Cl Matrix Elements

If the system contains symmetry, there are additional Cl matrix elements which become zero. The symmetry of a determinant is given as the direct product of the symmetries of the MOs. The Hamilton operator always belongs to the totally symmetric representation, thus if two determinants belong to different irreducible representations, the Cl matrix element is zero. This is again fairly obvious if the interest is in a state of a specific symmetry, only those determinants which have the correct symmetry can contribute. 

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)). 

This is an occupied-virtual off-diagonal element of the Fock matrix in the MO basis, and is identical to the gradient of the energy with respect to an occupied-virtual mixing parameter (except for a factor of 4), see eq. (3.67). If the determinants are constructed from optimized canonical HF MOs, the gradient is zero, and the matrix element is zero. This may also be realized by noting that the MOs are eigenfunctions of the Fock operator, eq. (3.41). 

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. 

In order to evaluate the Cl matrix elements one- and two-electron integrals over MOs are needed. These can be expressed in terms of the corresponding AO integrals and the 

---

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... 

Evaluation of the Cl matrix elements is somewhat difficult. Fortunately, most matrix elements are zero because of the orthogonality of the MO s. There are only three types of non-zero elements which are needed to be computed. 

Due to the size of the variational problem, a large Cl is usually not a practicable method for recovering dynamic correlation. Instead, one usually resorts to some form of treatment based on many-body perturbation theory where an explicit calculation of all off-diagonal Cl matrix elements (and the diagonalization of the matrix) are avoided. For a detailed description of such methods, which is beyond the scope of this review, the reader is referred to appropriate textbooks295. For the present purpose, it suffices to mention two important aspects. 

What is the Cl matrix element coupling lls22s2l and lls23s2l ... 

The advantage of the many-body approach over that represented by eqns. (4,49)-(4,57) is that it yields the CPMET equations directly in terms over orbitals instead of expressions containing Cl matrix elements. Moreover, the many-body approach is quite general and permits... 

The Cl matrix elements f/, , can be evaluated by the strategy employed for calculating... 

Thereby making it possible to express the Cl matrix elements in terms of MO integrals. There are, however, some general features which make many of the Cl matrix elements equal to zero. 

For an explicit calculation of the energy order. Slater rules yield the Cl matrix elements... 

The Hamilton operator (eq. (3.23)) does not contain spin, thus if two determinants have different total spin the corresponding matrix element is zero. This situation occurs if an electron is excited from an a spin-MO to a /3 spin-MO, such as the second S-type determinant in Figure 4.1. When the HF wave function is a singlet, this excited determinant is (part of) a triplet. The corresponding Cl matrix element can be written in terms of integrals over MOs, and the spin dependence can be separated out. If there are different numbers of a and [3 spin-MOs, there will always be at least one integral... 

The Hamilton operator consists of a sum of one-electron and two-electron operators, eq. (3.24). If two determinants differ by more than two (spatial) MOs there will always be an overlap integral between two different MOs which is zero (same argument as in eq. (3.28)). Cl matrix elements can therefore only be non-zero if the two determinants differ by 0, 1, or 2 MOs, and they may be expressed in terms of integrals of one- and two-electron operators over MOs. These connections are known as the Slater-Condon rules. If the two determinants are identical, the matrix element is simply the energy of a single determinant wave function, as given by eq. (3.32). For matrix elements between... 

The coupling coefficients for a particular choice of Cl expansion are obviously fixed by the expressions for the Cl matrix elements. For example the expressions in Appendix 20.B contain all the information to generate the coupling coefficients for Doubly-excited Cl for singlet states generated from a closed-shell determinant. Equally obviously, the expressions of Slater s Rules in Appendix 2. A imply the coupling coefficients for Cl using determinants. 

Of course, in using the idea of generating the Cl matrix elements from their defining rules we shall immediately come across the integral access problem each (MO) electron integral will appear many times in many elements of the Cl matrix, just as each (AO) electron-repulsion integral appears many times in many elements of G R) in an SCF calculation. We solve this problem in exactly the same way ... 

---