Main Encyclopedia of Chemical Physics and Physical Chemistry

Encyclopedia of Chemical Physics and Physical Chemistry

,
This three-volume encyclopedia offers extensive coverage in the closely related areas of chemical physics and physical chemistry. Focusing on both disciplines as interrelated, the Encyclopedia contains a wide variety of information to meet the special research needs of the chemical physicist and the physical chemist and, thus, enables specialists in both fields to conduct interdisciplinary research. Features include definitions of the scope of each subdiscipline and instructions on where to go for a more complete and detailed explanation. The Encyclopedia is also a helpful resource for graduate students because it gives a synopsis of the basics and an overview of the range of activities in which physical principles are applied to chemical problems.
Year: 2001
Edition: 1st
Publisher: Taylor & Francis
Language: english
Pages: 3076
ISBN 10: 0750303131
ISBN 13: 9780750303132
File: PDF, 68.35 MB
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-1-

A1.1 The quantum mechanics of atoms and
molecules
John F Stanton

A1.1.1 INTRODUCTION
At the turn of the 19th century, it was generally believed that the great distance between earth and the stars
would forever limit what could be learned about the universe. Apart from their approximate size and distance
from earth, there seemed to be no hope of determining intensive properties of stars, such as temperature and
composition. While this pessimistic attitude may seem quaint from a modern perspective, it should be
remembered that all knowledge gained in these areas has been obtained by exploiting a scientific technique
that did not exist 200 years ago—spectroscopy.
In 1859, Kirchoff made a breakthrough discovery about the nearest star—our sun. It had been known for some
time that a number of narrow dark lines are found when sunlight is bent through a prism. These absences had
been studied systematically by Fraunhofer, who also noted that dark lines can be found in the spectrum of
other stars; furthermore, many of these absences are found at the same wavelengths as those in the solar
spectrum. By burning substances in the laboratory, Kirchoff was able to show that some of the features are
due to the presence of sodium atoms in the solar atmosphere. For the first time, it had been demonstrated that
an element found on our planet is not unique, but exists elsewhere in the universe. Perhaps most important,
the field of modern spectroscopy was born.
Armed with the empirical knowledge that each element in the periodic table has a characteristic spectrum, and
that heating materials to a sufficiently high temperature disrupts all interatomic interactions, Bunsen and
Kirchoff invented the spectroscope, an instrument that atomizes substances in a flame and then records their
emission spectrum. Using this instrument, the elemental composition of several compounds and minerals were
deduced by measuring the wavelength of radiation that they emit. In addition, this new science led to the
discovery of elements, notably caesium and rubidium.
Despite the enormous benefits of the fledgling field of spectroscopy for chemistry, the underlying physical
processes were completely unknown a century ago. It was believed that the characteristic frequencies of
elements were caused by (nebulously defined) vibrations of the atoms, but even a remotely satisfactory
quantitative theory proved to be elusive. In 1885, the Swiss mathematician Balmer noted that wavelengths in
the visible region of the hydrogen atom emission spectrum could be fitted by the empirical equation

(A1.1.1)

where m = 2 and n is an integer. Subsequent study showed that frequencies in other regions of the hydrogen
spectrum could be fitted to this equation by assigning different integer values to m, albeit with a different
value of the constant b. Ritz noted that a simple modification of Balmer’s formula
(A1.1.2)

-2-

succeeds in fitting all the line spectra corresponding to different values of m with only the single constant RH.
Although this formula provides an important clue regarding the underlying processes involved in
spectroscopy, more than two decades passed before a theory of atomic structure succeeded in deriving this
equation from first principles.
The origins of line spectra as well as other unexplained phenomena such as radioactivity and the intensity
profile in the emission spectrum of hot objects eventually led to a realization that the physics of the day was
incomplete. New ideas were clearly needed before a detailed understanding of the submicroscopic world of
atoms and molecules could be gained. At the turn of the 20th century, Planck succeeded in deriving an
equation that gave a correct description of the radiation emitted by an idealized isolated solid (blackbody
radiation). In the derivation, Planck assumed that the energy of electromagnetic radiation emitted by the
vibrating atoms of the solid cannot have just any energy, but must be an integral multiple of hν, where ν is the
frequency of the radiation and h is now known as Planck’s constant. The resulting formula matched the
experimental blackbody spectrum perfectly.
Another phenomenon that could not be explained by classical physics involved what is now known as the
photoelectric effect. When light impinges on a metal, ionization leading to ejection of electrons happens only
at wavelengths (λ = c/ν, where c is the speed of light) below a certain threshold. At shorter wavelengths
(higher frequency), the kinetic energy of the photoelectrons depends linearly on the frequency of the applied
radiation field and is independent of its intensity. These findings were inconsistent with conventional
electromagnetic theory. A brilliant analysis of this phenomenon by Einstein convincingly demonstrated that
electromagnetic energy is indeed absorbed in bundles, or quanta (now called photons), each with energy hν
where h is precisely the same quantity that appears in Planck’s formula for the blackbody emission spectrum.
While the revolutionary ideas of Planck and Einstein forged the beginnings of the quantum theory, the physics
governing the structure and properties of atoms and molecules remained unknown. Independent experiments
by Thomson, Weichert and Kaufmann had established that atoms are not the indivisible entities postulated by
Democritus 2000 years ago and assumed in Dalton’s atomic theory. Rather, it had become clear that all atoms
contain identical negative charges called electrons. At first, this was viewed as a rather esoteric feature of
matter, the electron being an entity that ‘would never be of any use to anyone’. With time, however, the
importance of the electron and its role in the structure of atoms came to be understood. Perhaps the most
significant advance was Rutherford’s interpretation of the scattering of alpha particles from a thin gold foil in
terms of atoms containing a very small, dense, positively charged core surrounded by a cloud of electrons.
This picture of atoms is fundamentally correct, and is now learned each year by millions of elementary school
students.
Like the photoelectric effect, the atomic model developed by Rutherford in 1911 is not consistent with the
classical theory of electromagnetism. In the hydrogen atom, the force due to Coulomb attraction between the
nucleus and the electron results in acceleration of the electron (Newton’s first law). Classical electromagnetic
theory mandates that all accelerated bodies bearing charge must emit radiation. Since emission of radiation
necessarily results in a loss of energy, the electron should eventually be captured by the nucleus. But this
catastrophe does not occur. Two years after Rutherford’s gold-foil experiment, the first quantitatively
successful theory of an atom was developed by Bohr. This model was based on a combination of purely
classical ideas, use of Planck’s constant h and the bold assumption that radiative loss of energy does not occur
provided the electron adheres to certain special orbits, or ‘stationary states’. Specifically, electrons that move
in a circular path about the nucleus with a classical angular momentum mvr equal to an integral multiple of
Planck’s constant divided by 2π (a quantity of sufficient general use that it is designated by the simple symbol
) are immune from energy loss in the Bohr model. By simply writing the classical energy of the orbiting
electron in terms of its mass m, velocity v, distance r from the nucleus and charge e,

-3-

(A1.1.3)

invoking the (again classical) virial theorem that relates the average kinetic (〈T〉) and potential (〈V〉) energy of
a system governed by a potential that depends on pairwise interactions of the form rk via
(A1.1.4)

and using Bohr’s criterion for stable orbits
(A1.1.5)

it is relatively easy to demonstrate that energies associated with orbits having angular momentum
hydrogen atom are given by

in the

(A1.1.6)

with corresponding radii
(A1.1.7)

Bohr further postulated that quantum jumps between the different allowed energy levels are always
accompanied by absorption or emission of a photon, as required by energy conservation, viz.
(A1.1.8)

or perhaps more illustratively
(A1.1.9)

precisely the form of the equation deduced by Ritz. The constant term of equation (A1.1.2) calculated from
Bohr’s equation did not exactly reproduce the experimental value at first. However, this situation was quickly
remedied when it was realized that a proper treatment of the two-particle problem involved use of the reduced
mass of the system µ ≡ mmproton/(m + mproton), a minor modification that gives striking agreement with
experiment.

-4-

Despite its success in reproducing the hydrogen atom spectrum, the Bohr model of the atom rapidly
encountered difficulties. Advances in the resolution obtained in spectroscopic experiments had shown that the
spectral features of the hydrogen atom are actually composed of several closely spaced lines; these are not
accounted for by quantum jumps between Bohr’s allowed orbits. However, by modifying the Bohr model to

allow for elliptical orbits and to include the special theory of relativity, Sommerfeld was able to account for
some of the fine structure of spectral lines. More serious problems arose when the planetary model was
applied to systems that contained more than one electron. Efforts to calculate the spectrum of helium were
completely unsuccessful, as was a calculation of the spectrum of the hydrogen molecule ion ( ) that used a
generalization of the Bohr model to treat a problem involving two nuclei. This latter work formed the basis of
the PhD thesis of Pauli, who was to become one of the principal players in the development of a more mature
and comprehensive theory of atoms and molecules.
In retrospect, the Bohr model of the hydrogen atom contains several flaws. Perhaps most prominent among
these is that the angular momentum of the hydrogen ground state (n = 1) given by the model is ; it is now
known that the correct value is zero. Efforts to remedy the Bohr model for its insufficiencies, pursued
doggedly by Sommerfeld and others, were ultimately unsuccessful. This ‘old’ quantum theory was replaced in
the 1920s by a considerably more abstract framework that forms the basis for our current understanding of the
detailed physics governing chemical processes. The modern quantum theory, unlike Bohr’s, does not involve
classical ideas coupled with an ad hoc incorporation of Planck’s quantum hypothesis. It is instead founded
upon a limited number of fundamental principles that cannot be proven, but must be regarded as laws of
nature. While the modern theory of quantum mechanics is exceedingly complex and fraught with certain
philosophical paradoxes (which will not be discussed), it has withstood the test of time; no contradiction
between predictions of the theory and actual atomic or molecular phenomena has ever been observed.
The purpose of this chapter is to provide an introduction to the basic framework of quantum mechanics, with
an emphasis on aspects that are most relevant for the study of atoms and molecules. After summarizing the
basic principles of the subject that represent required knowledge for all students of physical chemistry, the
independent-particle approximation so important in molecular quantum mechanics is introduced. A significant
effort is made to describe this approach in detail and to communicate how it is used as a foundation for
qualitative understanding and as a basis for more accurate treatments. Following this, the basic techniques
used in accurate calculations that go beyond the independent-particle picture (variational method and
perturbation theory) are described, with some attention given to how they are actually used in practical
calculations.
It is clearly impossible to present a comprehensive discussion of quantum mechanics in a chapter of this
length. Instead, one is forced to present cursory overviews of many topics or to limit the scope and provide a
more rigorous treatment of a select group of subjects. The latter alternative has been followed here.
Consequently, many areas of quantum mechanics are largely ignored. For the most part, however, the areas
lightly touched upon or completely absent from this chapter are specifically dealt with elsewhere in the
encyclopedia. Notable among these are the interaction between matter and radiation, spin and magnetism,
techniques of quantum chemistry including the Born–Oppenheimer approximation, the Hartree–Fock method
and electron correlation, scattering theory and the treatment of internal nuclear motion (rotation and vibration)
in molecules.

-5-

A1.1.2 CONCEPTS OF QUANTUM MECHANICS
A1.1.2.1 BEGINNINGS AND FUNDAMENTAL POSTULATES

The modern quantum theory derives from work done independently by Heisenberg and Schrödinger in the
mid-1920s. Superficially, the mathematical formalisms developed by these individuals appear very different;
the quantum mechanics of Heisenberg is based on the properties of matrices, while that of Schrödinger is
founded upon a differential equation that bears similarities to those used in the classical theory of waves.
Schrödinger’s formulation was strongly influenced by the work of de Broglie, who made the revolutionary

hypothesis that entities previously thought to be strictly particle-like (electrons) can exhibit wavelike
behaviour (such as diffraction) with particle ‘wavelength’ and momentum (p) related by the equation λ = h/p.
This truly startling premise was subsequently verified independently by Davisson and Germer as well as by
Thomson, who showed that electrons exhibit diffraction patterns when passed through crystals and very small
circular apertures, respectively. Both the treatment of Heisenberg, which did not make use of wave theory
concepts, and that of Schrödinger were successfully applied to the calculation of the hydrogen atom spectrum.
It was ultimately proven by both Pauli and Schrödinger that the ‘matrix mechanics’ of Heisenberg and the
‘wave mechanics’ of Schrödinger are mathematically equivalent. Connections between the two methods were
further clarified by the transformation theory of Dirac and Jordan. The importance of this new quantum theory
was recognized immediately and Heisenberg, Schrödinger and Dirac shared the 1932 Nobel Prize in physics
for their work.
While not unique, the Schrödinger picture of quantum mechanics is the most familiar to chemists principally
because it has proven to be the simplest to use in practical calculations. Hence, the remainder of this section
will focus on the Schrödinger formulation and its associated wavefunctions, operators and eigenvalues.
Moreover, effects associated with the special theory of relativity (which include spin) will be ignored in this
subsection. Treatments of alternative formulations of quantum mechanics and discussions of relativistic
effects can be found in the reading list that accompanies this chapter.
Like the geometry of Euclid and the mechanics of Newton, quantum mechanics is an axiomatic subject. By
making several assertions, or postulates, about the mathematical properties of and physical interpretation
associated with solutions to the Schrödinger equation, the subject of quantum mechanics can be applied to
understand behaviour in atomic and molecular systems. The first of these postulates is:
1. Corresponding to any collection of n particles, there exists a time-dependent function Ψ(q1,
q2, . . ., qn; t) that comprises all information that can be known about the system. This function
must be continuous and single valued, and have continuous first derivatives at all points where
the classical force has a finite magnitude.
In classical mechanics, the state of the system may be completely specified by the set of Cartesian particle
coordinates ri and velocities dri/dt at any given time. These evolve according to Newton’s equations of
motion. In principle, one can write down equations involving the state variables and forces acting on the
particles which can be solved to give the location and velocity of each particle at any later (or earlier) time t′,
provided one knows the precise state of the classical system at time t. In quantum mechanics, the state of the
system at time t is instead described by a well behaved mathematical function of the particle coordinates qi
rather than a simple list of positions and velocities.

-6-

The relationship between this wavefunction (sometimes called state function) and the location of particles in
the system forms the basis for a second postulate:
2. The product of Ψ (q1, q2, . . ., qn; t) and its complex conjugate has the following physical
interpretation. The probability of finding the n particles of the system in the regions bounded by
the coordinates
and
at time t is proportional to the integral

(A1.1.10)

The proportionality between the integral and the probability can be replaced by an equivalence if the
wavefunction is scaled appropriately. Specifically, since the probability that the n particles will be found
somewhere must be unity, the wavefunction can be scaled so that the equality
(A1.1.11)

is satisfied. The symbol dτ introduced here and used throughout the remainder of this section indicates that
the integral is to be taken over the full range of all particle coordinates. Any wavefunction that satisfies
equation (A1.1.11) is said to be normalized. The product Ψ*Ψ corresponding to a normalized wavefunction is
sometimes called a probability, but this is an imprecise use of the word. It is instead a probability density,
which must be integrated to find the chance that a given measurement will find the particles in a certain region
of space. This distinction can be understood by considering the classical counterpart of Ψ*Ψ for a single
particle moving on the x-axis. In classical mechanics, the probability at time t for finding the particle at the
coordinate (x′) obtained by propagating Newton’s equations of motion from some set of initial conditions is
exactly equal to one; it is zero for any other value of x. What is the corresponding probability density function,
P(x; t) Clearly, P(x; t) vanishes at all points other than x′ since its integral over any interval that does not
include x′ must equal zero. At x′, the value of P(x; t) must be chosen so that the normalization condition
(A1.1.12)

is satisfied. Functions such as this play a useful role in quantum mechanics. They are known as Dirac delta
functions, and are designated by δ(r – r0). These functions have the properties
(A1.1.13)
(A1.1.14)
(A1.1.15)

-7-

Although a seemingly odd mathematical entity, it is not hard to appreciate that a simple one-dimensional
realization of the classical P(x; t) can be constructed from the familiar Gaussian distribution centred about x′
by letting the standard deviation (σ) go to zero,
(A1.1.16)

Hence, although the probability for finding the particle at x′ is equal to one, the corresponding probability
density function is infinitely large. In quantum mechanics, the probability density is generally nonzero for all
values of the coordinates, and its magnitude can be used to determine which regions are most likely to contain
particles. However, because the number of possible coordinates is infinite, the probability associated with any
precisely specified choice is zero. The discussion above shows a clear distinction between classical and
quantum mechanics; given a set of initial conditions, the locations of the particles are determined exactly at all
future times in the former, while one generally can speak only about the probability associated with a given
range of coordinates in quantum mechanics.

To extract information from the wavefunction about properties other than the probability density, additional
postulates are needed. All of these rely upon the mathematical concepts of operators, eigenvalues and
eigenfunctions. An extensive discussion of these important elements of the formalism of quantum mechanics
is precluded by space limitations. For further details, the reader is referred to the reading list supplied at the
end of this chapter. In quantum mechanics, the classical notions of position, momentum, energy etc are
replaced by mathematical operators that act upon the wavefunction to provide information about the system.
The third postulate relates to certain properties of these operators:
3. Associated with each system property A is a linear, Hermitian operator .
Although not a unique prescription, the quantum-mechanical operators can be obtained from their classical
counterparts A by making the substitutions x → x (coordinates); t → t (time); pq → -i ∂/∂q (component of
momentum). Hence, the quantum-mechanical operators of greatest relevance to the dynamics of an n-particle
system such as an atom or molecule are:

Dynamical variable A

Classical quantity

Quantum-mechanical operator

Time

t

t

Position of particle i

ri

ri

Momentum of particle i

mivi

–i ∇i

Angular momentum of particle i

mivi × ri

–i ∇i × ri

V(q, t)

V(q, t)

Kinetic energy of particle i
Potential energy

-8-

where the gradient

(A1.1.17)

and Laplacian

(A1.1.18)

operators have been introduced. Note that a potential energy which depends upon only particle coordinates
and time has exactly the same form in classical and quantum mechanics. A particularly useful operator in
quantum mechanics is that which corresponds to the total energy. This Hamiltonian operator is obtained by
simply adding the potential and kinetic energy operators

(A1.1.19)

The relationship between the abstract quantum-mechanical operators and the corresponding physical
quantities A is the subject of the fourth postulate, which states:
4. If the system property A is measured, the only values that can possibly be observed are those
that correspond to eigenvalues of the quantum-mechanical operator .
An illustrative example is provided by investigating the possible momenta for a single particle travelling in
the x-direction, px. First, one writes the equation that defines the eigenvalue condition
(A1.1.20)

where λ is an eigenvalue of the momentum operator and f(x) is the associated eigenfunction. It is easily
verified that this differential equation has an infinite number of solutions of the form
(A1.1.21)

with corresponding eigenvalues
(A1.1.22)

-9-

in which k can assume any value. Hence, nature places no restrictions on allowed values of the linear
momentum. Does this mean that a quantum-mechanical particle in a particular state ψ(x; t) is allowed to have
any value of px? The answer to this question is ‘yes’, but the interpretation of its consequences rather subtle.
Eventually a fifth postulate will be required to establish the connection between the quantum-mechanical
wavefunction ψ and the possible outcomes associated with measuring properties of the system. It turns out
that the set of possible momenta for our particle depends entirely on its wavefunction, as might be expected
from the first postulate given above. The infinite set of solutions to equation (A1.1.20) means only that no
values of the momentum are excluded, in the sense that they can be associated with a particle described by an
appropriately chosen wavefunction. However, the choice of a specific function might (or might not) impose
restrictions on which values of px are allowed.
The rather complicated issues raised in the preceding paragraph are central to the subject of quantum
mechanics, and their resolution forms the basis of one of the most important postulates associated with the
Schrödinger formulation of the subject. In the example above, discussion focuses entirely on the eigenvalues
of the momentum operator. What significance, if any, can be attached to the eigenfunctions of quantummechanical operators? In the interest of simplicity, the remainder of this subsection will focus entirely on the
quantum mechanics associated with operators that have a finite number of eigenvalues. These are said to have
a discrete spectrum, in contrast to those such as the linear momentum, which have a continuous spectrum.
Discrete spectra of eigenvalues arise whenever boundaries limit the region of space in which a system can be.
Examples are particles in hard-walled boxes, or soft-walled shells and particles attached to springs. The
results developed below can all be generalized to the continuous case, but at the expense of increased
mathematical complexity. Readers interested in these details should consult chapter 1 of Landau and Lifschitz
(see additional reading).

It can be shown that the eigenfunctions of Hermitian operators necessarily exhibit a number of useful
mathematical properties. First, if all eigenvalues are distinct, the set of eigenfunctions {f1, f2 ··· fn} are
orthogonal in the sense that the integral of the product formed from the complex conjugate of eigenfunction
and eigenfunction k (fk) vanishes unless j = k,

(A1.1.23)

If there are identical eigenvalues (a common occurrence in atomic and molecular quantum mechanics), it is
permissible to form linear combinations of the eigenfunctions corresponding to these degenerate eigenvalues,
as these must also be eigenfunctions of the operator. By making a judicious choice of the expansion
coefficients, the degenerate eigenfunctions can also be made orthogonal to one another. Another useful
property is that the set of eigenfunctions is said to be complete. This means that any function of the
coordinates that appear in the operator can be written as a linear combination of its eigenfunctions, provided
that the function obeys the same boundary conditions as the eigenfunctions and shares any fundamental
symmetry property that is common to all of them. If, for example, all of the eigenfunctions vanish at some
point in space, then only functions that vanish at the same point can be written as linear combinations of the
eigenfunctions. Similarly, if the eigenfunctions of a particular operator in one dimension are all odd functions
of the coordinate, then all linear combinations of them must also be odd. It is clearly impossible in the latter
case to expand functions such as cos(x), exp(–x2) etc in terms of odd functions. This qualification is omitted in
some elementary treatments of quantum mechanics, but it is one that turns out to be important for systems
containing several identical particles. Nevertheless, if these criteria are met by a suitable function g, then it is
always possible to find coefficients ck such that

-10(A1.1.24)

where the coefficient cj is given by

(A1.1.25)

If the eigenfunctions are normalized, this expression reduces to
(A1.1.26)

When normalized, the eigenfunctions corresponding to a Hermitian operator are said to represent an
orthonormal set.
The mathematical properties discussed above are central to the next postulate:
5. In any experiment, the probability of observing a particular non-degenerate value for the
system property A can be determined by the following procedure. First, expand the
wavefunction in terms of the complete set of normalized eigenfunctions of the quantummechanical operator, ,

(A1.1.27)

The probability of measuring A = λk, where λk is the eigenvalue associated with the normalized
eigenfunction φk, is precisely equal to
. For degenerate eigenvalues, the probability
of observation is given by ∑ck2, where the sum is taken over all of the eigenfunctions φk that
correspond to the degenerate eigenvalue λk.
At this point, it is appropriate to mention an elementary concept from the theory of probability. If there are n
possible numerical outcomes (ξn) associated with a particular process, the average value 〈ξ〉 can be calculated
by summing up all of the outcomes, each weighted by its corresponding probability
(A1.1.28)

-11-

As an example, the possible outcomes and associated probabilities for rolling a pair of six-sided dice are

Sum

Probability

2
3
4
5
6
7
8
9
10
11
12

1/36
1/18
1/12
1/9
5/36
1/6
5/36
1/9
1/12
1/18
1/36

The average value is therefore given by the sum
.
What does this have to do with quantum mechanics? To establish a connection, it is necessary to first expand
the wavefunction in terms of the eigenfunctions of a quantum-mechanical operator ,

(A1.1.29)

We will assume that both the wavefunction and the orthogonal eigenfunctions are normalized, which implies
that

(A1.1.30)

Now, the operator is applied to both sides of equation (A1.1.29), which because of its linearity, gives
(A1.1.31)

-12-

where λk represents the eigenvalue associated with the eigenfunction φk. Next, both sides of the preceding
equation are multiplied from the left by the complex conjugate of the wavefunction and integrated over all
space

(A1.1.32)

(A1.1.33)
(A1.1.34)

The last identity follows from the orthogonality property of eigenfunctions and the assumption of
normalization. The right-hand side in the final result is simply equal to the sum over all eigenvalues of the
operator (possible results of the measurement) multiplied by the respective probabilities. Hence, an important
corollary to the fifth postulate is established:
(A1.1.35)

This provides a recipe for calculating the average value of the system property associated with the quantummechanical operator , for a specific but arbitrary choice of the wavefunction Ψ, notably those choices which
are not eigenfunctions of .
The fifth postulate and its corollary are extremely important concepts. Unlike classical mechanics, where
everything can in principle be known with precision, one can generally talk only about the probabilities
associated with each member of a set of possible outcomes in quantum mechanics. By making a measurement
of the quantity A, all that can be said with certainty is that one of the eigenvalues of will be observed, and its
probability can be calculated precisely. However, if it happens that the wavefunction corresponds to one of
the eigenfunctions of the operator , then and only then is the outcome of the experiment certain: the
measured value of A will be the corresponding eigenvalue.
Up until now, little has been said about time. In classical mechanics, complete knowledge about the system at
any time t suffices to predict with absolute certainty the properties of the system at any other time t′. The
situation is quite different in quantum mechanics, however, as it is not possible to know everything about the
system at any time t. Nevertheless, the temporal behavior of a quantum-mechanical system evolves in a well
defined way that depends on the Hamiltonian operator and the wavefunction Ψ according to the last postulate
6. The time evolution of the wavefunction is described by the differential equation

(A1.1.36)

The differential equation above is known as the time-dependent Schrödinger equation. There is an interesting
and

-13-

intimate connection between this equation and the classical expression for a travelling wave

(A1.1.37)

To convert (A1.1.37) into a quantum-mechanical form that describes the ‘matter wave’ associated with a free
particle travelling through space, one might be tempted to simply make the substitutions ν = E/h (Planck’s
hypothesis) and λ = h/p (de Broglie’s hypothesis). It is relatively easy to verify that the resulting expression
satisfies the time-dependent Schrödinger equation. However, it should be emphasized that this is not a
derivation, as there is no compelling reason to believe that this ad hoc procedure should yield one of the
fundamental equations of physics. Indeed, the time-dependent Schrödinger equation cannot be derived in a
rigorous way and therefore must be regarded as a postulate.
The time-dependent Schrödinger equation allows the precise determination of the wavefunction at any time t
from knowledge of the wavefunction at some initial time, provided that the forces acting within the system are
known (these are required to construct the Hamiltonian). While this suggests that quantum mechanics has a
deterministic component, it must be emphasized that it is not the observable system properties that evolve in a
precisely specified way, but rather the probabilities associated with values that might be found for them in a
measurement.
A1.1.2.2 STATIONARY STATES, SUPERPOSITION AND UNCERTAINTY

From the very beginning of the 20th century, the concept of energy conservation has made it abundantly clear
that electromagnetic energy emitted from and absorbed by material substances must be accompanied by
compensating energy changes within the material. Hence, the discrete nature of atomic line spectra suggested
that only certain energies are allowed by nature for each kind of atom. The wavelengths of radiation emitted
or absorbed must therefore be related to the difference between energy levels via Planck’s hypothesis, ∆ E =
hν = hc/λ.
The Schrödinger picture of quantum mechanics summarized in the previous subsection allows an important
deduction to be made that bears directly on the subject of energy levels and spectroscopy. Specifically, the
energies of spectroscopic transitions must correspond precisely to differences between distinct eigenvalues of
the Hamiltonian operator, as these correspond to the allowed energy levels of the system. Hence, the set of
eigenvalues of the Hamiltonian operator are of central importance in chemistry. These can be determined by
solving the so-called time-independent Schrödinger equation,
(A1.1.38)

for the eigenvalues Ek and eigenfunctions ψk. It should be clear that the set of eigenfunctions and eigenvalues
does not evolve with time provided the Hamiltonian operator itself is time independent. Moreover, since the

eigenfunctions of the Hamiltonian (like those of any other operator) form a complete set, it is always possible
to expand the exact wavefunction of the system at any time in terms of them:
(A1.1.39)

-14-

It is important to point out that this expansion is valid even if time-dependent terms are added to the
Hamiltonian (as, for example, when an electric field is turned on). If there is more than one nonzero value of
cj at any time t, then the system is said to be in a superposition of the energy eigenstates ψk associated with
non-vanishing expansion coefficients, ck. If it were possible to measure energies directly, then the fifth
postulate of the previous section tells us that the probability of finding energy Ek in a given measurement
would be
.
When a molecule is isolated from external fields, the Hamiltonian contains only kinetic energy operators for
all of the electrons and nuclei as well as terms that account for repulsion and attraction between all distinct
pairs of like and unlike charges, respectively. In such a case, the Hamiltonian is constant in time. When this
condition is satisfied, the representation of the time-dependent wavefunction as a superposition of
Hamiltonian eigenfunctions can be used to determine the time dependence of the expansion coefficients. If
equation (A1.1.39) is substituted into the time-dependent Schrödinger equation

(A1.1.40)

the simplification
(A1.1.41)

can be made to the right-hand side since the restriction of a time-independent Hamiltonian means that ψk is
always an eigenfunction of H. By simply equating the coefficients of the ψk, it is easy to show that the choice
(A1.1.42)

for the time-dependent expansion coefficients satisfies equation (A1.1.41). Like any differential equation,
there are an infinite number of solutions from which a choice must be made to satisfy some set of initial
conditions. The state of the quantum-mechanical system at time t = 0 is used to fix the arbitrary multipliers ck
(0), which can always be chosen as real numbers. Hence, the wavefunction Ψ becomes

(A1.1.43)

Suppose that the system property A is of interest, and that it corresponds to the quantum-mechanical operator
. The average value of A obtained in a series of measurements can be calculated by exploiting the corollary
to the fifth postulate

(A1.1.44)

-15-

Now consider the case where is itself a time-independent operator, such as that for the position, momentum
or angular momentum of a particle or even the energy of the benzene molecule. In these cases, the timedependent expansion coefficients are unaffected by application of the operator, and one obtains

(A1.1.45)

As one might expect, the first term that contributes to the expectation value of A is simply its value at t = 0,
while the second term exhibits an oscillatory time dependence. If the superposition initially includes large
contributions from states of widely varying energy, then the oscillations in 〈A〉 will be rapid. If the states that
are strongly mixed have similar energies, then the timescale for oscillation in the properties will be slower.
However, there is one special class of system properties A that exhibit no time dependence whatsoever. If (and
only if) every one of the states ψk is an eigenfunction of , then the property of orthogonality can be used to
show that every contribution to the second term vanishes. An obvious example is the Hamiltonian operator
itself; it turns out that the expectation value for the energy of a system subjected to forces that do not vary
with time is a constant. Are there other operators that share the same set of eigenfunctions ψk with , and if
so, how can they be recognized? It can be shown that any two operators which satisfy the property
(A1.1.46)

for all functions f share a common set of eigenfunctions, and A and B are said to commute. (The symbol [ , ]
meaning
–
, is called the commutator of the operators and .) Hence, there is no time dependence for
the expectation value of any system property that corresponds to a quantum-mechanical operator that
commutes with the Hamiltonian. Accordingly, these quantities are known as constants of the motion: their
average values will not vary, provided the environment of the system does not change (as it would, for
example, if an electromagnetic field were suddenly turned on). In nonrelativistic quantum mechanics, two
examples of constants of the motion are the square of the total angular momentum, as well as its projection
along an arbitrarily chosen axis. Other operators, such as that for the dipole moment, do not commute with the
Hamiltonian and the expectation value associated with the corresponding properties can indeed oscillate with
time. It is important to note that the frequency of these oscillations is given by differences between the
allowed energies of the system divided by Planck’s constant. These are the so-called Bohr frequencies, and it
is perhaps not surprising that these are exactly the frequencies of electromagnetic radiation that cause
transitions between the corresponding energy levels.
Close inspection of equation (A1.1.45) reveals that, under very special circumstances, the expectation value
does not change with time for any system properties that correspond to fixed (static) operator representations.
Specifically, if the spatial part of the time-dependent wavefunction is the exact eigenfunction ψj of the
Hamiltonian, then cj(0) = 1 (the zero of time can be chosen arbitrarily) and all other ck(0) = 0. The second
term clearly vanishes in these cases, which are known as stationary states. As the name implies, all
observable properties of these states do not vary with time. In a stationary state, the energy of the system has a
precise value (the corresponding eigenvalue of ) as do observables that are associated with operators that
commute with . For all other properties (such as the position and momentum),

-16-

one can speak only about average values or probabilities associated with a given measurement, but these
quantities themselves do not depend on time. When an external perturbation such as an electric field is applied
or a collision with another atom or molecule occurs, however, the system and its properties generally will
evolve with time. The energies that can be absorbed or emitted in these processes correspond precisely to
differences in the stationary state energies, so it should be clear that solving the time-independent Schrödinger
equation for the stationary state wavefunctions and eigenvalues provides a wealth of spectroscopic
information. The importance of stationary state solutions is so great that it is common to refer to equation
(A1.1.38) as ‘the Schrödinger equation’, while the qualified name ‘time-dependent Schrödinger equation’ is
generally used for equation (A1.1.36). Indeed, the subsequent subsections are devoted entirely to discussions
that centre on the former and its exact and approximate solutions, and the qualifier ‘time independent’ will be
omitted.
Starting with the quantum-mechanical postulate regarding a one-to-one correspondence between system
properties and Hermitian operators, and the mathematical result that only operators which commute have a
common set of eigenfunctions, a rather remarkable property of nature can be demonstrated. Suppose that one
desires to determine the values of the two quantities A and B, and that the corresponding quantum-mechanical
operators do not commute. In addition, the properties are to be measured simultaneously so that both reflect
the same quantum-mechanical state of the system. If the wavefunction is neither an eigenfunction of nor ,
then there is necessarily some uncertainty associated with the measurement. To see this, simply expand the
wavefunction ψ in terms of the eigenfunctions of the relevant operators

(A1.1.47)
(A1.1.48)

where the eigenfunctions and
of operators and , respectively, are associated with corresponding
eigenvalues and . Given that ψ is not an eigenfunction of either operator, at least two of the coefficients
ak and two of the bk must be nonzero. Since the probability of observing a particular eigenvalue is
proportional to the square of the expansion coefficient corresponding to the associated eigenfunction, there
will be no less than four possible outcomes for the set of values A and B. Clearly, they both cannot be
determined precisely. Indeed, under these conditions, neither of them can be!
In a more favourable case, the wavefunction ψ might indeed correspond to an eigenfunction of one of the
operators. If
, then a measurement of A necessarily yields , and this is an unambiguous result.
What can be said about the measurement of B in this case? It has already been said that the eigenfunctions of
two commuting operators are identical, but here the pertinent issue concerns eigenfunctions of two operators
that do not commute. Suppose is an eigenfunction of . Then, it must be true that

(A1.1.49)

-17-

If is also an eigenfunction of , then it follows that
, which contradicts the
assumption that and do not commute. Hence, no nontrivial eigenfunction of can also be an eigenfunction
of . Therefore, if measurement of A yields a precise result, then some uncertainty must be associated with B.
That is, the expansion of ψ in terms of eigenfunctions of (equation (A1.1.48)) must have at least two nonvanishing coefficients; the corresponding eigenvalues therefore represent distinct possible outcomes of the
experiment, each having probability
. A physical interpretation of
is the process of measuring the
value of A for a system in a state with a unique value for this property . However
represents a
measurement that changes the state of the system, so that if after we measure B and then measure A, we would
no longer find as its value:
.
The Heisenberg uncertainty principle offers a rigorous treatment of the qualitative picture sketched above. If
several measurements of A and B are made for a system in a particular quantum state, then quantitative
uncertainties are provided by standard deviations in the corresponding measurements. Denoting these as σA
and σB, respectively, it can be shown that

(A1.1.50)

One feature of this inequality warrants special attention. In the previous paragraph it was shown that the
precise measurement of A made possible when ψ is an eigenfunction of necessarily results in some
uncertainty in a simultaneous measurement of B when the operators and do not commute. However, the
mathematical statement of the uncertainty principle tells us that measurement of B is in fact completely
uncertain: one can say nothing at all about B apart from the fact that any and all values of B are equally
probable! A specific example is provided by associating A and B with the position and momentum of a
. If
particle moving along the x-axis. It is rather easy to demonstrate that [px, x] = – i , so that
the system happens to be described by a Dirac delta function at the point x0 (which is an eigenfunction of the
position operator corresponding to eigenvalue x0), then the probabilities associated with possible momenta
can be determined by expanding δ(x–x0) in terms of the momentum eigenfunctions A exp(ikx). Carrying out
such a calculation shows that all of the infinite number of possible momenta (the momentum operator has a
continuous spectrum) appear in the wavefunction expansion, all with precisely the same weight. Hence, no
particular momentum or (more properly in this case) range bounded by px + dpx is more likely to be observed
than any other.
A1.1.2.3 SOME QUALITATIVE FEATURES OF STATIONARY STATES

A great number of qualitative features associated with the stationary states that correspond to solutions of the
time-independent Schrödinger can be worked out from rather general mathematical considerations and use of
the postulates of quantum mechanics. Mastering these concepts and the qualifications that may apply to them
is essential if one is to obtain an intuitive feeling for the subject. In general, the systems of interest to chemists
are atoms and molecules, both in isolation as well as how they interact with each other or with an externally
applied field. In all of these cases, the forces acting upon the particles in the system give rise to a potential
energy function that varies with the positions of the particles, strength of the applied fields etc. In general, the
potential is a smoothly varying function of the coordinates, either growing without bound for large values of
the coordinates or tending asymptotically towards a finite value. In these cases, there is necessarily a
minimum value at what is known as the global equilibrium position (there may be several global minima that
are equivalent by symmetry). In many cases, there are also other minima

-18-

(meaning that the matrix of second derivatives with respect to the coordinates has only non-negative
eigenvalues) that have higher energies, which are called local minima. If the potential becomes infinitely large
for infinite values of the coordinates (as it does, for example, when the force on a particle varies linearly with
its displacement from equilibrium) then all solutions to the Schrödinger equation are known as bound states;
that with the smallest eigenvalue is called the ground state while the others are called excited states. In other
cases, such as potential functions that represent realistic models for diatomic molecules by approaching a
constant finite value at large separation (zero force on the particles, with a finite dissociation energy), there
are two classes of solutions. Those associated with eigenvalues that are below the asymptotic value of the
potential energy are the bound states, of which there is usually a finite number; those having higher energies
are called the scattering (or continuum) states and form a continuous spectrum. The latter are dealt with in
section A3.11 of the encyclopedia and will be mentioned here only when necessary for mathematical reasons.
Bound state solutions to the Schrödinger equation decay to zero for infinite values of the coordinates, and are
therefore integrable since they are continuous functions in accordance with the first postulate. The solutions
may assume zero values elsewhere in space and these regions—which may be a point, a plane or a three- or
higher-dimensional hypersurface—are known as nodes. From the mathematical theory of differential
eigenvalue equations, it can be demonstrated that the lowest eigenvalue is always associated with an
eigenfunction that has the same sign at all points in space. From this result, which can be derived from the
calculus of variations, it follows that the wavefunction corresponding to the smallest eigenvalue of the
Hamiltonian must have no nodes. It turns out, however, that relativistic considerations require that this
statement be qualified. For systems that contain more than two identical particles of a specific type, not all
solutions to the Schrödinger equation are allowed by nature. Because of this restriction, which is described in
subsection (A1.1.3.3) , it turns out that the ground states of lithium, all larger atoms and all molecules other
than , H2 and isoelectronic species have nodes. Nevertheless, our conceptual understanding of electronic
structure as well as the basis for almost all highly accurate calculations is ultimately rooted in a single-particle
approximation. The quantum mechanics of one-particle systems is therefore important in chemistry.
Shapes of the ground- and first three excited-state wavefunctions are shown in figure A1.1.1 for a particle in
one dimension subject to the potential
, which corresponds to the case where the force acting on the
particle is proportional in magnitude and opposite in direction to its displacement from equilibrium (f ≡ –∇V =
–kx). The corresponding Schrödinger equation

(A1.1.51)

can be solved analytically, and this problem (probably familiar to most readers) is that of the quantum
harmonic oscillator. As expected, the ground-state wavefunction has no nodes. The first excited state has a
single node, the second two nodes and so on, with the number of nodes growing with increasing magnitude of
the eigenvalue. From the form of the kinetic energy operator, one can infer that regions where the slope of the
wavefunction is changing rapidly (large second derivatives) are associated with large kinetic energy. It is
quite reasonable to accept that wavefunctions with regions of large curvature (where the function itself has
appreciable magnitude) describe states with high energy, an expectation that can be made rigorous by
applying a quantum-mechanical version of the virial theorem.

-19-

Figure A1.1.1. Wavefunctions for the four lowest states of the harmonic oscillator, ordered from the n = 0
ground state (at the bottom) to the n = 3 state (at the top). The vertical displacement of the plots is chosen so
that the location of the classical turning points are those that coincide with the superimposed potential
function (dotted line). Note that the number of nodes in each state corresponds to the associated quantum
number.
Classically, a particle with fixed energy E described by a quadratic potential will move back and forth
between the points where V = E, known as the classical turning points. Movement beyond the classical
turning points is forbidden, because energy conservation implies that the particle will have a negative kinetic
energy in these regions, and imaginary velocities are clearly inconsistent with the Newtonian picture of the
universe. Inside the turning points, the particle will have its maximum kinetic energy as it passes through the
minimum, slowing in its climb until it comes to rest and subsequently changes direction at the turning points
(imagine a marble rolling in a parabola). Therefore, if a camera were to take snapshots of the particle at
random intervals, most of the pictures would show the particle near the turning points (the equilibrium
position is actually the least likely location for the particle). A more detailed analysis of the problem shows
that the probability of seeing the classical particle in the neighbourhood of a given position x is proportional to
. Note that the situation found for the ground state described by quantum mechanics bears very little
resemblance to the classical situation. The particle is most likely to be found at the equilibrium position and,
within the classically allowed region, least likely to be seen at the turning points. However, the situation is
even stranger than this: the probability of finding the particle outside the turning points is non-zero! This
phenomenon, known as tunnelling, is not unique to the harmonic oscillator. Indeed, it occurs for bound states
described by every potential

-20-

that tends asymptotically to a finite value since the wavefunction and its derivatives must approach zero in a

smooth fashion for large values of the coordinates where (by the definition of a bound state) V must exceed E.
However, at large energies (see the 29th excited state probability density in figure A1.1.2, the situation is
more consistent with expectations based on classical theory: the probability density has its largest value near
the turning points, the general appearance is as implied by the classical formula (if one ignores the
oscillations) and its magnitude in the classically forbidden region is reduced dramatically with respect to that
found for the low-lying states. This merging of the quantum-mechanical picture with expectations based on
classical theory always occurs for highly excited states and is the basis of the correspondence principle.

Figure A1.1.2. Probability density (ψ*ψ) for the n = 29 state of the harmonic oscillator. The vertical state is
chosen as in figure A1.1.1, so that the locations of the turning points coincide with the superimposed potential
function.
The energy level spectrum of the harmonic oscillator is completely regular. The ground state energy is given
by hν, where ν is the classical frequency of oscillation given by

(A1.1.52)

-21-

although it must be emphasized that our inspection of the wavefunction shows that the motion of the particle
cannot be literally thought of in this way. The energy of the first excited state is hν above that of the ground
state and precisely the same difference separates each excited state from those immediately above and below.
A different example is provided by a particle trapped in the Morse potential

(A1.1.53)

originally suggested as a realistic model for the vibrational motion of diatomic molecules. Although the
wavefunctions associated with the Morse levels exhibit largely the same qualitative features as the harmonic
oscillator functions and are not shown here, the energy level structures associated with the two systems are
qualitatively different. Since V(x) tends to a finite value (De) for large x, there are only a limited number of
bound state solutions, and the spacing between them decreases with increasing eigenvalue. This is another
general feature; energy level spacings for states associated with potentials that tend towards asymptotic values
at infinity tend to decrease with increasing quantum number.
The one-dimensional cases discussed above illustrate many of the qualitative features of quantum mechanics,
and their relative simplicity makes them quite easy to study. Motion in more than one dimension and
(especially) that of more than one particle is considerably more complicated, but many of the general features
of these systems can be understood from simple considerations. While one relatively common feature of
multidimensional problems in quantum mechanics is degeneracy, it turns out that the ground state must be
non-degenerate. To prove this, simply assume the opposite to be true, i.e.
(A1.1.54)
(A1.1.55)

where E0 is the ground state energy, and
(A1.1.56)

In order to satisfy equation (A1.1.56), the two functions must have identical signs at some points in space and
different signs elsewhere. It follows that at least one of them must have at least one node. However, this is
incompatible with the nodeless property of ground-state eigenfunctions.
Having established that the ground state of a single-particle system is non-degenerate and nodeless, it is
straightforward to prove that the wavefunctions associated with every excited state must contain at least one
node (though they need not be degenerate!), just as seen in the example problems. It follows from the
orthogonality of eigenfunctions corresponding to a Hermitian operator that

(A1.1.57)

-22-

for all excited states ψx. In order for this equality to be satisfied, it is necessary that the integrand either
vanishes at all points in space (which contradicts the assumption that both ψg and ψx are nodeless) or is
positive in some regions of space and negative in others. Given that the ground state has no nodes, the latter
condition can be satisfied only if the excited-state wavefunction changes sign at one or more points in space.
Since the first postulate states that all wavefunctions are continuous, it is therefore necessary that ψx has at
least one node.
In classical mechanics, it is certainly possible for a system subject to dissipative forces such as friction to
come to rest. For example, a marble rolling in a parabola lined with sandpaper will eventually lose its kinetic
energy and come to rest at the bottom. Rather remarkably, making a measurement of E that coincides with

Vmin (as would be found classically for our stationary marble) is incompatible with quantum mechanics.
Turning back to our example, the ground-state energy is indeed larger than the minimum value of the
potential energy for the harmonic oscillator. That this property of zero-point energy is guaranteed in quantum
mechanics can be demonstrated by straightforward application of the basic principles of the subject. Unlike
nodal features of the wavefunction, the arguments developed here also hold for many-particle systems.
Suppose the total energy of a stationary state is E. Since the energy is the sum of kinetic and potential
energies, it must be true that expectation values of the kinetic and potential energies are related according to

(A1.1.58)

If the total energy associated with the state is equal to the potential energy at the equilibrium position, it
follows that
(A1.1.59)

Two cases must be considered. In the first, it will be assumed that the wavefunction is nonzero at one or more
points for which V > Vmin (for the physically relevant case of a smoothly varying and continuous potential,
this includes all possibilities other than that in which the wavefunction is a Dirac delta function at the
equilibrium position). This means that 〈V〉 must also be greater than Vmin thereby forcing the average kinetic
energy to be negative. This is not possible. The kinetic energy operator for a quantum-mechanical particle
moving in the x-direction has the (unnormalized) eigenfunctions
(A1.1.60)

where

(A1.1.61)

and α are the corresponding eigenvalues. It can be seen that negative values of α give rise to real arguments
of the exponential and correspondingly divergent eigenfunctions. Zero and non-negative values are associated
with constant and oscillatory solutions in which the argument of the exponential vanishes or is imaginary,
respectively. Since divergence of the actual wavefunction is incompatible with its probabilistic interpretation,
no contribution from negative α eigenfunctions can appear when the wavefunction is expanded in terms of
kinetic energy eigenfunctions.

-23-

It follows from the fifth postulate that the kinetic energy of each particle in the system (and therefore the total
kinetic energy) is restricted to non-negative values. Therefore, the expectation value of the kinetic energy
cannot be negative. The other possibility is that the wavefunction is non-vanishing only when V = Vmin. For
the case of a smoothly varying, continuous potential, this corresponds to a state described by a Dirac delta
function at the equilibrium position, which is the quantum-mechanical equivalent of a particle at rest. In any
event, the fact that the wavefunction vanishes at all points for which V ≠ Vmin means that the expectation
value of the kinetic energy operator must also vanish if there is to be no zeropoint energy. Considering the
discussion above, this can occur only when the wavefunction is the same as the zero-kinetic-energy
eigenfunction (ψ = constant). This contradicts the assumption used in this case, where the wavefunction is a

delta function. Following the general arguments used in both cases above, it is easily shown that E can only be
larger than Vmin, which means that any measurement of E for a particle in a stationary or non-stationary state
must give a result that satisfies the inequality E > Vmin.

A1.1.3 QUANTUM MECHANICS OF MANY-PARTICLE SYSTEMS
A1.1.3.1 THE HYDROGEN ATOM

It is admittedly inconsistent to begin a section on many-particle quantum mechanics by discussing a problem
that can be treated as a single particle. However, the hydrogen atom and atomic ions in which only one
electron remains (He+, Li2+ etc) are the only atoms for which exact analytic solutions to the Schrödinger
equation can be obtained. In no cases are exact solutions possible for molecules, even after the Born–
Oppenheimer approximation (see section B3.1.1.1) is made to allow for separate treatment of electrons and
nuclei. Despite the limited interest of hydrogen atoms and hydrogen-like ions to chemistry, the quantum
mechanics of these systems is both highly instructive and provides a basis for treatments of more complex
atoms and molecules. Comprehensive discussions of one-electron atoms can be found in many textbooks; the
emphasis here is on qualitative aspects of the solutions.
The Schrödinger equation for a one-electron atom with nuclear charge Z is

(A1.1.62)

where µ is the reduced mass of the electron–nucleus system and the Laplacian is most conveniently expressed
in spherical polar coordinates. While not trivial, this differential equation can be solved analytically. Some of
the solutions are normalizable, and others are not. The former are those that describe the bound states of oneelectron atoms, and can be written in the form
(A1.1.63)

where N is a normalization constant, and Rnll(r) and Yl,m (θ, φ) are specific functions that depend on the
quantum numbers n, l and ml. The first of these is called the principal quantum number, while l is known as
the angular momentum, or azimuthal, quantum number, and ml the magnetic quantum number. The quantum
numbers that allow for normalizable wavefunctions are limited to integers that run over the ranges

-24-

(A1.1.64)
(A1.1.65)
(A1.1.66)

The fact that there is no restriction on n apart from being a positive integer means that there are an infinite
number of bound-state solutions to the hydrogen atom, a peculiarity that is due to the form of the Coulomb
potential. Unlike most bound state problems, the range of the potential is infinite (it goes to zero at large r, but
diverges to negative infinity at r = 0). The eigenvalues of the Hamiltonian depend only on the principal

quantum number and are (in attojoules (10–18 J))

(A1.1.67)

where it should be noted that the zero of energy corresponds to infinite separation of the particles. For each
value of n, the Schrödinger equation predicts that all states are degenerate, regardless of the choice of l and ml.
Hence, any linear combination of wavefunctions corresponding to some specific value of n is also an
eigenfunction of the Hamiltonian with eigenvalue En. States of hydrogen are usually characterized as ns, np,
nd etc where n is the principal quantum number and s is associated with l = 0, p with l = 1 and so on. The
functions Rnl(r) describe the radial part of the wavefunctions and can all be written in the form
(A1.1.68)

where ρ is proportional to the electron–nucleus separation r and the atomic number Z. Lnl is a polynomial of
order n – l – 1 that has zeros (where the wavefunction, and therefore the probability of finding the electron,
vanishes—a radial node) only for positive values of ρ. The functions Yl,ml(θ, φ) are the spherical harmonics.
The first few members of this series are familiar to everyone who has studied physical chemistry: Y00 is a
constant, leading to a spherically symmetric wavefunction, while Y1,0, and specific linear combinations of
Y1,1 and Y1,–1, vanish (have an angular node) in the xy, xz and yz planes, respectively. In general, these
functions exhibit l nodes, meaning that the number of overall nodes corresponding to a particular ψnlml is
equal to n – 1. For example, the 4d state has two angular nodes (l = 2) and one radial node (Lnl(ρ) has one
zero for positive ρ). In passing, it should be noted that many of the ubiquitous qualitative features of quantum
mechanics are illustrated by the wavefunctions and energy levels of the hydrogen atom. First, the system has a
zero-point energy, meaning that the ground-state energy is larger than the lowest value of the potential (–∞)
and the spacing between the energy levels decreases with increasing energy. Second, the ground state of the
system is nodeless (the electron may be found at any point in space), while the number of nodes exhibited by
the excited states increases with energy. Finally, there is a finite probability that the electron is found in a
classically forbidden region in all bound states. For the hydrogen atom ground state, this corresponds to all
electron–proton separations larger than 105.8 pm, where the electron is found 23.8% of the time. As usual,
this tunnelling phenomenon is less pronounced in excited states: the corresponding values for the 3s state are
1904 pm and 16.0%.
The Hamiltonian commutes with the angular momentum operator z as well as that for the square of the
angular momentum 2. The wavefunctions above are also eigenfunctions of these operators, with eigenvalues
and
. It should be emphasized that the total angular momentum is
,
and not a simple

-25-

integral multiple of as assumed in the Bohr model. In particular, the ground state of hydrogen has zero
angular momentum, while the Bohr atom ground state has L = . The meaning associated with the ml quantum
number is more difficult to grasp. The choice of z instead of x or y seems to be (and is) arbitrary and it is
illogical that a specific value of the angular momentum projection along one coordinate must be observed in
any experiment, while those associated with x and y are not similarly restricted. However, the states with a
given l are degenerate, and the wavefunction at any particular time will in general be some linear combination
of the ml eigenfunctions. The only way to isolate a specific ψnlml (and therefore ensure the result of measuring
Lz) is to apply a magnetic field that lifts the degeneracy and breaks the symmetry of the problem. The z axis

then corresponds to the magnetic field direction, and it is the projection of the angular momentum vector on
this axis that must be equal to ml .
The quantum-mechanical treatment of hydrogen outlined above does not provide a completely satisfactory
description of the atomic spectrum, even in the absence of a magnetic field. Relativistic effects cause both a
scalar shifting in all energy levels as well as splittings caused by the magnetic fields associated with both
motion and intrinsic properties of the charges within the atom. The features of this fine structure in the energy
spectrum were successfully (and miraculously, given that it preceded modern quantum mechanics by a decade
and was based on a two-dimensional picture of the hydrogen atom) predicted by a formula developed by
Sommerfeld in 1915. These interactions, while small for hydrogen, become very large indeed for larger atoms
where very strong electron–nucleus attractive potentials cause electrons to move at velocities close to the
speed of light. In these cases, quantitative calculations are extremely difficult and even the separability of
orbital and intrinsic angular momenta breaks down.
A1.1.3.2 THE INDEPENDENT-PARTICLE APPROXIMATION

Applications of quantum mechanics to chemistry invariably deal with systems (atoms and molecules) that
contain more than one particle. Apart from the hydrogen atom, the stationary-state energies cannot be
calculated exactly, and compromises must be made in order to estimate them. Perhaps the most useful and
widely used approximation in chemistry is the independent-particle approximation, which can take several
forms. Common to all of these is the assumption that the Hamiltonian operator for a system consisting of n
particles is approximated by the sum

(A1.1.69)

where the single-particle Hamiltonians i consist of the kinetic energy operator plus a potential (
) that does not explicitly depend on the coordinates of the other n – 1 particles in the
system. Of course, the simplest realization of this model is to completely neglect forces due to the other
particles, but this is often too severe an approximation to be useful. In any event, the quantum mechanics of a
system described by a Hamiltonian of the form given by equation (A1.1.69) is worthy of discussion simply
because the independent-particle approximation is the foundation for molecular orbital theory, which is the
central paradigm of descriptive chemistry.
Let the orthonormal functions χi(1), χj(2), . . ., χp(n) be selected eigenfunctions of the corresponding singleparticle Hamiltonians 1, 2, . . ., n, with eigenvalues λi, λj, . . ., λp. It is easily verified that the product of
these single-particle wavefunctions (which are often called orbitals when the particles are electrons in atoms
and molecules)

-26(A1.1.70)

satisfies the approximate Schrödinger equation for the system

(A1.1.71)

with the corresponding energy

(A1.1.72)

Hence, if the Hamiltonian can be written as a sum of terms that individually depend only on the coordinates of
one of the particles in the system, then the wavefunction of the system can be written as a product of
functions, each of which is an eigenfunction of one of the single-particle Hamiltonians, hi. The corresponding
eigenvalue is then given by the sum of eigenvalues associated with each single-particle wavefunction χ
appearing in the product.
The approximation embodied by equation (A1.1.69), equation (A1.1.70), equation (A1.1.71) and equation
(A1.1.72) presents a conceptually appealing picture of many-particle systems. The behaviour and energetics
of each particle can be determined from a simple function of three coordinates and the eigenvalue of a
differential equation considerably simpler than the one that explicitly accounts for all interactions. It is
precisely this simplification that is invoked in qualitative interpretations of chemical phenomena such as the
inert nature of noble gases and the strongly reducing property of the alkali metals. The price paid is that the
model is only approximate, meaning that properties predicted from it (for example, absolute ionization
potentials rather than just trends within the periodic table) are not as accurate as one might like. However, as
will be demonstrated in the latter parts of this section, a carefully chosen independent-particle description of a
many-particle system provides a starting point for performing more accurate calculations. It should be
mentioned that even qualitative features might be predicted incorrectly by independent-particle models in
extreme cases. One should always be aware of this possibility and the oft-misunderstood fact that there really
is no such thing as an orbital. Fortunately, however, it turns out that qualitative errors are uncommon for
electronic properties of atoms and molecules when the best independent-particle models are used.
One important feature of many-particle systems has been neglected in the preceding discussion. Identical
particles in quantum mechanics must be indistinguishable, which implies that the exact wavefunctions ψ
which describe them must satisfy certain symmetry properties. In particular, interchanging the coordinates of
any two particles in the mathematical form of the wavefunction cannot lead to a different prediction of the
system properties. Since any rearrangement of particle coordinates can be achieved by successive pairwise
permutations, it is sufficient to consider the case of a single permutation in analysing the symmetry properties
that wavefunctions must obey. In the following, it will be assumed that the wavefunction is real. This is not
restrictive, as stationary state wavefunctions for isolated atoms and molecules can always be written in this
way. If the operator Pij is that which permutes the coordinates of particles i and j, then indistinguishability
requires that

(A1.1.73)

-27-

for any operator (including the identity) and choice of i and j. Clearly, a wavefunction that is symmetric with
respect to the interchange of coordinates for any two particles
(A1.1.74)

satisfies the indistinguishability criterion. However, equation (A1.1.73) is also satisfied if the permutation of
particle coordinates results in an overall sign change of the wavefunction, i.e.
(A1.1.75)

Without further considerations, the only acceptable real quantum-mechanical wavefunctions for an n-particle
system would appear to be those for which
(A1.1.76)

where i and j are any pair of identical particles. For example, if the system comprises two protons, a neutron
and two electrons, the relevant permutations are that which interchanges the proton coordinates and that
which interchanges the electron coordinates. The other possible pairs involve distinct particles and the action
of the corresponding Pij operators on the wavefunction will in general result in something quite different.
Since indistinguishability is a necessary property of exact wavefunctions, it is reasonable to impose the same
constraint on the approximate wavefunctions φ formed from products of single-particle solutions. However, if
two or more of the χi in the product are different, it is necessary to form linear combinations if the condition
Pijψ = ± ψ is to be met. An additional consequence of indistinguishability is that the hi operators
corresponding to identical particles must also be identical and therefore have precisely the same
eigenfunctions. It should be noted that there is nothing mysterious about this perfectly reasonable restriction
placed on the mathematical form of wavefunctions.
For the sake of simplicity, consider a system of two electrons for which the corresponding single-particle
states are χi, χj, χk, . . ., χn, with eigenvalues λi, λj, λk, . . ., λn. Clearly, the two-electron wavefunction φ = χi
(1)χi(2) satisfies the indistinguishability criterion and describes a stationary state with energy E0 = 2λi.
However, the state χi(1)χj(2) is not satisfactory. While it is a solution to the Schrödinger equation, it is neither
symmetric nor antisymmetric with respect to particle interchange. However, two such states can be formed by
taking the linear combinations
(A1.1.77)
(A1.1.78)

which are symmetric and antisymmetric with respect to particle interchange, respectively. Because the
functions χ are orthonormal, the energies calculated from φS and φA are the same as that corresponding to the
unsymmetrized product state χi(1)χj(2), as demonstrated explicitly for φS:

-28-

(A1.1.79)

It should be mentioned that the single-particle Hamiltonians in general have an infinite number of solutions,
so that an uncountable number of wavefunctions ψ can be generated from them. Very often, interest is
focused on the ground state of many-particle systems. Within the independent-particle approximation, this
state can be represented by simply assigning each particle to the lowest-lying energy level. If a calculation is

performed on the lithium atom in which interelectronic repulsion is ignored completely, the single-particle
Schrödinger equations are precisely the same as those for the hydrogen atom, apart from the difference in
nuclear charge. The following lithium atom wavefunction could then be constructed from single-particle
orbitals
(A1.1.80)

a form that is obviously symmetric with respect to interchange of particle coordinates. If this wavefunction is
used to calculate the expectation value of the energy using the exact Hamiltonian (which includes the explicit
electron–electron repulsion terms),
(A1.1.81)

one obtains an energy lower than the actual result, which (see (A1.1.4.1)) suggests that there are serious
problems with this form of the wavefunction. Moreover, a relatively simple analysis shows that ionization
potentials of atoms would increase monotonically—approximately linearly for small atoms and quadratically
for large atoms—if the independent-particle picture discussed thus far has any validity. Using a relatively
simple model that assumes that the lowest lying orbital is a simple exponential, ionization potentials of 13.6,
23.1, 33.7 and 45.5 electron volts (eV) are predicted for hydrogen, helium, lithium and beryllium,
respectively. The value for hydrogen (a one-electron system) is exact and that for helium is in relatively good
agreement with the experimental value of 24.8 eV. However, the other values are well above the actual
ionization energies of Li and Be (5.4 and 9.3 eV, respectively), both of which are smaller than those of H and
He! All freshman chemistry students learn that ionization potentials do not increase monotonically with
atomic number, and that there are in fact many pronounced and more subtle decreases that appear when this
property is plotted as a function of atomic number.
There is evidently a grave problem here. The wavefunction proposed above for the lithium atom contains all
of the particle coordinates, adheres to the boundary conditions (it decays to zero when the particles are
removed to infinity) and obeys the restrictions P12φ = P13φ = P23φ = ±φ that govern the behaviour of the
exact wavefunctions. Therefore, if no other restrictions are placed on the wavefunctions of multiparticle
systems, the product wavefunction for lithium

-29-

must lie in the space spanned by the exact wavefunctions. However, it clearly does not, because it is proven in
subsection (A1.1.4.1) that any function expressible as a linear combination of Hamiltonian eigenfunctions
cannot have an energy lower than that of the exact ground state. This means that there is at least one
additional symmetry obeyed by all of the exact wavefunctions that is not satisfied by the product form given
for lithium in equation (A1.1.80).
This missing symmetry provided a great puzzle to theorists in the early part days of quantum mechanics.
Taken together, ionization potentials of the first four elements in the periodic table indicate that
wavefunctions which assign two electrons to the same single-particle functions such as

(A1.1.82)

(helium) and

(A1.1.83)

(beryllium, the operator

produces the labelled χaχaχbχb product that is symmetric with respect to

interchange of particle indices) are somehow acceptable but that those involving three or more electrons in
one state are not! The resolution of this zweideutigkeit (two-valuedness) puzzle was made possible only by the
discovery of electron spin, which is discussed below.
A1.1.3.3 SPIN AND THE PAULI PRINCIPLE

In the early 1920s, spectroscopic experiments on the hydrogen atom revealed a striking inconsistency with the
Bohr model, as adapted by Sommerfeld to account for relativistic effects. Studies of the fine structure
associated with the n = 4 → n = 3 transition revealed five distinct peaks, while six were expected from
arguments based on the theory of interaction between matter and radiation. The problem was ultimately
reconciled by Uhlenbeck and Goudsmit, who reinterpreted one of the quantum numbers appearing in
Sommerfeld’s fine structure formula based on a startling assertion that the electron has an intrinsic angular
momentum independent of that associated with its motion. This idea was also supported by previous
experiments of Stern and Gerlach, and is now known as electron spin. Spin is a mysterious phenomenon with
a rather unfortunate name. Electrons are fundamental particles, and it is no more appropriate to think of them
as charges that resemble extremely small billiard balls than as waves. Although they exhibit behaviour
characteristic of both, they are in fact neither. Elementary textbooks often depict spin in terms of spherical
electrons whirling about their axis (a compelling idea in many ways, since it reinforces the Bohr model by
introducing a spinning planet), but this is a purely classical perspective on electron spin that should not be
taken literally.
Electrons and most other fundamental particles have two distinct spin wavefunctions that are degenerate in the
absence of an external magnetic field. Associated with these are two abstract states which are eigenfunctions
of the intrinsic spin angular momentum operator z
(A1.1.84)

-30-

The allowed quantum numbers ms are and – , and the corresponding eigenfunctions are usually written as α
and β, respectively. The associated eigenvalues and – give the projection of the intrinsic angular momentum
vector along the direction of a magnetic field that can be applied to resolve the degeneracy. The overall spin
angular momentum of the electron is given in terms of the quantum number s by
. For an electron, s
= . For a collection of particles, the overall spin and its projection are given in terms of the spin quantum
numbers S and MS (which are equal to the corresponding lower-case quantities for single particles) by
and
, respectively. S must be positive and can assume either integral or half-integral values,
and the MS quantum numbers lie in the interval

(A1.1.85)

where a correspondence to the properties of orbital angular momentum should be noted. The multiplicity of a
state is given by 2S + 1 (the number of possible MS values) and it is customary to associate the terms singlet
with S = 0, doublet with
, triplet with S = 1 and so on.
In the non-relativistic quantum mechanics discussed in this chapter, spin does not appear naturally. Although

Dirac showed in 1928 that a fourth quantum number associated with intrinsic angular momentum appears in a
relativistic treatment of the free electron, it is customary to treat spin heuristically. In general, the
wavefunction of an electron is written as the product of the usual spatial part (which corresponds to a solution
of the non-relativistic Schrödinger equation and involves only the Cartesian coordinates of the particle) and a
spin part σ, where σ is either α or β. A common shorthand notation is often used, whereby
(A1.1.86)
(A1.1.87)

In the context of electronic structure theory, the composite functions above are often referred to as spin
orbitals. When spin is taken into account, one finds that the ground state of the hydrogen atom is actually
doubly degenerate. The spatial part of the wavefunction is the Schrödinger equation solution discussed in
section (A1.1.3.1), but the possibility of either spin α or β means that there are two distinct overall
wavefunctions. The same may be said for any of the excited states of hydrogen (all of which are, however,
already degenerate in the nonrelativistic theory), as the level of degeneracy is doubled by the introduction of
spin. Spin may be thought of as a fourth coordinate associated with each particle. Unlike Cartesian
coordinates, for which there is a continuous distribution of possible values, there are only two possible values
of the spin coordinate available to each particle. This has important consequences for our discussion of
indistinguishability and symmetry properties of the wavefunction, as the concept of coordinate permutation
must be amended to include the spin variable of the particles. As an example, the independent-particle ground
state of the helium atom based on hydrogenic wavefunctions
(A1.1.88)

must be replaced by the four possibilities
(A1.1.89)

-31-

(A1.1.90)
(A1.1.91)
(A1.1.92)

While the first and fourth of these are symmetric with respect to particle interchange and thereby satisfy the
indistinguishability criterion, the other two are not and appropriate linear combinations must be formed.
Doing so, one finds the following four wavefunctions
(A1.1.93)
(A1.1.94)
(A1.1.95)
(A1.1.96)

where the first three are symmetric with respect to particle interchange and the last is antisymmetric. This
suggests that under the influence of a magnetic field, the ground state of helium might be resolved into
components that differ in terms of overall spin, but this is not observed. For the lithium example, there are

eight possible ways of assigning the spin coordinates, only two of which
(A1.1.97)
(A1.1.98)

satisfy the criterion Pijφ = ± φ. The other six must be mixed in appropriate linear combinations. However,
there is an important difference between lithium and helium. In the former case, all assignments of the spin
variable to the state given by equation (A1.1.88) produce a product function in which the same state (in terms
of both spatial and spin coordinates) appears at least twice. A little reflection shows that it is not possible to
generate a linear combination of such functions that is antisymmetric with respect to all possible interchanges;
only symmetric combinations such as
(A1.1.99)

can be constructed. The fact that antisymmetric combinations appear for helium (where the independentparticle ground state made up of hydrogen 1s functions is qualitatively consistent with experiment) and not for
lithium (where it is not) raises the interesting possibility that the exact wavefunction satisfies a condition more
restrictive than Pijψ = ±ψ, namely Pijψ = –ψ. For reasons that are not at all obvious, or even intuitive, nature
does indeed enforce this restriction, which is one statement of the Pauli exclusion principle. When this idea is
first met with, one usually learns an equivalent but less general statement that applies only within the
independent-particle approximation: no two electrons can have the same quantum numbers. What does this
mean? Within the independent-particle picture of an atom, each single-particle wavefunction, or orbital, is
described by the quantum numbers n, l, ml and (when spin is considered) ms.

-32-

Since it is not possible to generate antisymmetric combinations of products if the same spin orbital appears
twice in each term, it follows that states which assign the same set of four quantum numbers twice cannot
possibly satisfy the requirement Pijψ = –ψ, so this statement of the exclusion principle is consistent with the
more general symmetry requirement. An even more general statement of the exclusion principle, which can be
regarded as an additional postulate of quantum mechanics, is
The wavefunction of a system must be antisymmetric with respect to interchange of the
coordinates of identical particles γ and δ if they are fermions, and symmetric with respect to
interchange of γ and δ if they are bosons.
Electrons, protons and neutrons and all other particles that have s = are known as fermions. Other particles
are restricted to s = 0 or 1 and are known as bosons. There are thus profound differences in the quantummechanical properties of fermions and bosons, which have important implications in fields ranging from
statistical mechanics to spectroscopic selection rules. It can be shown that the spin quantum number S
associated with an even number of fermions must be integral, while that for an odd number of them must be
half-integral. The resulting composite particles behave collectively like bosons and fermions, respectively, so
the wavefunction symmetry properties associated with bosons can be relevant in chemical physics. One
prominent example is the treatment of nuclei, which are typically considered as composite particles rather
than interacting protons and neutrons. Nuclei with even atomic number therefore behave like individual
bosons and those with odd atomic number as fermions, a distinction that plays an important role in rotational
spectroscopy of polyatomic molecules.

A1.1.3.4 INDEPENDENT-PARTICLE MODELS IN ELECTRONIC STRUCTURE

At this point, it is appropriate to make some comments on the construction of approximate wavefunctions for
the many-electron problems associated with atoms and molecules. The Hamiltonian operator for a molecule is
given by the general form

(A1.1.100)

It should be noted that nuclei and electrons are treated equivalently in , which is clearly inconsistent with the
way that we tend to think about them. Our understanding of chemical processes is strongly rooted in the
concept of a potential energy surface which determines the forces that act upon the nuclei. The potential
energy surface governs all behaviour associated with nuclear motion, such as vibrational frequencies, mean
and equilibrium internuclear separations and preferences for specific conformations in molecules as complex
as proteins and nucleic acids. In addition, the potential energy surface provides the transition state and
activation energy concepts that are at the heart of the theory of chemical reactions. Electronic motion,
however, is never discussed in these terms. All of the important and useful ideas discussed above derive from
the Born–Oppenheimer approximation, which is discussed in some detail in section B3.1. Within this model,
the electronic states are solutions to the equation

-33-

(A1.1.101)

where the nuclei are assumed to be stationary. The electronic energies are given by the eigenvalues (usually
augmented by the wavefunction-independent internuclear repulsion energy) of . The functions obtained by
plotting the electronic energy as a function of nuclear position are the potential energy surfaces described
above. The latter are different for every electronic state; their shape gives the usual information about
molecular structure, barrier heights, isomerism and so on. The Born–Oppenheimer separation is also made in
the study of electronic structure in atoms. However, this is a rather subtle point and is not terribly important in
applications since the only assumption made is that the nucleus has infinite mass.
Although a separation of electronic and nuclear motion provides an important simplification and appealing
qualitative model for chemistry, the electronic Schrödinger equation is still formidable. Efforts to solve it
approximately and apply these solutions to the study of spectroscopy, structure and chemical reactions form
the subject of what is usually called electronic structure theory or quantum chemistry. The starting point for
most calculations and the foundation of molecular orbital theory is the independent-particle approximation.
For many-electron systems such as atoms and molecules, it is obviously important that approximate
wavefunctions obey the same boundary conditions and symmetry properties as the exact solutions. Therefore,
they should be antisymmetric with respect to interchange of each pair of electrons. Such states can always be
constructed as linear combinations of products such as
(A1.1.102)

The χ are assumed to be spin orbitals (which include both the spatial and spin parts) and each term in the
product differs in the way that the electrons are assigned to them. Of course, it does not matter how the
electrons are distributed amongst the χ in equation (A1.1.102), as the necessary subsequent
antisymmetrization makes all choices equivalent apart from an overall sign (which has no physical
significance). Hence, the product form is usually written without assigning electrons to the individual orbitals,
and the set of unlabelled χ included in the product represents an electron configuration. It should be noted that
all of the single-particle orbitals χ in the product are distinct. A very convenient method for constructing
antisymmetrized combinations corresponding to products of particular single-particle states is to form the
Slater determinant

(A1.1.103)

-34-

where the nominal electron configuration can be determined by simply scanning along the main diagonal of
the matrix. A fundamental result of linear algebra is that the determinant of a matrix changes sign when any
two rows or columns are interchanged. Inspection of equation (A1.1.103) shows that interchanging any two
columns of the Slater determinant corresponds to interchanging the labels of two electrons, so the Pauli
exclusion principle is automatically incorporated into this convenient representation. Whether all orbitals in a
given row are identical and all particle labels the same in each column (as above) or vice versa is not
important, as determinants are invariant with respect to transposition. In particular, it should be noted that the
Slater determinant necessarily vanishes when two of the spin orbitals are identical, reflecting the alternative
statement of the Pauli principle—no two electrons can have the same quantum number. One qualification
which should be stated here is that Slater determinants are not necessarily eigenfunctions of the S2 operator,
and it is often advantageous to form linear combinations of those corresponding to electron configurations that
differ only in the assignment of the spin variable to the spatial orbitals. The resulting functions φ are
sometimes known as spin-adapted configurations.
Within an independent-particle picture, there are a very large number of single-particle wavefunctions χ
available to each particle in the system. If the single-particle Schrödinger equations can be solved exactly,
then there are often an infinite number of solutions. Approximate solutions are, however, necessitated in most
applications, and some subtleties must be considered in this case. The description of electrons in atoms and
molecules is often based on the Hartree–Fock approximation, which is discussed in section B3.1 of this
encyclopedia. In the Hartree–Fock method, only briefly outlined here, the orbitals are chosen in such a way
that the total energy of a state described by the Slater determinant that comprises them is minimized. There
are cogent reasons for using an energy minimization strategy that are based on the variational principle
discussed later in this section. The Hartree–Fock method derives from an earlier treatment of Hartree, in
which indistinguishability and the Pauli principle were ignored and the wavefunction expressed as in equation
(A1.1.102). However, that approach is not satisfactory because it can lead to pathological solutions such as
that discussed earlier for lithium. In Hartree–Fock theory, the orbital optimization is achieved at the expense
of introducing a very complicated single-particle potential term νi. This potential depends on all of the other
orbitals in which electrons reside, requires the evaluation of difficult integrals and necessitates a selfconsistent (iterative) solution. The resulting one-electron Hamiltonian is known as the Fock operator, and it
has (in principle) an infinite number of eigenfunctions; a subset of these are exactly the same as the χ that
correspond to the occupied orbitals upon which it is parametrized. The resulting equations cannot be solved
analytically; for atoms, exact solutions for the occupied orbitals can be determined by numerical methods, but

the infinite number of unoccupied functions are unknown apart from the fact that they must be orthogonal to
the occupied ones. In molecular calculations, it is customary to assume that the orbitals χ can be written as
linear combinations of a fixed set of N basis functions, where N is typically of the order of tens to a few
hundred. Iterative solution of a set of matrix equations provides approximations for the orbitals describing the
n electrons of the molecule and N – n unoccupied orbitals.
The choice of basis functions is straightforward in atomic calculations. It can be demonstrated that all
solutions to an independent-particle Hamiltonian have the symmetry properties of the hydrogenic
wavefunctions. Each is, or can be written as, an eigenfunction of the z and 2 operators and involves a radial
part multiplied by a spherical harmonic. Atomic calculations that use basis sets (not all of them do) typically
choose functions that are similar to those that solve the Schrödinger equation for hydrogen. If the complete set
of hydrogenic functions is used, the solution to the basis set equations are the exact Hartree–Fock solutions.
However, practical considerations require the use of finite basis sets; the corresponding solutions are therefore
only approximate. Although the distinction is rarely made, it is preferable to refer to these as self-consistent
field (SCF) solutions and energies in order to distinguish them from the exact Hartree–Fock results. As the
quality of a basis is improved, the energy approaches that of the Hartree–Fock solution from above.

-35-

In molecules, things are a great deal more complicated. In principle, one can always choose a subset of all the
hydrogenic wavefunctions centred at some point in space. Since the resulting basis functions include all
possible electron coordinates and Slater determinants constructed from them vanish at infinity and satisfy the
Pauli principle, the corresponding approximate solutions must lie in the space spanned by the exact solutions
and be qualitatively acceptable. In particular, use of enough basis functions will result in convergence to the
exact Hartree–Fock solution. Because of the difficulties involved with evaluating integrals involving
exponential hydrogenic functions centred at more than one point in space, such single-centre expansions were
used in the early days of quantum chemistry. The main drawback is that convergence to the exact Hartree–
Fock result is extraordinarily slow. The states of the hydrogen molecule are reasonably well approximated by
linear combinations of hydrogenic functions centred on each of the two nuclei. Hence, a more practical
strategy is to construct a basis by choosing a set of hydrogenic functions for each atom in the molecule (the
same functions are usually used for identical atoms, whether or not they are equivalent by symmetry). Linear
combinations of a relatively small number of these functions are capable of describing the electronic
distribution in molecules much better than is possible with a corresponding number of functions in a singlecentre expansion. This approach is often called the linear combination of atomic orbitals (LCAO)
approximation, and is used in virtually all molecular SCF calculations performed today. The problems
associated with evaluation of multicentre integrals alluded to above was solved more than a quarter-century
ago by the introduction of Gaussian—rather than exponential—basis functions, which permit all of the
integrals appearing in the Fock operator to be calculated analytically. Although Gaussian functions are not
hydrogenic functions (and are inferior basis functions), the latter can certainly be approximated well by linear
combinations of the former. The ease of integral evaluation using Gaussian functions makes them the standard
choice for practical calculations. The importance of selecting an appropriate basis set is of great practical
importance in quantum chemistry and many other aspects of atomic and molecular quantum mechanics. An
illustrative example of basis set selection and its effect on calculated energies is given in subsection
(A1.1.4.2). While the problem studied there involves only the motion of a single particle in one dimension, an
analogy with the LCAO and single-centre expansion methods should be apparent, with the desirable features
of the former clearly illustrated.
Even Hartree–Fock calculations are difficult and expensive to apply to large molecules. As a result, further
simplifications are often made. Parts of the Fock operator are ignored or replaced by parameters chosen by
some sort of statistical procedure to account, in an average way, for the known properties of selected

compounds. While calculating properties that have already been measured experimentally is of limited
interest to anyone other than theorists trying to establish the accuracy of a method, the hope of these
approximate Hartree–Fock procedures (which include the well known Hückel approximation and are
collectively known as semiempirical methods) is that the parametrization works just as well for both
unmeasured properties of known molecules (such as transition state structures) and the structure and
properties of transient or unknown species. No further discussion of these approaches is given here (more
details are given in section B3.1 and section B3.2); it should only be emphasized that all of these methods are
based on the independent-particle approximation.
Regardless of how many single-particle wavefunctions χ are available, this number is overwhelmed by the
number of n-particle wavefunctions φ (Slater determinants) that can be constructed from them. For example,
if a two-electron system is treated within the Hartree–Fock approximation using 100 basis functions, both of
the electrons can be assigned to any of the χ obtained in the calculation, resulting in 10,000 two-electron
wavefunctions. For water, which has ten electrons, the number of electronic wavefunctions with equal
numbers of α and β spin electrons that can be constructed from 100 single-particle wavefunctions is roughly
1015! The significance of these other solutions may be hard to grasp. If one is interested solely in the
electronic ground state and its associated potential energy surface (the focus of investigation in most quantum
chemistry studies), these solutions play no role whatsoever within the HF–SCF approximation. Moreover, one
might think (correctly) that solutions obtained by putting an electron in one of the

-36-

unoccupied orbitals offers a poor treatment of excited states since only the occupied orbitals are optimized.
However, there is one very important feature of the extra solutions. If the HF solution has been obtained and
all (an infinite number)