Slater Matrix Element Rules

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 ... 

One Must be Able to Evaluate the Matrix Elements Among Properly Symmetry Adapted N-Electron Configuration Eunctions for Any Operator, the Electronic Hamiltonian in Particular. The Slater-Condon Rules Provide this Capability... 

II. The Slater-Condon Rules Give Expressions for the Operator Matrix Elements Among the CSFs... 

To form the Hk,l matrix, one uses the so-ealled Slater-Condon rules whieh express all non-vanishing determinental matrix elements involving either one- or two- eleetron operators (one-eleetron operators are additive and appear as... 

The Slater-Condon rules give the matrix elements between two determinants... 

As a first step in applying these rules, one must examine > and > and determine by how many (if any) spin-orbitals > and > differ. In so doing, one may have to reorder the spin-orbitals in one of the determinants to aehieve maximal eoineidenee with those in the other determinant it is essential to keep traek of the number of permutations ( Np) that one makes in aehieving maximal eoineidenee. The results of the Slater-Condon rules given below are then multiplied by (-l) p to obtain the matrix elements between the original > and >. The final result does not depend on whether one ehooses to permute ... 

Onee maximal eoineidenee has been aehieved, the Slater-Condon (SC) rules provide the following preseriptions for evaluating the matrix elements of any operator F + G eontaining a one-eleetron part F = Zi f(i) and a two-eleetron part G = Zij g(i,j) (the Hamiltonian is, of eourse, a speeifie example of sueh an operator the eleetrie dipole... 

In this form, it is elear that E is a quadratie funetion of the Cl amplitudes Cj it is a quartie funetional of the spin-orbitals beeause the Slater-Condon rules express eaeh <

Cl matrix element in terms of one- and two-eleetron integrals < > and... 

The Slater-Condon rules allow one to express the Hamiltonian matrix elements appearing here as... 

The amplitude for the so-ealled referenee CSF used in the SCF proeess is taken as unity and the other CSFs amplitudes are determined, relative to this one, by Rayleigh-Sehrodinger perturbation theory using the full N-eleetron Hamiltonian minus the sum of Foek operators H-H as the perturbation. The Slater-Condon rules are used for evaluating matrix elements of (H-H ) among these CSFs. The essential features of the MPPT/MBPT approaeh are deseribed in the following artieles J. A. Pople, R. Krishnan, H. B. Sehlegel, and J. S. Binkley, Int. J. Quantum Chem. 14, 545 (1978) R. J. Bartlett and D. M. Silver, J. Chem. Phys. 3258 (1975) R. Krishnan and J. A. Pople, Int. J. Quantum Chem. 

As a result, the exaet CC equations are quartic equations for the ti , ti gte. amplitudes. Although it is a rather formidable task to evaluate all of the eommutator matrix elements appearing in the above CC equations, it ean be and has been done (the referenees given above to Purvis and Bartlett are espeeially relevant in this eontext). The result is to express eaeh sueh matrix element, via the Slater-Condon rules, in terms of one- and two-eleetron integrals over the spin-orbitals used in determining , ineluding those in itself and the Virtual orbitals not in . 

The benzene molecule can now be treated very simply by the Slater method, with the help of the rules formulated by one of us4 for finding the matrix elements occurring in the secular equation. The bonds between the six eigenfunctions can be drawn so as to give the independent canonical structures shown in Fig. 1. Any other... 

Since Doo and are constructed with the same set of orthonormal spinorbitals, the two first matrix elements can easily rewritten, according to the Slater s rules [13], as ... 

The determinential wave functions shown in equations (42)-(44) have the correct normalization for many-electron Sturmians (i.e. the normalization required by equation (6)). To see this, we can make use of the Slater-Condon rules, which hold for the diagonal matrix elements of... 

To determine the image, the first step is to determine the distribution of tunneling current as a function of the position of the apex atom. We set the center of the coordinate system at the nucleus of the sample atom. The tunneling matrix element as a function of the position r of the nucleus of the apex atom can be evaluated by applying the derivative rule to the Slater wavefunctions. The tunneling conductance as a function of r, g(r), is proportional to the square of the tunneling matrix element ... 

Using the Slater-Conden rules to obtain the matrix elements between configurations we get f <1o2IzI2o2> <1o2IzI1o2o> 1... 

Matrix elements of V among determinental wavefunctions constructed from the SCF spin-orbitals <i I V I k> can be expressed, using the Slater-Condon rules, in terms of matrix elements over the full Hamiltonian H... 

To solve Eq. (7.11), we need to know how to evaluate matrix elements of the type defined by Eq. (7.12). To simplify matters, we may note that the Hamiltonian operator is composed only of one- and two-electron operators. Thus, if two CSFs differ in their occupied orbitals by 3 or more orbitals, every possible integral over electronic coordinates hiding in the r.h.s. of Eq. (7.12) will include a simple overlap between at least one pair of different, and hence orthogonal, HF orbitals, and the matrix element will necessarily be zero. For the remaining cases of CSFs differing by two, one, and zero orbitals, the so-called Condon-Slater rules, which can be found in most quantum chemistry textbooks, detail how to evaluate Eq. (7.12) in terms of integrals over the one- and two-electron operators in the Hamiltonian and the HF MOs. 

A somewhat special case is the matrix element between the HF determinant and a singly excited CSF. The Condon-Slater rules applied to this situation dictate that... 

Figure 7.4 Structure of the CI matrix as blocked by classes of determinants. The HF block is the (1,1) position, the matrix elements between the HF and singly excited determinants are zero by Brillouin s theorem, and between the HF and triply excited determinants are zero by the Condon-Slater rules. In a system of reasonable size, remaining regions of the matrix become increasingly sparse, but the number of determinants in each block grows to be extremely large. Thus, the (1,1) eigenvalue is most affected by the doubles, then by the singles, then by the triples, etc...

The matrix elements r o are quite straightforward to evaluate. Before leaving them, however, it is worthwhile to make some qualitative observations about them. First, the Condon-Slater rules dictate that for the one-electron operator r, the only matrix elements that survive are those between determinants differing by at most two electronic orbitals. Thus, only absorptions generating singly or doubly excited states are allowed. 

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