Slater determination

The so-ealled Slater-Condon rules express the matrix elements of any one-eleetron (F) plus two-eleetron (G) additive operator between pairs of antisymmetrized spin-orbital produets that have been arranged (by permuting spin-orbital ordering) to be in so-ealled maximal eoineidenee. Onee in this order, the matrix elements between two sueh Slater determinants (labelled >and are summarized as follows ... 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

Expanding the Slater determinants and integrating out the spin part and collecting terms that are the same under exchange of electeon indices, we have... 

To this pom t, th e basic approxmi alien is th at th e total wave I lnic-tion IS a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ah miiio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ah mitio calculation. However, there are two main things to be considered m the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222). will require the few esl possible term s for an accurate representation of a molecular orbital. The second one is the speed of tW O-electron integral calculation. 

V i hen the Slater determinant is expanded, a total of N1 terms results. This is because there are N different permutations of N eleefrons. For example, for a three-electron system with spin orbitals X2 and xs the determinant is... 

For any sizeable system the Slater determinant can be tedious to write out, let alone the equivalent full orbital expansion, and so it is common to use a shorthand notation. Various notation systems have been devised. In one system the terms along the diagonal of the matrix are written as a single-row determinant. For the 3x3 determinant we therefore have ... 

VVc can now see why the normalisation factor of the Slater determinantal wavefunction is I v/N . If each determinant contains N terms then the product of two Slater determinants, ldeU rminant][determinant], contains (N ) terms. However, if the spin orbitals form an oi lhonormal set then oidy products of identical terms from the determinant will be nonzero when integrated over all space. We Ccm illustrate this with the three-electron example, k ljiiiidering just the first two terms in the expansion we obtain the following ... 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

In accordance with the variation theorem we require the set of coefficients that gives the lowest-energy wavefunction, and some scheme for changing the coefficients to derive that wavefunction. For a given basis set and a given functional form of the wavefunction (i.e. a Slater determinant) the best set of coefficients is that for which the energy is a minimum, at which point... 

In order to calculate higher-order wavefunctions we need to establish the form of the perturbation, f. This is the difference between the real Hamiltonian and the zeroth-order Hamiltonian, Remember that the Slater determinant description, based on an orbital picture of the molecule, is only an approximation. The true Hamiltonian is equal to the sum of the nuclear attraction terms and electron repulsion terms ... 

Equation (3.74) is the exact exchange energy (obtained from the Slater determinant the Kohn-Sham orbitals), is the exchange energy under the local spin densit) ... 

Manipulating linear eombinations for Li, one soon diseovers that the only one that satisfies the indistinguishability prineiple for eleetrons 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 faetorial series... 

The top row of the Slater determinant shows no preferenee for any spinorbital [Pg.269]

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

The anti symmetrized orbital produet A (l)i(l)2Cl)3 is represented by the short hand (1>1(1>2(1>3 I and is referred to as a Slater determinant. The origin of this notation ean be made elear by noting that (1/VN ) times the determinant of a matrix whose rows are labeled by the index i of the spin-orbital (jii and whose eolumns are labeled by the index j of the eleetron at rj is equal to the above funetion A (l)i(l)2Cl)3 = (1/V3 ) det(( )i (rj)). The general strueture of sueh Slater determinants is illustrated below ... 

We substitute these expressions into the Slater determinants that form the singlet and triplet states and eolleet terms and throw out terms for whieh the determinants vanish. 

To express, in terms of Slater determinants, the wavefunctions corresponding to each of the states in each of the levels, one proceeds as follows ... 

One more quantum number, that relating to the inversion (i) symmetry operator ean be used in atomie eases beeause the total potential energy V is unehanged when all of the eleetrons have their position veetors subjeeted to inversion (i r = -r). This quantum number is straightforward to determine. Beeause eaeh L, S, Ml, Ms, H state diseussed above eonsist of a few (or, in the ease of eonfiguration interaetion several) symmetry adapted eombinations of Slater determinant funetions, the effeet of the inversion operator on sueh a wavefunetion P ean be determined by ... 

In faet, the Slater determinants themselves also are orthonormal funetions of N eleetrons whenever orthonormal spin-orbitals are used to form the determinants. 

The full N terms that arise in the N-eleetron Slater determinants do not have to be treated explieitly, nor do the N (N + l)/2 Hamiltonian matrix elements among the N terms of one Slater determinant and the N terms of the same or another Slater determinant. 

The resultant family of six eleetronie states ean be deseribed in terms of the six eonfiguration state funetions (CSFs) that arise when one oeeupies the pair of bonding a and antibonding a moleeular orbitals with two eleetrons. The CSFs are eombinations of Slater determinants formed to generate proper spin- and spatial symmetry- funetions. 

The spin- and spatial- symmetry adapted N-eleetron funetions referred to as CSFs ean be formed from one or more Slater determinants. For example, to deseribe the singlet CSF eorresponding to the elosed-shell orbital oeeupaney, a single Slater determinant... 

Also, the Ms = 1 eomponent of the triplet state having aa orbital oeeupaney ean be written as a single Slater determinant ... 

However, to deseribe the singlet CSF and Ms = 0 triplet CSF belonging to the aa oeeupaney, two Slater determinants are needed ... 

To simplify the analysis of the above CSFs, the familiar homonuelear ease in whieh z = 1.0 will be examined first. The proeess of substituting the above expressions for a and a into the Slater determinants that define the singlet and triplet CSFs ean be illustrated as follows ... 

---