Perturbation Energy Expansions

Dyall and Lenthe [52] have recently derived and implemented an alternative perturbation expansion approach for the lORA equation. They analyzed the perturbation-energy expansions for U H- g d for neutral uranium up to third-order. 

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 symmetric perturbation equations [Eq. (275)] give the following energy expansion ... 

Applying a modified Rayleigh-Schrodinger perturbation theory, one obtains the following expressions for the first four coefficients in the expansion of the perturbation energy. For charmonium, this gives (Cizek and Vinette,... 

This equation includes the first derivative of the energy with respect to the parameter a, Eq. It is also an equation with a very real correspondence to first-order perturbation theory, and that suggests how best to use it. Indeed, the general procedure being outlined here differs from a perturbation expansion in only one minor way. A perturbation expansion is in terms of powers of one or more parameters. The derivative expansion is a Taylor-series-type expansion that has each nth power series term divided by n. That factor converts perturbative energy corrections into energy derivatives. So, Eqn. (30) is conveniently rearranged, just as is usually done in an elementary introduction to perturbation theory ... 

As discussed in detail in Refs. 77 and 82, for example, this expansion is not N-fold (where N is the number of electrons in the system) for the lower perturbational orders, but truncates to include only modest excitation levels. For example, the first-order wavefunction, which may be used to compute both the second- and third-order energies, contains contributions from doubly excited determinants only, whereas the second-order wavefunction, which contributes to the fourth- and fifth-order perturbed energies, contains contributions from singly, doubly, triply, and quadruply excited determinants. Furthermore, the sum of the zeroth- and first order energies is equal to the SCF energy. This determinantal expansion of the perturbed wavefunctions suggests that we may also decompose the cluster operators, T , by orders of perturbation theory ... 

This completes a literate program for evaluating third order ring energies in the many-body perturbation theory expansion for closed-shell systems. 

The RS formulas for the energy expansion are well known and are given in many places (e.g., Ref. 22). A thorough development of the wave-reaction operator perturbation theory has been presented by Low-din.23 Using conventional first quantized operators, we may write down the expressions for the nth-order energy E(n), for instance, as... 

The dependence of rag on the internal coordinates is not restricted by requirements other than the center of mass conditions (2.4) and that Eq. (2.6) is invertible. In expressing the rag functions we may therefore also consider how the final Hamiltonian is influenced, so that we obtain an operator of optimum suitability characterized by e.g. rapid convergence of the perturbing terms. In this respect there are two particular concerns, the vibration-rotation interaction and the potential energy expansion. 

Table 4 Comparison of small-box perturbation and precise numerical energies for the ns (n = 1, 2, 3, 4,5) levels of a hydrogen atom at the centre of a spherical cavity of radius R. For each value of R the first row refers to the perturbation energy whereas the second refers to the precise numerical energy. The final row of the table specifies the estimated radius of convergence, RIC, of the perturbation theory expansion (see text)... 

Table 8 Maximum box radii Rpj and Rpa( e for, respectively, direct perturbation theory expansion and Pade approximant, which guarantee a relative accuracy of at least 10 6 in the energy...

The estimated radius of convergence of each series is indicated in the final rows of Tables 4-6 and they would appear to agree well with the behaviour of the actual numerical energy values obtained from summation of the perturbation series. From Table 9 it seems that a is (slowly) approaching 0.5 as i increases. If we choose a = 0.5 and solve 2 linear equations for A.o 2 and A.r we find very little difference from the results in Table 9 for example, for i = 50 we obtain Ao 2 = 36.49 with a = 0.5 as opposed to Ao 2 = 36.45 with a = 0.53, so it would appear that Eis(A) has a square root branch point close to A. = 6.04. Other series Eni(X) also behave in a similar manner, with square root branch points, which would appear to be a general feature of perturbation theory expansions for linear operators [42],... 

When also higher-order energy corrections are desired, the next 0, term in the perturbation theory expansion can be obtained from the upper half of the perturbation theory expansion as... 

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