Perturbation expansions Rayleigh-Schrodinger

The Rayleigh-Schrodinger perturbation expansion for the exact wave function to the first order is given by... 

Ahlrichs R (1973) Comments on the convergence of the ordinary Rayleigh- Schrodinger perturbation expansion. Chem Phys Lett 18 67-68... 

Explicitly, the first few terms in the Rayleigh-Schrodinger perturbation expansion may be written in the form ... 

In order to develop the Rayleigh-Schrodinger perturbation expansion for the energy and the wave function, we define the resolvent... 

Table 1 Convergence of the Rayleigh-Schrodinger perturbation expansion for the interaction of two ground-state helium atoms at R = 1.0 and 5.6 bohr. The variational interaction energies of the fully symmetric state are equal to -1.01904 hartree and -166.5301 hartree, respectively. The Coulomb energy for R = 5.6 bohr is equal to -77.4764 /ihartree. The B66 and B71 basis sets were used for distances 1.0 and 5.6 bohr, respectively. E(n) denotes the sum of perturbation corrections up to and including the nth-order and S(n) the percent error of E(n) with respect to the variational interaction energy. Energies are in hartree for R = 1 bohr, and in /ihartree for R = 5.6 bohr.

Table 4 Convergence of the symmetrized Rayleigh-Schrodinger perturbation expansion for the interaction of a ground-state helium atom with a hydrogen molecule at R = 6.5 bohr. The B83 basis set was used. (n) and 6(n) are defined as in Table 1. Energies are in /ihartree.

The Rayleigh-Schrodinger perturbation expansion is obtained by ordering E in powers of A... 

The Lennard-Jones, Brillouin, Wigner perturbation expansion is not a simple power series in A since each depends on the exact energy, S. Each energy coefficient in the Rayleigh-Schrodinger perturbation expansion consists of a principal term of the form... 

It is this effective Hamiltonian which semiempirical methods try to approximate more or less successfully. Therefore, an ab initio evaluation of is certainly desirable for an analysis of semiempirical methods. The computational implementation of Eqs. (7)-(9) requires a number of approximations [103] Use of a large but finite basis set, a Rayleigh-Schrodinger perturbation expansion of the inverse matrix in Eq. (9) around zero-order energies, and a quasidegenerate perturbation treatment through second or third order. This leads to an energy-independent Hamiltonian which includes nonclassical effective many-... 

The Rayleigh-Schrodinger perturbation expansion can be derived from the Brillouin-Wigner perturbation theory by expanding the denominator factors in the latter. The expressions for the energy coefficient in the Brillouin-Wigner perturbation theory expansion have the form... 

The generalized Bloch equation in both (2.195) and (2.199) is completely equivalent to the original Schrbdinger equation for the states in the model space (P. It can be employed to generate a Rayleigh-Schrodinger perturbation expansion as we shall now show. 

Equations (2.222) and (2.229) are the first two terms in the general Rayleigh-Schrodinger perturbation expansion for the multi-reference case. 

Introducing the expansion (D.7) (or (D.8)) and collecting terms of the same order in A we can obtain the Rayleigh—Schrodinger perturbation expansion for the ith state. 

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