Fock operator perturbed

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... 

A Moeller-Plesset Cl correction to v / is based on perturbation theory, by which the Hamiltonian is expressed as a Hartree-Fock Hamiltonian perturbed by a small perturbation operator P through a minimization constant X... 

So far, we ve presented only general perturbation theory results.We U now turn to the particular case of Moller-Plesset perturbation theory. Here, Hg is defined as the sum of the one-electron Fock operators ... 

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). 

Here in eq. (38) "EpqfpQN a.pag is new Hartree-Fock operator for a new fermions (25), (26), operator Y,pQRsy>pQR a Oq 0s%] is a new fermion correlation operator and Escf is a new fermion Hartree-Fock energy. Our new basis set is obtained by diagonalizing the operator / from eq. (36). The new Fermi vacuum is renormalized Fermi vacuum and new fermions are renormalized electrons. The diagonalization of/ operator (36) leads Jo coupled perturbed Hartree-Fock (CPHF) equations [ 18-20]. Similarly operators br bt) corresponds to renormalized phonons. Using the quasiparticle canonical transformations (25-28) and the Wick theorem the V-E Hamiltonian takes the form... 

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

Another way that additional configurations can be added to the the ground-state wave function is by the use of Moller-Plesset perturbation theory (MPPT). As it happens, a Hamiltonian operator constructed from a sum of Fock operators has as its set of solutions the HF single determinantal wave function and all other determinantal wave... 

Moller-Plesset perturbation theory (MPPT) aims to recover the correlation error incurred in Ilartree- Fock theory for the ground state whose zero-order description is ,. The Moller-Plesset zero-order Hamiltonian is the sum of Fock operators, and the zero-order wave functions are determinantal wave functions constructed from HF MOs. Thus the zero-order energies are simply the appropriate sums of MO energies. The perturbation is defined as the difference between the sum of Fock operators and the exact Hamiltonian ... 

NBO analysis can be used to quantify this phenomenon. Since tire NBOs do not diagonalize the Fock operator (or tire Kohn-Sham operator, if the analysis is carried out for DFT instead of HF), when the Fock matrix is formed in the NBO basis, off-diagonal elements will in general be non-zero. Second-order perturbation tlieory indicates that these off-diagonal elements between filled and empty NBOs can be interpreted as the stabilization energies... 

A similar scheme can be exploited to compute the additional polarizability a(—wa wa) in this case the time-dependent problem to solve is determined by the Fock operator in which the external perturbation is the polarization P"" and thus the dipole-like operator to be included is m only. The resulting polarizability is now ... 

The projection operator formulation of the Hartree-Fock problem can be used for constructing a perturbation procedure for determining the electronic structure in terms of the latter.20 The simplest formulation departs from the Hartree-Fock equation for the projection operator to the occupied MOs eq. (1.152). Let us assume that the bare perturbation (see below) concerns only the one-electron part of the Fock operator so that ... 

Any Fock operator can be represented as a sum of the symmetric one and of a perturbation which includes both the dependence of the matrix elements on nuclear shifts from the equilibrium positions and the transition to a less symmetric environment due to the substitution. To pursue this, we first introduce some notations. Let hi be the supervector of the first derivatives of the matrix of the Fock operator with respect to nuclear shifts Sq counted from a symmetrical equilibrium configuration. By a supervector, we understand here a vector whose components numbered by the nuclear Cartesian shifts are themselves 10 x 10 matrices of the first derivatives of the Fock operator, with respect to the latter. Then the scalar product of the vector of all nuclear shifts 6q j and of the supervector hi yields a 10 x 10 matrix of the corrections to the Fockian linear in the nuclear shifts ... 

This does not form the entire ( dressed ) perturbation because, in case the electron density changes to the first order in the above perturbation, the Fock operator acquires additional perturbation through the variation of its self-energy part, which leads to the self-consistent perturbation. Thus the perturbed Fock operator can be written as ... 

Inserting the expansion eq. (4.40) rewritten in terms of matrices V in the energy expression eq. (4.31) with the perturbed Fock operator eq. (4.39) yields a DMM model of the CC of an arbitrary symmetry since the transition densities V take account of all possible perturbations of the electronic structure, keeping the CLS a separate entity. The series eq. (4.40) in fact appears by expanding the closed expression for the projection operator ... 

A remarkable feature is that the derivative of the one-electron part of the Fock operator with respect to the symmetry adapted nuclear shift Sqr < (an operator acting on the one-electron states in the CLS carrier space) itself transforms according to the irreducible representation T and its row 7. That means that applying the deformation T7) to a complex results in a perturbation of the Fock operator having the same symmetry 1 7. This allows us to write the vibronic operator in a symmetry-adapted form ... 

A general approach to the intramonomer correlation problem is known as the many-electron (or many-body) SAPT method88,141 213-215. In this method the zeroth-order Hamiltonian H0 is decomposed as H0 = F + W, where F = FA + FB is the sum of the Fock operators, FA and FB, of monomer A and B, respectively, and W is the intramonomer correlation operator. The correlation operator can be written as W = WA + WB, where Wx = Hx — Fx, X = A or B. The total Hamiltonian can be now be represented as H = F + V + W. This partitioning of H defines a double perturbation expansion of the wave function and interaction energy. In the SRS theory the wave function is obtained by expanding the parametrized Schrodinger equation as a power series in and A,... 

The first-order MCSCF response equations were first derived by Dalgaard and Jdrgensen (1978). For geometrical perturbations these equations were derived by Osamura et al. (1982a) using a Fock-operator approach. 

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