Download Atomic Orbitals - Stephen Berry

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Prep.&
under the aponrorrhip of
The Advisory Council on College Chemistry
R. Stephen Berry1
The University of Chicago
Chicago, Illinois
Atomic Orbitals
Today the teaching of chemistry probably leans more heavily on the theory of atomic structure than on any other single pillar. We use the concepts of shell structure to develop the periodic table
and all the characteristic physical and chemical properties associated with chemical periodicity. The Bohr
model of atomic shells gives us a universal tool to
estimate and exhibit the magnitudes characteristic of
virtually all atomic and molecular properties. Orbitals
are a t the base of our interpretations of molecular
structure, and we almost always express these orbitals
in terms of, or a t least in relation to, the orbitals of the
constituent atoms. We even can begin to use orbital
concepts to interpret many reaction mechanisms.
The subject is, in fact, so integrated into our whole
approach to chemistry that we are astonished when a
freshman comes to us from a high school chemistry
course that did not interpret chemistry in terms of
orbitals.
I. Atomic Orders of Magnitude and the Bohr Alom
The Bohr-Sommerfeld theory of atomic planetary
orbits was the first quantitative statement of atomic
shell structure, and is still the source of much of our
intuition about atoms. Even more important, this
model was the statement of the postulate of stationary
states, a statement that simply defied the laws of classical physics: an electron in a "Bohr atom" remains in
orbit forever, and does not spiral in toward the nucleus.
Classically, the angular acceleration of such a bound
electron would force it to radiate its energy away slowly
and eventually fall into its attracting nucleus. With
the postulate of stationary states and the postulate of
the quantization of action, the theory of circular Bohr
orbits can be developed in afew lines of algebra see references (21-34; also see Appendix). From it, we develop
a powerful little table (see Table 1)of numerical expressions for the radius, velocity, energy, and period of the
circular orbit characterized by two numbers: 2, the
number (or effective number) of positive unit charges
attracting the electron of interest, and n, the quantum
number (or effective quantum number) characterizing
that particular stable orbit. (We shall say more about
Table 1.
Quantity
Expressions for Characteristic Properties of
Circular Bohr Orbits
Expression
Zs
Energy, E
Radius, r
Velocity, V
Period
2h'
n'
tLZ
nz
e2m X
ez
t- X
Z
2rha
n8
X Za
Value
109,687 cm-I
or 13.58 ev
or 2.178 X 10-LLerg
n2
0.529 X
X
em
z
2 188 X lo8 X
l* S X
Z
;cm/sec
n3
X Z, see
tL = Planck's constant h divided by 2r, e = charge on the e l e c
tron, m = mass of the electron, and Ze = nuclear charge.
effective charges and effective quantum numbers further along.) The Bohr-Sommerfeld model is admittedly inconsistent with classical mechanics and it
gives some results that do not agree well with experiment, or that simulv are w r o n ~ . Nevertheless
~
its
Alfred P. Sloan Fellow.
' One example is the nonzero angular momentum of the ground
state of hydrogen, and the properties such as magnetic moments
associated with angular momentum. The theory also prohibits
the electron from entering the nucleus; electrons actually can
penetrate nuclei. The ability of electrons to approach and
penetrate nuclei to varying degrees is the reason that proton magnetic resonance lines occur a t s. varietu of energies in a given
field. Without this property, nuclear magnetic resonance would
not he anything like the powerful analytical tool that i t actually
is.
Volume 43, Number 6, June 7 966
/
283
utility as a tool for estimating orders of magnitude is
universally recognized, and it is surely the source of
much of our visceral intuition about atomic ~ t r u c t u r e . ~
With it, we can say, for example, that phenomena occurring in times longer than about 10-l5 sec can best be
described in terms of the time-averaged distribution
of an electron and not by a moving point-charge,
simply because the classical electron would go through
many orbits during the event. Phenomena requiring
less than lo-" see, say, are not well described by
average distributions of charge in atoms. By the same
token phenomena involving energies much greater than
1&20 ev are necessarily quite disruptive to the outer
shells of atoms, but energies a hundredfold smaller do
not affectthese shells very much.
Naturally, the classical estimates we have just made
have their parallels in quantum mechanics, in terms of
wave structure and the uncertainty principle. We shall
examine these; hut first let us spend a moment examining the representation of waves and the basis of the
wave theory of matter.
The historical background of wave-particle dualitythe particle-like properties of light exhibited (for
example) in the photoelectric effect, their parallel in
the wave character of particles as suggested by deBroglie
and demonstrated by diffraction of electrons, and finally
the flowering of quantitative quantum m e c h a n i c ~ i sa
familiar and rather romantic subject. The entire
period of its invention and development was so short
that many, perhaps most, of its major figures contributed to its very origins and were still aliveand active
when it had become a mature and universally accepted
cornerstone of physical science. Moreover the logic of
its growth and the relationships between different
theories and between experiment and theory is exceedingly clear. And fortunately it is very well
documented; consequently we shall make no attempt to
discuss this background here. A few of the author's
own favorite references are given in the bibliography.
II. Matter Waves
Let us examine some of the relationships associated
with matter waves and with waves in general. For,
after all, atomic orbitals are nothing more than the
forms msumed by the standing waves characteristic of
single electrons hound in an atomic potential field.
We can discuss them i~?&igentlyand correctly only if
we are ready to recognize and use their wave-like
properties.
To begin, let us distinguish standing waves from
running waves. The latter are worth a little of our
attention here for two reasons. First, running waves
are less cumbersome in a development of the relationship between the quantum conditions and the wave
equation. Second, the transition between running and
standing waves gives us a natural and mathematically
simple example of the principle of superposition, in its
most precise way-in terms of the interaction of two
readily visualized waves.
8 It is worth noting that occasionally, respectable attempts are
made to find classical models that fit better with quantum
mechanics than the Bohr atom. One in fact appeared in preM., Phy8 Rev. Letters,
liminary form quite recently [GRYZINSKI,
14,1059 (1965)l.
284
/
Journd of Chemical Education
Traveling waves are appropriate for describing
free particles; running waves, for describing bound
particles. Any mathematical function of two or more
variables-the time, t, and the spatial coordinate, x,
for example--describes a running wave if the independent variable or variables can he written in one particular form. If a function f(x,t) can be written as f(z)
where z = kx - wt-that is, if x and t are always related
so that the real independent variation off always can
be given in terms of such a a-then the function f is a
traveling wave. It need not be a periodic function like
siu[A(kx - at)], hut it may be. The crucial property
is this: a given point on a plot of f(z), say f(ao), corresponds to an infinite number of pairs of values of x
and t , corresponding to the solutions of kx - at = a.
with k and w fixed by f. Since t moves inexorably forward, the value of x that keeps the quantity kx - ot,
called the phase, equal to zo must also move inexorably
forward, a t just the velocity w/lc, the phase velocity.
Note that we have not introduced any explicit concept
of wavelength or frequency because we have not yet
discussed periodic waves.
Periodic waves, and sinusoidal waves in particular,
play a unique role in wave mechanics. Their mathematical simplicity, and the fact that combinations of
sine and cosine waves can be used to represent any
smooth function to an arbitrary degree of accuracy,
make sinusoidal waves attractive. However, the
property that makes them so important for atomic
physics and chemistry is the fact that they give the
exact representation of the simplest stationary states
that occur in quantum mechanics, the states of a free
particle. Let us see how these waves are a consequence
of the Einstein relation between energy E and frequency
v in cycles/sec [ ( 2 ~ ) - ' X o radians/sec],
E
= hv,
(1)
pA = h
(2)
The deBroglie relation
between momentum p and wavelength A, and the
classical expression for the total energy of a free particle, is simply the kinetic energy:
E
= pn/2m
(3)
From relations (I), (Z), and (3) we obtain the connection between the frequency and wavelength of matter
waves. tbthdispersion relation
often written in terms of 1c
=
2s/A instead of A, so that
where 15 = h/2a. This is quite a diierent thing from
the relation for light waves
Free matter waves have a frequency that varies inversely as the square of the wavelength, not as the first
power--or alternatively, the phase velocity of matter
waves in free space varies inversely with wavelength.
If matter waves can be described by a wave equation,
then we can infer that equation from their dispersion
relation, eqns. (4a) and (4b). A wave equation descrihing a wave function fi is a relationship between the
time and space derivatives of J.. What is that relationship?
I n the function J.(x,t), the quantity u is a frequency,
and must be associated with the time t to give a diiensionless variable of the form vt. Similarly X must be
associated with x to give a dimensionless variable
x/X, or lcx. Since v appears to the first power, the wave
equation must involve only the first derivative of J.
with respect to time. The wavelength X appears as
C2,
SO that it must he the second derivative of J.(x,t)
with respect to x which is related to the first time derivative:
More specifically, using eqn. (4a), we have
What about the numerical constant of proportionality?
If it were m1, then J. would be a real exponential of the
form e x p [ i (2svt + kx)]. This is a formal solution to
the equation but is not quite acceptable in polite
physical company because of its annoying property of
becoming infinite when the exponent becomes large.
A physically allowable free-state solution is one whose
amplitude is at least bounded everywhere; this is
readily achieved if our constant is i = 47.
We have
which has the solutions
+(z,t) = A e W - 2 4
(6)
(We choose +i and not -i so that positive k corresponds to a wave moving to m as t increases.) The
form (6) for J. allows us to identify three differentiating
operations with the evaluation of physical quantities:
+
so that the time rate of change of J. for any x is a
constant proportional to the energy of the state:
next, the slope of the function J. is a constant proportional to the momentum:
giving us the momentum p, and finally, we recognize
that the identity
is just equivalent to the dispersion relation, and
amounts to writing
E = p2/2m
I n the general case, we identify ikb/bt with the total
E-the sum of T, the kinetic, and V, the potential,
contributions.
Finally we conclude this little exposition of running
waves by writing the running wave, eqn. (6), in its
equivalent forms
+ ( z , t ) = A [cos ( k z
- 2 r u t ) + i sin ( k z - 2 r v l ) l
(9a)
It is the last of these which is most important for our
understanding of standing waves and atomic orbitals.
Standing waves are waves that oscillate in time but
whose crests and troughs remain fixed in space. For
example $(x,t) = f(x) sin(2s vt) is such a function; at
any point xo, $(x,t) oscillates between f(xa) and -f(xo).
The function (9) is not a standing wave, but a superposition of them. We can rewrite (9) as the sum of
+
+
+ ( z , t ) = A [cos Zsvt cos k z
i eos 2 n d sin kz
sin Zrvt sin k z - i sin 2sut cos k z ]
four separate standing waves having just the right
phases to give one real and one imaginary running
wave moving together, as (9a) displays them.
The mathematics that make a superposition of
standing sine and cosine waves into a traveling wave
are quite clearly exhibited in eqns. (9) and (10). At
this point one could discuss the more philosophical
aspects of superposition. One might, for example, ask
about the probability of finding the electron in a sin kx
distribution, if we know that the wave function is J.(x,t)
of eqns. (9) and (10). We shall not pursue this point
here [cf. Reference (5)1.
The point we must make now is this: the proportionality of v and E, and therefore the proportionality
of the first time derivative and E, require that all
stationary (constant E ) solutions of the quantum mechanical wave equation for matter waves have a factor
e ~ l r i .u ~ I n other words, the functions representing all
the stationary states of an electron (or of a complex
system) contain a complex oscillating factor containing
the time. If the system is free, then the spatial part
may be complex also, and the function J. can be a running wave.
The foregoing aside on running and standing waves
has served two functions. It has developed in a sort of
painless way a primitive example of the mathematical
statement of the superposition principle. This principle is the very basic quantum mechanical concept that
there are always alternative and equivalent descriptions
of an electron wave. No one description necessarily
tells us explicitly all the properties of the electron that
we might want to know, or makes apparent all the
useful ways of interpreting the physical properties of a
wave function.
The concept of superposition will become a very
important one in our discussion of alternative representations of orbitals; of hybridization; and of valence
bond, molecular orbital, and mixed representations.
The basic concept to be grasped now is the existence of
equivalent descriptions, any one of which can be obtained from any other by a re-expression or transformation no more subtle or complicated in principle than the
transformation that gives us eqn. (9) as an alternative
to eqn. (6).
The second main reason we have dwelt on the concept of a wave is to develop the time dependence of an
electron wave in its simplest example. Now, as we
proceed into a discussion of bound states and of atomic
orbitals, we will be using a form and a physical picture
that will let us drop the explicit time dependence of our
wave function. Nevertheless, all along, it is important
to remember that every wave has a time dependence,
that electron waves describing states of constant energy
have factors
and therefore have real and imaginary
Volume 43, Number 6, June 1966
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285
parts whose amplitudes oscillate sinusoidally in time
about mean values of zero.
We conclude this general discussion by amplifying
briefly the physical and mathematical notions associated with the concept of a stationary state, and with the
idea that a wave function should have a factor e"".'
Let us generalize eqns. (5) and (7) a bit for our later use
by supposing that E is not necessarily p2/2m hut may
also contain some potential energy V(x) that may or
may not vary with distance hut does not vary with
time. Then, instead of eqn. ( 5 ) , the general statement
of (7) is
or, by way of defining the Hamiltonian X,
If the Hamiltonian X does not contain time explicitly,
and ours clearly does not, then X acts only on the space
variable of $(x,t) and not on t. But the partial time
derivative acts only on the time part. These two
conditions can be satisfied only if itia$/at and X J . are
one and the same constant multiple of J. itself. The
multiplicative factor is obviously just E, from our
previous discussion. But this implies that the energy
E is constant i n time, i.e., that the energy is stationary,
or that the system is in a statiaar?~
state. "Stationary"
in this usage does not imply that the wave function is
constant, but only that the energy is constant and the
wave function is a periodic traveling or standing
wave. The second implication follows from the foregoing because the form of eqn. (11) implies that J.(x,t)
can he written as a product +(x)t(t), and that
waves within some region (not arbitrary) appropriate
to each individual problem. All other formal solutions
would in some way fail to satisfy the various conditions.
Ill. Discrete Stationary States for Single
Particles: One-Electron Systems
Orbitols in O n e Electron Atoms
Under what circumstances do we find discrete
quantized states for a single particle? These circumstances are the kind that lead to the quantized states of
the particle in a square box, the harmonic oscillator, the
rigid rotor, and the one-electron atom or molecule. The
circumstances require that the potential energy V have
a dip or well of some sort, so that V(m), its value at
infinity, is higher than its value somewhere in the well.
If there are any states of the particle whose energy E is
less than V(m), then these states must be discretely
quantized. For example, the simple harmonic oscillator
with V = kx2/2 has infinite V(m), so that all its states
are quantized. The hydrogen atom has V(m) = 0,
conventionally, so that any state of negative energy, E
< 0,must be discretely quantized.
Graphically, the discrete quantization is a generalization of the discrete quantization of the oscillatory states
of a rope with hoth ends fastened, or of a particle in a
box. For the rope or particle-in-box, a continuum of
oscillations is possible if only one end is held or if one end
of the box is open. However, if both ends are fastened or
closed, respectively, the only oscillations are those that
give constant displacements of zero at hoth ends (see
Fig. 1). If the rope's length is L, these have the spatial
form A sin (nrrx/L) (or some combination of these)
where n is any positive integer. The generalization
and
The function t(t) is simply e-""';
+(x) is just a function of x independent of time, so that J.(x,t) is necessarily a wave with period Elti.
Where does the idea of discrete quantized states enter
our physics? So far, all we have done applies to continuous distributions of states. The discrete quantiz*
tion is, in essence, a result only of the introduction of
finite boundary conditions. So long as we make no
restriction on how the wave function behaves as it goes
off to
m, there is no quantization. (We require
only that the free functions remain bounded by some
upper and lower limit.) However as soon as we introduce "finity" boundary conditions-like saying that
+(x) must vanish a t the walls of a box located a t *a, or
that +(x) must correspond to a function on a ring and
+(x) = +(x+211), or that +(x) must go to zero exponentially as x approaches hoth * m -any such conditions immediately remove the possibility of a continuum
of E values and of a corresponding continuum of states.
[A detailed discussion of this is given in Ref. (,$).I I n
essence, the imposition of houndary conditions a t both
ends of the range, plus the conditions that the houndstate wave have no kinks or discontinuities and he
quadratically integrable, eliminates all possible functions except those having an integral number of half-
*
nodal point
(b)
Figure 1.
I.) Rope with a free end; the dirplocement of the end of the
rope con hove any volue. The oniy conditions are that ot one end (x = 01,
the dirplasement y(O1 = 0, and the1 the other end g e h no further from
x = 0,y = 0 than 1, the lengthof the rope-i.e., ylendl
1.
5
(bl Rope with both ends swashed, the conditions y101 = 0 and yIL1 = 0
allow the rope oniy a discrete (but infinitel number of vibrational stoles,
nomely thwe with 0,1.2.
nodes between the ends.
...
comes when we replace the two rigid fastenings with exponential decreases to zero a t both ends, + m , for all
three Cartesian coordinates-or tie the two ends of the
rope together.
It is worth noting that a central potential may be
attractive and still not be able to s u ~ n o rbound
t
states.
In the region where E < V, the wave function is curving
away from the axis, so that i t must be leaving the axis
as it enters the potential well. I n the region where E >
V, and the wave function is curving back toward zero
amplitude, its curvature is always proportional to the
depth of the well below its energy value. If the well is
both shallow and narrow, the wave may be unable to
bend back to re-enter the forbidden region with its
slope inclining toward the axis, the slope required to
make the function die exponentially as it penetrates
the region where E < V . If no wave can re-enter
properly, then no wave can correspond to a bound state.
Figure 2 illustrates this behavior; we can picture it in
terms of trying to fasten a very stiff rope to two hooks
in a tight space.
Bound States in a Potential
Figure 2. Three types of behavior; (a)bound, quantized rtote with curvature just suitable far matching both decaying curves with the & w r o i d d
curve; ib) and (4 phygically impossible situations, corresponding to no true
V is only o dying exponential on one
states; the wove function with E
ride, and grows exponentidly on the other. Arrows mark points where
E = V a n d cvrvotvrechonger sign.
<
The exponential decrease is a very simple consequence
of the mathematics whenever the total energy E is less
than the potential V. We refer again to Ref. ( 2 ) ,
pp. 51-58, for a particularly clear exposition of this
topic. It is worth noting one point now that will become
especially important in the last section of this paper,
dealing with electron correlation. Whenever E < V,
the kinetic energy is necessarily negative,and the momentum, being p2/2m, becomes imaginary. This seems
strange and formal, but it is just as strange, formal, and
above all nonclassical to have E < V a t all. The existence of the wave function in nonclassical regions is
one of the most important physical differences between
classical and quantum mechanic^.^
With our conclusions from the previous section and
the paragraphs just preceding, we have a basis for a
clear but still qualitative picture of a stationary state of
a singleelectron atom. The electron is described by a
complex wave in three dimensions; the wave's real and
imaginary parts oscillate sinusoidally in time, 90' out
of phase, with a frequency v = E/h. The energy E
of the electron is negative, and can only assume certain discrete values, and the wave itself goes to zero
exponentially as r, the electron's position vector, goes
to infinity. Such a function is the simplest example of
an atomic orbital.
Quantum Numbers and Constants of the Motion
So far we have characterized an atomic orbital by
one number only, the energy. We all know perfectly
well that there are other characteristic numbers or
quantum numbers associated with orbitals. Let us
inquire into their origin.
That E is a good quantum number is a consequence
of the conservation of energy. This is a rather trivial
statement when it is put this way, but we can say it
slightly differently: if X, the Hamiltonian of a system,
does not change explicitly with time, then the energy of
that system will be a constant or a good quantum number. That means that if the potential and the parameters such as mass, and the universal constants do
not depend on time, so that X a t t, is identical with X
a t ts then E will be a good quantum number.
The other familiar quantum numbers of atomic
orbitals represent other physical constants of the
motion associated with other invariances of a Hamiltonian, i.e., of the physical description of a system. The
quantum number 1 of a particle is a good quantum
number when the total orbital angular momentum of
the particle is a constant. This comes about if, and only
if, the Hamiltonian of the system is spherically symmetric. Since the kinetic energy depends only on
momentum and makes no reference to any spatial coordinate, i t is as symmetrical in space as it can be;
we need only examine the potential energy for its
symmetry. If it depends only on r, the distance of the
particle from the origin, and not on any angle, then
naturally the potential energy is spherically symmetric.
Suppose we are studying an electron bound by a
potential V. The electron has some instantaneous
angular momentum. We now seize the apparatus that
produces V, and rotate the apparatus to a new angular
orientation without translating it a t all. If the potential
were not spherically symmetric, such a motion would
clearly disturb the electron, in general, but if V were
spherically symmetric, rotating the apparatus would
leave the electron entirely unaffected. Specifically, if
V is not spherically symmetric, the rotation would in
general introduce a torque on the electron and change
its orbital angular momentum. If V is spherically
symmetric, then the electron's orbital angular momentum is unchanged by the operation applied to V.
I n other words the orbital angular momentum is a constant of the motion.
For the sphere, or for a spherically symmetric system,
Q mmacroseopie example of the penetration of a. quantum
mechanical wave through a classiedly forbidden barrier zone i3
the Josephson effert. In this phenomenon current emtrrien in
semi- and super-conductors are able to penetrate layers of insnlator between two semiconditcting or mpereondneting bodies.
Volume 43, Number 6, June 1966
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287
re-orientation into any angle is a symmetry operation:
re-orientation leaves the system in a condition indistinguishable from its initial condition. Symmetry
operations never come singly (except those for the most
unsymmetrical things we can think of, things that can
only be left alone) and do not come in arbitrary combinations; in general one operation followed by another
is equivalent to a third operation. The set of all operations associated with a particular symmetry type is the
grmp of operations for that symmetry. The set of all
rotations constitutes the simplest full symmetry group
for a spherical system. The rotations about the figure
axis constitute a symmetry group for a cylinder or a
helix.
Just as the total orbital angular momentum is a
constant if V is spherically symmetrical, the angular
momentum component about the figure axis of a cylindrical system is a good quantum number. The invariance of the cylindrical system with respect to any
rotation about its axis implies the existence of a constant
of its motion, the corresponding component of orbital
angular momentum.
A special case of a system with cylindrical symmetry
is the sphere--and, indeed, a spherical system has a
constant orbital angular momentum component along
an arbitrarily chosen axis, as well as a constant total
orbital angular momentum. This quantum number is
m,, of course. (We shall sometimes use m for m,.)
Why can we not h d the angular momentum components along three axes of a sphere, instead of just one?
At the risk of being too brief, we can answer this just
by saying that such knowledge would localize the
particle's orientation too much, to the point of violating
the uncertainty principle.
I n general, in a spherically symmetric potential, the
energy of a hound electron depends on its angular
momentum-i.e.,
on 1, as well as on the principal
quantum number n, but never on m. (The exception is
the Coulomb potential, for which all states of the same
n are of equal energy, regardless of I.) There are 21 1
values form, for any 1, so that there are 21 1different
degenerate orbitals having the same n but different m
+
+
"-1
[n2or
+ I), for the Coulomb or hydrogen-like
These 21 + 1 different states transform into
C
(21
1-0
case].
mixtures of each other if we redefine the orientation of
the coordinate axes of our spherical system. But no
matter how we re-orient the coordinates, a given set of
21
1 wave functions transform only into each other,
and never into any other functions. This is exactly
analogous to the way sin no and cos n8 can be transformed by addition and subtraction into en' and e-"O
or mixtures of these, but never into anything containing
e-"n+l)e. The set of 21
1 functions is called a basis
for a representation of the rotation group.
I n fields of lower symmetry than spherical, the
orbital angular momentum is no longer quantized.
However, the symmetry invariances may remain in
part. Sometimes we can suppose that 1is a good, constant quantum number but m is not (Russell-Saunders
coupling: there, the same thing happens to spin; S is
a constant but M , is spoiled as a quantum number).
In other cases, even 1 is not constant. It is very often
useful to start with a set of free atom functions, either
orbitals or many-electron state functions, with all the
+
+
288
/
Journal of Chemicol Education
degeneracy appropriate to a spherical potential; then
one asks what happens to this particular set of orbitals
if the symmetry of the potential is lowered to something
less than spherical, like octahedral or simple threefold
for example. One uses the symmetry properties of the
wave functions to determine their behavior in a t best a
semiquantitative way. This can be an exceedingly
powerful way of elucidating chemical problems. The
most famous application of this sort is of course crystal
and ligand field theory in its phenomenological form.
We shall return to this topic for some examples, but we
shall not try to develop the entire theory of group
representations in atomic physics and chemistry.
Several references a t various levels are given in the
bibliography. The pertinent conclusion for us now is
this: if the symmetry of the potential V is lowered from
spherical to some new form, then some but not all the
degeneracies of the spherical case may remain. The
rather straightforward algebra of group representations
lets us determine very easily exactly how any given basis
set of 21
1 degenerate functions from the spherical
case will split into smaller sets in the new and lower
symmetry. The answers are expressed in terms depending only on 1 and on a smaller number (sometimes only
one) of angle-independent quantities that can be treated
as empirical parameters or can be evaluated from atomic
wave functions. We shall return briefly to this subject
later.
+
The Forms of One-Electron W a v e Functions
The mathematics of a spherically symmetric oneelectron problem lead directly to a set of standing waves
that we can understand and describe very easily, a t
least in part. Putting together the pieces of the foregoing discussion, we can tell just what to expect. In general, the characteristic stationary waves must describe
constant-energy states whose total orbital angular
momentum is characterized by 1, and these waves must
come in degenerate sets of 21
1. If we make use of
all the spatial constants of motion, then these functions
correspond to the states of definite mi. We are free to
describe a system of energy E in terms of any combination of m, eigenfunctions. As we have indicated
previously, it is sometimes useful to suppose that the
appropriate wave function is not an eigenfunction of
any one component of angular momentum, but has
some other property, like directionality.
The solution of the spherically symmetric Schrodinger equation appears almost as soon as we convert
the Hamiltonian into a form that can be written as a
sum. In the sum, one set of terms contaiosall the radial
dependence including the entire potential, and the other,
all the angular part, which is only kinetic. (We let
3, = rZT, and 3 0 , =
~ r2 X l(1
1)RZ/2mr2,or rZ X
Tn,* rotational energy.)
+
+
+
r'X = [3,
+V(T)]
r-dependent
only
f
%.p
(14)
angledependent
only
We say X is separable when it can he so expressed, as
a sum of independent terms. Using the same reasoning
that gave us equations (12) and (13) from (ll), we
obtain a product form for fi(r, 8, q ) :
and one equation for each of the three factors R(r), O(8)
and %(q). The function R(r) must depend on the
specific problem. It is known, naturally, for the
Coulomb potential V(r) = -e2/r, and for the general
Ar-" potential as well. I n the next section we shall
see how such a problem arises and is treated in manyelectron atoms. The Coulomb solutions are given by
Laguerre functions; some of the lower ones are shown
in elementary texts.
Basically, all the radial solutions R&) for bound
states have certain properties in common. They all go
exponentially to zero as r goes to infinity; the function of
lowest energy is nodeless and each successive higher
function with the same 1has one more node than the one
before it, and the functions are orthogonal and can
always be normalized in the sense that
6
R,:*(r) R,,i(r) r 2 d ~=
S,.,'
(i.e., 0 if n f n', and 1 if n = n')
These conditions have the usual consequence that each
function reaches its maximum amplitude in its outermost lobe, and that each successively higher function
has its outer lobe a t larger r than the one before. This
is all clear and obvious in the case of the hydrogenic
functions, but is is worth noting that these characteristics are quite general. I n the next section we show
some radial functions for atoms.
The angular functions are the very well known spherical harmonics, the forms taken by standing waves in
any spherical problem. For example the standing
waves on a flooded planet exhibit exactly the same
angular behavior as do atomic orbitals hut occur only
on a single spherical surface, so they are perhaps easier
to visualize than the wave function of an atomic electron. This example is developed in Ref. (3).
Table 2.
Some Lower Spherical Harmonics Y d 8 . d
YP=
=
&
1
eos 8
d-functions: 1 = 2
~
~
=
-
415
1
8s
yZs =
sin 0 cos 0 ec's
$33%
IV.
Atomic Orbitals in Many-Electron Atoms
The Many-Elecfron Problem
s-function: 1 = 0
pfunctions: 1
First, we see that each Yim(8,u)is a polynon~ialof degree 1 in sin 8 and cos 8. These can be expressed, alternately, as trigonometric functions of multiples of 8.
Either form lets us visualize the 8-dependence explicitly.
We could combine functions like Y I 1and Y1-I to get
real (i.e., not complex) comhinationsvarying as sin 8 cos
a (i.e., as x) and as sin 0 sin q (i.e., as y).
Another point to recognize about the spherical
harmonics is a symmetry property with respect to the
origin. The functions with even 1 all contain only
even powers of sin 8 and cos 8, and the odd 1 functions,
only odd power. If we change s to -x, y to -y, and
z to - z (i.e., invert the coordinate system through the
origin), then Y's of even 1 go into themselves (even or
gerade functions) while Y's of odd 1 go into their negatives (odd or ungerade functions).
Third, as the number of angular nodes increases with
I, so each lobe becomes more directional or pointed.
Very high 1 functions, with large numbers of angular
nodes, describe electrons that are nearly classical in
their orbital rotations. For, according to the Correspondence Principle, when the wave length of an electron wave is small compared with the dimensions of the
volume in which the electron moves, the particle
ceases to show its wave character and behaves like a particle.
By contrast, the radial parts of the one-electron bound
state wave functions retain quantum character even for
high quantum numbers because the outermost lobes are
always both the largest and the most spread out. The
inner parts of highly excited states, the parts at low r
values, do have rapid oscillations. Consequently, in
states of high n, electrons behave classically in their
radial coordinate when they come near the nucleus but
as waves when they move to large r and their local
momentum is small.
=
4z
sin 20 e'v
-
,in2
6
=
!+5
2
(1 32%
eels
28) e2tq
Table 2 represents some of the analytical expressions
for the spherical harmonics Ytm(8, q) = O(O)%(q).
Certain general properties are worth explicit mention.
Up to this stage, we have examined the wave functions for a single electron moving in a potential field,
especially a spherical potential. But most atoms do not
consist of a single electron moving in a potential. Let us
now explore the way one can develop well-defined wave
functions for single electrons in many-electron systems.
Then we can look at some of the properties of these
orhitals and see how they are used in some representative chemical problems. In the next and final section
we can look at the limitations of the orbital method,
and see how its limitations affect our interpretations of
physical problems and how we can try to overcome these
limitations.
Suppose we consider first the two-electron case, the
helium atom. The lowest state of He must have both
electrons in 1s orbitals, we say. But the Hamiltonian of
the system contains potential terms -ez/r,, -eZ/rz, and
e2/rlz--that is, not only is the nuclear attraction for each
electron part of the potential; the electron-electron
repulsion is part also. Obviously a t any instant neither
electron moves in a spherically-symmetric potential.
Then how can we possibly refer to a 1s orbital, much
less assign both electrons to it?
The rationale for assigning the electrons in a qualitative building-up model of the atom and the method by
Volume 43, Numher 6, June 1966
/
289
which we calculate the shapes of orbitals depend on the
notion that we can find some effective potential V(r,)
for each electron. Each V(rj) must have spherical
symmetry and approximate the true potential felt by
electron j. Could such an effective potential be found,
and could one solve the resulting equations to find
orbitals for a many-electron atom?
Both questions have been answered with an almost
positive "yes." The proper effective potential was
developed by Hartree, Fock, and Slater in the early
1930's for atoms with closed shells or with one electron
or one hole in a closed shell. Their original method
left some ambiguities about the best way of defining
V(rJ for arbitrary open-shell atoms. Then, the method
was exteuded by several people so that potentials could
be defined for open shells. (These can probably never
be made to approximate real potentials as well as the
Hartree-Fock potentials for closed-shell atoms. This is
inherent in the slipperiness of an open shell system-the
electrons tend to move simultaneously from one m,
state to another within a given open shell of lixed 1.
The mathematics of open shells tells us this when it
requires that a single stationary state be a superposition of several specific assignments of electrons to mistates, i.e., to be a multideterminant function, if we
may use a term to be defined later.)
The Hartree-Fock method consists of using a
Hamiltonian for each electron that contains V(r,) defined by the Hartree-Fock prescription, which we shall
describe shortly, and solving all the equations together
for all the electrons of an atom. The original solutions
for HartreeFocB orbitals were obtained by numerical
integration and were therefore tabulated functions with
no analytical expressions. Numerical solutions are
still obtained and, at their most refined level, are probably still the most accurate. Analytical approximations for Hartree-Fock orbitals can be obtained with
high-speed computers; they are extremely useful and
often are very close approximations to the best functions we have.
Note that when the method reaches the stage of computation, the Hartree-Fock equations are ordinary, not
partial differential equations. This is because the
Hartree-Fock potential is chosen to be spherical, so that
the orbitals must he spherical harmonics multiplied by
radial functions. These radial functions are the solutions of the differential equations. The symmetry of
the problem allows it to be reduced this way by letting
us use our general knowledge about spherical systems
to go most of the way.
The seemingly mysterious potentials of the HartreeFoclr equations are defined this way: the potential for
each electron is the mean potential due to the nucleus
and all the other electrons-more specifically, it was
shown to be the root-mean-square potential. The original form used by Hartree utilized a simple Coulomb
field based on Born's probabilistic interpretation of the
wave function. If $(r) is the amplitude of the wave
function at r, then '$(r) or $*(r) $(r) is the intensity of the wave there, a quantity everywhere positive and whose space integral is normalized to unity.
These properties led to the identification of $(r)i2 as the
density of probability, or, in our minds replacing a time
average with a space average, led to the identification
of l$(r)i2 as the mean charge density at r. Hence each
290
/
Journal of Chemical Education
electron must feel the Coulomb field of the spherical
average or the spherically symmetric sum of the charge
densities defined by all the i$(r),2's for all the other
electrons.
But, we ask, how can we find one orbitd unless we
know all the others? The answer to this question is the
crux of the success of the method. We can start with a
guess for all the Vsof the system, with a most outlandish
collection of orbitals if we wish. We use all but one of
these to determine the potential for the last electron,
and then solve the differential equation for this last
electron's orbital, $(N). Then we use the new orbital
with all the originals but one, say $(N-,), to determine a
refined version of the deleted $(N-I) from the original
set. The two new orbitals $(N) and $(N-l) plus the old
ones give us a third new one, $(N. 2), and we continue
until we've determined an orbital for each electron.
Then we start through the process again, refining our
first $(,,, $(,-I),
. . . , $cl) revisionsinto second revisions,
and then go through again and again until we find that
the orbitals are not changed by further recycling. The
potential field defined by this series of operations is now
consistent with the orbitals that it determines and that
determine it. It is called a self-consistent field or SCF,
in this case a Hartree SCF.
Exclusion Principle and Many-Electron Wave Functions
We must interject a brief acknowledgment of the
existence of electron spin and a review of its role in
many-electron problems. (It could have been part of
our discussion of symmetry; cf. References (36) and
(27).) And we must introduce the Pauli Exclusion
Principle, too. The former adds the quantum number
112, to our set n, 1 and m, for a oneelectron function;
strictly, it adds s as well, but since all the electrons we
know haves = '/s, we don't bother carrying it explicitly.
We do have to pay attention to m, (=
the component of s along one chosen axis, in any many-electron or magnetic field problem. Them, quantum number's values do contribute very much to magnetic properties; more germane for this context, spin adds one
extra degree of freedom to each orbital.
A spatial function $(r) with an assigned m, of +'/2
is said to be a spin orbital with or spin, and if it is necessary to designate the spin state explicitly, is usually
written either as $(r) a or as $(r) alone. If m, is
assigned as
one commonly writes $(r)B or &(r).
Sometimes one need only say that $ should stand for
the entire spin orbital.
The Pauli Exclusion Principle is the statement that
no two electrons may be assigned to the same spin
orbital. This can be restated many other ways, perhaps less concisely: no more than two electrons can be in
the same orbital, and if two are in the same one-electron
state in coordinate space, they must have different spin
states; or, no two electrons in a system can have the
same values for all their quantum numbers. If two
electrons with the same spin are forced together in
space, they must go into different energy states. One
may remain in a low-energy state, but the other must go
to at least the next higher energy orbital. Such a restriction is equivalent to the requirement that the two
electrons stay apart in momentum space if their wave
functions are close together in coordinate space. The
measure of this requirement is closely related to the
uncertainty relation ApAq 2 R/2 for any single coordinate of one particle. The requirement, stated in
terms of the six-dimensional space of three spatial
coordinates and three momentum coordinates is simply
that each electron needs a volume AT = AxAyAz
Ap,Ap,Ap, = h3.
Electrons are identical particles, so that no physical
property can be affected if we rename or renumber them.
For example (and not an accidental example), if we deal
with a wave function q(rl, . . ., rN)for N electrons, then
-?(r,, . . . , r N ) =must be unaffected if we interchange two
electrons; moreover the wave function *(rl, . . . , rN)
must go back into itself if we interchange the same pair
of electrons twice. These two properties imply that
interchanging two electrons must either leave Y unchanged ( q is symmetric) or, at most, change it into its
negative ( q is antisymmetric). Now \E must always he
zero if two electrons are assigned to the same spin
orbital, according to the Pauli Principle. This is just
what the situation would be if Y were to change into
- q whenever a pair of electrons was interchanged. If
electrons 1 and 2 were arbitrarily assigned to the same
spin orbital, then the identity of electrons and their
spin orbitals would imply that -?(r,, r,, . . , r =
( r r , . . ., rw) but also our guessed permutation
property would require that q(r,, r2, . . ., rN) =
-V(r%, rl, . . ., rN),so that this Y would necessarily be
zero. We have never seen a nonzero Y in nature with
two electrons in the same spin orbital; we can infer,
therefore, that the antisymmetric choice correctly
describes the behavior of real many-electron functions.
This argument is not meant as a rigorous proof of the
antisymmetric property of many-electron functions but
only as a demonstration of the relationship between the
Exclusion Principle and the property of antisymmetry.
The Form of Many-Electron Orbital W a v e Functions
Now we can go on to consider the form of manyelectron wave functions and the relationship of this form
to atomic orbitals, and then we shall return to the
Hartree-Fock problem.
If we assume that we can find appropriate self-consistent potential fields V(rJ for each electron of an
atom, then the Hamiltonian of the atom can be written
as a sum of Hamiltonians, each one containing the
kinetic energy T, and potential energy V(r,) for only
one electron. That is,
x =
C xi
sll electrons
j
Once again, as withequations (12) and (13) and with (14)
and (15), we have a separable Hamiltonian. This time
it is separable into its one-electron terms. As before,
whenever the Hamiltonian is separable, the wave
function can be expressed as a product of functions,
each of which is the solution of its own equation:
XAr;)
=
v&;)
(17)
That is, we may write
*(r,,.
.. r ~ =
) +(11(11).. .ILIN~IN)
(18)
indicate the J t h spin orbital.
where we let each
Equation (18) is a very useful form and tells us very
explicitly something about the kind of atomic wave function we get if we start with the orbital concept. This
equation says that in this picture, the electrons have
probability amplitudes, and therefore probability distributions, that are independent of the position of any
other electron. The only effect elertron 1 has on
electron 2 is through mutual effect of their average
potentials of interaction with each other and to a lesser
degree, with the other electrons.
Expression (18) cannot present the oomplete picture
even in terms of orbitals because it does not have the
property of being antisymmetric with respect to exchange of any pair of electrons. If we exchange electrons 1 and 2, and require that the function q be repaired so that it changes sign with this exchange, then
we can do it this way: replace (18) with
Now we can make any other pair interchange and subtract the corresponding new functions from each of the
two terms in (18a) to get to a more repaired state.
Eventually we'll reach some of the rearrangements
again; in fact we can construct all the permutations of
the N electrons among the N spin orbitals &,). If we
continue to change the sign when we make a pair exchange, we will eventually construct the totally antisymmetric function that can be constructed with the
spin orbitals $ill, . . . , fiIwl. The factor 1 / 4 2 in (18a)
was added to keep the function normalized. When we
have all N permutations, the normalizing factor is
I/.\/#!!, instead of 1 / 4 2 .
I t was pointed out by J. C. Slater that the construction of a completely antisymmetric function from a set
of products, adding and subtracting as we have just
described, is exactly equivalent to making a determinant
out of the J.ln(rk) spin orbitals. One lets the number
(J) of the spin orbital be the column index and the electron number k be the row index, or vice versa.
I n this way we have
This expression is the Slater determinant, or the determinantal function based on the spin orbitals $(I,, . . . ,
\I.(N). Because Slater determinants are so commonly
used, and because it is really redundant to write more
than the principal diagonal or the N spin orbitals themselves, once we know that a determinant is meant, we
frequently use a shorthand:
Even the normalizing factor is left implicit. Another
common notation applicable even if \Ir is not a product
function uses the antisymmetrizing operator (2 (which
is usually defined to do the normalizing also). We say
simply that if (2 acts on the function (IS), it generates
the function (19). Although this seems like an arbitrary and rather useless formal definition, it is possible
to write out explicit prescriptions for (2, and to use this
Volume 43, Number 6, June 1966
/
291
shorthand as a powerful tool for dealing with manyelectron problems.
It turns out that a single determinant function like
(19) is a very suitable form for a Hartree-Fock wave
function whenever all the spin orbitals of a given n and 1
are either full or empty-i.e., in the closed shell cases
like He, Be, Ne, or Zn+2. The form is also fine when
there is one electron outside a closed shell structure or
one hole in a closed shell system. I n other cases, if we
try to use a single Slater determinant to define the wave
function of N electrons, we find we are in trouble.
Except for special cases, we cannot obtain functions
with quantized total orbital angular momentum L or
total spin S with single determinant functions. Physically, in the proper functions with quantized L and 8,
the inl and in, values of the individual electrons in the
open shells are no longer good quantum numbers. The
correct function based on orbitals must be written as a
sum of two or more determinants. Here are two
examples. The lowest triplet state of He has orbitals
1s and 2s each singly occupied. I n the notation of
(19a), we have
The extra 1 / 4 2 is another normalizing factor. The
two assignments in (20) differ only in assignments of
m,. I n effect, the electrons flip each other's spins. A
more drastic reorganization takes place in the 8Pground
term of the carbon atom:
Similarly the 2p2, 'D, and '8 levels of carbon also involve 1 and s reorganization (we omit 1s and 2s electrons in writing).
P(p:'D,Mfi = 0,M I= 0) =
1 1212~0%l
12p+%I
+
- l%+2~-
I I,
*(p,= '8,Mr. = 0, M. = 0) =
--431 [ PP+% I - 1 2 x 2 ~ l-
and
-
12~02x1I
In this example, the 2p electrons change both their m.
values and their m:s.
In all cases of this sort, the
changes are not random but occur in concert. Here,
by the way, we have our first explicit inkling that there
might be a limitation to the orbital picture, just because in the second of these examples, we cannot assign
a specific set of four quantum numbers to the
electrons. We can specify only n and I for each electron.
The set of states defined by assigning n and I for each
orbital is called a configuration. A real state is based on
a single configuration just so far as each one-electron
nand 1are true constants of the system.
If,in addition to all then's and l's, we specify L and S,
then we are said to specify a term, like ls22s22p2to give
aP of carbon. From any single configuration we can
always find one or more terms.
What physical effects result because we must use
the antisymmetrized function (19) instead of the simple
product (18)? We have discussed how the antisymmetry property was associated with the Exclusion
Principle and with the fact that the electrons can't all
292
/
Journol of Chemical Education
condense in the 1s orbital of an atom.
The antisymmetry property exhibits its effects on the
forms of atomic orbitals in a very explicit way. This
is through the famous exchange interaction which Fock
added to Hartree's simple average Coulomb interaction
in the self-consistent field. I n Hartree's original expressions, the potential energy of electron j in an atom
had the form
Fock showed that an antisymmetric function like
(19) introduced a similar-looking but physically different
term, due to single pair permutations:
Because of the additional pair exchange, these always appear in closed shell cases as the combination
FJK - G J K . Also the G's are zero if J.c,l and $yK1 have
different m,, while the F's are not. The effect of GJKis
to reduce the effectof Coulomb repulsion between electrons of the same spin in $qJ,and J.(K). This in turn
permits the orbitals to be closer to the atomic nucleus
and therefore to be more strongly bound. The improvements in energies with the exchange interaction
included are quite dramatic; the effects on wave functions are sometimes subtle. Figure 3 shows a comparison of typical Hartree and Hartree-Fock radial functions.
Hartree-Fock orbitals have been computed numerically to high accuracy for some atoms, and approximately
for the entire periodic table. They have also been
computed in analytic form to rather high accuracy for a
number of atoms. The very first analytic form for the
radial wave function, historically, was the simple exponential, introduced in 1930 by Frenkel. The most
popular extension of this form is a monomial multiplying an exponential:
The parameter 3 is fixed by normalization, so that
only a is a free parameter. These are the Slater type
orbitals; Slater's original set of orbitals was based on a
simple set of rules for choosing optimum ru's [J. C.
SLATER,
Phys. Rev. 36,507 (1930)l. The most popular
of the modern accurate analytical forms is a sum of
functions like (24), often with several terms of the same
n. The other popular form in current use is based on a
sum of Gaussians, which replace ar or (24) with j3rBr2.
These have the advantage that they lead to simpler
integrals than do exponentials in r, but more of them
are required to reach a given level of accuracy. References to some of the more extensive tables and sources
are given in the bibliography (4f-44).
The Hartree-Fock orbitals, as we said, represent the
best ae+lectron functions based on the mean field of all
the electrons, in the sense of giving the best one-electron energies in this mean field. Because of this energy
property, Hartree-Fock orbital energies give very good
estimates of ionization potentials (and electron affinities, if we start with Z 1electrons around a nucleus of
charge +Ze). The orbital energy of the highest filled
orbital is approximately equal to the ionization energy.
I t was also shown that Hartree-Fock orbitals give good
estimates of all other atomic properties that involve interactions of a potential field without inducing transitions. By "good," we mean accurate to about the same
percentage accuracy as the total energy. Typical
properties that are evaluated well by Hartree-Fock fnnctions are charge distributions, electric field gradients a t
nuclei (the source of nuclear quadrupole coupling in
chlorine compounds), and in certain cases like the alkali
atom principal series, spectral line frequencies and intensities. Other properties that are likely to be less
accurately described by Hartree-Fock functions are
properties like spin-spin and spin-orbit coupling, that
depend on electron-electron interactions, and spectral
frequencies and transition probabilities that involved
more than a simple s-to-p transition, like the forbidden
atmospheric line of oxygen.
One property for which there is a significant number
of calculated and experimental values is the ordinary
electric dipole polarizability, the induced dipole per
unit of applied electric field. Table 3 contains some of
the calculated (Hartree-Fock) and experimental polarizahilities for several atoms and ions. It appears that
the lower the polarizability, the lower the relative error
of the calculation, which is probably because the species
with high polarizabilities have many low-lying excited
states including doubly-excited states or states that are
not included in the Hartree-Fock calculation.
+
Table 3.
Atom or
Ion
Experimental and Theoretical (Hartree-Fock)
Dipole Polarizabilities
Theoretical (A3)
Experjmentd~
(Aa)
in a free atom. I n a molecule, one can use the same
hybrids, or the closely related equivalent orbitals, or
any of several other similar localized orbitals; they vary
a little in their definitions and manner of construction
but prove to be quite similar in the long run. We shall
here restrict ourselves to atoms, sidestepping a lengthy
discussion of these various localized molecular orbitals.
Let us first examine the mathematics relating the
Hartree-Fock orbitals to the localized orbitals. This
way we can see how the best approximations to oneelectron states of constant energy are related to the best
approximations to localized orbitals for electron pairs.
Then we shall compare the two types of orbitals.
The approach is based on a transformation like one to
which we alluded in our discussion of spherical harmonics. The localized orbitals appear when we superpose the standing waves that are Hartree-Fock orbitals.
We quote and then use a theorem about our determinantal functions like (19): we can make any unitary
transformation we like of the filled orbitals and leave
the value of the determinantal function (19) unaffected.
Recall that a unitary transformation is like a pure rotation of a coordinate system: it preserves all the lengths
(normalization) and angles (orthogonality) when it
transforms the old coordinates (functions) into the new
ones. This means that if we mix the individual factors
+(rJ with each other by adding and subtracting them
in any size pieces, so long as we maintain the orthogonality and normalization of the new $'s, the determinant (19) formed from the new +'s will be equal to
the old determinant, for any fixed choice of electron
coordinates. We can mix functions with 1 = 1 and
mr's of +1 and - 1 to get functions that look like sin 0
cos p and sin 0 sin p in their angular dependence; that
is we can go from
R(n)R(n) sin Ole+ sin B d w
R(r,)R(rl) sin A e - i ~ ,sin B&
to
R(?,)R(m) sin 8, eos o, sin 82 sin m R(r,)R(m) sin 8, sin
q: sin
& cos oz
by letting
COHEN,H. D., J . Chem. Phys. 43, 3558 (1965).
L:ANGHOFF,
P. W., AND HURST,R. P., P h ~ s Rev.
.
139, A1415
(1965).
Experimental values compiled by A. Dn~GanNo,Advan. Phys.
11, 281 (1962).
Transformations of Orbitals: Hybrids and Ligand Field
Orbitals
One of the most striking properties of molecules is
the relative rigidity of their structures. If we had no
information about their structures, we might well
attribute much looser structures to them, structures
more like liquid drops. It has been one aim of theoretical chemistry to understand the directional character
of chemical bonds, and to do this in terms of atomic
orbitals, if possible.
One of the most attractive ways to interpret the
directional character of bonds is based on using directional atomic orbitals. These are the hybrid orbitals
=
.@
[Rfr) sin B sin ql
4a
We have transformed from complex orbitals with
quantized mr into directional orbitals looking like cos 0
but with respect to the z and y axes.
The orbitals can be made still more directional if we
are willing to spoil not only mrbut also 1 itself as a good
one-electron quantum number. We can mix a spherically-symmetric s-function with a cosine-like p-function;
if we mix them by adding equal parts of each, then on
the positive z-axis, where cos 0 has its maximum, the pwave will add its maximum to the outer (positive) lobe
of the s-function and reinforce the total wave amplitude. On the negative z-axis, the p-wave will tend to
Volume 43, Number 6, June 1966
/
293
cancel the s-function, or, in optical terms, there will be
maximum destructive interference of the waves. If
the radial parts of the s and p-waves are very similar, as
they are if n = 2 for both waves, then the reinforcement
and cancellation is very significant. I n fact, for this
case, the p-wave slightly more than cancels the s-wave
on the negative z-axis. Written out, we have (using the
normalizing factors from Table 2),
With equal parts of the normalized s and p-functions,
we have constructed an sp hybrid. If we had subtracted the p-function from the s-function, we would
have simply changed z for -z and put the reinforcement onto the negative z lobe, and the cancellation on
the positive side.
I n this example, we constructed two equivalent sp
hybrids. We could do the same to construct three
equivalent sp2hybrids from two p's and an s. Let us
do this example with p, and p,-orbitals, keeping to the
x, y plane, and use the fact that p-orbitals are likevectors in their angular behavior. First, one may be
chosen as 2/3 pU:
energy that we can find for it, so long as each orbital is
orthogonal to the other orbitals of the same atom that
have still lower energies. The only condition the
Hartree-Fock orbitals satisfy, other than the minimum
energy and orthogonality conditions, is the normalization condition, to conserve our scale of probability. It
was shown only recently that localized orbitals, even
the most localized orbitals in the sense of maximum
intra-orbital interaction of electron pairs, can also be
solutions of well-defined and rigorous variational equations. This is an important point historically and
didactically. It seemed for quite some time that since
there were physically-based and mathematically sound
equations for Hartree-Fock orbitals, they rested on a
firm basis while the localized orbitals were in a more
vulnerable situation, without any rigorous physical
basis. They were always widely used, hut with some
trepidation about their physical interpretation. I n
any event, the variational equations for localized
orbitals necessarily carry other conditions than the
energy condition of the Hartree-Fock equations. There
is avariety of conditions one can use, like the one requiring that the electron-electron mean interaction within
an orbital be maximized (and therefore the interorbital mean interaction becomes minimized).
Now, the other two functions must contain
A; they
must contain equal parts of J.nz with opposite signs and
these parts must contain all of J.,,
(i.e., the sums of the
squares of the coefficients of each component orbital
must add to 1); and they must contain all the rest
of J.vn divided evenly and with the same sign, so
that both of them point toward the negative y direction.
This fixes the other two as
and
r(Bohr radii)
For convenience in illustration, one sometimes implicitly supposes that Rz0(r),the 2s radial function, is
identical with the 2p radial function Ral(r), so that the
angular dependence of the hybrid orbital can be plotted
unambiguously in one or more planes. I n truth, Rzl(r)
and Rza(r)are not generally identical, as Figure 3 shows.
What relationship exists between the localized hybrid orbitals or localized orbitals in general, and the
Hartree-Fock orbitals that we examined previously?
First of all, the two sets give the same total energy of
any atom, by virtue of our quoted theorem. Second,
they can be obtained from similar but by no means
identical sets of equations. The Hartree-Fock orbitals
are in one sense the best orbitals we know how to findthe energetic sense. The Hartree-Fock equations and all
the fundamental equations we have for deriving orbitals
ab initio (not just by transforming them into each
other) are variational equations. They say that the
one-electron energy of each orbital is the lowest possible
294
/
Journal of Chemicol Education
Figure 3. Radio1 functions RIr) for neutral carbon in in ground state.
The solid curves ore Hortree-Fosk functionr UUCYS. A., Proc. Roy. Soc.
(London1 A173, 5 9 (1939)l. m d the dotted curve is a Hortree 2p function
[TORRANCE, C. C., Phyr. Rev. 46, 3 8 4 1193411. The Hortree Is and 2s
functions are very rimiior to the Hortree-Fock functions .how" here. Note
the rimilority of the outer parts of the 2s and 2 p functions. (The functions
plotted are mluolly R = r
ro that
R'dr = 1 .I
+,
JP
The equations for localized orbitals are somewhat less
tractable than those for Hartree-Fock orbitals. If we
are only interested in visualizing and describing localized
orbitals qualitatively, it is relatively easy and satisfactory to use existing Hartree-Fock orbitals to make
hybrids within a specific shell. Symmetry alone fixes all
the mixing coefficients for us and we can derive them
with a little algebra, just as we did in the previous section. However if we want the localized orbitals that are
best by some criterion like the one we just cited, then
we are forced to do a moderate amount of computation.
The variational equations for localized orbitals are more
cumbersome than the Hartree-Fock equations, so that
i t may be best to find the Hartree-Fock orbitals first and
scramble them to meet the extra conditions.
Naturally, since localized orbitals must meet extra
conditions, they cannot be the best orbitals in the
orbital energy sense. Nor can they satisfy the symmetry conditions, the conditions that lead to the spherical harmonic form for each Hartree-Fock orbital. Nor
can they have quantized angular momentum. (It may
be that for open shell atoms, the energetically-best
Hartree-Foclc orbitals don't have quantized angular
momenta either. This question is currently unsolved.)
Moreover the fact that localized orbitals do not give the
best one-electron energies means that they do not give
good values for ionization potentials, a t least from oneelectron energies. Of course if we compare the total
energy of atom and corresponding positive ion to get the
ionization potential, it makes absolutely no difference
whether we use the Hartree-Fock functions or some
local set obtained by scrambling the Hartree-Fock
functions. However, the Hartree-Fock orbital energies
are generally better approximations to ionization
potentials than the total energy differences.
It may turn out that localized orbitals are more u s e
ful than Hartree-Fock orbitals in other ways. One
hope is that they will offer a straightforward way to
evaluate correlation energies. Correlation energy in an
atom or molecule is the difference between the actual
total (usually with relativist,ic contributions subtracted)
and the Hartree-Fock total energy. It represents the
extra energy of the system coming from the fact that
electrons really interact with real electrons, not with
mean fields of other electrons. It would be helpful
if it turns out that we can calculate correlation effects
by looking a t electrons just two at a time. If that is the
case, then localized orbitals are likely to be extremely
useful for this purpose. As yet, we cannot say for sure.
V.
Beyond Orbitals-The
Correlation Problem
As we just said, the Hartree-Fock energy differs from
the true energy of an atom or molecule by a relativistic
contribution that need not concern us here: and a
correlation contribution. The origin of the correlation
energy is quite clear and we can even attribute to it a
precise potential field, the Jluctuation potential. This
total field is the difference between the mean field
and the exact field felt by an electron. An example
is shown in Figure 4. In a two-electron atom one can
visualize this field clearly. It acts between two electrons but depends somewhat on the distances between
electrons and nuclei. It is highly repulsive and very
short-ranged, stronger when the two electrons are far
from the nucleus than when they are close. When one
electron comes sufficiently close to the other, the
electrons' potential energy becomes greater than their
potential and, correspondingly, their kinetic energy
'This is not to say that relativistic effects are negligible. In
terms of the total energy, relativistic effects become important
even in the second-row elements. However, these contributions
occur primarily in the inner shells, part,icols;rlp because of the
high kinetic energies associated with the inner shells. As a result,
the relativistic effects have little direct effect on the chemical
behavior of atoms.
must become negative and their momenta, imaginary.
This is a two-particle, three-dimensional example of
the phenomenon we discussed in connection with the
existence of quantized bound states of a single particle.
7
7
I
6-
54-
I
I
I
I
I
Bcryilium 1s electrons
I
-
- fluct~ationpotential
*en by electron 2
ex~cttw-elsctmnWtentiol
e2/r,, when electron 1 is of
-
3averope potentiol seen by
2-
-3
-2
-I
t
Figure 4. The fluctuation potentiol for I s electrons in beryllium (after ref.
26). The fluctuation potentiol is the difference between the exact and
mean potentidr. It is o function of n 0 %well as of rs; the value r~ = 0.27
a,"., the most proboble Is rodiur, w m chosen. The figure described potentiolr dong the linecontaining the nucleus and electron l .
We should interject a comment on the quantitative
importance of the correlation problem. These energies
in atoms are just the size of chemically important energy
differences-roughly 1 or 2 ev per pair of opposite-spin
electrons in the valence shell and higher for shells nearer
the nucleus. Moreover correlation is the primary
source of London forces, and therefore of the cohesive
energy of molecular crystals. I n other words, even
though we may get qualitative, graphic, and clear notions about electronic wave functions from orbital
descriptions, and even though we may find that orbital
calculations are useful for correlating chemical phenomena, we must be exceedingly cautious about using
any wave function for quantitatiu~purposes unless it contains correlation effectsor unless we can show that the
correlation effects drop out in our particular problem.
The tendency of one electron to repel another and to
force the wave functions of the two electrons to have low
amplitudes when the electrons are near gives rise to the
concept of the correlation hole. This hole takes the
form of a region around each electron, the region where
the fluctuation potential is very large, where no other
electron is likely to be. The Pauli Exclusion Principle
establishes this hole moderately well for electrons of the
same spin, but it has no effect on the spatial distribution of electrons with opposite spins, so does not help
to introduce any correlation effect. I n this case the
correlation hole must be a pure Coulomb hole and can
only be introduced into a wave function by inclusion of
specific terms above and beyond the Hartree-Fock function. I n other words we can improve on the HartreeFock function by superposing additional terms, which
are the result of the fluctuation potential, to account for
electron correlation.
What does the fluctuation potential do to our orbital
concept? One thing it cannot do. I t cannot spoil
Volume 43, Number 6, June 1966
/
295
the overall symmetry or angular momentum of our total
wave function. It can, however, spoil virtually all the
other bases of our orbital picture. First of all, it obviously spoils any quantization of the angular momentum and energy of individual electrons. This means,
in turn, that individual electrons cannot he described
by specified quantum numbers n and 1. This is turn
means that we cannot, strictly, specify the configuration of an atom or molecule. The entire structure
of our atomic physics seems momentarily to be crashing
down.
I n fact, the situation is not disastrous. To a reasonable degree of approximation, we can specify atomic
configurations and the n and 1 quantum numbers
of individual electrons. (We generally cannot specify
individual m(s and m,'s though; the electrons' interactions do spoil their orientations.) At least we can
specify the most likely or most important atomic configuration, or one-electron n,l and orbital energy.
This brings us directly to the problem of how one
can actually go beyond the orbital picture, and how one
can get a better wave function than the Hartree-Fock
function. There are two principal approaches. The
first historically and the most extensively explored is
the method known as configuration interaction. The
second has several variations, but all are based essentially on the concept of cluster expansions.
Configuration interaction is, as its name suggests, the
addition to the Hartree-Fock N-electron function of one
or more similar functions having different assignments
of the individual n's and 1's of the electrons. These new
functions correspond then to configurations differing
from the Hartree-Foclc configuration. For example the
beryllium Hartree-Fock configuration is obviously
1~2292. We could imagine adding to this some lsZ2s2p
and ls22s3s. However neither of these will do: the
first is necessarily an overall P state (total L = 1) while
our Hartree-Fock and true functions are both S states,
and in the long run, we can only build the latter with S
configurations. The second, ls22s3s, does not affect
construction of the lowest state provided we have really
used Hartree-Fock functions. Strictly, the ls22s3s can
enter to a very slight degree, but for a first and very
good approximation, we can neglect it. I n fact
we are justified in saying that neither it, nor any other
configuration that differs from the one of interest in the
quantum numbers of a single electron, will contribute
to configuration mixing with the configuration of interest. Inotherwords the ls22s2configurationberyllium
mixes with ls22p2and ls23s2but not with 1sZ2s2pin the
overall zero-angular momentum IS state of the atom.
What physical effect is associated with this configuration interaction? The basic idea is relatively easy to
see. Let us neglect the two 1s electrons. The 2s radial
function of Be looks roughly like that of Figure 3 and
the angular function is of course constant. I n the 2sZ
configuration, the radial and angular coordinates of one
electron are entirely unrelated to the radial and angular
coordinates of the other. Now, if we add some 2p2
character to the total wave function, we introduce some
correlation between the two sets of angular coordinates.
The 2p2 part has the effect of accounting partially for
the polarization of one electron by the field of the other.
This occurs because the two electrons' p functions reinforce their s functions in different angular regions.
296
/
Journal o f Chemical Education
When one electron's probability amplitude is large in a
particular direction as a result of constructive interference of its s and p parts, the other electron's prohability amplitude is largest about 90' away, due to its
own simultaneous interferences. The admixture of 3sZ
has a similar effect on the radial variables. I n effect,
mixing 3sZwith 2s2 introduces some in-out correlation
(cf. References 52-55).
Configuration interaction is a formally exact method
that must converge to the exact wave function of the
Hamiltonian of choice. I t is computationally straightforward and we are learning to interpret the physical
significance of its various sorts of terms. In general, it
is a very slowly converging proce~s.~The beryllium
atom was treated with 37 conf?gurations, for one of the
most accurate calculations available on any atom more
complex than helium. Configuration interaction is
probably the method of choice at present if one simply
wants a very accurate wave function of a moderately
complex atom and cares very little about computational
efficiency.
The other general approach to correlation has its
origins in the fact that one can treat the interaction of a
single pair of electrons. One can get a much more
accurate representation of helium-like a t o m than one
gets from the orbital model alone. This better rellresentation comes from the addition of terms in the twoelectron wave function that contain the interelectronic
distance explicitly. One of the simplest examples is a
that can be included
dying exponential factor eas a factor only for a small range of the inter-electronic
distance r12 around the value rlz = 0. A much more
elaborate one is the wave function of He originally
introduced by Hylleraas, which contains a number of
terms in r12is its most highly developed form.
The general approach to correlation through pair
interactions can be paraphrased somewhat generally
this way. I n atoms, it may be valid to suppose that the
order in which one should calculate interactions is:
""?
''
(1) interaction of each electron with the mean field of all the
others (HartreeFock);
(2) deviations from (1) due to simple pair interactions
(binary encounters in two electrons in the HartreeFock field of
the rest);
(3) deviations from (1) and (2) due to three-body encounters; etc.
If binary encounters are much more important than
three-body effects, then we should be able to treat part
(2) by using methods similar to those developed for the
two-electron problem. This approach n-as developed
by Szasz, Tsang, and Sinanoglu, and is perhaps most
graphically described in an article by Sinanoglu (51).
The results indicate that for atoms up to beryllium,
the cluster expansion approach is very efficient and
physically clear. It is too early to tell whether it will be
useful for calculation of the wave functions of more com-
6 I t is possible to choose a. set of orbitals that maximizes the
rate at which a. configuration interaction series converges to the
exact wave functions. These orbitals are the natural spin orbitals
introduced by LBwdin znd developed by him, Shull, Gilbert,
Wahl, and others. I t seems at present that the natural spin
orbital expansion might lead to quantitatively useful wave functions with about three configurations per electron pair. This is
the most optimistic estimate of the efficiencyof the configuration
interaction expansion.
plex systems. I t may be that tho three-, four-, and nbody terms will be as important as the pair terms just
because of their numbers, even though they may be
small individually. There is some reason to fear that
this could happen simply because the Coulonlb correlation hole can be comparable to the size of an atom.
This means that the electrons might be best thought of
as though they are constantly in collision with all the
rest, at least those with the same principal quantum
number. On the other hand estimates of the pair
correlation energies of a number of atoms suggest that
higher contributions are probably small.
I n fairness to the many people actively working on
the problem of electron correlation, it should he said
that most of these workers recognize that both configuration interaction and cluster expansions may have
their own domains of utility. Just where these domains
extend and overlap, and whether they cover all the
problems of chemical interest, is still a very open question.
Bibliography
Within each section, the order is approximately from
the most fundamental and/or diacult to the most
elementary. The parenthetic numbers 1 4 and G indicate in a rough way the level at which the author of
this article has found the book useful or would guess it
to be useful. The sections are ordered from the most
general to the most specific.
We should refer the reader to Val. 2 of Reference (9)
for a particularly extensive bibliography, especially of
atomic theory.
Quontum Mechonicol Foundations
(1) Dmnc, P. A. M., "The Principles of Quantum Mechanics,"
Oxford University press, New York, 1947.
The elassio treatise, its first two chapters often lend
clarity to the broadest fundament& of the quantum
mechanical viewpoint, even at the intermediate level.
(3,4,G)
(2) KRAMERS,
H. A,, "Quantum Mechanics," IntersciencePublishers, (division of John Wiley & Sons, Inc.), New
York, 1957.
Extremely clear basic treatise with particularly elegant
and efficient mathematics. Treatment of matter waves
is especially useful at a relatively early level (2, for the
mathematically prepared). Otherwise (3,4,G).
(3) KAUZMANN,
W., 'iQuantum Chemistry," Academic Press,
Inc., New York, 1957.
Particularly useful because of its extensive development
of the mathematics of waves in a way that gives a lot of
physical insight. In all, a good reference far self-teaching as well as a good text. (2-G)
R. P., LEIGHTON,
R. B., AND SANDS,M., '*The
(4) FEYNMAN,
Feynman Lectures on Physics. Quantum Mechanics,"
Addison-Wesley Publishing Co., Reading, Mass., 1965.
The book speaks for itself. A pleasure to read, and a.
fimt-class book; to be used at thelowest possible level,
before bad habits set in. (1-G)
(5) S I ~ R ~ IC.N W.,
,
'Tntraduction to Quantum Mechanics,"
Henry Holt and Co., New York, 1959.
This is a straightforward and very clear elementary
text, with many illustrations m d diagrams.
(6) HEISENBERG,
W., "The Physical Principles of Quantum
Theow." Dover Publications. Inc.. 1930.
page Appendix. (1 or 2-G).
( 7 ) HINSHELWOOD,
C. N., "The Struclure of Physical Chemistry," Oxford University Press, 1951.
Parts I11 and IV present a qualitative and well orranieed storv that c a i be undekood bv a well-educated
freshman. An excellent overview. (1-3)
Atoms: General Monographs and Texts
(8) CONDON,
E. U., AND SHOILTLEY,
G. H., "Theory of Atomic
Spectra," Cambridge Univemity Press, 1953. [First
printing, 1935.1
Long the single most important work in the field, it is
fundamental, comprehensive (for its time), and an
excellent source for study if you are willing to work at it.
(4,G)
(9) SLATER,JOHNC., "Quantum Theory of Atomic Structure,"
McGraw-Hill Book Co., 1960, 2 velum%?.
The only competition for Candon and Shortley in scope
and level. Shter's style is less ten-e than Condon and
Shortley's. This book places more emphasis on the
mathematics and computation and less on the physics
than Condon and Shortley. It has a large number of
useful tables and probably the world's most extensive
selected bibliography of (primarily) theoretical treatments of atoms. (3,4,G)
E. E., "Quantum Mechanic8
(10) BETHE,H. A,, AND SALPETER,
of One- and Two-Electron Atoms." Academic P m s . New
York, 1957.
Concise and broad survey of the area a t a relatively
high level, intended primarily a? a definitive reference.
(~.,.,
4 C,)
(11) Reviews of Modern Physics, 35, No. 3, July (1963). Proeeedings of the Internetional Symposium on Atomic and
Molecular Quantum Mechanics, Sanibel Island, Florida,
January, 1963.
A very good representation of the state of the science
with a large variety of clearly presented material of
current interest. Contains extensive discussion following the papers and two "soft" articles, a memoir by
Egil Hylleraas and a review by J. C. Slater. Most of
the other articles are not elementary. (Some articles,
I-G; a few, 3 or 4,G; most, G.)
(12)J . C h m . Phys., Special Issue in Honor of R. S. Mulliken,
November 15, 1965.
Somewhat like the preceding reference, this too is
based on the proceedings of a Srtnibel Island meeting.
It has same review and survey articles, one or two almost historical in their approach, and many other
articles that represent the state of the art fairly well.
(Like the preceding reference, mostly G, with same
I-G)
B., editors, r'M~lecular
(13) LOWDIN,P.-O., AND PULLMAN,
Orbitals in Chemistry, Physics and Biology," Academic
Press, New York, 1964.
Much the same as the two preceding references but with
considerably more emphasis on molecules and molecular
orbitals and correspondingly less on atoms.
(14) F m o , U., AND FANO,L., "Basic Physics of Atoms and
Molecules," John Wiley & Sons, Inc., New York, 1959.
A clear, fundamental treatment, unique in its ability to
combine physical and mathematical ideas into a
coherent and comprehensive teaching book. This has
proved itself at all levels, inclnding freshman honors.
Highly recommended. ( 1 4 , G )
(15) SINANOGLU,
O., AND TUAN,D. F.-T., "Quantum Theory of
Atoms and Molecules," Ann. Rev. Phys. Chem.,15, 451
il!464\.
-,
Particularly useful as an introduction to the literature.
12
4 r.1
\-,-,-,
(16) BALLHAUSEN,
C. J., "Introduct~onto Ligand Field Theory,"
MeGraw-Hill Book Co., New York, 1962.
Chapter2 offers agentler introduction than Condon and
Shortley but leads you to almost the same point, at
least with regard to manipulation. Chapter 3 is a dear
and useful but rather brief treatment of symmetry
properties. (3,4,G)
(17) K~NDRATYEV,
V., "The Structure of Atoms and Molecules,"
translated from the Russian by G. YANHOVSBY,
P.
Noordhoff N. V. Groningen, The Netherlands.
Contains good descriptions of Bohr-Sommerfeld model
and considerable material about molecules.
(18) L I N N EJ.
~ W., "Wave Mechanics and Valemy," John
Wiley & Sons, Inc., New York, 1960.
A brief and elementary-to-intermediate discussion oriented toward interpretation of structure, with considerable emphasis on orbital concepts. (2-G)
Volume 43, Number 6, June 1966
/
297
(19) HERZBERO,
G., "Atomic Spectra and Atomic Structure,"
Prentice-Hall, Inc., New York, 1937.
(20)
(21)
(22)
(23)
(24)
(25)
An old but classic treatment that deals more with
atomic spectra than with orbitals.
J@RGENSEN.
CHR.KLIXBULL,"Orbitals in Atom and Malecules," Academic Press, 1962.
A short but densely packed intermediate-level book.
Emphasis is on symmetry and on coordination chemistry applicatiom. ( 2 4 )
GREENWOOD,
N. N., "Principles of Atomic Orbitals," Royal
Institute of Chemistry Monographs for Teachers, No. 8,
The Royal Institute of Chemistry, London, 1964.
Thorough elementary discussion. Possibly bard to
oblrtin locally.
GRAY, HARRYG., "Electrons and Chemical Bonding,"
W. A. Benjamin, Inc., 1964.
Elementary, but quite ready to use simple mrtthemstics; the emphasis in the purely atomic part is on
manipulations. (1,2)
RYSCHKEWITSCH,
G. E., "Chemical Bonding and the Geome
try of Molecules," Reinhold Pub. Corp., 1962.
An elementary survev, about half of which is devoted
to conventiond presentation of atomic structure. (1)
HOCHSTRASSER,
ROBIN M., "Behavior of Electrons in
Atoms," W. A. Banjamin, Inc., 1964.
A very elementary and expository text with essentially
no m a t h e m a h , it is probably most useful for people
who want to acquaint themselves with the general ideas,
concll~4ions,and vocabulary of electronic structure of
atoms, but have little or no intention of making use of
it. (1,2) See also reference (3).
THOMSON,
G., "The Atom," Oxford University Press, New
York, 1962.
A small readable qualitative discussion.
Symmetry
(26) WIGNER,E., ''Group Theory and Its Application to the
Quantum Mechanics of Atomic Spectra," Academic Press,
New York, 1959.
The standard and classic reference; a revised translation of the old edition in German. ( 3 or 4,G)
(27) HAMERMESH,
M., "Group Theory and Its Application to
Phvsicd Problems." Addison-Weslev Publishine Co..
Reading, Mass., 1962.
One of a small number of very good books for selfteaching as well as for course texts in this subject.
Considerably broader in scope than books such as the
following reference. (3,4,G)
(28) COTTON,
F. A,, "Chemical Applications of Group Theory,"
Interscience Publishers (division of John Wiley & Sons),
New York, 1963.
A popular and clear exposition of haw to use methods
of group theory in problem like ligand field theory.
(2-G) See also references (9), (14), and (16).
-
Correbtion
(29) LOWIN, P. O., "Correlation in Quantum Mechanics. I.
Review," Ad". in Chemical Physies II, p. 207 (1959).
An extensive review of the field. Some significant
advances have been made since this was written.
(4,G)
(30) Yosarzu~r,A,, "Correlation in Quantum Mechanics. 11.
Bibliography," Ado. in Chemical Physies 11,p. 323 (1959).
(31)
(32)
(33)
(34)
An annotated bibliography of atomic theory accompanying the previous reference.
SINANOGLU,
O., "Many-Electron Theory of Atoms and
Molecules," Pmc. Nat. Acad. Sci., 47, 1217 (1961).
A rather qualitative description of the physics and
formalism of the correlation problem, with emphasis an
cluster method. (3,4G).
LENNARD-JONES,
J. E., AND POPLE,J. A,, "The Spatial
Correlation of Electrons in Atoms and Molecules. I.
Helium and Similar Two-Electron Systems in Their
Ground States," Phil. Mag. 43, 581 (1962).
LENNARDJONES,
J. E., "The Spatial Correlation of Electrons in Atoms and Molecules. 11." PTOC.Nat. Amd.
Sei., 38.496 (1952). General discussions (3-G).
DICKENS,
P. G., AND LINNETT,
J . W., "Electron Correlation
and Chemical Consequence%" Quart. Rars., 11, 291
(1957).
298
/
lournol o f Chemical Education
A very useful and extremely clear discussion of the
physical phenomenon, with much discussion of very
simple models. Definitely recommended for anyone
trying to discuss physical origins of localization. (2-G)
See also Refs. (9), ( l l ) , (ld), and (14).
Localized and Direcfionol Orbitols
(35) LMNETT,
J. W., 'The Electronic Structure of Molecules, A
New Approach," John Wiley & Sons, Inc., New York,
1964. ? 3 - ~ )
(36) LINNETT,
J. W., Valency and the Chemical Bond," American Scientist, 53, 459 (1964). (I-G)
These t,wo references resent an ad hoe but somewhat
with some correlation effects.
(37) BENT,H. A,, "An Appraisal of Valence-Bond Structures and
Hybridization in Compounds of the First-Row Elements." Chem. Revs., 611 275 (1961).
An empirical and semiquantitative review; develops
structure bv
" usine orbital ideas as the oreanizine framework without delving into quantum mechanics of the
orbitals themselves. Bent has also written less comprehensive and more descriptive articles for J. CHEM.
EDUC.,e.g., 40, 446, 523 (1963); 42, 302, 348 (1965).
(2 or 3,4)
(38) OORYZLO,
E. A., AND PORTER,G. B., "Contour Surfaces for
Atomic and Molecular Orbitale," J. CHEM.EDnc., 40,256
-
(1963). (I-G)
(39) ADAMSON,
A. W., "Domain Representations of Orbitals,"
J. &EM. EDUC.,42, 141 (1965).
A device for visualizing orbitals, similar to those used
in solid state work.
(40) JOHNSON,
R. C., AND RETTEW,R. R., "Shapes of Atoms,"
J . CEEM.EDTIC., 42, 145 (1965).
These three are primarily useful as teaching aids for
their presentations of visualization devices. They contribute essentially no physical ideas, other than those
already inherent in the orbital concepts. (Ideas me
useful for 1,2,3).
(41) COHEN,I., AND BUSTARD,
T., "Atomic Orbitals: L i m i b
tions and Variations," J. CHEM.EDUC.,43, 187 (1966).
Tables
(42) MOORE,CHARLOTPI:
E., "Atomic Energy Levels," US.
National Bureau of Standards Circular 457, Vol. 1, 1949;
Vol. 2, 1952; Vol. 3, 1958.
The comprehensive tabulation of known and predicted
atomic energies. A must for research, and very useful
for examples.
(43) HERMAN,
F., AND SKILLMAN,
S., "Atomic Structure Cslculations," Prentice-Hall, Inc., Englewood Cliffs, N. J.,
,062
A=-".
Approximate orbital energies, orbitals, and total
potentials for the element^. A convenient source.
Analytic Hartwe-Foek Functions.
CLEMENTI,
E., ROOTAAAN,
C. C. J., AND YOSHIMINE,
M.,
Phys. Rev. 127, 1618 (1962).
Firsbrow atoms.
CLEMENTI,
E., J . Chem. Phys., 38, 996 (1962).
Ground and excited states of isoelectronic series with
2 to 10 electrons.
CLEMENTI,
E., AND MCLEAN,
A. D., Phys. Rev., 133, A419
(1964).
Li-, B-, C-, N; 0-, F-.
CLEMENTI,
E., J. Chem. Phys., 38, 1001 (1964).
11 to 18 electrons.
E., ET AL.,P h g ~Rev., 133, A1274 (1964).
CLEMENTI,
Ns-, Al-, Si-, P-, S-, C1-.
CLEMENTI,
E.. J. Chem. Phvs..
. . 41.. 295 (1964).
19 to 30 electrons.
CLEMENTI,
E., J. Chem. Phys., 41, 303 (1964).
31 to 36 electrons.
See also Reference (a), Vol. 1, Appendix 16.
Hisfory
Presentofions for
(45) GAMOW,
G., "Mr. Tcmpkins in Wonderland," Maemillan
Co., New York, 1940.
(46) GAMOW,
G., "Mr. Tompkins Explores the Atom," MacmilIan Co., New Yark, 1945.
(47) HOFFMAN,
B., "Strange Story of the Quantum," Dover,
New York, 2nd Rev Ed., 1959.
148) JAFFE.
B.. "Crucibleg." Fawcett Publications. New York.
. .
rev. ed.., -1960.
(49) SHAMOS,
M. H., ed~tor,"Great Experiments in Physics,"
Henry Holt and Co., New York, 1960. See also Ref. (25).
ZeP
ZeZ
rnvr
nfi
y = - = -
and r:
Appendix
This represents the briefest derivation of the circular orbit
Bohr atom known to this writer. We write three equations, in
terms of the mass m of the electron (strictly the reduced mass, but
we neglect this distinction), its charge -e, the charge +Ze of the
nucleus, the velocity v and the radius r.
(a) the definition of the total energy
of an electron in a Coulomb field;
(b) thearbit,rary condition that the orbit bestable-centrifugal
force equaling centripetal force,
The period r is
Tllc clergy Level.; call be dercn>oiwrl I,? diwct suhslitution for r
UNI u > n r u [ A l l h ~ itt i? wwtl~whikI U u e ( W 18, I d 1 w a purely
c l a 4 n l rdarion that sirnpllfivc ( k l ) , itarnvl\, t h : vlrial relnricm
for a l/r2 force:
rn"'
=
Ze'/r
or the potential energy is -2 X the kinetic energy, so that
E = - - mu2
2
and
(c) the quantum condition that the aetimz he quantized,
and from (A4) or (A5),
Here n is any positive integer and h is Planck's constant, to be
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