164 Computational Chemistry

If switching the coordinates of two of the particles leaves the function unchanged, it is said to be symmetric with respect to particle exchange, while if the function changes sign it is said to be antisymmetric with respect to particle exchange. Comparing the predictions of theory with the results of experiment shows [5] that electronic wavefunctions are actually antisymmetric with respect to exchange (such particles are called fermions, after the physicist Enrico Fermi; particles like photons whose wavefunctions are exchange-symmetric are called bosons, after the physicist S. Bose). Any rigorous attempt to approximate the wavefunction should use an antisymmetric function of the coordinates of the electrons 1 , 2 , . . . . n, but the Hartree product is symmetric rather than antisymmetric; e.g. if we approximate a helium atom wavefunction as the product of two hydrogen atom 1s orbitals, then if and then

These defects of the Hartree SCF method were corrected by Fock (section 4.3.4) and by Slater2 in 1930 [6], and Slater devised a simple way to construct a total wavefunction from one-electron functions (i.e. orbitals) such that will be antisymmetric to electron switching. Hartree’s iterative, average-field approach supplemented with electron

spin and antisymmetry leads to the HF equations.

5.2.3The HF equations

5.2.3.1Slater determinants

The Hartree wavefunction (above) is a product of one-electron functions called orbitals, or, more precisely, spatial orbitals: these are functions of the usual space coordinates x, y, z. The Slater wavefunction is composed, not just of spatial orbitals, but of spin orbitals. A spin orbital (spin) is the product of a spatial orbital and a spin function,

operator and eigenvalues, say the energy operator and energy eigenvalues, according

everywhere except at one value of its variable, where it spikes sharply, is a delta function

2John Slater, born Oak Park Illinois, 1900. Ph.D. Harvard, 1923. Professor of physics, Harvard, 1924– 1930; MIT 1930–1966; University of Florida at Gainesville, 1966–1976. Author of 14 textbooks, contributed to solid-state physics and quantum chemistry, developed X-alpha method (early density functional theory method). Died Sanibel Island, Florida, 1976.

Ab initio calculations 165

(invented by Dirac – section 4.2.3). Since the spin function (spin or ) describing

called an electron, and said to have up spin, and an electron with spin is called aelectron, and said to have down spin. Up and down electrons are often denoted by arrows and respectively. A nice, brief treatment of the delta function and of the mathematical treatment of the spin functions is given by Levine [7].

The Slater wavefunction differs from the Hartree function not only in being composed of spin orbitals rather than just spatial orbitals, but also in the fact that it is not a simple product of one-electron functions, but rather a determinant (section 4.3.3) whose elements are these functions. To construct a Slater wavefunction (Slater determinant) for a closed-shell species (the only kind we consider in any detail here), we use each of the occupied spatial orbitals to make two spin orbitals, by multiplying the spatial orbital by and, separately, by The spin orbitals are then filled with the available electrons. An example should make the procedure clear (Fig. 5.2). Suppose we wish to write a Slater determinant for a four-electron closed-shell system. We need two spatial molecular orbitals, since each can hold a maximum of two electrons; each spatial orbital

(spatial) is used to make two spin orbitals, (spatial)and(spatial)(alternatively, each spatial orbital could be thought of as a composite of two spin orbitals, which we are separating and using to build the determinant). Along the first (top) row of a determinant we write successively the firstspin orbital, the first spin orbital, the second spin orbital, and the second spin orbital, using up our occupied spatial (and thus spin) orbitals. Electron one is then assigned to all four spin orbitals of the first row – in a sense it is allowed to roam among these four spin orbitals [8]. The second row of

166 Computational Chemistry

the determinant is the same as the first, except that it refers to electron two rather than electron one; likewise the third and fourth rows refer to electrons three and four, respectively. The result is the determinant of Eq. (5.10).

(The factor ensures that the wavefunction is normalized, i.e. that integrated over all space= 1 [9].) This Slater determinant ensures that there are no more than two electrons in each spatial orbital, since for each spatial orbital there are only two one-electron spin functions, and it ensures that is antisymmetric since switching two electrons amounts to exchanging two rows of the determinant, and this changes its sign (section 4.3.3). Note that instead of assigning the electrons successively to row 1, row 2, etc., we could have placed them in column 1, column 2, of Eq. (5.11) = of Eq. (5.10). Some authors use the row format for the electrons, others the column format.

Slater determinants enforce the Pauli exclusion principle, which forbids any two electrons in a system to have all quantum numbers the same. This is readily seen for an atom: if the three quantum numbers n, l and of (x, y, z) (section 4.2.6) and the spin quantumnumber of or were all the same for any electron, two rows (or columns, in the alternative formulation) would be identical and the determinant, hence the wavefunction, would vanish (section 4.3.3).

For 2n electrons (we are limiting ourselves for now to even-electron species, as the theory for these is simpler) the general form of a Slater determinant is clearly the 2n × 2n determinant

The Slater determinant for the total wavefunction of a 2n-electron atom or molecule is a 2n × 2n determinant with 2n rows due to the 2n electrons and 2n columns due to the 2n spin orbitals (you can interchange the row/column format); since these are closed-shell species, the number of spatial orbitals is half the number of electrons. We use the lowest n occupied spatial orbitals (the lowest 2n spin orbitals) to make the determinant.

Ab initio calculations 167

The determinant (= total molecular wavefunction ) just described will lead to (remainder of section 5.2) n occupied, and a number of unoccupied, component spatial molecular orbitals These orbitals from the straightforward Slater determinant are called canonical (in mathematics the word means “in simplest or standard form”) molecular orbitals. Since each occupied spatial can be thought of as a region of space which accommodates a pair of electrons, we might expect that when the shapes of these orbitals are displayed (“visualized”; section 5.5.6) each one would look like a bond or a lone pair. However, this is often not the case; e.g. we do not find that one of the canonical MOs of water connects the O with one H, and another canonical MO connects the O with another H. Instead most of these MOs are spread over much of a molecule – delocalized (lone pairs, unlike conventional bonds, do tend to stand out). However, it is possible to combine the canonical MOs to get localized MOs which look like our conventional bonds and lone pairs. This is done by using the columns (or rows) of the Slater to create a with modified columns (or rows): if a column/row of a determinant is multiplied by k and added to another column/row, the determinant is unchanged (section 4.3.3). We see that if this is applied to the Slater determinant, we will get a “new” determinant corresponding to exactly the same total wavefunction, i.e. to the same molecule, but built up from different component occupied MOs The newand the new are no less or more correct than the previous ones, but by appropriate manipulation of the columns/rows the can be made to correspond to our ideas of bonds and lone pairs. These localized MOs are sometimes useful.

5.2.3.2Calculating the atomic or molecular energy

The next step in deriving the HF equations is to express the energy of the molecule or atom in terms of the total wavefunction the energy will then be minimized with respect to each of the component molecular (or atomic; an atom is a special case of a molecule) spin orbitals and (cf. the minimization of energy with respect to basis function coefficients in section 4.3.3). The derivation of these equations involves considerable algebraic manipulation, which is at times hard to follow without actually writing out the intermediate expressions. The procedure has been summarized by Pople and Beveridge [10], and a less condensed account is given by Lowe [11].

It follows from the Schrödinger equation that the energy of a system is given by

This is similar to Eq. (4.40) in chapter 4, but here the total wavefunction has been specified, and allowance has been made for the possibility of being a complex function by utilizing its complex conjugate this ensures that E, the energy of the atom or molecule, will be real. If is complex then will not be a real number, while will, as must be the case for a probability. Integration is with respect to three spatial coordinates and one spin coordinate, for each electron. This is symbolized by which means dx dy dz d so for a 2n-electron system these integrals are actually 4 × 2n-fold, each electron having its set of four coordinates. We assume the use of orthonormal functions (section 4.3.4), since this makes several integrals disappear in the derivation of the energy. Working with the

Ab initio calculations 169

The operator is so called because it leads to the electronic energy of a single electron moving simply under the attraction of a nuclear “core”, with all the other electrons stripped away; is the electronic energy of, for example, H, or (of course, it is different for these various species). Note that (1) represents the kinetic energy of electron 1 plus the potential energy of attraction of that electron to each of the nuclei the 1 in parentheses in these equations is just a label showing that the same electron is being considered in and (we could have used, say, 2 instead). The integration in Eq. (5.18) is respect to spatial coordinates only, not because spin coordinates have been “integrated out”: on integration, i.e. summation over the discrete spin variable, these give 0 or 1 [7,10,12]. We are left with the three integration variables (x, y, z) and so the integral is threefold.

J is called a coulomb integral; it represents the electrostatic (i.e. coulombic) repulsion between an electron in and one in i.e. between the charge clouds of orbitals and This may be clearer if one considers the integral as a sum of potential energy terms involving repulsion between infinitesimal volumeelements (Fig. 5.3). The 1 and 2 are just labels showing we are considering two electrons. The integrals J and

K allow each electron to experience the average electrostatic repulsion of a charge cloud due to all the other electrons. Since J represents potential energy corresponding to a destabilizing electrostatic repulsion, it is positive. As for in Eq. (5.18), the integration is with respect to spatial coordinates because the spin coordinates have been integrated out. There are six integration variables, x, y, z for electron and x, y, z for electron and so the integral is sixfold. Note that the ab initio coulomb

170 Computational Chemistry

integral J is not the same as what we called a coulomb integral in simple Hückel theory; that was (Eq. (4.61)) and represents at least crudely the energy of an electron in the orbital (section 4.3.4). The ab initio coulomb integral can also be written

but unlike (5.20) this does not notationally emphasize the repulsion (invoked by the operator) between electrons 1 and 2 (on the left and right, respectively, of

in Eq. (5.20)).

K is called an exchange integral; mathematically, it arises from Slater determinant expansion terms that differ only in exchange of electrons. Note that the terms on either side of differ by exchange of electrons. It is often said to have no simple physical interpretation, but looking at Eq. (5.17), we can regard K as a kind of correction to J, reducing the effect of J (both J and K are positive, with K smaller), i.e. reducing the electrostatic potential energy due to the mutual charge cloud repulsion referred to in connection with J. This reduction in repulsion arises because as particles with an antisymmetric wavefunction, two electrons cannot occupy the same spin orbital (roughly, cannot be at the same point in space), and can occupy the same spatial orbital only if they have opposite spins, so two electrons of the same spin avoid each other more assiduously than expected only from the coulombic repulsion taken into account by J. We could consider the summed 2J–K terms of Eq. (5.17) to be the true coulombic repulsion, corrected for electron spin, i.e. corrected for the Pauli exclusion principle effect. The J and K interactions are shown in Fig. 5.4 for a four-electron molecule,