Large-order shifted1/Nexpansions

Abstract

The hypervirial perturbative method is used to obtain large-order shifted 1 / N expansions, where N is the number of spatial dimensions. Numerical investigation shows that the usually chosen shift is useful only when low-order expansions are considered. An appropriate order-dependent shift proves to lead to highly accurate large-order expansions. The class of power-law potentials is discussed as a0 illustrative example.

The determination of transition state properties in the ethane and ethene ion dissociation

“…The fact that we do not find a significant J dependence of the dissociation energy is in agreement with a complex elimination reaction governed by a tight TS. In contrast to ab initio work by Herbst et al [25] we conclude that the energy of this TS is probably higher than the energy of the products. However, the energetic difference between the energy of the TS and the products may be quite small.…”

“…In both studies a TS connecting the educts and the product in reaction (3) was located. In the earlier work the energy of that TS was found to be higher [24] than that of the educts, in the more recent work at a higher level of theory it was found to be lower [25] in energy. The latter situation would imply that reaction (3) should proceed without an activation barrier.…”

Calculation of the bound states of power-law and logarithmic potentials through a generalized pseudospectral method

“…[17] e Exact value, as quoted in ref. [17] calculation reproduces the exact results very well. The variational as well as the numerical results exist for all of these states excepting ν = 2.5 and 3.5 For the ν = 4 case, several accurate calculations are available in the literature including the large-order shifted 1/N expansion [17] and the present result is in complete agreement with the most accurate "exact" result of [17].…”

“…[18] c Numerical results, from [16] d Ref. [17] e Exact values, as quoted in [17] Table VI displays the computed eigenvalues for selected n and ℓ quantum numbers of the logarithmic potential. The numerical results [4] exist for all the states with n, ℓ ≤ 4, while for states with n ≤ 4, ℓ ≤ 3 and ℓ = 4; n = 3, 4, the shifted 1/N expansion [16] as well as the variational results [16] are also available.…”

Anharmonic oscillators energies via artificial perturbation method

“…One has therefore to sum up such series [7][8][9][10]. Hence, apparently artificial perturbation recipes have been devised and shown to be ways to make progress [2,3,[11][12][13][14][15][16]. Without being exhaustive, several eligible methods have been used to calculate the eigenvalues and eigenfunctions for…”