Expectation Values of One-Index Transformed Hamiltonians

Appendix F. Expectation Values of One-Index Transformed Hamiltonians 

The expectation value of the one-index transformed Hamiltonian can be written in the following alternative ways (Helgaker, 1986) 

If the transformation matrix is antisymmetric the Hamiltonian expectation value may be further rewritten as 

In some cases the AO basis is more convenient, in particular 

In the first expression the integrals are in the covariant AO representation (in which they are calculated), and the one-index transformed density elements are in the contravariant representation (obtained from the MO basis in usual one- and two-electron transformations). The second expression is useful whenever the transformation matrix is calculated directly in the covariant AO representation and requires the transformation of the Fock matrix to the contravariant representation. The last expression is convenient when the number of perturbations is large, since it avoids the transformation of the covariant AO Fock matrix to the MO or contravariant AO representations. 

---

In the MCSCF case the undifferentiated Fock matrix is symmetric since the orbital optimization ensures that ffj = 2(FfJ — FfJ) = 0. The Fock matrix also appears in the calculation of expectation values of one-index transformed Hamiltonians (see Appendix F). 

---