Rogers Computational Chemistry Using the PC

each other and that each new value of RIJ provides a new solution to the Schroedinger equation. In a simple diatomic molecule, for example, we expect that the locus of energies at different values of R will pass through a minimum at the equilibrium bond length Req.

Returning to the electronic equation, we make the standard orbital assumption that the molecular orbital is a product of single electron orbitals

Each orbital is calculated for one electron moving in an average field of the nuclei and all other electrons. The field that influences electron 1 in helium, for example, is

where UðriÞ is the effective or average potential that electron i experiences from all nuclei I and all electrons j 6¼i. When this Hamiltonian is used in an eigenvalue equation where ei designates an orbital energy, the result is the Hartree equation for electron i

For the helium atom ground state, which we shall later generalize to many electron atoms and molecules,

ð

¼ cð Þcð Þ ^ cð Þ cð Þ t ð Þ

EHe r1 r2 H r1 r2 d 9-11

for normalized wave functions. The integral over all space dt in Eq. (9-11) is a double integral over dr1 and dr2, and the Hamiltonian is given by Eq. (9-2), which assumes the Born–Oppenheimer approximation with Z ¼ 2. The five terms in Eq. (9-2) yield three integrals

The first two integrals are simplified by the fact that orthonormal functions yield

Under this simplification, the three integrals (9-12a, b, and c) are called I1, I2, and J12. Now,

Using more flexible trial functions (polynomials in r1, r2, and r12 perhaps) for cðriÞ, one can calculate very accurate energies for He by the SCF method. This gives us confidence that we can make a valid generalization to larger systems.

Exchange Symmetry

Strangely enough, the universe appears to be comprised of only two kinds of particles, bosons and fermions. Bosons are symmetrical under exchange, and fermions are antisymmetrical under exchange. This bit of abstract physics relates to our quantum molecular problems because electrons are fermions.

By Max Born’s postulate, the product of cðxÞ and its complex conjugate c ðxÞ times an infinitesimal volume element d3x is proportional to the probability that a particle will be in the volume element d3x

A two-particle wave function cðx1; x2Þ

gives the probability that two particles will occupy volume elements d3x1 and d3x2 simultaneously.

For the probable location of two particles to be identical under the exchange operation, x1x2 ! x2x1, the wave function before exchange must be exactly equal to the wave function after exchange times a phase factor eiy

Equality between the 1, 2 wave function and the modulus of the 2, 1 wave function, jcðx2; x1Þj, shows that they have the same curve shape in space after exchange as they did before, which is necessary if their probable locations are to be the same. The phase factor orients one wave function relative to the other in the complex plane, but Eq. (9-17) is simplified by one more condition that is always true for particle exchange. When exchange is carried out twice on the same particle pair, the operation must produce the original configuration of particles

There are only two ways this can be true. Either the phase factor is 1 or it is 1, that is,

The first allowable equation holds for bosons, and the second holds for fermions. If we form a linear combination of wave functions for bosons,

everything is OK because, taking into account particle indistinguishability, this is the same as

but if we do the same thing with fermions,

and we have lost the orbital. Wave functions can be identical for bosons but not for fermions. This is the origin of the statement that no two fermions can occupy the same orbital. It is a generalization of Pauli’s original exclusion principle for electrons, which are fermions.

Electron Spin

All of our orbitals have disappeared. How do we escape this terrible dilemma? We insist that no two electrons may have the same wave function. In the case of electrons in spatially different orbitals, say, 1s and 2s orbitals, there is no problem, but for the two electrons in the 1s orbital of the helium atom, the space orbital is the same for both. Here we must recognize an extra dimension of relativistic space-time

that adds a new quantum number s ¼ 12 for what is commonly called the electron spin.

If two electrons occupy the same space orbital but have different spins, we can write

indicating that the 1s parts of the orbital cð1; 2Þ are the same but the spins are different, a for one-electron and b for the other. We can also write

for the same configuration with the spins reversed.

These equations are legitimate spinorbitals, but neither is acceptable because they both imply that we can somehow ‘‘label’’ electrons, a for one and b for the other. This violates the principle of indistinguishability, but there is an easy way out of the problem; we simply write the orbitals as linear combinations

Neither of these equations tells us which spin is on which electron. They merely say that there are two spins and the probability that the 1, 2 spin combination is a, b is equal to the probability that the 2, 1 spin combination is a, b. The two linear combinations cð1; 2Þ cð2; 1Þ are perfectly legitimate wave functions (sums and differences of solutions of linear differential equations with constant coefficients are also solutions), but neither implies that we know which electron has the ‘‘label’’ a or b.

Now that we have selected two wave functions that do not violate the principle of indistinguishability, let us look at their exchange properties. The linear combinations are

which is acceptable for fermions but not for bosons. Because electrons are fermions, we are driven to the conclusion that the linear combination (9-24b) is the only combination that properly describes the ground-state 1s orbital of helium. This is true of higher atomic and molecular orbitals as well.

Slater Determinants

While idly dreaming over these equations (theoreticians call it ‘‘working’’) we might happen to notice that the linear combination (9-24b) we have selected for ground-state helium is the same as the expansion of a 2 2 determinant

which, with slight notational changes, is Eq. (8-52). Might it be a general principle that legitimate wave functions which obey the Pauli principle for electrons are expanded determinants?

Consider lithium. We know from a century of empirical chemistry that one electron in the active metal Li is very different from those in inert helium He. Reasoning from the pattern of hydrogen energy levels developed by Bohr, we do the reasonable thing and put one electron in the high-energy 2s orbital of Li and leave the other two electrons in the 1s orbital. This agrees with the empirical evidence that ground-state Li, like all alkali metals, loses one electron easily, but only one, to go to the single positive ion Liþ.

Manipulating linear combinations for Li, one soon discovers that the only one that satisfies the indistinguishability principle for electrons is the expansion of a

Slater determinant

Moreover, there are 2 terms in the expansion of the Slater determinant for He but there are 6 terms for Li. Looking at beryllium, we find 24 terms. This is the beginning of the factorial series

1!; 2!; 3!; 4!; . . . ¼ 1; 2; 6; 24; . . .

When we square the wave function, we expect to find a probability P ¼ 1 over all

space, so the n! terms in the expanded determinant must be multiplied by the factor p

1= n! to obtain the determinantal wave function normalized to 1.

In summary:

1.The top row of the Slater determinant shows no preference for any spinorbital fi over any other; the electron may be in any one of them with equal

270 COMPUTATIONAL CHEMISTRY USING THE PC

probability. The same is true for electron 2 as shown in the second row, and so on. The Slater determinant

‘‘mixes’’ probabilities for all electrons in all orbitals equally to find the molecular orbital cð1; 2; . . . ; nÞ. The radius vector from the nucleus to the electron ri in Eq. (9-29) will usually be represented by its scalar magnitude ri.

2.The Slater determinant changes sign on exchange of any two rows (electrons), so it satisfies the principle of antisymmetrical fermion exchange.

3.In short, the Slater determinantal molecular orbital and only the Slater determinantal molecular orbital satisfies the two great generalizations of quantum chemistry, uncertainty (indistinguishability) and fermion exchange antisymmetry.

4.We shall assume antisymmetrized orbitals from this point on when we write cð1; 2; . . . ; nÞ.

Exercise 9-1

Show that the atomic determinantal wave function

is normalized if the 1s orbitals are normalized.

Solution 9-1

In the notation of Eq. (9-29), f1ðr1Þ ¼ 1 s. If the 1s orbitals are normalized, then the

spinorbitals 1 sað1Þ, etc. are normalized because a and bpare normalized. If we take just

the expanded determinant for two electrons without 1= 2, the normalization constant, and (omitting complex conjugate notation for the moment) integrate over all space

ð ð

cð1; 2Þcð1; 2Þdx1 dx2

ð ð

½1 sað1Þ1 sbð2Þ 1 sbð1Þ1 sað2Þ&½1 sað1Þ1 sbð2Þ 1 sbð1Þ1 sað2Þ&dx1 dx2

we get a sum of four integrals

ð ð ð ð

1 sað1Þ1 sbð2Þ1 sað1Þ1 sbð2Þdx1 dx2 1 sbð1Þ1 sað2Þ1 sað1Þ1 sbð2Þdx1 dx2

ð ð ð ð

1 sað1Þ1 sbð2Þ1 sbð1Þ1 sað2Þdx1 dx2 þ 1 sbð1Þ1 sað2Þ1 sbð1Þ1 sað2Þdx1 dx2

Exercise 9-3

Generalize the solution of Exercise 9-1 to the case of a many-electron wave function [Eq. (9-29)] yielding Pm permutations.

Solution 9-3

Each of the products 1 sað1Þ1 sbð2Þ and 1 sbð1Þ1 sað2Þ is a different permutation of electrons over orbitals, call them P1 and P2, respectively. Resuming conventional complex conjugate notation, call the permutations over the complex conjugate c ð1; 2Þ Q1 and Q2, respectively. The integration Ð Ð c ð1; 2Þcð1; 2Þdx1dx2 can be written

ð ð

½Q1 Q2&½P1 P2&dt

where dt designates integration over all space available to the system. This integral produces a sum

ð ð ð ð ð ð ð ð

Q1P1 dt Q2P1 dt Q1P2 dt þ Q2P2 dt

After regrouping, we found that the first and fourth integrals with P and Q the same

In the general case, expanding determinant (9-29) gives

ðð

X

cð1; 2; . . . ; nÞ ¼ cP f1ðP1Þf2ðP2Þ . . . fMðPMÞdt

where M ¼ n! and cP is the number of exchanges necessary to go from P1 to some other permutation. For example, for permuting books, RGB ! GRB ! GBR involves two exchanges. The number of exchanges determines the sign of each term in the sum: the term is positive if cP is even and negative if it is odd.

The complex conjugate of cð1; 2; . . . ; nÞ can be written

ðð

X

c ð1; 2; . . . ; nÞ ¼ cQ f1ðQ1Þf2ðQ2Þ . . . fMðQMÞdt

whereupon the multiple integral

ðð

c ð1; 2; . . . ; nÞcð1; 2; . . . ; nÞdt

produces a sum of possibly very many terms

ðð

X X

cP cQ ½f1ðQ1Þf2ðQ2Þ . . . fM ðQMÞ&½f1ðP1Þf2ðP2Þ . . . fM ðPMÞ&dt

Those terms with identical P and Q integrate to 1 and those with nonidentical P and Q integrate to 0. There are n! nonzero integrals yielding a sum of n! terms. The normal-

ð1= n!Þð1= n!Þn! ¼ 1.

The Hartree--Fock Equation

In later work, both Hartree and, independently, Fock (1930) used antisymmetrized orbitals in what we now know as the Hartree–Fock equation, an extension of the Hartree equation. When we treat many-electron atoms or molecules by the variational method using antisymmetrized orbitals, a new term, Kij, appears in the energy equation, Eq. (9-14)

due to the possibility of exchanging electrons between different spinorbitals fi

and fj.

Picking the helium atom again as our prototypical system (Atkins and Friedman, 1997),

in the excited state where I is the one-electron orbital energy and J is the Coulomb energy. The new integral K12 is similar to J12 in Eqs. (9-12c) and (9-14) except that the electrons have been exchanged ðr1 ! r2 and r2 ! r1Þ on the right of integral (9-30). The integral Kij is the exchange integral corresponding to the linear combination

for He in the excited state, Heexc. It appears on antisymmetrization of the atomic orbital; therefore, it is a quantum mechanical term, as distinct from I and J, which are classical. The sign shows that the exchange integral can enhance or diminish stability. In helium, the magnitude of K is about 101 that of the excitation energy from the 1s orbital to the 2s orbital (Levine, 1991) (Fig. 9-2).

To see how and under what conditions stability is enhanced or diminished, we need to consider the symmetry of the orbital (9-32). Electrons in the antisymmetric orbital c have a zero probability of occurring at the node in c where r1 ¼ r2. Electron mutual avoidance of the node due to spin correlation reduces the total energy of the system because it reduces electron repulsion energy due to charge

correlation. Do not confuse the two correlations; spin correlation is a fundamental characteristic of bosons and fermions. Chemists call it the Pauli exclusion principle. It operates independently of charge correlation, but it influences the potential energy of charge correlation according to whether electrons are close together or far apart due to their spin. When the electrons in Heexc avoid each other due to spin correlation, the energy is decreased

When electrons attract each other, as in the symmetric spin case, the total energy is increased due to increased charge repulsion

assuming S ¼ I. Therefore, by the method given in the section on the secular matrix in Chapter 6, the elements of the determinant must be

When an antisymmetrized orbital is used in place of a single orbital for manyelectron systems, the energy of the ground state is ‘‘better,’’ that is, lower, than the Hartree energy by the exchange energy. This is consistent with (but does not prove) the qualitative idea that replacing an orbital with a new orbital that better represents the physics of the system lowers its calculated energy. In this case, the antisymmetrized orbital better represents the true orbital (even though we do not know exactly what that is) because the Pauli principle is a valid part of the physics of electron interaction.

For the record, we should point out that the equations developed in this chapter are extensions of the nonrelativistic, time-independent Schroedinger equation. The Pauli principle arises from a relativistic treatment of the problem, but we shall follow the custom of most chemists and accept it as a postulate, ‘‘proven’’ because it gives the right answers.