Generalized Fock operator

Here, Flffl are matrix elements of a zeroth-order Hamiltonian, which is chosen as a one-electron operator in the spirit of MP2. is an overlap matrix The excited CFs are not in general orthogonal to each other. Finally, Vf)(i represents the interaction between the excited function and the CAS reference function. The difference between Eq. [2] and ordinary MP2 is the more complicated structure of the matrix elements of the zeroth-order Hamiltonian in MP2 it is a simple sum of orbital energies. Here H is a complex expression involving matrix elements of a generalized Fock operator F combined with up to fourth-order density matrices of the CAS wave function. Additional details are given in the original papers by Andersson and coworkers.17 18 We here mention only the basic principles. The zeroth-order Hamiltonian is written as a sum of projections of F onto the reference function 0)... 

Here the generalized Fock operator/with matrix elements/ appears for the first time. It looks familiar and resembles the Fock operator of Hartree-Fock theory. However, now the yf are matrix elements of the exact one-particle density matrix... 

The energy (to be evaluated from Eq. (51)) appears as a constant term, consistent with the fact that the expectation values of the last two terms with respect to fk automatically vanish. The fP are the matrix elements of the generalized Fock operator, Eq. (52). 

The success of the MP2 method for closed shell HF reference functions makes it interesting to try to develop a Hamiltonian that has the MP2 case as the limit when there are no active orbitals. For this purpose a generalized Fock operator was defined ... 

The systematic error caused by the definition of the zeroth order Hamiltonian, as described above, leads to too low relative energies for systems with open shells. A consequence is that dissociation and excitation energies will be too low because the dissociated or excited state has usually more open shell character than the reference state. Is there a way we can remedy this systematic error The diagonal elements of the generalized Fock operator can for an active orbitals be estimated as ... 

The zeroth-order Hamiltonian of the CASPT2 theory is a projected generalized Fock operator, and the orthogonal functions are internally contracted excitations. In this case the resolvent operator is not diagonal, and thus its inverse is determined in an iterative process. To be able to perform the inversion of the resolvent operator efficiently,... 

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... 

Here, an effective one-electron operator matrix has Fock and energy-dependent, self-energy terms. Prom this matrix expression, one may abstract one-electron equations in terms of the generalized Fock and energy-dependent, self-energy operators ... 

From equations (33) and (35), a general HPHF Fock operator for determining the a, orbitals of excited states can be extracted after some straightforward transformations ... 

Let us make clear now the correspondence between our treatment here and Erdahl s 1978 treatment [4, Sec. 8]. Erdahl works in general Fock space and his operators conserve only the parity of the number of nuclei. He exhibits two families of operators that are polynomials in the annihilation and creation operators containing a three-body and a one-body term. Generic instances of these operators are denoted y and w. The coefficients are real, and Erdahl stresses that this is essential for his treatment. The one-body term is otherwise unrestricted, but the three-body term must satisfy conditions to guarantee that y+y or H +w does not contain a six-body term. For the first family the conditions amount to the three-body term being even under taking the adjoint, and for... 

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... 

Since the many-electron Hamiltonian, the effective one-electron Hamiltonian obtained in Hartree-Fock theory (the Fock operator ), and the one- and two-electron operators that comprise the Hamiltonian are all totally symmetric, this selection rule is extremely powerful and useful. It can be generalized by noting that any operator can be written in terms of symmetry-adapted operators ... 

The Fock operator for the valence orbitals may therefore be approximated by a simple generalization of equation (22),... 

There are n spatial orbitals ij/ since we are considering a system of 2n electrons and each orbital holds two electrons. The 1 in parentheses on each orbital emphasizes that each of these n equations is a one-electron equation, dealing with the same electron (we could have used a 2 or a 3, etc.), i.e. the Fock operator (Eq. 5.36) is a one-electron operator, unlike the general electronic Hamiltonian operator of Eq. 5.15, which is a multi-electron operator (a 2n electron operator for our specific case). The Fock operator acts on a total of n spatial orbitals, the ij/1, Jj2,, i// in Eq. 5.35. 

The self-consistent nature of the Fock operator is sometimes modeled in the methods without interaction by the schemes relating the ionization potentials of each AO with the overall electronic population (or effective charge) of a given atom and sometimes with the orbital populations of the AOs centered on the considered atom. This generally leads to the expressions of the form [55] ... 

The Hartree-Fock operator fi(k) is generally expressed as a sum of the two terms,... 

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,... 

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