Fock operator unperturbed

This method [ ] uses the single-configuration SCF process to detennine a set of orbitals ( ).]. Then, using an unperturbed Flamiltonian equal to the sum of the electrons Fock operators // = 2 perturbation... 

So far the theory has been completely general. In order to apply perturbation theory to the calculation of correlation energy, the unperturbed Hamilton operator must be selected. The most common choice is to take this as a sum over Fock operators, leading to Mdller-Plesset (MP) perturbation theory. The sum of Fock operators counts the (average) electron-electron repulsion twice (eq. (3.43)), and the perturbation becomes... 

The use of the Hartree-Fock model allows the perturbation-theory equations (1.2)-(1.5) to be conveniently recast in terms of underlying orbitals (,), orbital energies (e,), and orbital occupancies (n,). Such orbital perturbation equations will allow us to treat the complex electronic interactions of the actual many-electron system (described by Fock operator F) in terms of a simpler non-interacting system (described by unperturbed Fock operator We shall make use of such one-electron perturbation expressions throughout this book to elucidate the origin of chemical bonding effects within the Hartree-Fock model (which can be further refined with post-HF perturbative procedures, if desired). 

This Fock operator is used to define the unperturbed Hamiltonian of Mpller-Plesset perturbation theory (MPPT) ... 

Then we know that for the unperturbed Fock operator... 

Then we assume that the projection operator for the exact Fock operator is close to the unperturbed one so that the expansion... 

We restrict ourselves here to results that were obtained with the Mj ller-Plesset partitioning of the Hamiltonian, which means that the Hartree-Fock operator was extracted from the Hamiltonian as the "unperturbed" operator and the rest of the Hamiltonian was taken as the perturbation. The formula... 

Perturbative approaches to the electron correlation problem have proved to be successful even when calculating second-order corrections only but finer results require fourth order energy corrections, as we will see later in this text. The reliability of a perturbation expansion greatly depends on the partitioning of the exact Hamiltonian H = H0 + W to an unperturbed part H0 and a perturbation W. Good quality approximations are to be expected if the N-particle operator H0 is chosen as the sum of equivalent one-particle Fock operators... 

The coefficients are the eigenvector components of the Fock operator of the unperturbed (Est = — 0) polymer). The new matrices A OT are also cyclic hypermatrices and can be block-diagonalized in the same way as before. Therefore, one obtains finally again equation (86) but on the r.h.s. we now have... 

The many-electron hamiltonian can be partitioned into an unperturbed part consisting of the Fock operator in diagonal form and a perturbation consisting of modified interaction terms. One can write... 

The solution mechanism described is quite general. The most common choice for the reference of the unperturbed Hamiltonian operator is the sum over Fock operators. This is known explicitly as Mpller-Plesset (MP) perturbation theoryl AS]... 

It can be seen that correction terms to the unperturbed Fock operator involve generalized one- and two-electron integrals with the effective interaction kernel, G(r, r ) in place of the usual direct Coulomb interaction r —... 

The M0ller-Plesset perturbation theory [26] corresponds to the application of the stationary perturbation theory to the calculation of the correlation energy using the Hartree-Fock Slater determinant as the zeroth order wavefunction. These methods are denoted MPn where n is the order of the perturbative corrections included. In the M0ller-Plesset method, the unperturbed Hamiltonian operator is chosen as a sum of Fock operators... 

All the calculations reported in this work were done on a DEC 20-60 using a modified version of the GAUSSIAN 80 series of program (6). Standard ST0-3G minimal basis set (7) was considered. Polarizabilities were calculated by the finite-field SCF method of Cohen and Roothaan (8) which is virtually equivalent to the analytic Coupled Hartree-Fock scheme. A term yf, describing the interaction between the electric field, E, and the molecule is added to the unperturbed molecular Hamiltonian, H y is the total dipole moment of the molecule. At the Hartree-Fock level, the electric field appears explicitly in the one-electron part of the modified Fock operator, F( ),... 

Following Moller and Plesset we can take as the unperturbed Hamiltonian Hq the sum of the Fock operators,... 

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 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) ... 

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