Read an Excerpt
Electronic Structure and the Properties of Solids
The Physics of the Chemical Bond
By Walter A. Harrison Dover Publications, Inc.
Copyright © 1989 Walter A. Harrison
All rights reserved.
ISBN: 978-0-486-14178-7
CHAPTER 1
The Quantum-Mechanical Basis
SUMMARY
This chapter introduces the quantum mechanics required for the analyses in this text. The state of an electron is represented by a wave function ψ. Each observable is represented by an operator O. Quantum theory asserts that the average of many measurements of an observable on electrons in a certain state is given in terms of these by ∫ Ψ*OΨd3r. The quantization of energy follows, as does the determination of states from a Hamiltonian matrix and the perturbative solution. The Pauli principle and the time-dependence of the state are given as separate assertions.
In the one-electron approximation, electron orbitals in atoms may be classified according to angular momentum. Orbitals with zero, one, two, and three units of angular momentum are called s, p, d, and f orbitals, respectively. Electrons in the last unfilled shell of s and p electron orbitals are called valence electrons. The principal periods of the periodic table contain atoms with differing numbers of valence electrons in the same shell, and the properties of the atom depend mainly upon its valence, equal to the number of valence electrons. Transition elements, having different numbers of d orbitals or f orbitals filled, are found between the principal periods.
When atoms are brought together to form molecules, the atomic states become combined (that is, mathematically, they are represented by linear combinations of atomic orbitals, or LCAO's) and their energies are shifted. The combinations of valence atomic orbitals with lowered energy are called bond orbitals, and their occupation by electrons bonds the molecules together. Bond orbitals are symmetric or nonpolar when identical atoms bond but become asymmetric or polar if the atoms are different. Simple calculations of the energy levels are made for a series of nonpolar diatomic molecules.
1-A Quantum Mechanics
For the purpose of our discussion, let us assume that only electrons have important quantum-mechanical behavior. Five assertions about quantum mechanics will enable us to discuss properties of electrons. Along with these assertions, we shall make one or two clarifying remarks and state a few consequences.
Our first assertion is that
(a) Each electron is represented by a wave function, designated as ψ(r). A wave function can have both real and imaginary parts. A parallel statement for light would be that each photon can be represented by an electric field E(r, t). To say that an electron is represented by a wave function means that specification of the wave function gives all the information that can exist for that electron except information about the electron spin, which will be explained later, before assertion (d). In a mathematical sense, representation of each electron in terms of its own wave function is called a one-electron approximation.
(b) Physical observables are represented by linear operators on the wave function. The operators corresponding to the two fundamental observables, position and momentum, are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1-1)
where h is Planck's constant. An analogous representation in the physics of light is of the observable, frequency of light; the operator representing the observable is proportional to the derivative (operating on the electric field) with respect to time, [partial derivative]/[partial derivative][t. The operator r in Eq. (1-1) means simply multiplication (of the wave function) by position r. Operators for other observables can be obtained from Eq. (1-1) by substituting these operators in the classical expressions for other observables. For example, potential energy is represented by a multiplication by V(r). Kinetic energy is represented by (h2/2m)[nabla]2. A particularly important observable is electron energy, which can be represented by a Hamiltonian operator:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(1-2)
The way we use a wave function of an electron and the operator representing an observable is stated in a third assertion:
(c) The average value of measurements of an observable O, for an electron with wave function ψ, is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(1-3)
(If ψ depends on time, then so also will O.) Even though the wave function describes an electron fully, different values can be obtained from a particular measurement of some observable. The average value of many measurements of the observable O for the same ψ is written in Eq. (1-3) as O. The integral in the numerator on the right side of the equation is a special case of a matrix element; in general the wave function appearing to the left of the operator may be different from the wave function to the right of it. In such a case, the Dirac notation for the matrix element is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(1-4)
In a similar way the denominator on the right side of Eq. (1-3) can be shortened to <ψ|ψ>. The angular brackets are also used separately. The bra<1| or ψ1| means ψ1(r)*; the ket |2> or |[ψ2> means ψ2(r). (These terms come from splitting the word "bracket.") When they are combined face to face, as in Eq. (1-4), an integration should be performed.
Eq. (1-3) is the principal assertion of the quantum mechanics needed in this book. Assertions (a) and (b) simply define wave functions and operators, but assertion (c) makes a connection with experiment. It follows from Eq. (1-3), for example, that the probability of finding an electron in a small region of space, d3r, is ψ*(r)ψ (r)d3r. Thus ψ*ψ is the probability density for the electron.
It follows also from Eq. (1-3) that there exist electron states having discrete or definite values for energy (or, states with discrete values for any other observable). This can be proved by construction. Since any measured quantity must be real, Eq. (1-3) suggests that the operator O is Hermitian. We know from mathematics that it is possible to construct eigenstates of any Hermitian operator. However, for the Hamiltonian operator, which is a Hermitian operator, eigenstates are obtained as solutions of a differential equation, the time-independent Schroedinger equation,
Hψ(r) = Eψ(r).
(1-5)
where E is the eigenvalue. It is known also that the existence of boundary conditions (such as the condition that the wave functions vanish outside a given region of space) will restrict the solutions to a discrete set of eigenvalues E, and that these different eigenstates can be taken to be orthogonal to each other. It is important to recognize that eigenstates are wave functions which an electron may or may not have. If an electron has a certain eigenstate, it is said that the corresponding state is occupied by the electron. However, the various states exist whether or not they are occupied.
We see immediately that a measurement of the energy of an electron represented by an eigenstate will always give the value E for that eigenstate, since the average value of the mean-squared deviation from that value is zero:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(1-6)
We have used the eigenvalue equation, Eq. (1-5), to write H|ψ> = E|ψ> The electron energy eigenstates, or energy levels, will be fundamental in many of the discussions in the book. In most cases we shall discuss that state of some entire system which is of minimum energy, that is, the ground state, in which, therefore, each electron is represented by an energy eigenstate corresponding to the lowest available energy level.
In solving problems in this book, we shall not obtain wave functions by solving differential equations such as Eq. (1-5), but shall instead assume that the wave functions that interest us can be written in terms of a small number of known functions. For example, to obtain the wave function ψ for one electron in a diatomic molecule, we can make a linear combination of wave functions ψ1 and ψ2, where 1 and 2 designate energy eigenstates for electrons in the separate atoms that make up the molecule. Thus,
ψ(r) = u1ψ1(r) + u2ψ2(r),
(1-7)
where u1 and u2 are constants. The average energy, or energy expectation value, for such an electron is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(1-8)
The states comprising the set (here, represented by |ψ1> and |ψsub> 2> ) in which the wave function is expanded are called basis states. It is customary to choose the scale of the basis states such that they are normalized; that is, <ψ1| ψ1> = <ψ2|ψ2> = 1. Moreover, we shall assume that the basis states are orthogonal: <ψ1|ψ2> = O. This may in fact not be true, and in Appendix B we carry out a derivation of the energy expectation value while retaining overlaps in <ψ1 |ψ2.>. It will be seen in Appendix B that the corrections can largely be absorbed in the parameters of the theory. In the interests of conceptual simplicity, overlaps are omitted in the main text, though their effect is indicated at the few places where they are of consequence.
We can use the notation Hij = <ψi| H|[ψj> ; then Eq. (1-8) becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(1-9)
(Actually, by Hermiticity, H21 = H*12, but that fact is not needed here.)
Eq. (1-7) describes only an approximate energy eigenstate, since the two terms on the right side are ordinarily not adequate for exact description. However, within this approximation, the best estimate of the lowest energy eigenvalue can be obtained by minimizing the entire expression (which we call E) on the right in Eq. (1-9) with respect to u1 and u2. In particular, setting the partial derivatives of that expression, with respect to u*1 and u*2, equal to zero leads to the two equations
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1-10)
(In taking these partial derivatives we have treated u1, u*1, u2, and u*2 as independent. It can be shown that this is valid, but the proof will not be given here.) Solving Eqs. (1-10) gives two values of E. The lower value is the energy expectation value of the lowest energy state, called the bonding state. It is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(1-11)
An electron in a bonding state has energy lowered by the proximity of the two atoms of a diatomic molecule; the lowered energy helps hold the atoms together in a bond. The second solution to Eqs. (1-10) gives the energy of another state, also in the form of Eq. (1-7) but with different u1 and u2. This second state is called the antibonding state. Its wave function is orthogonal to that of the bonding state; its energy is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(1-12)
We may substitute either of these energies, Eb or Ea, back into Eqs. (1-10) to obtain values for u1 and u2 for each of the two states, and therefore, also the form of the wave function for an electron in either state.
A particularly significant, simple approximation can be made in Eqs. (1-11) or (1-12) when the matrix element H12 is much smaller than the magnitude of the difference |H11 - H22|. Then, Eq. (1-11) or Eq. (1-12) can be expanded in the perturbation H12 (and H21) to obtain
E1 ≈ H11 + H12H21/ H11 - H22,
(1-13)
for the energy of a state near H11; a similar expression may be obtained for an energy near H22. These results are part of perturbation theory. The corresponding result when many terms, rather than only two, are required in the expansion of the wave function is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(1-14)
Similarly, for the state with energy near H11, the coefficient u2 obtained by solving Eq. (1-10) is
u2 = E - H11/H12 u1 ≈ H21u1/H11 - H22.
(1-15)
The last step uses Eq. (1-13). When H21 is small, u2 is small, and the term u2 ψ2(r) in Eq. (1-7) is the correction to the unperturbed state, ψ1(r), obtained by perturbation theory. The wave function can be written to first order in the perturbation, divided by H11 - H22, and generalized to a coupling with many terms as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(1-16)
The perturbation-theoretic expressions for the electron energy, Eq. (1-14), and wave function, Eq. (1-16), will be useful at many places in this text.
All of the discussion to this point has concerned the spatial wave function ψ(r) of an electron. An electron also has spin. For any ψ(r) there are two possible spin states. Thus, assertion (a) set forth earlier should be amended to say that an electron is described by its spatial wave function and its spin state. The term "state" is commonly used to refer to only the spatial wave function, when electron spin is not of interest. It is also frequently used to encompass both wave function and electron spin.
In almost all systems discussed in this book, there will be more than one electron. The individual electron states in the systems and the occupation of those states by electrons will be treated separately. The two aspects cannot be entirely separated because the electrons interact with each other. At various points we shall need to discuss the effects of these interactions.
In discussing electron occupation of states we shall require an additional assertion-the Pauli principle:
(d) Only two electrons can occupy a single spatial state; these electrons must be of opposite spin. Because of the discreteness of the energy eigenstates discussed above, we can use the Pauli principle to specify how states are filled with electrons to attain a system of lowest energy.
(Continues...)
Excerpted from Electronic Structure and the Properties of Solids by Walter A. Harrison. Copyright © 1989 Walter A. Harrison. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
