@ARTICLE{Amos:1972:CVP, author = "A. T. Amos", title = "A change of variable for the perturbation parameter in Rayleigh-Schrodinger perturbation theory", year = "1972", journal = "Int. J. Quantum Chem.", volume = "6", pages = "125", abstract = "The Feenberg-Goldhammer change of scale, whereby the H/sub 0/ in perturbation theory is replaced by (1/ mu )H/sub 0/ with mu a scaling parameter, is shown to be equivalent to a change of variable for the perturbation parameter. A more general change of variable is shown to lead to a perturbation series with perturbation energies E/sub 3/, ..., E/sub 2n+1/ equal to zero. The resulting energy through (2n+1) the order has the same form as that found from the Brillouin-Wigner series by different methods." } @ARTICLE{Bhattacharyya:1982:FSH, author = "K. Bhattacharyya", title = "Feenberg scaling and higher-order invariants in Rayleigh-Schrodinger perturbation theory", year = "1982", journal = "Int. J. Quantum Chem.", volume = "21", pages = "857", abstract = "It is shown that the determination of a unique scaling parameter, based on scale-invariant forms for energy in the scaled zero-order Hamiltonian approach of Feenberg (1958), is not possible because the higher-order invariants themselves are nonunique." } @ARTICLE{Bhattacharyya:1982:GET, author = "K. Bhattacharyya", title = "Generalized Euler transformation in extracting useful information from divergent (asymptotic) perturbation series and the construction of Pade approximants", year = "1982", journal = "Int. J. Quantum Chem.", volume = "22", pages = "307", abstract = "Euler transformation for accelerating convergence of a series is considered in the context of handling divergent (asymptotically convergent) perturbation series. A generalized (parametrized) version of this transformation is developed, based on the conjecture of Dalgarno and Stewart (1960), which works better. Viewed from this standpoint, the Pade approximants follow as a special case of the parametrized Euler transformation (PET), as is the case with the mu transformation procedure of Feenberg in a perturbative context. The PET is shown to serve as a more general method of handling a divergent series and is able to appreciate the construction and convergence behavior of specific sequences of Pade approximants. The role of parametrization in the context of the Z/sup -1/ perturbation theory of atoms is also noted and the workability of the adopted strategy is demonstrated by choosing some specific test cases." } @ARTICLE{Schmidt:1993:FSA, author = "C. Schmidt and M. Warken and N. C. Handy", title = "The Feenberg series. An alternative to the Moller-Plesset series", year = "1993", journal = "Chem. Phys. Lett.", volume = "211", pages = "272", abstract = "The Feenberg series is reintroduced as a perturbation series with improved convergence characteristics for quantum-chemistry calculations. It is defined through a scaling of H/sub 0/, the parameter being defined such that the third- order correction is zero. The Feenberg series is suggested as an improvement to the Moller-Plesset series, the fourth-order energy being trivially obtained in terms of MP2, MP3 and MP4 energies. Benchmark studies are reported, where it is found that if MP3[left angle bracket]0, then the Feenberg fourth-order energy is better than the MP energy at fifth order, and if MP3[right angle bracket]0, the perturbation series is smoothed." } @ARTICLE{Zhi:1996:SOM, author = "Zhi He and D. Cremer", title = "Sixth-order many-body perturbation theory. IV. Improvement of the Moller- Plesset correlation energy series by using Pade, Feenberg, and other approximations up to sixth order", year = "1996", journal = "Int. J. Quantum Chem.", volume = "59", pages = "71", abstract = "Three different ways of getting reliable estimates of full configuration interaction (FCI) correlation energies are tested, namely (a) by Pade approximants [k, k] and [k, k-1], (b) by using extrapolation formulas, and (c) by Feenberg scaling of Moller-Plesset (MP) correlation energies. By using MPn energies up to sixth order, i.e., MP2, MP3, MP4, MP5, and MP6, it was possible to test the convergence behavior of the Pade series [1, 0], [1,1], [2, 1], [2, 2] and the Feenberg series up to sixth order where in the latter case a scaling factor lambda /sup 5 /(scaling of the second-order wave function, FE2) rather than the previously tested lambda /sup 3 /(scaling of the first-order wave function, FE1) was considered. Investigation of 26 different correlation energies for systems with monotonic convergence in the MPn series (class A systems) or initially oscillatory convergence behavior (class B systems) indicates that Pade approximants lead in some cases to reasonable estimates of FCI correlation energies, but in other cases, in particular for class B systems, they give too negative correlation energies. Both monotonic and oscillatory behavior for the Pade series is observed where it is possible to predict its convergence behavior on the basis of calculated MPn energies. The best estimates of the FCI correlation energy are obtained by FE2 scaling. At sixth-order FE2, values for atoms and molecules with equilibrium geometry differ on the average by just 0.146 mhartree from FCI correlation energies. The FE2 correlation energies all converge monotonically. Also, FE2 scaling reduces the exaggeration of MP6 correlation energies for class B systems. However, surprisingly good estimates of FCI energies are also obtained by simple extrapolation formulas based on MP4, MP5, and MP6 correlation energies." } @ARTICLE{Homeier:1996:CEE, author = "H. H. H. Homeier", title = "Correlation energy estimators based on Moller-Plesset perturbation theory", year = "1996", journal = "Theochem", volume = "366", pages = "161", abstract = "Some methods for the convergence acceleration of the Moller-Plesset perturbation series for the correlation energy are discussed. The order-by-order summation is less effective than the Feenberg series. The latter is obtained by renormalizing the unperturbed Hamilton operator by a constant factor that is optimized for the third order energy. In the fifth order case, the Feenberg series can be improved by order-dependent optimization of the parameter. Alternatively, one may use Pade approximants or a further method based on effective characteristic polynomials to accelerate the convergence of the perturbation series. Numerical evidence is presented that, besides the Feenberg- type approaches, suitable Pade approximants, and also the effective second order characteristic polynomial, are excellent tools for correlation energy estimation." } @ARTICLE{Cremer:1997:PFC, author = "D. Cremer and Zhi He", title = "Prediction of full CI energies with the help of sixth-order Moller-Plesset (MP6) perturbation theory", year = "1997", journal = "Int. J. Quantum Chem.", volume = "398-399", pages = "7", abstract = "The sixth-order Moller-Plesset (MP6) correlation energy is analysed by using first- and second-order cluster operators and distinguishing between connected and disconnected operator products. Each product is described by simplified Brandow diagrams that help to characterize the associated energy contributions in terms of orbital relaxation, pair correlation, three-electron correlation or four-electron correlation effects. The importance of the various correlation terms and their coverage at MP2, MP3, MP4, MP5, and MP6 are analysed to understand and to predict the convergence behaviour of the MPn series, which strongly depends on the electronic structure of the atoms and molecules investigated. Adjusting existing extrapolation procedures to the convergence behaviour of the MPn series leads to improved predictions of full CI (FCI) energies based on MP6 correlation energies. The best results are obtained by a combination of first-order and second-order Feenberg scaling, which produces the results of higher order Feenberg scaling. The mean absolute deviation of predicted FCI energies from exact values is found to be 0.07 mhartree for atoms and molecules in their equilibrium geometry and 1.03 mhartree for molecules with stretched geometries and, thereby, considerable multi-reference character. Reasonable FCI energies can also be obtained with approximate MP6 methods, the most economic method of which is MP6(M7) which scales with O(M/sup 7/) (M is the number of basis functions). Mean absolute deviations of FCI energies based on MP6(M7) are 0.40 and 1.88 mhartree for equilibrium and stretched geometries, respectively." } @ARTICLE{Forsberg:2000:CBM, author = "B. Forsberg and Zhi He and Yuan He and D. Cremer", title = "Convergence behavior of the Moller-Plesset perturbation series: use of Feenberg scaling for the exclusion of backdoor intruder states", year = "2000", journal = "Int. J. Quantum Chem.", volume = "76", pages = "306", abstract = "The convergence behavior of the Moller-Plesset (MP) perturbation series is investigated utilizing MP correlation energies up to order 65 calculated at the full CI (FCI) level. Fast or slow convergence, initial oscillations, or divergence of the MPn series depend on the electronic system investigated and the basis set used for the FCI calculation. Initial oscillations in the MPn series are observed for systems with electron clustering due to the fact that MP theory exaggerates electron-correlation effects at even orders and corrects this at odd orders. In such cases, it is important that the s, p basis is first saturated before diffuse functions are added. With a VDZ+diff basis, too much weight is given to high-order correlation effects described by pentuple and higher excitations, which leads to the formation of artificial intruder states and to the divergence of the MPn series. This can be corrected by extending to VQZ or VPZ basis sets before one adds diffuse functions. Alternatively, one can use m-order Feenberg scaling to exclude backdoor intruder states from the convergence region of the MPn series. For all cases considered, divergence of: the MP n series caused by unbalanced basis sets including diffuse functions can be suppressed by Feenberg scaling. Also, initial oscillations of the MPn series can be dampened and convergence acceleration of the MPn series achieved if the appropriate order of Feenberg scaling is determined for the problem in question. The relationship between electronic structure, basis set, and convergence of the MPn series is discussed." } @ARTICLE{Moller:1934:NAT, author = "Chr. M{\o}ller and M. S. Plesset", title = "Note on an approximation treatment for many-electron systems", year = "1934", journal = "Phys. Rev.", volume = "46", pages = "618", abstract = "A perturbation theory is developed for treating a system of n electrons in which the Hartree-Fock solution appears as the zero-order approximation. It is shown by this development that the first order correction for the energy and the charge density of the system is zero. The expression for the second-order correction for the energy greatly simplifies because of the special property of the zero-order solution. It is pointed out that the development of the higher approximation involves only calculations based on a definite one-body problem." } @ARTICLE{Goldhammer:1956:RBW, author = "Paul Goldhammer and Eugene Feenberg", title = "Refinement of the {Brillouin - Wigner} perturbation method", year = "1956", journal = "Phys. Rev.", volume = "101", pages = "1233", abstract = "New variational parameters are introduced into the wave function employed in the Brillouin-Wigner perturbation method, and determined to minimize the total energy. The original and modified procedures are illustrated by a numerical example." } @ARTICLE{Feenberg:1956:IPB, author = "Eugene Feenberg", title = "Invariance property of the {Brillouin - Wigner} perturbation series", year = "1956", journal = "Phys. Rev.", volume = "103", pages = "1116", abstract = "The complete perturbation series for the energy is invariant under the operation of adding a velocity-dependent interaction to the zeroth order Hamiltonian and subtracting the same quantity from the perturbation operator. The same invariance property appear to hold also for an optimum formulation of the terminated energy series generated by the nth order approximation to the wave function. An explicit proof is given for the first- and second-order wave functions and also for the complete energy series. Variational procedures for determining (a) the optimum velocity dependence of the zeroth order Hamiltonian and (b) the optimum uniform displacement of the zeroth order energy spectrum are discussed in relation to the invariant formulations." } @ARTICLE{Knowles:1985:CHO, author = "P. J. Knowles and K. Somasundram and N. C. Handy", title = "The calculation of higher-order energies in the many-body perturbation series", year = "1985", journal = "Chem. Phys. Lett.", volume = "113", pages = "8", abstract = "Perturbation series through 8th order...variety of basis sets..." }