonding
t
II
K
»
> .:■■:■.■.
■'■-■:■'■ ' : + ■>••••.
','./■ •••
:'■:■[ \i^
N.,
s*^i
.a
2^ + ^ + ft
y,
ir N
A
•"*-:
y« _ >t
Figure 7-10 The ir-orbital combinations in N0 2 .
Angular Triatomic Molecules
I 5 I
N orbitals
NO, orbitals
O orbitals
n
n
' i'
\ \
\ \
2 ^>--\>-'
.-'"[— 55
o-
Figure 7-11 Relative orbital energies in N0 2 .
I 5 i
Electrons and Chemical Bonding
1-6 GROUND STATE OF NO 2
There are 17 valence electrons in N0 2 (five from nitrogen, six from
each oxygen) to place in the molecular orbitals given in Fig. 7-11.
The ground state is
(2 Ja ) 2 C^) 2 C^) 2 C^7C^) 2 C2?, ) 2 (2^) 2 CO 2 C^) s = I
Since there is one unpaired electron, the NOa molecule is paramag-
netic. Electron-spin resonance measurements have confirmed that
the unpaired electron in the ground state of N0 2 is in a c orbital.
The ground-state electronic configuration gives two er bonds and
one 7r bond. It is instructive to compare the molecular-orbital bond-
ing scheme with two possible equivalent valence-bond structures
that can be written for NO2 (see Fig. 7-12). The resonance between
structures I and II spreads out the one ir bond over the three atoms,
an analogy to the it bonding molecular orbital (see Fig. 7-10). The
unpaired electron is in an sp 2 hybrid orbital, which is similar to <r x .
The lone pair in the 2p v system goes from O a to Oj, an analogy to the
two electrons in the ir y molecular orbital (see Fig. 7-10).
The N — O bond length in N0 2 is 1.20 A. This compares with an
N — O distance of 1.13 A in NO. The molecular-orbital bonding
^
if
JTa'^Oh
<o
1 n
Figure 7-12 Valence-bond structures for NO2.
Angular Triatomic Molecules
153
Table 7-2
Properties of Angular Triatomic Molecules 8
AB 2
molecule
B-A-B
angle, deg
Bond
Bond
length, A
Bond energies,
heal/mole
H 2
105
HO-H
O-H
0.958
117.5(DE)
110.6(BE)
H 2 S
92
H-SH
H-S
1.334
90(DE)
83(BE)
H 2 Se
91
H-Se
1.47
66(BE)
H 2 Te
90
H-Te
57(BE)
HOC1
113
HO- CI
60(DE)
HOBr
HO- Br
56(DE)
HOI
HO- 1
56(DE)
OF 2
102
O-F
1.41
45.3(BE)
OCl 2
115
O-Cl
1.68
49(BE)
cio 2
117
OC1-0
Cl-O
1.484
57(DE)
60(BE)
Br0 2
O-BrO
Br-O
70(DE)
60(BE)
N0 2
132
O-NO
1.20
72(DE)
NOC1
116
Cl-NO
1.95
37(DE)
NOBr
117
Br-NO
2.14
28(DE)
so 2
120
S-O
1.43
119(BE)
SeCl 2
Se-Cl
58(BE)
3
117
O-O
1.278
NCV
115
N-0
1.24
a Data from T. L. Cottrell, The Strengths of Chemical Bonds, Butter-
worths, London, 1958, Table 11.5.1; L. E. Sutton (ed.), "Interatomic Dis-
tances," Special Publication No . 11, The Chemical Society, London, 1958.
154 Electrons and Chemical Bonding
scheme predicts 1| -k bonds for NO, and only \ for the NO in N0 2 ;
thus a longer NO bond in N0 2 is expected. The O — NO bond-disso-
ciation energy is 72 kcal/mole.
Bond properties for a number of angular triatomic molecules are
given in Table 7-2.
SUPPLEMENTARY PROBLEMS
1. Describe the electronic structures of the following molecules:
(a) 3 ; (b) C10 2 ; (c) C10 2 +; (d) OF 2 .
2. What structure would you expect for the amide ion? for SC1 2 ?
XeF 2 ?
VIII
Bonding in Organic Molecules
8-1 INTRODUCTION
Carbon atoms have a remarkable ability to form bonds with
hydrogen atoms and other carbon atoms. Since carbon has one
Is and three If valence orbitals, the structure around carbon tor full
a bonding is tetrahedrai Qf). We discussed the bonding in CH 4 ,
a simple tetrahedrai molecule, in Chapter V . By replacing one hydro-
gen in CH 4 with a — CH 3 group, the C 2 H 6 (ethane) molecule is ob-
tained- The C 2 H 6 molecule contains one C — C bond, and the struc-
ture around each carbon is tetrahedrai (j£ 8 ), as shown in Fig. 8-1.
By continually replacing hydrogens with — CH 3 groups, the many
hydrocarbons with the full sp s ^--bonding structure at each carbon are
obtained.
H
ti
Vrf
*&f
H
sp C sf
s?c sp
H
Figure 8-1 Valence-bond structure for CsHe.
J 55
i 5 6
Electrons and Chemical Bonding
In many organic molecules, carbon uses only three or two of its
four valence orbital s for tr bonding. This leaves one or two 2p orbit-
als for x bonding. The main purpose of this chapter is to describe
bonding in some of the important atomic groupings containing
carbon with ir valence orbitals.
It is common practice to describe the a bonding of carbon in
organic molecules in terms of the hybrid-orbital picture summarized
in Table 8-1. The t bonding will be described in terms of molecular
orbitals, and the energy-level schemes 'will refer only to the energies
of the ir molecular orbitals. This is a useful way of handling the
electronic energy levels, since the c bonding orbitals are usually con-
siderably more stable than the -n bonding orbitals. Thus the chemi-
cally and spectroscopically "active" electrons reside in the tt molec-
ular orbitals.
8—2 C2H4
The structure of ethylene, C2H4, is shown in Fig. 8-2. The mole-
cule is planar, and each carbon is bonded to two hydrogens and to the
other carbon. With three groups attached to each carbon, we use a
set of sp 2 hybrid orbitals for a bonding.
■MNWriMBpk . Ci i ?*
Figure 8-2 Coordinate system for C^H^,
Bonding in Organic Molecules
c 57
Table 8-1
Hybrid- Orbital Picture for <j Bonding of
Carbon In Organic Molecules
Number of atoms
bound to carbon
a Bond
orbitals
sP 3
sP 2
sP
Structure
around carbon
tetrahedral
trigonal planar
linear
H
O
I
■+:'
1
'".'f»
. H
/
/
-;>
- -,,
1
£•£
-
;~
?■.£?
/ j.
~"
_ _ _
I "■•
~t~
i
i':+-
:■*
'.-•
•■£:;
I.*,'
i
.+.
Figure 8-3 Boundary surfaces of the ir molecular orbitals
of C 2 H 4 .
158 Electrons and Chemical Bonding
This leaves each carbon with a 2p orbital, which is perpendicular
to the plane of the molecule. We form bonding and antibonding
molecular orbitals with the 2p x valence orbitals, as follows:
4fo»-) = ^=Q Xa + xO (8-1)
V2
*0*) = ~h<x a - xi) (8-2)
V2
The boundary surfaces of the ir b and w* MO's are shown in Fig. 8-3.
8-3 ENERGY LEVELS IN C 2 H 4
The energies of the ir b and tt* MO's are obtained just as were the
energies of the a h and <r* MO's of H 2 (Section 2-4):
EIW)] = SK^ytoK**) dr = \fQca + xt)3CCx a + Xh ) dr
= ic + Pec (8-3)
EtyQir*')] = \f(x a - x b )3C(x - xO dr = q c ~ fi ec (8-4)
Thus we have the same type of energy-level scheme for the -w molec-
ular orbitals of ethylene as we had for the a molecular orbitals of the
hydrogen molecule. The diagram for C2H4 is shown in Fig. 8-4.
8-4 GROUND STATE OF C 2 H 4
There are twelve valence electrons in C2H4, eight from the two
carbons (2s 2 2p 2 ~) and one from each hydrogen. Ten of these electrons
are used in a bonding, as shown in Fig. 8-5- Two electrons are left
Bonding in Organic Molecules
x 59
to place in the ir molecular orbitals. The ground state is (ir*) 2 ,
which gives one tt bond. The usual pictures of the bonding in C2H4
are shown in Fig, 8-6.
carbon- w orbitai
7r molecular orbitals carbon^ ir orbital
for C 2 H (
O—
■ <?r + fit.
Figure 8-4 Relative n orbital energies in C2H4.
h .:':■:
€..,
^m:M<M*m* W
''•Vi\-v^* C ,*; : ':vV '*';.V-!. ••'.:;: -yi-^'^ V.-/.^;.--'*" C . ;; -": , '.'; ; '*
5 o-bonding pairs = 10 electrons
Figure 8-5 The a bonding structure of C2H4.
■H-.
160 Electrons and Chemical Bonding
8-5 BENT-BOND PICTURE OP C2H4
The C=C bond can be formulated as involving two equivalent
"bent" bonds, rather than one cr and one v bond. One simple way
to construct equivalent bent bonds is to linearly combine the o* and
ir h molecular orbitals of C2H4 as follows:
fe = ^WOJO + K^F)} C&-5)
*•" V5 WC«* - ^ x « ^ CM)
The equivalent orbitals ^1 and ^ 2 are shown in Fig. 8-7. If the a b
orbitals used are derived from carbon sf orbitals (Section 8-2), the
H— C— H and H— C— C bond angles should be 120°.
X / H
vCrtL"
«=., xv, yx
H H
(a) simple picture
lines, indicate electron-pair bonds
XAAX
(6) a—is bond orbital picture
Figure 8-6 Common representations of the bonding in C S H4.
Bonding in Organic Molecules 161
Using only valence-bond ideas, we can formulate the bonding in
C 2 H 4 as involving four sp s orbitals on each carbon. Two of the sj?
orbitals are used to attach two hydrogens, and two are used to bond
to the other carbon in the double bond. Thus, C2H4 would be repre-
sented as shown in Fig. 8-8. This model predicts an H — C — H angle
of 109°28' and an H— C=C angle of 125°16'.
The observed H — C — H angle in C2H4 is 117°. Since the molecule
is planar, the H — C=C angle is 121.5°. These angles are much closer
in size to the 120° angle between equivalent sp* hybrid orbitals than
they are to the tetrahedral hybrid-orbital predictions. Howeyer,
certain other molecules containing the C=C group have X — G=C
C ' I V: , . . ;. , ..'. 111 c
H / ..-Jigs*"-; -. o >h
C ii'i ' ■ I .- . ..- c
Figure 8-7 Equivalent orbitals in CjHj, constructed from the
o* and v h orbitals.
i6z
Electrons and Chemical Bonding
Figure 8-8 Equivalent orbitals in C7H4, using sp' orbitals on
each carbon.
angles in the neighborhood of 125°.
The multiple bonds in molecules such as N 2 , H 2 CO, and QH 2 can
be formulated either as equivalent bent bonds or as a combination of
a and ir bonds. For a more complete discussion of equivalent orbit-
als, the reader is referred elsewhere. 1
8-6 BOND PROPERTIES OF THE C=C GROUP
There are two kinds of bonds in C 2 H 4 , C=C and C — H. Thus we
must know the value of BE(C — H) in order to obtain the value of
BE(C=C) from the process
H H
\ /
G=C
/ \
H H
•C + C + H+H+H + H (8-7)
1 J. A. Popie, Quart, ftw., XI, 273 (1957); L. Pauling, Mature of tie Chemical Bond,
Cornell University Press, Ithaca, N.Y., I960, p. 138 ff.
Bonding in Organic Molecules
163
The value of BE(C — H) used to calculate bond energies such as C=C,
C=0, etc., is 98.7 kcal/mole, which is very nearly the BE(C — H) in
CH4. Bond energies and bond lengths for a number of important
groups are given in Table 8-2. The values are averaged from several
compounds unless otherwise indicated.
The average G=C bond energy is 145-8 kcal/mole, a value almost
twice as large as the C — C bond energy of 82.6 kcal/mole. The
C=C bond length is 1.35 A, which is shorter than the 1.54 A C — C
bond distance.
Table 8-2
Bond Properties of Organic Groups 8
Bond
Bond energy,
Bond
length, A
kcal/mole
C-H
1.08
98.7
C-C
1.54
82.6
C=C
1.35
145.8
C=C
1.21
199.6
C-C(inC 2 H 6 )
1.543
83(DE)
C=C (in C 2 H 4 )
1.353
125(DE); 142.9(BE)
C=C (in C 2 H 2 )
1.207
230(DE); 194.3(BE)
C-N
1.47
72.8
C=N
147
C^N
1.14
212.6
c-o
1.43
85.5
C=0 (in aldehydes)
1.22
176
C=0 (in ketones)
1.22
179
C=0 (in H 2 CO)
1.21
166
C-F(in CF 4 )
1.36
116
C-Si [in SKCiyj
1.93
72
C-S(in C 2 HgSH)
1.81
65
C=S (in CS 2 )
1.55
128
C-Cl
1.76
81
C-Br
1.94 (in CHgBr)
68 (in C 2 H 5 Br)
C-I (in CH3I)
2.14
51
"Data from T. L. Cottrell, The Strengths of Chemical Bonds , Butter-
worths, London, 1958, Table 11.5.1.
164 Electrons and Chemical Bonding
8-7 THE VALUE OF f} oe IN C 2 H 4
The first excited state of C2H4 occurs upon excitation of an electron
from tt 6 to ir*, giving the configuration (y)Gr*)- We see that the
difference in energy between it 6 and ir* is —2/3. Absorption of light
at the 1650 A wavelength causes the ir b — ► it* excitation to take place.
Since 1650 A is equal to 60,600 cm -1 or 174 kcal/mole, we have
— 2/8 cc = 60,600 cm- 1 or 174 kcal/rnole
and
fe= -30,300 cm- 1 or -87 kcal/mole (8-8)
8-8 H 2 CO
The simplest molecule containing the C=0 group is formaldehyde,
H 2 CO. The a bonding in H 2 CO can be represented as involving sp 2
orbitals on carbon. This leaves one 2p orbital on carbon for ir bond-
Figure 8-9 Orbitals in the H s CO molecule.
Bonding in Organic Molecules 165
ing to the oxygen, as shown in Fig. 8-9. The ir molecular orbitals
are:
$(jf) = &xc + CWo (8-9)
$>«*) = Qxc - Qxq (8-10)
Since oxygen is more electronegative than carbon, we expect (Cs>) 2 >
(Ci) 2 and (C 3 ) 2 > (C 4 ) 2 . Since the oxygen 2p s orbital is used in a
bonding, we have the lp y orbital remaining as a nonbonding MO of
the v type. The energy-level scheme expected for the ir molecular
orbitals of H 2 CO is shown in Fig. 8-10.
8-9 GROUND STATE OF H 2 CO
There are twelve valence electrons in H 2 CO, two from the hydro-
gens, four from carbon, and six from oxygen (ls i 2-p € ). Six of these
electrons are involved in a bonding, and two are in the oxygen Is
orbital as a lone pair. This leaves four electrons for the t orbitals
shown in Fig, 8-10. The ground state is (ir/) 2 ^) 2 . There is one
carbon ir-orbital x-molecular orbitals oxygen ff-orbitals
for H 2 CO
s
IP, / <t
\
/-y^i ^ 2p; o n
\
Figure 8-10 Relative tr orbital energies in HsCO.
1 66 Electrons and Chemical Bonding
carbon-oxygen it bond, along with the <r bond, giving an electronic
structure that is commonly represented as shown in Fig. 8-11.
The carbonyl (0=O) group is present in many classes of organic
compounds, among them aldehydes, ketones, esters, acids, and
amides. The simplest ketone is acetone, (CH^OMZ). The 0=0
bond energy in H;CO is 166 kcal/mole. As C — H bonds are replaced
by C — C bonds, the C=0 bond energy increases. The average C=0
bond energy for aldehydes is 176 kcal/mole; for ketones it is 179
kcal/mole. Each of these average values is more than twice the 85.5
kcal/mole value for the C — O bond energy. The average C=^0 bond
length is 1.22 A, which lies between OsO (R = 1.13 A) and C— O
(R = 1.43 A).
c ^^^ o ;
m
Figure 8-11 Common representations of the bonding in
H a CO.
Bonding in Organic Molecules
167
8-10 THE 72— »7T TRANSITION EXHIBITED BY
THE CARBONYL GROUP
The excitation of an electron from w y to ir x * occurs with absorption
of light in the 2700-3000 A wavelength region. Thus the carbonyl
group exhibits a very characteristic absorption spectrum. Since the
transition is from a nonbonding t orbital to an antibonding w orbital,
it is commonly called an n — > ir* transition.
8-11 C 2 H 2
The structure of acetylene, C 2 H 2 , is shown in Fig. 8-12. The <r
bonding involves sp hybrid orbitals on the carbons, leaving each
carbon with two mutually perpendicular 2p orbitals for ir bonding.
The v molecular orbitals are the same as those for a homonuclear
diatomic molecule:
^(nf) = ^7=C x a + Xb)
V2
(8-11)
i'ClTx*') = —f^Xa — Xb)
V2,
(8-12)
(8-13)
_. c
H
Figure 8-12 Coordinate system for C2H2.
1 68 Electrons ani Chemical Bonding
K*v*') = ^j«-Jd (8-14)
The energies of the t molecular orbitals are shown in Fig. 8-13-
8-12 GROUND STATE OF C 2 H 2
There are ten valence electrons in C2H2. Six are required for a
bonding, and the othef four give a ground state (*/)*(*/)*. Thus
we have three carbon-carbon bonds, one <r bond, and two it bonds.
The common bonding pictures for C 2 Hj are shown in Fig. 8-14.
The bond energy of the GsC group, 199-6 kcal/mole, is larger than
that of C — C or C=C, but smaller than that of CsQ. The
bond length is 1.21 A, shorter than either C=C or C — C.
8-13 CH 3 CN
The nitrile group, Gs=N, is another important functional group
in organic chemistry. The simple compound CH3CN is called ace-
tonitrile; its structure is shown in Fig. 8-15. The ir bonding in the
Cs=N group is very similar to the sr bonding in 0=C. The usual
bonding pictures are also shown in Fig. 8-15.
carbon, Trorbitals jr-molecular orbitals for C 2 H S carbons i-orbitals
/
1
Figure 8-13 Relative orbital energies in C s Hs,
Bonding in Organic Molecules
H c c H
169
W
H -irQf- C t-Q^p- C Jp-Qb-H
ID
Figure 8-14 Common representations of the bonding in QH^.
H«
*">
n ;
H
(<0
Figure 8-15 Common representations of the bonding in
CH S CN.
170 Electrons and Chemical Bonding
The C=N bond energy, 212.6 kcal/mole, is larger than that of
G^C. The C=N bond length is about 1.14 A.
8-14 C 6 H 6
The planar structure of benzene (CeHe) is shown in Fig. 8-16.
Each carbon is bonded to two other carbons and to one hydrogen.
Thus we use xp 2 hybrid orbitals on the carbons for cr bonding. Each
carbon has a 2p orbital for x bonding, also shown in Fig. 8-16. With
six ir valence orbitals, we need to construct six it molecular orbitals
for C ( H 6 . The most stable bonding orbital concentrates electronic
density between each pair of nuclei:
&vf) = ^(fc + * + {, + & + 6 + Kf) (8-15)
The least stable antibonding orbital has nodes between the nuclei:
<K>s*) = ~7&* ~ **■+•*> — ft + &■■— %3 (8-16)
,t. I
\ i -■■*■■}
+ x 1 t m
H K *( .'•■ • ■ ;'"V C H
m
Figure 8—16 Structure and the -w valence orbitals of CsHs.
Bonding in Organic Molecules 171
The other molecular orbitals 1 have energies between ir b and t*:
W) = -^7=(2^ + Zb ~ Kc - 2Zd -Ze+Zf) (8-17)
2V3
K*f) = \(.Za +%,-&- *0 (8-18)
t^*) = -i<2fc - Zi - Zc + 1*4 ~Ze~zi) (8-19)
2V3
iKV) = \(Xa -Ki+Kd- Ze~) (8-20)
The molecular orbitals for benzene are shown in Fig. 8-17.
8-15 MOLECULAR-ORBITAL ENERGIES IN C 6 H 6
The most stable orbital in benzene is ^(tti 6 ). The energy of this
MO is calculated below :
J5[#V)] = /VOrrWOn 6 ) dr
= \f(Xo. + Zb + Zc + Zd + Ze + Zf)W
X (&, + Zb + Zc + Zd + Ze + Zf) dr
= \{6 ic + 12ft c + 2/Za^Zc dr + IfZa^Zd dr + IfZa&Ze
Xdr+ IfZiSCZd dr + IfZb^Ze dr + Ifz^Zf dr
+ 2fZc3CZe dr + Iftc^Zf dr + IftMZf dr] (8-21)
In other words, on expansion of the integral, we obtain six cou-
lomb integrals (such as SZa^-Za dr) and twelve exchange integrals
involving adjacent p orbitals (such as SZoH^Zb dr); the other integrals
are exchange integrals involving nonadjacent p orbitals (such as
fZa&Zc dr). We expect these integrals to be much smaller than the
regular /3's. If we adopt the frequently used Hiickel approximation
in which such integrals are taken to be zero, we have
E[W)l = I* + Wcc (8-22)
The energy-level scheme for CeH 6 is shown in Fig. 8-18.
1 The rules for constructing the benzene molecular orbitals are straightforward, but
require symmetry and orthogonality principles that have not been presented in this
book.
:•*■•■
■.•.v:'.*'!?-' <■
-o
/ .'•. Si... : \^
x 3 *
v O
x,»
°- &&$*&.
/ 'fHi#
top view
Xl*
Figure 8-17 Top view of the boundary surfaces of the C 6 H 6
molecular orbitals.
X2*
i7z
Bonding in Organic Molecules iji
7r-mokcular orbital energies in QH,
JTj*
Q~ ~ ■?<■ - 2 &<
JTl*
OCF 1
-OO
zLq.
ii — i
f. +,
& +
Figure 8-18 Relative energies of the x MO's in CeH s .
PROBLEM
8-1 . Show that ^(WD and #£*#) ate degenerate in energy, with
E = q c + fc- Show that i^On*) and ^(^2*) are degenerate in energy,
with E = q , — ft c . Show that the energy of ^(tt 3 *) is q c — 2/3«.
8-16 GROUND STATE OF C 6 H 6
There are a total of thirty valence electrons in benzene. Twenty-
four are used in <t bonding (six C — C, six C — H bonds), leaving six
for the Tr-molecular-orbital levels shown in Fig. 8-18. This gives the
ground state C^X^X 71 "^) 2 , and a total of three w bonds. Each
carbon-carbon bond consists of one full a bond and half a x bond.
The 0=C bond length in C 6 H 6 , 1.397 A, lies between the C — C and
C^C bond lengths.
The common bonding pictures of benzene are shown in Fig. 8-19.
8-17 RESONANCE ENERGY IN C 6 H 6
Benzene is actually more stable than might be expected for a system
of six C — C single bonds and three C — C -ir bonds. This added sta-
bility is due to the fact that the electrons in the three ?r bonds are
delocalized over all six carbons. This is evident both from the molec-
174 Electrons and Chemical Banding
Kekule structures
Dewar structures
simple MO picture
Figure 8-19 Common representations of the bonding in
CeHe.
ular orbitals shown in Fig. 8-17 and from the valence-bond struc-
tures shown in Fig. 8-19.
In the MO view, the total gain in C 6 H 6 stability due to ir bonding is
calculated in units of j6 M as follows:
2 electrons (in #$) X 2$, = 4ft c
2 electrons (in wf) X & e = 2ft
2 electrons (in tt 3 *) X ft. = 2ft,
total 8ft c
If we did not allow the derealization of electrons in C«H«, we would
have a system of three isolated double bonds (only one of the Kekule
structures shown in Fig. 8-19). Let us calculate the w bonding sta-
bility of three isolated double bonds.
An electron in the sr* orbital of C=C is more stable than an electron
in a carbon 2p atomic orbital by one ft c unit (see Section 8-3). With
six electrons in isolated tt 6 orbitals, we have 6X&= 6/3^. The de-
localization of three it bonds in C 6 H e gives an added stability of
8ft„ — 6ft = 2ft c . This is the calculated resonance energy in ben-
zene.
Bonding in Organic Molecules 175
The so-called experimental resonance energy of benzene is obtained
by totaling the bond energies of the C — C, C=C, and C — H bonds
present and comparing the total with the experimentally known
value for the heat of formation of C 6 H 6 . The difference indicates
that benzene is about 40 kcal/mole more stable than the sum of the
bond energies for a system of six C — H, three C — C, and three isolated
G=C units would suggest.
The value of p cc derived from the experimental resonance energy is
therefore —20 kcal/mole. This value differs substantially from the
value of —87 kcal/mole obtained from the absorption spectrum of
C2H4. It is a general result that the resonance-energy /3's are much
smaller than the spectroscopic /3's.
SUPPLEMENTARY PROBLEMS
1. Calculate the energies of the -w molecular orbitals for C2H2.
2. Give the "bent-bond" descriptions of C 2 H 2 ; of H 2 CO; of HCN.
TX
Bonds Involving &
Valence Orbitals
9-1 INTRODUCTION
There are many structures in which the central atom requires one
or more d valence orbitals to complete a set of a bonding orbitals.
The most important of these structures are square planar, trigonal
bipyramidal, square pyramidal, and octahedral; examples are shown
in Fig. 9-1, Transition-metal ions have available a very stable set of
d valence orbitals. The bonding in complexes formed between tran-
sition-metal ions and a large number of molecules and other ions un-
doubtedly involves d orbitals. In this chapter we shall describe the
bonding between metal ions and ligands 1 in certain representative
metal complexes.
9-2 THE OCTAHEDRAL COMPLEX Ti(H 2 0) 6 3+
The Ti 3+ ion forms a stable complex ion with six water molecules.
The structure around the Ti 8+ ion is octahedral, as shown in Fig. 9-2.
1 Groups attached to metal ions in complexes are called ligands.
I76
Bonis Involving d Valence Orlitals
177
3 +
NH 3
H 3 N
H 3 N NH 3
NH 3
NH 3
octahedral
O
C
CI
_ _ __,F
F F
square pyramidal
OC
i
^-CO / p-
CI
-CI
C
o
CI
trigonal bipyramidal
J^. V
NC _ CN
NC^-- _ CN
square planar
Figure 9-1 Examples of structures in which d orbitals are
used in bonding.
178 Electrons and Chemical Bonding
I
I
1
.1
I
OH,
I/"
- H 2 Ti — Ti — -<7 2 oH 2
• ff > I
H,0 <r*
X OH 2
x
I
I
♦
Figure 9-2 Coordinate system for Ti(H 2 0) 6 3+ .
The titanium has five 3d, one As, and three 4p valence orbitals that
can be used in constructing molecular orbitals. Each water molecule
must furnish one a valence orbital, which, in accord with the dis-
cussion in Chapter VIII, is approximately an sp 3 hybrid orbital. We
shall not specify the exact s and p character of the water a valence
orbital, however, but simply refer to it as a.
The metal orbitals that can form <r molecular orbitals are 3^x 2 -!, 2 ,
3^s 2 , As, Ap x , Ap v , and Ap s . Since the sign of the As orbital does not
change over the boundary surface, the proper linear combination of
ligand orbitals for As is
<T\ + 0-2 + ff3 + 0-4 + C5 + 06 (9-1)
This is shown in Fig. 9-3. The wave function for the molecular
orbital involving the metal As orbital is therefore
lAOO = c i4s + £2(0-1 + o- 2 + o 3 + o 4 + 0-5 + o- 6 ) (9-2)
We find the other molecular orbitals by matching the metal-orbital
Bonis Involving d Valence Orhi tals 179
OH,
; :+.
,OH a
+
-H,0 T> OH 2 -
H 5
OH,
1
1
1
4i + (71 + CT2 + 0-3 4" <Fi "I" CS -f" <T6
Figure 9-3 Overlap of the titanium 4.s orbital with the cr
orbitals of the water molecules.
lobes with ligand <r orbitals that have the proper sign and magnitude.
This procedure is shown in Fig. 9-4. The wave functions are:
iKO = CsAp x + c*Oi - ff3 ) (9-3)
lK>„) = c-Ap v + €i(a, - at) (9-4)
$ GO = M& + c 4 (o- 5 - <r 6 ) (9-5)
f (av^) = (%3«&?.y + ca(ffi — (ra + 0-3 — «a) (9-6)
Tp(v/) = v-tbd^- + fs(2<r5 + 2o- 8 — <t\ — 1T2 — as — <0
(9-7)
9-3 ENERGY LEVELS IN Ti(H 2 0) 6 S +
Figure 9-4 shows 4p x , 4p y , and 4p s to be equivalent in an octahedral
complex; on this basis the a x , a y , and <x g molecular orbitals are de-
generate in energy. Although it is not obvious from Fig. 9-4, the
i8o
Electrons and Chemical Bonding
3(/ I i_ S ! + cri " tT2 + era —
FLO*
H.O
OH,
H,()
,Oll
W§0 OH a
OH s
3d g l + 2(76 "(- 2(Tb — <7i -^ (7a — (73 -- ff4
CH, - 2^ - ^ - /)
Figure 9-4 Overlap of the titanium '3d and 4p <j orbitals with
the it orbitals of the water molecules.
Bonis Involving d Valence Orbitals 181
3<ix 2 -j/ 2 and 3<4 2 orbitals are also equivalent in an octahedral complex,
and (7^-5,2 and oy> are degenerate in energy. We shall solve a problem
at the end of this chapter to prove the equivalence of 34c 2 -j/ 2 and
3d,?. Finally, we see that, including the <r s orbital, there are three
sets of a molecular orbitals in an octahedral complex: as; a x , <r y , <t z ;
and ffx 1 ^, <5£-
We have used all but three of the metal valence orbitals in the a
molecular orbitals. We are left with 3d xz , 3d yz , and 3d xy . These
orbitals are situated properly for 7r bonding in an octahedral com-
plex, as will be discussed later. However, since water is not a good
it bonding ligand, we shall consider that the 3d xz , 3d yz , and 3d xy orbit-
als are essentially nonbonding in Ti(H 2 0) 6 3+ . The three d* orbitals
are clearly equivalent in an octahedral complex, and we have the
degenerate set: ir xs , w yz , ir xy .
In order to construct an energy-level diagram for Ti(H20)6 3+ , we
must know something about the relative energies of the starting
orbitals 3d, 4s, 4p, and cth 2 o. In this case, <th 2 o is more stable than
any of the metal valence orbitals. This is fairly general in metal
complexes, and in energy-level diagrams the ligand a valence orbitals
are shown to be more stable than the corresponding metal valence
orbitals. It is also generally true that the order of increasing energy
for the metal valence orbitals in transition-metal complexes is
nd < + 1> < (n + 1>.
The energy-level diagram for Ti(H 2 0) 6 3+ is shown in Fig. 9-5-
There are three sets of bonding orbitals and three sets of antibonding
orbitals. The virtually nonbonding ir(df) orbitals are less stable than
the bonding aQf) set but more stable than the antibonding er(d~) set.
The relative energies of the three bonding a sets are not known. The
order given in Fig. 9-5 was obtained from a calculation that is
beyond the level of our discussion.
9~4 GROUND STATE OF Ti(H 2 0) 6 3 +
We must count every electron in the valence orbitals used to con-
struct the diagram in Fig. 9-5- The complex is considered to be com-
posed of Ti 3+ and six water molecules. Each of the six c valence
orbitals of the water molecules furnishes two electrons, for a total of
i8i
Ti orbitals
Electrons and Chemical Bonding
Ti(H,0) a 3+ orbitals H a O orbitals
Figure 9-5 Relative orbital energies in TifHaOJs 54 .
twelve. Since the electronic structure of Ti 3+ is (3^)', we have a
total of thirteen electrons to place in the molecular orbitals shown
in Fig. 9-5- The ground state of Ti(H 2 0)6 s+ is therefore
WX^^X^-wX^Bfl)'
s=h
Bonis Involving d Valence Orhitah
183
There is one unpaired electron in the ir(d~) level. Consistent with
this ground state, Ti(H 2 0)e 3+ is paramagnetic, with $ = |.
The electrons in u bonding orbitals are mainly localized on the
water molecules, since the a- valence orbital of H 2 is more stable
than the metal orbitals. The nonbonding and antibonding orbitals,
on the other hand, are mainly located on the metal. We shall focus
our attention in the sections to follow on the molecular orbitals that
are mainly based on the metal and derived from the 3d valence
orbitals.
9-5 THE ELECTRONIC SPECTRUM OF Ti(H.20)(; 3 +
The difference in energy between <r*(i) and ir(_d~) is called A or
lQDq. Excitation of the electron in #(d) to ff*G0 occurs with ab-
sorption of light in the visible region of the spectrum, and Ti(H20) 6 3+
is therefore colored reddish- violet. The electronic spectrum of
Ti(H 2 0) 6 3+ is shown in Fig. 9-6. The maximum absorption occurs
at 4930 A, or 20,300 cm -1 . The value of the splitting A is usually ex-
pressed in cm -1 units; thus we say that Ti(HsO) 6 3+ has a A of 20,300
cm _1 .
The colors of many other transition-metal complexes are also due
to such "d-d" transitions.
?, cm '
Figure 9-6 The absorption spectrum of Ti(HjO)6 3+
visible region.
the
184 Electrons and Chemical Bonding
9-6 VALENCE-BOND THEORY FOR Ti(H 2 Q) 6 3+
The localized bonding scheme for Ti(H 2 0)6 3+ is obtained by first
constructing six equivalent hybrid orbitals that are octahedrally
directed. We use the six a valence orbitals of Ti for this purpose:
34z 2 _/, 3d z *, 4s, 4p x , 4p y , and 4p%. Thus we want to construct six d 2 sp s
hybrid orbitals, each with one-third d character, one-sixth s char-
acter, and one-half p character.
Referring back to Fig. 9-2, let us form linear combinations of
the d, s, and p valence orbitals that direct large lobes at the six
ligands. We first construct the orbitals that are directed toward
ligands © and ©. We shall call these orbitals ^5 and \[/ t , respec-
tively. The metal orbitals that can <r bond with © and © are 34, 2 ,
4s, and 4p z . Choosing the coefficients of the 3<4 2 , 4s, and 4p z orbitals
so that 1^6 and ^ 6 have the desired d, s, and p character, we obtain the
following hybrid-orbital wave functions :
ft = 4= 3*' + -7= 4s + "7= 4 ^ (9-8)
V3 V6 Vl
#6=4= 3* + -7= 4j - -7= 4J>, (9-9)
V3 V6 Vl
The positive coefficient of 4p z in ^5 directs a large lobe toward ©,
and the negative coefficient of 4p z in ^ directs a large lobe toward ©.
The orbitals directed toward © and ® are constructed from the
34 t ?-?/ 2 ,3<4 2 , 4s and 4p x metal orbitals. The orbitals directed toward ©
and © are constructed from the 34c 2 -!, 2 , 3<i 2 , 4s, and 4p v orbitals.
The coefficients of 4s and 4p pose no problem, but we have to divide
the one-third d character in each hybrid orbital between 3d z % and
3<4 2 -V- We see from Eqs. (9-8) and (9-9) that we have "used up"
two-thirds of the 3d z orbital in fa and ^ 6 . Thus we must divide the
remaining one-third equally among \f/\, */% i/% and ^ 4 . This means
that each of i/'i, $2, $3, and ^ 4 has one-twelfth 3d z 2 character and one-
fourth 3d /_/ character. Choosing the signs of the coefficients so
that a large lobe is directed toward each ligand in turn, we have:
1 11
i 1 = I 34w 7= 3d/ + - 7 =4s+-=4p x (9-10)
Vl2 V6 Vl
Bonds Involving d Valence Orhitds
1 11
^2 = — | 34^-/ — —7= 3<&" -\ — J= 4s + — -p 4ft,
V12 V6 V2
f 3 = | 34^-b 2
12 V6 V2
fc = -| Hf_^ ~ vff 3 ^ ! + ^ 4j ~ ^ 4 A
185
(9-11)
(9-12)
(9-13)
These six localized d 2 sp s orbital s are used to form electron-pair bonds
with the six water molecules. The valence-bond description of the
ground state of Ti(H 2 0) e 3+ is shown in Fig. 9-7. The unpaired elec-
- h 2 o ■ *fo — ~ •-% ~ — - — \ .;y>; OH,-
o o
o
3d
is
*f
. 11 11
4 t
XXX
1 1
1 1 1
000
H, H„ H„
Figure 9-7 Valence-bond representations of Ti(HjO)j ;,+ .
1 86
Electrons and Chemical Bonding
tron is placed in one of the d orbitals that has not been used to con-
struct hybrid bond orbitals. This simple valence-bond orbital dia-
gram is also shown in Fig. 9-7.
9-7 CRYSTAL-FIELD THEORY FOR Ti(H20) 6 3+
In the crystal-field-theory formulation of a metal complex, we
consider the ligands as point charges or point dipoles. The crystal-
field model is shown in Fig. 9-8. The point charges or point dipoles
constitute an electrostatic field, which has the symmetry of the com-
plex. The effect of this electrostatic field on the energies of the
metal d orbitals is the subject of our interest. /
Let us examine the energy changes in the 2>d orbitals of Ti 3+ that
result from placement in an octahedral field of point dipoles (the
water molecules). First, all the d orbital energies are raised, owing
to the proximity of the negative charges. More important, however,
the two orbitals (3t4 2 , 3^-j, 2 ) that point directly at the negative
charges are raised higher in energy than the three orbitals (j>d xz ,
o
/
M
o
-*-y
/
o
Figure 9-8 An octahedral field of point charges.
Bonds Involving d Valence Orhitals 187
/boooo-(
OOOOCH^
A
free ion octahedral crystal field
Figure 9-9 Splitting of the metal d orbitals in an octahedral
crystal field.
"idyz, 3^) that are directed at points between the negative charges.
Thus we have a splitting of the five d orbitals in an octahedral crystal
field as shown in Fig. 9-9. It is convenient to use the group-theoret-
ical symbols for the split d levels. The 1d^ and 3<4 2 -/ orbitals form
the degenerate set called e, and the 3dm, 7>dy Z , and 3^ orbitals form
the degenerate set called t%. The separation of e and t% is again desig-
nated A or \§D<j..
The one d electron in Ti 3+ is placed in the more stable h orbitals in
the ground state. The excitation of this electron from h to t is re-
sponsible for the spectral band shown in Fig. 9-6.
9-8 RELATIONSHIP OF THE GENERAL MOLECULAR-ORBITAL
TREATMENT TO THE VALENCE-BOND AND CRYSTAL-FIELD THEORIES
The valence-bond and crystal-field theories describe different parts
of the general molecular-orbital diagram shown in Fig. 9-5- The <r
bonding molecular orbitals are related to the six d 2 sf bonding orbit-
als of the valence-bond theory. The valence-bond theory does not
include the antibonding orbitals, and therefore does not provide an
explanation for the spectral bands of metal complexes. The fa and
e levels of the crystal-field theory are related to the icQf) and <r*(d)
molecular orbitals. A diagram showing the relationship between
the three theories is given in Fig. 9-10.
i88
Electrons and Chemical Bonding
crystal-field splitting related to splitting between <r*{d) and r(d) molecular orbitals
]r,A-\_A— t 71
valence bond orbitals related to o-bonding molecular orbitals
fff!
L L L L L L
Figure 9-10 Comparison of the three theories used to de-
scribe the electronic structures of transition-metal complexes.
9-9 TYPES OF T BONDING IN METAL COMPLEXES
The d xz , d yz , and i m orbitals may be used for w bonding in octa-
hedral complexes. Consider a complex containing six chloride
ligands. Each of the d T orbitals overlaps with four ligand ir orbitals,
as shown in Fig. 9-11. In the bonding orbital, some electronic
charge from the chloride is transferred to the metal. We call this
ligand-to-metal (L — * M) ir bonding. The x orbitals based on the
metal are destabilized in the process and are made antibonding.
If the complex contains a diatomic ligand such as CN _ , two types
of it bonding are possible. Recall from Chapter II that CN~ has
filled ir* and empty ir* molecular orbitals, as shown in Fig. 9-12.
The occupied ir* orbitals can enter into L — » M ir bonding with the
~bd xz , 3dy Z , and 3^ orbitals. In addition, however, electrons in the
metal ir(<T) level can be delocalized into the available tt* (CN~)
orbitals, thus preventing the accumulation of too much negative
Bonis Involving d Valence Orbitals
189
- . ® +
-a-
*$£.
+
:<■;{
Figure 9-11 Overlap of a d r orbital with four ligand x
orbitals in an octahedral complex.
charge on the metal. This type of bonding removes electronic den-
sity from the metal and is called metal-to-ligand (M. — ► IS) w bonding.
It is also commonly called back donation or back bonding. Back dona-
tion stabilizes the ir(d) level and makes it less antibonding. Both
types of ir bonding between a d T orbital and CN~ are shown in Fig.
9-12.
9-10 SQUARE-PLANAR COMPLEXES
A simple square-planar complex is PtCl 4 2_ . The coordinate system
that we shall use to discuss the bonding in PtCL 2- is pictured in
Fig. 9-13.
The metal valence orbitals suitable for a molecular orbitals are
5<4 2 V> 54s 2 , 6s, 6px, and 6f y . Of the two d <r valence orbitals, it is
clear that 5^ 2 V interacts strongly with the four ligand er valence
orbitals and that 5«4 2 interacts weakly (most of the 5d z i orbital is
directed along the £ axis).
The 5d xz , 5d yz , and 54tj, orbitals are involved in t bonding with the
ligands. The 5^ w orbital interacts with -k valence orbitals on all four
190 Electrons and Chemical Bonding
— ±±i-.[ *$# :.i— <- c — ■■■-' N
w b ^d r
Figure 9-12 Types of x bonding between CN~ and a metal
d T orbital.
ligands, whereas 5d xz and 5d yz are equivalent and interact with only-
two ligands. The overlap of the metal 'yd orbitals with the valence
orbitals of the four ligands is shown in Fig. 9-14.
We can now construct an approximate energy-level diagram for
PtCU 2- . We shall not attempt to pinpoint all the levels, but instead
Bonds Involving d Valence Orhitah iqi
ci
/
/
/
a *j
(74 « - .
CI — — — Pt ■ ■ CI *-y
°i.
I
CI I
/
Figure 9-13 Coordinate system for PtCl4 ;
to recognize a few important regions of energy. A simplified energy-
level diagram for PtCU 2- is shown in Fig. 9-15- The most stable
orbitals are a bonding and are located on the chlorides. Next in
order of stability are the t molecular orbitals, also mainly based on
the four chlorides. The molecular orbitals derived from the 5d
valence orbitals are in the middle of the diagram. They are the anti-
bonding partners of the a and 7r bonding orbitals just described.
We can confidently place the strongly antibonding a*J--J- highest.
We can also place ir xy * above Tr* xz , yz , since 3d xy interacts with all four
ligands (see Fig. 9-14). The weakly antibonding a z 2 * is believed to
lie between ir xy * and ir* xz , yz . However, regardless of the placement of
o-j 2 *, the most important characteristic of the energy levels in a
square-planar complex is that one A level has very high energy
whereas the other four are much more stable and bunched together.
Since Pt 2+ is 5d s and since the four chlorides furnish eight a and
sixteen w electrons, the ground state of PtCl 4 2 ~ is
(^K^^X^X^*) 1 s = o
The complex is diamagnetic since the eight metal valence electrons
ICJ1
Electrons and Chemical Bonding
d ti — Wi — (72 — (73 — ff4
/
.CI
-j 1 + ffi — ff2 + o-a — <T4
■Cts-*-s
W-
/
a
%pCl - y
S#*C1— y
5^^iB
-Qr l -
XI..
/
m- ,,, ci-
:; -jf:r'v'*
4, (4 ligand jr orbitals)
rf = d zl (2 iigand )r orbitals)
figure 9-14 Overlap of the metal d valence orbitals with the
ligand valence orbitals in a square-planar complex.
Bonis Involving d Valence Orhitals
193
Pt orbitals
KXDOV
PtCl 4 orbitals
CI orbitals
^~ooo- (
-OOOOCR
Figure 9-15 Relative orbital energies in PtCU 2- .
are paired in the more stable d levels. It is easy to see from the
energy-level diagram that the best electronic situation for a square-
planar complex is d a . This observation is consistent with the fact
that the d* metal ions, among them Ni 2 +, Pd 2 +, Pt 2+ , and Au 8+ ,
form a great number of square-planar complexes.
i 9 4
Electrons anH Chemical Bonding
9-11 TETRAHEDRAL COMPLEXES
A good example of a tetrahedral metal complex is VOi, the co-
ordinate system for which is shown in Fig. 9-16. We have already
discussed the role of s and p valence orbitals in a tetrahedral mole-
cule (Chapter V). The 4s and 4p orbitals of vanadium can be used to
form u molecular orbitals. The J>d xz , J>d yz , and 34^ orbitals are also
situated properly for such use. In -valence-bond language, sd* and
sp 3 hybrid orbitals are both tetrahedrally directed. The 34c 2 -s, 2 and
3>d/ orbitals interact very weakly with the ligands to form v molec-
ular orbitals.
The simplified molecular-orbital energy-level diagram for VCU is
shown in Fig. 9-17. Again we place the stable <r bonding levels
lowest, with the ir levels, localized on the chlorides, next. The anti-
bonding molecular orbitals derived from the J>d valence orbitals are
split into two sets, those based on ^d xz , J>d yz , and 3<&j, being less stable
than those based on 3<4 ! and 3*4 2 V- ^ e sna U designate A ( as the
difference in energy between u*(d~) and ir*0O in a tetrahedral com-
plex.
With eight <r and sixteen -it valence electrons from the four chlorides
*~y
Figure 9-16 Coordinate system for VCli.
Bonis Involving d Valence Orbitals icjc
V orbitals VC1, orbitals CI orbitals
"a* <r y " <r s *
,^-000,
— OOO— *'' r££— '
t
4j W
t
t
r / m
v
— O — ; *
• \
\ \
( ' ff «* <T M * ff w « i I
\ !
i «
, !,*" A, « 11
.,"' A,
« M
I } I \ \\
\& i\
t«' i , r I rV" —
1 ,) I ir orbitals , ,<
V \ *
V 4
V 3
ft
[ p- 6 orbitals
Figure 9-17 Relative orbital energies in VClj
and with one valence electron from V 4+ (3^ x ), the ground state of
VC1 4 is
The paramagnetism of VCI4 is consistent with the ground state, there
being one unpaired electron.
196
Electrons and Chemical Bonding
Table 9-1
Values of A for Representative Metal Complexes
Octahedral
complexes
A, cm'
Octahedral
complexes
A, cm' 1
Ti(H 2 0) 6 3+
20,300 a
TiF 6 3 -
17,000 b
V(H 2 0) 6 3+
17,850 a
V(H 2 0) 6 2+
12,400°
Cr(H 2 0) 6 3+
17,400°
Cr(NH 3 ) 6 3+
21,600°
Cr(CN) 6 3 -
26,600 d
Cr(CO) 6
34,150 d
Fe(CN) 5 3 -
35,000 d
Fe(CN) 6 4 -
33,800 d
Co(H 2 0) 6 3+
Co(NH 3 ) 6 3+
CoJCN). 3 "
Co(H 2 <0) 6 2+
Ni(H 2 0) 6 2t
Ni(NH 3 ) 6 2+
RhCl 6 3 -
Rh(NH 3 ) 6 3+
RhBr 6 3 -
IrCle 3 -
Ir(NH 3 ) 6 3+
18,200°
22,900°
34,800 d
9,300°
8,500°
10,800°
20,300°
34^100°
19,000°
25,000°
40,000 e
Tetrahedral complexes
A, cm'
VC1 4
CoCl/-
CoBr 4 2 -
Col/-
Co(NCS) 4 2
9000 a
3300 f
2900'
2700 f
4700 f
Square -planar
Total
*%vi*A,r *s —£SMA,tKA.r X "^ (/Ctrl-
complexes 5 A v cm' 1 A 2 , cm' 1 A 3 , cm' 1 A, cm' 1
PdCl 4 2 ~
19,150
6200
1450
26,800
PdBr 4 2 -
18,450
5400
1350
25,200
RC1 4 2 -
23,450
5900
4350
33,700
RBr 4 2 -
22,150
6000
3550
31,700
Ni(CN) 4 2 "
24,950
9900
650
35,500
(Footnotes appear on next page)
Bonis Involving d Valence Orbitals iay
Excitation of the electron in ir*(d) to a*(cf) is accompanied by light
absorption, with a maximum at 9000 cm -1 . Thus A ( for VC1 4 is
9000 cm- 1 .
9-12 THE VALUE OF A
The splitting of the molecular orbitals derived from metal d val-
ence orbitals involves a quantity that is of considerable interest when
discussing the electronic structures of metal complexes. The A values
for a representative selection of octahedral, square-planar, and tetra-
hedral complexes are given in Table 9-1 ■ The value of A depends on a
number of variables, the most important being the geometry of the
complex, the nature of the ligand, the charge on the central metal
ion, and the principal quantum number n of the d valence orbitals.
We shall discuss these variables individually.
Geometry of the Conrplex
By extrapolating the data in Table 9-1, we may estimate that,
other things being equal, the total J-orbital splitting decreases as
follows:
square planar > octahedral > tetrahedral
1.3A A 0.45Ao
In the molecular-orbital theory, the //-orbital splitting is interpreted
as the difference between the strengths of a and -zr bonding as meas-
ured by the difference in energy between the a* and -k (or tt*) molec-
ular orbitals. The tetrahedral splitting is smallest because the d
orbitals are not involved in strong a bonding. In both octahedral
and square-planar complexes, d orbitals are involved in strong a bond-
a C. J. Ballhausen, Introduction to Ligand Field Theory, McGraw-Hill,
New York, 1962, Chap. 10.
b H. Bedon, S. M. Horner, and S. Y. Tyree, Inorg. Chem., 3, 647 (1964).
C C. K. J0rgensen, Absorption Spectra and Chemical Bonding, Pergamon,
London, 1962, Table 11.
d H. B. Gray and N. A. Beach, J. Am. Chem. Soc, 85, 2922 (1963).
e H. B. Gray, unpublished results.
£ Averaged from values in Ref. c and in F. A. Cotton, D. M. L. Good-
game, and M. Goodgame, J. Am. Chem. Soc, 83, 4690 (1961).
g H. B. Gray and C. J. Ballhausen, J. Am. Chem. Soc, 85, 260 (1963).
iq8 Electrons and Chemical Bonding
ing, but the total square-planar splitting (Ai + A 2 + A 3 ) will always
be larger than the octahedral splitting since the d xz and d yz orbitals
interact with only two ligands in a square-planar complex (as op-
posed to four in an octahedral complex; see Fig. 9-11).
Nature of the Ligand: the Spectrochemical Series
The spectrochemical series represents the ordering of ligands in
terms of their ability to split the a*(jf) and Tr(cf) molecular orbitals.
Complexes containing ligands such as CN~ and CO, which are high
in the spectrochemical series, have A values in the range of 30,000
cm^ 1 . At the other end of the series, Br~ and I~ cause very small
splittings — in many cases less than 10,000 cm -1 . We have already
discussed the important types of metal-ligand bonding in transition-
metal complexes. The manner in which each type affects the value
of A is illustrated in Fig. 9-18. We see that a strong ligand-to-metal
<r interaction destabilises <r*(cT), increasing the value of A. A strong
L — » M it interaction destabilizes ir(_d~), decreasing the value of A. A
strong M — > L ir interaction stabilizes ir(d~), increasing the value of A.
It is striking that the spectrochemical series correlates reasonably
metal orbitals
molecular
A
' A
orbita
Is
(L-
V
/ \
ligand orbitals
7T*
d
/
/
■»M)
/
/
M -> L)
\(L -> M)
7T
\
Figure 9-18 The effect of interaction of the ligand a, it, and
x* orbitals on the value of A.
Bonis Involving d Valence Orhitals icjcj
well with the Tr-bonding abilities of the ligands. The good ir-
accepor ligands (those capable of strong M — > L tt bonding) cause
large splittings, whereas the good ■w-donor ligands (those capable of
strong L — > M t bonding) cause small splittings. The ligands with
intermediate A values have little or no 7r-bonding capabilities.
The spectrochemical-series order of some important ligands is
indicated below:
-CO, — CN~ > — N0 2 " > o-phen 1 > NH 3 > OH 2 > OH", F~
weak it donors
7r acceptors ' non-7r-bonding
> SCN-, CI- > Br- > I-
7r donors
Charge on the Central Metal Ion
In complexes containing ligands that are not good ir acceptors, A
increases with increasing positive charge on the central metal ion.
A good example is the comparison between V(H 2 0) 6 2+ , with A =
11,800 cm" 1 , and V(H 2 0) 6 8 +, with A = 17,850 cm- 1 . The increase in
A in these cases is interpreted as a substantial increase in a bonding on
increasing the positive charge of the central metal ion. This would
result in an increase in the difference in energy between o*(d~) and
In complexes containing good 7r-acceptor ligands, an increase in
positive charge on the metal does not seem to be accompanied by a
substantial increase in A. For example, both Fe(CN) 6 4 " and
Fe(CN) 6 3 - have A values of approximately 34,000 cm- 1 . In the
transition from Fe(CN) 6 4_ to Fe(CN) 6 s ", the -*(£) level is destabilized
just as much as the <r*(i) level, probably the result of a decrease in
M — > L t bonding when the positive charge on the metal ion is in-
creased.
1 «-phen is
zoo Electrons and Chemical Bonding
Principal Quantum Number of the d Valence Orbitals
In an analogous series of complexes, the value of A varies with n
in the d valence orbitals as follows: 3d < Ad < 5d. For example, the
A values for Co(NH 3 ) 6 3 +, Rh(NH 3 ) 6 8+ , and Ir(NH 3 ) 6 3 + are 22,900,
34,100, and 40,000 cm -1 , respectively. Presumably the 5d and Ad
valence orbitals are better than the 3d in <r bonding with the ligands.
9^13 THE MAGNETIC PROPERTIES OF COMPLEXES: WEAK- AND
STRONG-FIELD LIGANDS
We shall now consider in some detail the ground-state electronic
configurations of octahedral complexes containing metal ions with
more than one valence electron. Referring back to Fig. 9-5, we see
that metal ions with one, two, and three valence electrons will have
the respective ground-state configurations ic(d~), S = \; [Tr(jf)\ 2 , S =
1; and [ir(d)f, ■$"=•§-. There are two possibilities for the metal d A
configuration, depending on the value of A in the complex. If A is
less than the energy required to pair two d electrons in the irQf) level,
the fourth electron will go into the a*(d) level, giving the configura-
tion |V(/0] 3 [a-*00] 1 anc ^ ^ our un P arre d electrons (S = 2). Ligands
that cause such small splittings are called weak-field ligands.
On the other hand, if A is larger than the required pairing energy,
the fourth electron will prefer to go into the more stable ir(d) level
and pair with one of the three electrons already present in this level.
The ground-state configuration of the complex in this situation is
WCcOY, with only two unpaired electrons (S = 1). Ligands that
cause splittings large enough to allow electrons to preferentially
occupy the more stable ir(d~) level are called strong-field ligands.
It is clear that, in filling the r(_d) and <r*(tf) levels, the configura-
tions d*, d & , d 6 , and d 7 can have either of two possible values of S,
depending on the value of A in the complex. When there is such a
choice, the complexes with the larger S values are called high-spin
complexes, and those with smaller S values are called low-spin com-
plexes. The paramagnetism of the high-spin complexes is larger
than that of the low-spin complexes. Examples of octahedral com-
plexes with the possible [ir(jt)] x [a*(jf)] y configurations are given in
Table 9-2.
Bonds Involving d Valence Orbitals
Table 9-2
Electronic Configurations of Octahedral Complexes
2.01
Electronic configuration
Electronic structure
of the
metal ion
of the complex
Example
3d 1
[TT(d)] 1
Ti(H 2 0) 6 3+
3d 2
[77(d)] 2
V(H 2 0) 6 3+
3d 3
[77(d)] 3
Cr(H 2 0) 6 3+
3d 4
low- spin
[77(d)] 4
Mn(CN) 6 3 -
high- spin
[V(d)] 3 [a*(d)]
Cr(H 2 0) 6 2 *
3d 5
low -spin
h(d)f
Fe(CN) 6 3 -
high- spin
[77(d)] 3 [o*(d)f
Mn(H 2 0) 6 2+
3d 6
low- spin
[ir(d)] e
Co(NH 3 ) 6 3+
high- spin
[77(d)] 4 [oHd)T
CoF 6 3 "
3d 7
low -spin
[n(d)] e [o*(d)]
Co(N0 2 ) 6 4 -
high- spin
[77(d)] 5 [ff*(d)] 2
Co(H 2 0) 6 2+
3d 8
[77(d)] 6 [a*(d)f
Ni(NH 3 ) 6 2+
3d 9
[77(d)] 6 [a*(d)] 3
Cu(H 2 0) 6 2 *
The first-row transition-metal ions that form the largest number of
stable octahedral complexes are Cr 3 +(i 3 ), Ni 2+ (// 8 )> and Co s+ (^ 6 ; low-
spin). This observation is consistent with the fact that the MO
configurations [?t(d^)] s and.[7r(/)] 6 take maximum advantage of the
more stable ir(d) level. The [-r(i)] 6 [cr*(^)] 2 configuration is stable for
relatively small A values.
The splitting for the tetrahedral geometry is always small, and
no low-spin complexes are known for first-row transition-metal
ions. There are many stable tetrahedral complexes of Co 2+ (3^ 7 ),
among them CoCl 4 2 ^, Co(NCS)4 2- , and Co(OH) 4 2 ~. This is consistent
with the fact that the [K*(jf)} i [<j*Qf)Y' configuration makes maximum
use of the more stable ir*(d) level.
9-14 THE ELECTRONIC SPECTRA OF OCTAHEDRAL COMPLEXES
The Ti(H 2 0) 6 3+ spectrum is simple, since the only d-d transition
possible is wQd) — > c*00- We must now consider how many absorp-
2.01
Electrons and Chemical Bonding
tion bands can be expected in complexes containing metal ions with
more than one d electron. One simple and useful method is to calcu-
late the splitting of the free-ion terms in an octahedral crystal field.
As an example, consider the spectrum of V(H 2 0)(; 2 +.
The valence electronic configuration of V 2+ is 3^ 3 . The free-ion
terms for d s are obtained as outlined in Chapter I; they are 4 F, 4 P,
2 G, 2 D, and l S, the ground state being 4 i ? according to Hund's rules.
/
/:
r orbital
; ; . orbital
.;: ".k;:.
■iiy,.f
p v orbit
-TV
. orbitaf
'., .orbitals
|:&S/
p, orhhd
Figure 9-19 Splittings of the s, p, d, and / orbitals in an
octahedral crystal field.
A,,
^ orbital
/'*'
^-r_ tJ orbital
*r;- :
f orbitals
J,!
dL orbital
■>:•;*£
;•:;*:■:■:
;;^
d, t orbital
</. nrbitr.l
t-> orbitals
t" 4
"
if ! ■ :
r L i
-
■'fei
/.> - jjr»
i
a>
J" " f !
/,„-
* a orbitals
J''
/*(«■ - J«I
'-j-
"» orbM
Figure 9-19 (con rireued)
2.O3
2.0A Electrons and Chemical Bonding
Since transitions between states that have different S values are for-
bidden (referred to as spin-forbidden), we shall consider the splitting
of only the 4 F and 4 P terms in the octahedral field. In order to deter-
mine this splitting, we make use of the fact that the free-ion terms
and the single-electron orbitals with the same angular momentum
split up into the same number of levels in a crystal field. That is, a
D term splits into two levels, which we call T 2 and E, just as the d
orbitals split into h and e levels.
The s, p, d, and / orbitals are shown in an octahedral field in Fig.
9-19- The splittings we deduce from Fig. 9-19 are summarized in
Table 9-3- We see that the 4 _F term splits into three levels, 4 Ai, 4 T 2 ,
and 4 7i; the 4 P term does not split, but simply gives a 4 Ti level.
The energy-level diagram appropriate for a discussion of the spec-
trum of V(H 2 0) 6 2+ is shown in Fig. 9-20. The 4 P term is placed
higher than 4 F, following Hund's second rule. The 4 P term is known
to be 11,500 cm -1 above the 4 F term in the V 2+ ion. A calculation is
required in order to obtain the relative energies of the three levels
produced from the 4 F term. The results are given in Fig. 9-20 in
terms of the octahedral splitting parameter A.
The ground state of V(H 2 0) 6 2+ is 4 A% From the diagram, we see
that there are three transitions possible: 4 At — >■ 4 T 2 ; M 2 — » 4 Ti(F); and
M 2 -► ^(P). The spectrum of V(H 2 0) 6 2+ is shown in Fig. 9-21.
There are three bands, in agreement with the theoretical prediction.
Table 9-3
Splittings Deduced from Figure 9-19
Orbital
Number of
Level
Level
Set
levels
notation
degeneracy
s
1
ai
1
P
1
ti
3
d
2
e
3
2
f
3
«2
t-,
ti
1
3
3
Bonds Involving d Valence OrUtals
Z05
* P
E(*F - l P)
^AP)
Ti(P)
+ 3 A*
V "«■
\
\
\
\
\
\
+T
V
free ion octahedral field
Figure 9-20 Energy-level diagram for a d a metal ion in an
octahedral field.
Figure 9-21 Electronic absorption spectrum of V(H a O) ( t+ .
xo6 Electrons and Chemical Bonding
Table 9-4
Energy Expressions for the Three Possible Transitions of V(H 2 0) 9 2+
Transition . Energy
4 A 2 - 4 T 2 A
4 A 2 - *T,(F) I A
4 A Z - 4 T X (P) | A + ECF - 4 P)
According to the energy-level diagram, the energies of the transitions
are those listed in Table 9-4.
Assigning the first band at 12,300 crn^ 1 to the 4 /4 2 — > 4 T 2 transi-
tion, we obtain A = 12,300 cm^ 1 . Using A = 12,300 cm" 1 and
EQF - 4 P) = 11,500 cm^ 1 for V(H 2 0) 6 2 +, the other two transition
energies can be calculated and compared with experiment as shown
in Table 9-5-
The appropriate energy-level diagrams for several important d
electron configurations are given in Fig. 9-22.
Table 9-5
Comparison between Calculated and Observed Transition Energies
for V(H 2 0) a 2+
Energy
values,
cm l
Transition
Calculated
Observed a
*A 2 ~ 4 T 2
*A 2 - 'T.iF)
4 A 2 - 'TAP)
(12,300)
22,140
26,260
12,300
18,500
27,900
a C. K. J^rgensen, Absorption Spectra and Chemical Bonding, Pergamon,
London, 1962, p. 290.
3 A„
3 T,(P)
+ %A
free :
%m
octahedral field
+X*
-feA
S D
free ion
+ %A
-H*
octahedral field
-+H&
*S _.
5 Z)<.
ee ion
octahedral field
free ion
octahedral field
a 16
d e
*p
4 T,(F)
3 T,{P)
'4
•*•„<*}
-K&
/
ip <•--"
+ KA
T,(F)
* C--,
M„
+%A
^A
free ion
2 D C
octahedral field
•+fcA
free ion
octahedral field
free ion
octahedral field
■%A
free ion
d l «
octahedral field
Figure 9-22 Energy-level diagrams for the d" metal ions in
an octahedral field.
ZO7
2.o8 Electrons ani Chemical Bonding
PROBLEM
9-1. Show that the d£ and (4 J - B 2 orbitals are equivalent in an
octahedral complex.
Solution. We shall solve this problem by calculating the total over-
lap of the (42_„2 and d£ orbitals with their respective normalized
ligand-orbital combinations. The total overlap in each case,
•T04 s _v0 and S(d/), will be expressed in terms of the standard two-
atom overlap between dj and a Iigand a orbital, as shown in Fig.
9-23. This overlap is called SQr, <4). From Table 1-1, we see that
the angular functions for d^i-^ and d^ are
4f = <3e - r*) (9-14)
and
<&3_„2 = V37(X 2 - /) (9-15)
with c = v5/(4Vxr 2 ). The normalized combinations of Iigand
orbitals are
&*■■ — ~Gki + 2fc - Zi-Ki-Ks- zi) (9-16)
2V3
and
4*-^ Kzi-Z^ + ^-zd (9-17)
We first evaluate S(d x z_yi):
This integral is transformed into the standard two-atom overlap
integral S(jr t di) by rotating the metal coordinate system to coincide
in turn with the coordinate systems of ligands ®, ®, ®, and ®.
: .. ■'■ '.'. ;.'■. -'...-I-.'.. '■''\::'-..M : , ; :y.' :■'■'. '':.'.'■''■.' r— — .' ''; ' , '.:....- : v'.V i ."' L -
S{s;4s)
figure 9-23 Standard two-atom a overlap between a d and a
Iigand <r valence orbital.
Bonds Involving d Valence Orbitals
xo9
Using the coordinates shown in Fig. 9-24, we obtain the following
transformations :
Mto®
Mto©
M to ®
x f
/
-o
} J/
/! t
Mto®
z^> y
Z— ► x
K~* — x
z-+ — y
x-~> — z
x-+ y
x^ z
x-+ —x
y-* x
y^—z
y-+-y
y ->■ z
Thus we have:
V3
— c(x 2 - y'Ozi -
V3
(9-19)
V3
V3
(9-20)
V3
V5
-> ~yc(z 2 - j 2 >
(9-21)
V3
— ^c*- 2 - y>4 -
V3
-» — c(x* - Z*)"
(9-22)
«@»»y
Figure 9-24 Coordinate system for an octahedral complex.
xio Electrons and Chemical Bonding
Adding the four transformed terms, we have
SQd x ^ y i) = fVicO-Z 2 ~ x 2 - jy 2 > dr = fVScOz* ~ ? 2 > dr
= "ArO A) (9-23)
Next we evaluate S(d£) :
S(dj) = JX3S 2 - >" 2 >J-(2s; 6 + 2^ 6 - *i - * 2 - fc - ? 4 ) *
2V3
(9-24)
The integrals involving Zs, and Ze are simply two-atom overlaps, as
shown in Fig. 9-23. Thus we have
fcO? - r 2 >— =(2fc + 2z<d dr = —=S(aM (9-25)
2V3 V3
The integral involving Zi, Z2, Z>, an d Zt is transformed into S(a,d^),
using the transformation table that was used for SCd^yi). Thus
-<3* 2 - r^Zi -> -<3j 2 - r 2 > (9-26)
-cQz 2 - r 2 \ 2 -> -<3x 2 - r 2 > (9-27)
-<3^ 2 - r 2 > 3 -+ -c(3x 2 - r 2 > (9-28)
-cOz 2 - r 2 )^-* -c(3y - r 2 > (9-29)
Totaling the four transformed terms, we find
fcOz? - f 2 ) — 7=(-s:i - £2 - Zz - zO dr
2V3
1 1
-=:/c(6x 2 + 6j 2 - 4r 2 > Jr = -~=fcQ>Z 2 - r 2 > ^
2V3 V3"
= —JO A) (9-30)
V3
Finally, combining the results of Eqs. (9-25) and (9-30), we obtain
S(df) = —SQrj,} + -=S(<T,dJ = V^OA) (9-31)
V3 V3
Then
1(^2) = SQd^^yi) = V^OA) (9-32)
Bonis Involving d Valence Orhitals
m
Thus the total overlap of d x ^^ and d^ with properly normalized
ligand-orbital combinations is the same, and it follows that the
two orbitals are equivalent in an octahedral complex.
SUPPLEMENTARY PROBLEMS
1. Under what conditions are the molecular-orbital and valence-
bond descriptions ol the <r bonding in an octahedral complex equiva-
lent? Derive the valence-bond functions shown in Fig. 9-7 from the
general molecular-orbital functions.
2. Construct the molecular-orbital and valence-bond wave func-
tions for the a bonding in a square-planar complex. When are the
molecular-orbital and valence-bond descriptions the same?
3. Which complex has the larger A value, Co(CN) 6 3 ~ or
Co(NH 3 ) 6 3 +? Co(NH 8 ) 6 3 + or CoF 6 s ~? Co(H 2 0> 3 + or Rh(H 2 0) 6 3 +?
PdCl 4 2 ^ or PtCl 4 2 -? Ptl 4 2 - or PtQ 4 2 "? VC1 4 or GoGh 2 "? VC1 4 or
CoF 6 3 -? PdCi 4 2 ~ or RhQe 3 -? Co(H 2 0) 6 2 + or Co(H 2 0) 6 s +?
4. Give the number of unpaired electrons for each of the following
complexes: (a) VF 6 3 -; (f) FeCl 4 ~; (V) NiCh. 2 " (tetrahedral);
(J) PdCU*-; (e) Cu(NH 3 ) 4 2 +; (f) Fe(CN) 6 4 -; (g) Fe(CN) 6 3 -;
(J) TiVi~; (0 Ni(CN) 4 2 "; Q) RhCl«»-; (K) IrQ 6 2 -.
5. Explain why Zn 2+ is colorless in aqueous solution. Why is
Mn 2 + pale pink?
6. The spectrum of Ni(NH 3 ) 6 2+ shows bands at 10,750, 17,500,
and 28,200 cm -1 . Calculate the spectrum, using the appropriate dia-
gram in Fig. 9-22 and assuming that AJ5( 8 F — 3 P) = 15,800 cm -1 for
Ni 2+ . What are the assignments of the three bands?
7. Plot the energies of the four states arising from 3 F and 3 P in the
d 2 octahedral-field case (see Fig. 9-22) for A values up to 20,000 cm -1 .
Assume a reasonable value for AEQF — 3 P). Predict the general fea-
tures of the absorption spectra expected for d 2 ions in an octahedral
field for A values of 8,000, 12,000, and 18,000 cmr 1 .
Suggested Reading
C. J. Ballhausen, Introduction to Ligand-Field Theory, McGraw-Hill,
New York, 1962. An excellent treatment of electronic structure
of transition-metal complexes.
C.J. Ballhausen and H. B. Gray, Molecular-Orbital Theory, Benjamin,
New York, 1964^ More advanced than the present treatment.
E. Cartmell and G. W. A. Fowles, Valency and Molecular Structure,
2d ed., Butterworths, London, 1961.
F. A. Cotton, Chemical Applications of Group Theory, Wiley-Inter-
science, New York, 1963. The best place for a chemist to go to
learn how to use group theory.
C. A. Coulson, Valence, 2d ed., Oxford University Press, Oxford,
1961. Thorough treatments of molecular-orbital and valence-
bond theories.
H. Eyring, J. Walter, and G. E. Kimball, Quantum Chemistry, Wiley,
New York, 1960. Highly recommended.
G. Herzberg, Atomic Spectra and Atomic Structure, Dover, New York,
1944. Complete and rigorous treatment of the subject matter
presented in Chapter I.
J. W. Linnett, Wave Mechanics and Valency, Wiley, New York, I960.
Good discussion of diatomic molecules.
L. E. Orgel, An Introduction to Transition-Element Chemistry: Ligand-
Field Theory, Methuen, London, I960. Nonmathematical ap-
proach.
R. G. Parr, The Quantum Theory of Molecular Electronic Structure, Ben-
jamin, New York, 1963- Mathematical treatment of small
molecules and organic t orbital systems.
2.12.
Suggested Reading Z13
L. Pauling, The Nature of the Chemical Bond, Cornell University Press,
Ithaca, N.Y., I960. The classic book on valence-bond theory.
F. O. Rice and E. Teller, The Structure of Matter, Wiley, New York,
1949. A very readable account of quantum-mechanical methods.
A Final Message
It is currently popular in elementary courses to discuss chemical
bonding as if the subject were completely understood. My opinion
is that this approach is very dangerous and should be avoided. In
reality, our knowledge of the chemical bond is still at a primitive
stage of development. It is fair to admit that the approximate
theories at our disposal are able to correlate a large body of experi-
mental information, and that, therefore, we have provided a work-
able language for the "laws" of chemical bonding. However, the
theory which gives an exact accounting of the forces that hold atoms
together and allows an accurate prediction of all the properties of
polyatomic molecules is far in the future.
115
Append:
enaix
Atomic Orbital Ionization
Energies
Throughout the book we have presented molecular orbital energy
level schemes in a take-it-or-leave-it fashion. To better understand
the diagrams in this book, and to construct similar MO energy level
schemes, it is desirable to know the relative energies of the com-
bining valence orbitals. The orbital ionization energies which are
given in Table A-l were calculated at Columbia by Dr. Arlen Viste
and Mr. Harold Basch. They are the one-electron ionization energies
of the valence orbitals in the atoms given, calculated by finding the
average energies of both the ground-state and ionized-state con-
figurations (that is, the average energy of all the terms within a
particular configuration was calculated).
Table A-l follows on page 218.
xi7
Table A- 1
Orbital Ionization Energies
Atom configurations s or s 2 p n ; energies in 10 3 cm" 1
Atom Is 2s Ip 3s 3/> 4s ip
H
110
He
198
Li
44
Be
75
B
113
67
C
157
86
N
206
106
261
128
F
374
151
Ne
391
174
Na
42
Mg
62
Al
91
48
Si
121
63
P
151
82
S
167
94
CI
204
111
Ar
236
128
K
35
Ca
49
Zn
76
Ga
102
48
Ge
126
61
As
142
73
Se
168
87
Br
194
101
Kr 222 115
3d n 'Hs~3d n - 2 4s 3d n - 1 4s-»3d n - 1 3d n - l Ap—3d n - 1
Atom 3d 4s 4p
46 26
49 27
51 28
53 28
55 29
57 30
59 31
61 31
62 32
2.1 8
Sc
38
Ti
45
V
51
Cr
58
Mn
64
Fe
70
Co
76
Ni
81
Cu
86
Ind
.ex
A1(CH 3 ) 3 , 118
Alkali halides, 75
Alkaline-earth halides, 100
Angular momentum, 3, 14
total, 22
Angular wave function, 14
Atomic number, 22
Atoms, 1
many-electron, 20
Au 3 +, 193
Aufbau principle, 20
i8, 44
in C 2 H 4 , 164
in C 6 H 6 , 175
B 2 , 56
Back bonding (donation), 189, 190
Balmer series, 35
B(CH 3 ) 3 , 118
Be 2 , 56
BeH 2> 87
Bent bonds, 160
in C 2 H 2 , 160
BF 3 , 106
BN, 80
BO, 81
Bohr orbits, 22, 34
Bohr-Sommerfeld theory, 9
Bohr theory, 1
Bonds, 36
covalent, 37
electron pair (Lewis), 37, 39
Bond angle (see Bond properties)
Bond energy (see Bond properties)
Bond length (see Bond properties)
Bond properties, table of diatomic
molecules, heteronuclear, 82
homonuclear, 39
organic molecules, 163
tetrahedral molecules, 127
triatomic molecules, angular, 153
linear, 102
trigonal planar molecules, 118
trigonal pyramidal molecules, 138
Br 2 , 59
Q, 56
CaCl 2 , 102
Charge densities, 12
BeH 2 , 93
CH 3 CN, 168
CH 4 , 121, 155
C 2 H 2 , 167
C 2 H 4 , 156
C 2 H 6 , 155
C 6 H 6 , 170
119
zzo
Index
Cl 2 , 59
CIO4 2 ", 128
CN, 81
CN-, 81, 188
CO, 81
CO+, 81
CO s 2 ~ 117
Co 3+ , 201
C0CI4 2 -, 201
Complementarity principle, 11
Co(NCS) 4 2 ~, 201
Configuration interaction, o-(j-) —
aQO, 54, 55, 142
Co(OH) 4 2 ~, 201
Coordinate bond energy, 77
Coulomb energy, 45
Coulomb integral, 44, 171
Cr 8 +, 201
Crystal field theory, 186, 188
of effect of octahedral field on or-
bitals, 202
of octahedral field, 186
Cs 2 , 59
A, 183, 187
effect of, back bonding, 199
charge on metal, 199
geometry, 197
interaction of molecular orbitals,
198
n quantum number, 200
value of, 196
Diatomic molecules, 36
heteronuclear, 62
homonuclear, 36, 49
Diborane, 118
Dipole, 67
bond, 138, 139, 145
table of molecular dipoles for dia-
tomic, 70
triatomic angular, 148
trigonal pyramidal, 140
Dissociation energy, 100
Eigenfunctions, 13, 14
Eigenvalues, 13
Einstein equation, 9
Electron affinity, 33
Electron diffraction, 11
Electronegativity, 69, 71
Electron spin, 17, 20, 48
Electron waves, 9
Electrostatic energy, 73, 74, 103
of CaCl 2 , 103
Energy levels, 42
BeH 2 , 91
BF 3 , 113
CH 4 , 124
C2H2, 168
C 2 H 4 , 165
C 6 H 6 , 173
C0 2 , 99
diatomic molecules, heteronuclear,
79
homonuclear, 54
H 2 ,47
H 2 +, 45
H 2 CO, 165
H s O, 145
NH 3 , 139
N0 2 , 151
octahedral field, 203, 206
PtCl 4 , 2 -, 143
Ti(H 2 0) 6 3 +, 182
VCI4, 195
Excited state, atomic, 5
F 2 , 57
Index
XXI
Ground-state electronic configuration,
atomic, 5, 20, 26
molecular, B 2 , 56
Be 2 , 56
BeH 2 , 91, 93, 95
BF 3 , 114
BN, 80
BO, 81
Br 2 , 59
C 2 , 56
CH 4 , 122
C 2 H 2 , 168
CiR 4 , 159
C 6 H 6 , 173
Cl 2 , 59
CN, 81
CN~, 81
CO, 81
CO+, 81
C0 2 , 99
Cs 2) 58
F 2 , 57
H 2 , 46
H 2 +, 45
Is, 59
K 2 , 58
Li 2 , 55
LiH, 68
N 2 , 57
Na 2 , 58
Ne 2 , 58
NH 3 , 138
NO, 81
NO+, 81
N0 2 , 152
2 , 57
PtCl/ - , 191
Ti(H 2 0) 6 3 +, 182
VC1 4 , 195
Ground-state term, atomic, 25, 27, 35
molecular, 60
H 2( 61
2 , 61
Group-theoretical symbols, 187
H 2 , 36, 46, 47
H 2 +, 43, 47
Hamiltonian operator, 13
H 2 CO, 164
H 2 0, 142
Hiickel approximation, 171
Hund's rules, 25
Hybridization, d 2 sp 3 , 184
sd\ 194
sp, 55
in BeH 2 , 93 -
in C 2 H 2 , 167
sp 2 , 115, 116
in BF 3 , 114
in C 2 H 4 , 157
in C 6 H 6 , 170
in H 2 CO, 164
if, 126
bent bonds, 161
in CH 4 , 125, 155
in C 2 H 6 , 155
in H 2 CO, 164
in H 2 Q, 146
Hydrocarbons, 155
I 2 , 59
Interelectronic repulsion, 59, 135
in H 2 0, 144
in NH 3 , 135
Internuclear distance, 37
Ionic bonding, 73
in alkalai halides, 75
in LiH, 68
in triatomic molecules, 100
XXX
Index
Ionic resonance energy, 71
Ionization potentials, 6, 7, 27, 32, 44
orbital, 215
K 2 , 59
Kekule structure, 174
I — S terms, 22
Li 2> 55
Ligands, 176
LiH, 62
Linear combination of atomic orbit-
als, 38
London energy, 75
Lyman series, 8
Magnetic properties, 200
diamagnetism, 48, 191
high-spin complexes, 200
low-spin complexes, 200
magnetic moment, 48
paramagnetism, 48-, 183
strong-field ligands, 200
weak-field ligands, 200
Microstates, 23
■ Molecular orbitals, antibonding, 39
BeH 2 , 90
BF 3 , 107
bonding, 39
CH 4 , 121
C 2 H 2 , 167
C 2 H 4 , 157, 159
C 6 H 6 , 170
C0 2 , 98
coefficients, 66
of BeH 2> 89, 92 '
degenerate, 55
H 2) 45
H 2 +, 43
H 2 CO, 164
H 2 0, 142, 146
LiH, 65
NH 3 , 129
N0 2 , 148
octahedral complexes, 178
it, 50
ligand-to-metal jr bonding, 188,
190
metal-to-ligand x bonding, 189
a orbitals, 49, 53
square-planar, 190
tetrahedral, 194
Molecular orbital theory, 38
N 2 ,57
N 2 + 57
Na 2 , 59
Ne 2 , 58
NH 3 , 129
Ni 2 +, 193, 201
NO, 81
NO+, 81
NO,, 148
NO s -, 117
Node, 16
Normalization, 13
Nuclear charge, effective, 33
2 , 57
Octahedral complexes, 186
Orbitals, 14, 16, 20, 21
d, 14, 18, 176
/, 14, 18
p, 14, 17
s, 14, 16
valence, 39
Ind
2.Z3
Organic molecules, 125, 155
Overlap, 40, 42
of orbitals in, BeH 2 , 88
BF 3 , 108 ff.
CH 4 , 122, 123
C0 2 , 97
H 2 0, 143, 144
LiH, 63
NH,, 131 ff.
octahedral complexes, 184
square-planar complexes, 192
Ti(H 2 0) 6 3 +, 179, 180
standard two-atom, d-a, 207
p-p<y), 51
f-XO, 50, 114
s-f, 133
s-s, 50
Pauli principle, 20
Pd 2+ , 193
Photons, 9, 10
Planck's constant, 5
Pt 2 +, 193
PtCl 4 2 -, 189
Quantum assumption, 3
Quantum jump, 5
Quantum number, /, 14, 20, 22
mi, 14, 20, 22
m„ 14, 20, 22
it, 4, 14, 20
Term designation, 23
Term symbols, for linear molecules, 60
Tetrahedral metal complexes, 194
Tetrahedral molecules, 121, 137, 155
Ti(H 2 Oy+, 176
Transition metals, 176
Triatomic molecules, angular, 142
linear, 87
Trigonal planar molecules, 106
Trigonal pyramidal molecules, 130
Uncertainty principle, 11, 12
Valence-bond theory, BeH 2 , 93, 95
BF 3 , 115, 117
CH 4 , 125, 126
CH 3 CN, 164
C 2 H 2) 164
C 2 H 4 , 159
C 2 H 6 , 155
C 6 H 6 , 174
co 2 , 100
H 2 CO, 166
H 2 0, 146
NH 3 , 137
N0 2 , 152
octahedral complexes,
187, 182
Ti(H 2 OV+,
184, 185
van der Waals
energy, 73
, 103
VCh, 194
V(H 2 Oy + , 202
Radial wave function, 13
Rb 2 , 59
Wave function, 12,
angular, 14
radial, 13
13
Square-planar complexes, 189
Zeeman effect, 9
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