Matrix element between determinants

One particular advantage of Slater determinants constructed from orthonormsd spin-orbitals is that matrix elements between determinants over operators such as H sure very simple. Only three distinct cases arise, as is well known and treated elsewhere. It is perhaps not surprising that the simplest matrix element formulas should be obtained from the treatment that exploits symmetry the least, as only the fermion antisymmetry has been accounted for in the determinants. As more symmetry is introduced, the formulas become more complicated. On the other hand, the symmetry reduces the dimension of the problem more and more, because selection rules eliminate more terms. We consider here the spin adaptation of Slater determinants. 

The averaging of SCF energy expressions to impose symmetry and equivalence restrictions is a straightforward, if sometimes tedious, application of the Slater-Condon rules for matrix elements between determinants of orthonormal orbitals. This matter is discussed in detail elsewhere. The most general SCF programs can handle energy expressions of the form... 

In this section we will briefly outline the different steps in a Cl calculation. Before we do this we first need to recapitulate the main results for the Hamiltonian matrix elements between determinants. In this course it is assumed that the students have already seen a detailed derivation of the matrix elements and we will therefore only sketch this derivation here. 

With the further condition that the spin-orbitals are orthogonal the special cases, Slater s rules, for matrix elements between determinants are obtained from this formula by inspection. The general formula can be written... 

Exercise 6.11 Neglecting (352 terms, the matrix elements between determinants are as follows ... 

For the reader who is encountering second quantization for the first time, one can observe that application of the definition of Eq. (4) for CSF built from Slater determinants in the context of (Eq. 6) just reduces to the usual rules for matrix elements between determinants if K and L are determinants however the definitions of Eqs. (6) and (7) remain true irrespective of the nature of the CSF (and become very powerful when the CSF are chosen as spin eigenfunctions). 

Slater, J. C., Quantum Theory of Matter, 2nd ed., McGraw-Hill, New York, 1968. Chapter 11 discusses determinantal wave functions and derives expressions for matrix elements between determinants in a somewhat different way than we have done. 

The mixing of the two states depends on the off-diagonal element This matrix element is obtained by using the rules for evaluating matrix elements between determinants, and the result can be read directly from Tables 2.5 and 2.6. 

A VB calculation is just a configuration interaction in a space of AO or FO determinants, which are in general nonorthogonal to each other. It is therefore essential to derive some basic rules for calculating the overlaps and Hamiltonian matrix elements between determinants. The fully general rules have been described in detail elsewhere. Examples will be given here for commonly encovmtered simple cases. 

The matrix elements between the CSFs in the real basis involve two Hamiltonian matrix elements between determinants, Hpq and HpQ. However, HpQ is only nonzero for Mk P)-Mk Q) < 2, because otherwise the excitation between P and Q is more than a two-particle excitation. Thus it is only in the center blocks of the Hamiltonian that the linear combination of matrix elements needs to be taken. 

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 ... 

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 ... 

The matrix element between the HF and a singly excited determinant is a matrix element of the Fock operator between two different MOs (eq. (3.36)). 

The disappearance of matrix elements between the HF reference and singly excited states is known as Brillouins theorem. The HF reference state therefore only has nonzero matrix elements with doubly excited determinants, and the full Cl matrix acquires a block diagonal structure. 

The matrix elements between the HF and a doubly excited state are given by two-electron integrals over MOs (eq. (4.7)). The difference in total energy between two Slater determinants becomes a difference in MO energies (essentially Koopmans theorem), and the explicit formula for the second-order Mpller-Plesset correction is... 

In order to solve equation (4), the following matrix elements between Slater Determinants have to be considered ... 

Suppose a r-electron operator to be written as S2(r), with the r-dimensional vector r representing the coordinates of the canonically ordered r (matrix element between two Slater determinants can te written as ... 

The matrix elements between the five determinants can be easily evaluated 12) and are given by ... 

The Fock-space Hamiltonian H is equivalent to the configuration-space Hamiltonian H insofar as both have the same matrix elements between n-electron Slater determinants. The main difference is that H has eigenstates of arbitrary particle number n it is, in a way, the direct sum of aU // . Another difference, of course, is that H is defined independently of a basis and hence does not depend on the dimension of the latter. One can also define a basis-independent Fock-space Hamiltonian H, in terms of field operators [11], but this is not very convenient for our purposes. 

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