Time-dependent coupled perturbed Hartree-Fock equations

In the first method, Frankel s variational principle is used to give equations for the changes in the Hartree-Fock orbitals caused by an applied time dependent field. These equations are called the time-dependent coupled perturbed Hartree-Fock (TDCPHF) equations. They involve matrices known as the electric and magnetic Hessians which have coulomb and exchange contributions. The first-order TDCPHF equations can be used to calculate a and but in order to calculate Y it is necessary to use second-order TDCPHF. 

RPA, and CPHF. Time-dependent Hartree-Fock (TDHF) is the Hartree-Fock approximation for the time-dependent Schrodinger equation. CPHF stands for coupled perturbed Hartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

We saw in Section III that the polarization propagator is the linear response function. The linear response of a system to an external time-independent perturbation can also be obtained from the coupled Hartree-Fock (CHF) approximation provided the unperturbed state is the Hartree-Fock state of the system. Thus, RPA and CHF are the same approximation for time-independent perturbing fields, that is for properties such as spin-spin coupling constants and static polarizabilities. That we indeed obtain exactly the same set of equations in the two methods is demonstrated by Jorgensen and Simons (1981, Chapter 5.B). Frequency-dependent response properties in the... 

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