Hartree-Fock unperturbed states

Further we want to study the nonadiabatic corrections to the ground state. Therefore /o> will be the unperturbed ground state wave function (we shall use Hartree-Fock ground state Slater determinant -Fermi vacuum) and % ) will be boson ground state-boson vacuum 0). 

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) ... 

While the multiconfiguration methods lead to large and accurate descriptions of atomic states, formal insight that can lead to a productive understanding of structure-related reaction problems can be obtained from first-order perturbation theory. We consider the atomic states as perturbed frozen-orbital Hartree—Fock states. It is shown in chapter 11 on electron momentum spectroscopy that the perturbation is quite small, so it is sensible to consider the first order. Here the term Hartree—Fock is used to describe the procedure for obtaining the unperturbed determi-nantal configurations pk). The orbitals may be those obtained from a Hartree—Fock calculation of the ground state. A refinement would be to use natural orbitals. 

For the coupled cluster methods, which are non-variational, the initial values of the A s are nonzero, and 0) does not correspond to the unperturbed reference state but, in most applications, to the Hartree-Fock state. Tire initial values of the parameters are found in an iterative optimization of the coupled cluster state, and the time-dependent values of the parameters were determined from the coupled-cluster time-dependent Schrodinger equation by Koch and Jprgensen [35], The coupled cluster state is not norm conserving, but the inno roduct of the coupled cluster state vector CC(f)) and a constructed dual vector (CC(f) remains a constant of time... 

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... 

We saw above that the unperturbed functions iAf are all possible Slater determinants formed from n different spin-orbitals. Let i, j, k, I,. .. denote the occupied spin-orbitals in the ground-state Hartree-Fock function o, and let a, b,c,d,.. . denote the unoccupied (virtual) spin-orbitals. Each unperturbed wave function can be classified by the number of virtual spin-orbitals it contains this number is called the excitation level. Let denote the singly excited determinant that differs from d>o solely by replacement of the occupied spin-orbital m, by the virtual spin-orbital Let denote the doubly excited determinant formed from [Pg.541]

In this section, we consider spin-unrestricted MPPT, taking as our unperturbed state the spin-unrestricted Hartree-Fock wave function and as our zero-order Hamiltonian the Fock operator. A spin-restricted treatment suitable for closed-shell states is given in Section 14.4, following the discussion of CCPT in Section 14.3. 

The unperturbed Hamiltonian Ho is thus, apart from the trivial energy-shift term, simply the model Hamiltonian Hm for any number of particles moving independently in the Hartree-Fock field. Since, by (9.2.20), the expectation value of Hm in the reference state is orb. the unperturbed energy is... 

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