Perturbation series expansion

Of the many quantum chemical approaches available, density-functional theory (DFT) has over the past decade become a key method, with applications ranging from interstellar space, to the atmosphere, the biosphere and the solid state. The strength of the method is that whereas conventional ah initio theory includes electron correlation by use of a perturbation series expansion, or increasing orders of excited state configurations added to zero-order Hartree-Fock solutions, DFT methods inherently contain a large fraction of the electron correlation already from the start, via the so-called exchange-correlation junctional. 

E. J. Weniger, J. Cizek, and F. Vinette, Phys. Rev. Lett., 156, 169 (1991). Very Accurate Summation for the Infinite Coupling Limit of the Perturbation Series Expansions of Anhar-monic Oscillators. 

The hydrogen atom in a linear external field (charmonium) and in a quadratic external field (harmonium) has already been studied by Vrscay (1983, 1984, 1985). In both cases, Lie algebraic methods provided a considerable number of coefficients in the perturbation series expansion for small values of the coupling constant. However, these results were obtained for a... 

All the important contributions to the forces between molecules arise ultimately from the electrostatic interactions between the particles that make up the two molecules. Thus our main theoretical insight into the nature of intermolecular forces comes from perturbation theory, using these interactions as the perturbation operator H = Z e, /(4jtSor/y), where is the charge on particle i in one molecule, is the distance between particles i and / in different molecules, and 8q is permittivity of a vacuum. The definitions of the contributions, such as the repulsion, dispersion, and electrostatic terms, which are normally included in model potentials, correspond to different terms in the perturbation series expansion. 

In the spirit of the direct perturbation theory approach [8-23,38], equation (26) can be expanded in an infinite perturbation series expansion as... 

One complication in the literature concerning these properties is the multiple conventions for their definitions. In addition to the Taylor series convention used in this work (Eq. [4]), an alternate convention (designated with a tilde) is based on a perturbation series expansion given as follows ... 

Dalgarno A and Lewis J T 1956 The representation of long-range forces by series expansions. II. The complete perturbation calculation of long-range forces Proc. Phys. Soc. A 69 59... 

We then write the solution of equation B1.5.7 as a power series expansion hi temis of the strength X of the perturbation ... 

Truncating this series after the first derivative and integrating provides the basis for the hermodynamic integration approach. Moreover, if the Taylor series expansion is continued intil it converges then Equation (11.45) is equivalent to the thermodynamic perturbation brmula, so providing a link between the two approaches. In practice, it is always necessary... 

The approximate expression (10.50) for the nuclear energy levels E j is observed to contain the initial terms of a power series expansion in (n -I- ) and J J + 1). Only terms up to (n + ff and J(J + )f and the cross term in (n + )J(J 4-1) are included. Higher-order terms in the expansion may be found from higher-order perturbation corrections. 

When a molecule A is attacked by another molecule B, it will be perturbed in either its number of electrons NA or its external potential vA(r). At the very early stages of the reaction, the total electronic energy of A, EA can be expressed as a Taylor series expansion around the isolated system values NA and v jfr)... 

A power series expansion of the state energy E, computed in a manner consistent with how P is determined (i.e., as an expectation value for SCF, MCSCF, and Cl wavefunctions or as <IHIvP> for MPPT/MBPT or as <lexp(-T)Hexp(T)l> for CC wavefunctions), is carried out in powers of the perturbation V ... 

The parameter is introduced to keep track of the order of the perturbation series, as will become clear. Indeed, one can perform a Taylor series expansion of the perturbed wave functions and perturbed energies using X to keep track of the order of the expansions. Since the set of eigenfunctions of the unperturbed SE form a complete and orthonormal set, the perturbed wave functions can be expanded in terms of them. Thus,... 

The calculation of AH° and AS° values from the pK-temperature data in each solvent mixture was performed by the nonempirical method of Clarke and Glew (26) as simplified by Bolton (27). In this method the thermodynamic parameters are considered to be continuous, well-behaved functions of temperature, and their values are expressed as perturbations of their values at some reference temperature 0 by a Taylor s series expansion. The basic equation is ... 

Using either the exact form of the projection operator as a function of V and V or cutting the series expansion, allows us to construct different approximate variation schemes with matrix elements of V as variables. On the other hand, inserting the expansion for the projection operator eq. (1.107) in the Schrodinger equation for the projection operator eq. (1.95) with the perturbed Hamiltonian gives in the first order ... 

The utility of this method hinges on the assumption that a is small in some relative sense, and possibly also that k is small. With these assumptions the method bases itself on the expansion of the mean-field potential in a perturbation series of the form... 

Table 1-1. Convergence of the polarization expansion for the interaction of two ground-state helium atoms at R = 1 and 5.6 bohr, and of the lithium and hydrogen atoms in their ground states at R = 10 and 12 bohr. The Coulomb energies represent 53.50% (He2) R = 5.6bohr), 73.4% (LiH, R = 10 bohr), and 85.53% (LiH, R= 12 bohr) of the energies of the fully symmetric (Pauli forbidden) states. The quantity S(n) represents the percent error of the perturbation series through the nth-order with respect to the variational interaction energy of the Pauli forbidden state...

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