Rules for matrix elements

Thus in the lowest order approximation the angle x is eliminated from the off-diagonal matrix elements of [second and third of Eqs. (60)] it solely determines the selection rules for matrix elements of Hg with respect to nuclear basis functions. 

Slater-Condon rules for matrix elements between Slater determinants, c c... 

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... 

As was shown in chapter three we can compute the transition densities from the Cl coefficients of the two states and the Cl coupling coefficients. Matrix elements of two-electron operators can be obtained using similar expresssions involving the second order transition density matrix. This is the simple formalism we use when the two electronic states are given in terms of a common orthonormal MO basis. But what happens if the two states are represented in two different MO bases, which are then in general not oithonormal We can understand that if we realize that equation (5 8) can be derived from the Slater-Lowdin rules for matrix elements between Slater determinants. In order to be a little more specific we expand the states i and j ... 

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... 

It can indeed be shown that this form gives rise to the Slater-Condon rules for matrix elements between multi-electron states. Further expressions for two-particle operators may be found in the literature [8,9],... 

These rules are a consequence of the fact that the spin-orbit operator for the many-electron states is a sum of one-particle operators according to (5) and the Slater-Condon rules for matrix elements between states of such operators [121]. 

The selection rules for matrix elements of Hso are summarized as follows (Kayama and Baird, 1967), see also Eq. (3.4.46) ... 

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). 

The implementation of any CC scheme requires the derivation of explicit equations for energy and amplitudes. There are several possibilities to accomplish this task starting from the basic equations (7) and (8). Most straightforward is an algebraic approach, as followed by Pople and co-workers. 2 The usual Slater-Condon rules for matrix elements of the Hamiltonian are applied to derive the required expressions. A problem is that mutual cancellation of some terms, the so-called unlinked contributions, is easily overlooked. Furthermore, the derivation of the corresponding equations is... 

In conclusion, we note that the phase factors (1.2.3) were necessary to reproduce the Slater-Condon rules for matrix elements between Slater determinants. 

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