Condon-Slater rules

The Slater determinants represent something like the daily bread of quantum chemists. Our goal is to learn how to use the Slater determinants when they are involved in the calculation of the mean values or the matrix elements of some important operators. We will need this in the Hartree-Fock method, as well as in other important methods of quantum chemistry. 

Only the final results of the derivations presented in this Appendix are the most important. 

The operator A has some nice features. The most important is that, when applied to any function, it produces either a function that is antisymmetric with respect to the permutations of N elements, or zero. This means that A represents a sort of magic wand whatever it touches it makes antisymmetric or causes it disappear The antisymmetrizer is also idempotent, i.e. does not change any function that is already antisymmetric, which means A = A. 

Let us check that A is indeed idempotent. First we obtain  

Of course PP represents a permutation opera tor, which is then multiplied by its own parity - )P+P and there is a sum over such permutations at a given fixed P.  

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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 ... 

Calculate the energy (using Slater Condon rules) associated with the 2p valence electrons for the following states of the C atom. 

The notation used for the Slater Condon rules will be the same as used in the text (a.) zero (spin orbital) differenee ... 

These so-called interaction perturbations Hint are what induces transitions among the various electronic/vibrational/rotational states of a molecule. The one-electron additive nature of Hint plays an important role in determining the kind of transitions that Hint can induce. For example, it causes the most intense electronic transitions to involve excitation of a single electron from one orbital to another (recall the Slater-Condon rules). 

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... 

These density matriees are themselves quadratie funetions of the CI eoeffieients and they refleet all of the permutational symmetry of the determinental funetions used in eonstrueting F they are a eompaet representation of all of the Slater-Condon rules as applied to the partieular CSFs whieh appear in F. They eontain all information about the spin-orbital oeeupaney of the CSFs in F. The one- and two- eleetron integrals < I f I > and < (l)i(l)j I g I (l)ic(l)i > eontain all of the information about the magnitudes of the kinetie and Coulombie interaetion energies. 

For the N-eleetron speeies whose Hartree-Foek orbitals and orbital energies have been determined, the total SCF eleetronie energy ean be written, by using the Slater-Condon rules, as ... 

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. 

P I H I P >. It ean be shown (H. P. Kelly, Phys. Rev. 131, 684 (1963)) that this differenee allows non-variational teehniques to yield size-extensive energies. This ean be seen in the MPPT/MBPT ease by eonsidering the energy of two non-interaeting Be atoms. The referenee CSF is = Isa 2sa Isb 2sb the Slater-Condon rules limit the CSFs in P whieh ean eontribute to... 

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 Slater-Condon rules (Section 4.2.1) are incorporated in the operators. 

A different—and arguably better—approach is based on the realization that every A-representable g-density can be associated with an ensemble average of Slater determinants. (Recall Section III.E, and especially the discussion surrounding Eq. (54). For a g-density built from Slater determinants, it follows from the Slater-Condon rules that the one-electron contributions to the energy can be written... 

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... 

Again making use of the Slater-Condon rules, we find that... 

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