Fifth-order constant denominator perturbation therory within a localized bond model: Contributions from single and double excitations

A Fifth-order constant denominator perturbation treatment of all single and double excitations occuring in the third-order perturbation wave function is presented for the perturbation configuration interaction using localized orbitals (PCILO ) method. Contributions from triple and quadruple excitations which decay back to singles and doubles at third order are automatically included in this theory. This method is computationally very fast, with an execution speed proportional to N³, Where N is the number of orbitals present. A [2,1] Padé approximate involving only singles and doubles contributions through to fifth order is shown to be remarkably accurate.     

Electron correlation effects in the 4f14 shell

This paper serves a twofold purpose. First, Löwdin's inner projection in both nonperturbative and perturbative forms is applied to the quartic anharmonic oscillator. Inner projection with perturbation theory yields rational approximations to Brillouin–Wigner-type perturbation expansions. These lower bounds are compared with [N ? 1, N] Padé approximants to the Rayleigh–Schrödinger perturbation series for this problem. These Padés are also expressible as the even convergents, w_2N, of a Stieltjes-type continued fraction. The latter representation has certain advantages with respect to its Padé counterpart. Inner projection without perturbation theory provides significantly better results than the perturbative version. The application of inner projection techniques to a perturbed hydrogen atom is not straightforward. The usual problems associated with the continuum spectrum of hydrogen are present. By means of a nonunitary “tilting” transformation associated with the Lie group SO(4, 2), these problems may be bypassed. In the SO(4, 2)-reformulated eigenvalue problem, a reinterpretation of the basic variables, as developed by Silverstone and Moats, yields a new Hamiltonian that permits direct use of the inner projection method. This method has been applied to the ground state of the hydrogen atom in a magnetic field, using both four- and eight-dimensional basis manifolds. This represents the first application of inner projection to this problem.     

Generalized Euler transformation in extracting useful information from divergent (asymptotic) perturbation series and the construction of Padé approximants

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, which works better. Viewed from this standpoint, the Padé approximants follow as a special case of the parametrized Euler transformation (PET ), as is the case with the μ 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 Padé approximants. The role of parametrization in the context of the Z^?1 perturbation theory of atoms is also noted and the workability of the adopted strategy is demonstrated by choosing some specific test cases.     

An approximate variational perturbation method

The Goldhammer—Feenberg variational procedure is applied to the perturbation treatment of the configuration interaction matrix of Diner, Malrieu and Claverie (PCILO) to yield a scheme (PVCILO) that appears more accurate for the small systems investigated. In addition, PVCILO shows promise as executing as an N² method, where N is the number of basis bonds.     

Resolvent technique and Padé approximants in configuration interaction calculations

A perturbation approach based on resolvent technique and Padé approximants is proposed. The eigenvalue of interest is part of a solution of two nonlinear algebraic equations. The nonlinear equations are arrived at by considering two different expression of the expectation value of the resolvent of an outer projection of the Hamiltonian. The first expression is based on the spectral resolution of the resolvent, and the second one is obtained by a power series expansion analogous to that applied in the derivation of the energy expression in the Brillouin–Wigner perturbation theory. The truncated power series is extrapolated by Padé approximants of type II. The method is tested on a CI calculation of the energy of the lowest ¹Σ state of the B₂ molecule.     

Configuration interaction calculations on the nitrogen molecule

A contracted Gaussian basis set capable of describing about 63% of the correlation energy of N₂ has been used in a detailed configuration-interaction calculation. Second-order perturbation theory overestimated the correlation energy by 23–50% depending on how H⁰ was chosen. Pair-pair interaction affects the correlation energy by about 20% while quadruple excitations have an 8% effect.     

Length dependence of excitation energies in linear polyenes: Localized and delocalized descriptions

The length dependence of the lowest allowed transition energy of linear polyenes is studied using delocalized SCF and localized excitonic approaches. Within the PPP SCF approximations the calculated transition energies converge to a finite values as N^?1 as the number of double bonds (N) becomes large, when the excited state contains all singly excited configurations. On the other hand, the fully localized excitonic method at the level of single excitations, although it predicts a gap in the excitation spectrum of an infinite polyene, gives results which converge to this value as N^?2. The inclusion of double and triple excitations into the excitonic method by means of perturbation theory does not appear to change this behavior. The reasons for the discrepancy between the two approaches is analyzed. Experimentally, the transition energies in solution converge to a finite value as N^?1 to a good degree of approximation. If it is assumed that the solvent shift is constant for long polyenes, the available experimental results favour the delocalized approach as the starting point in describing the length dependence of the excitation energy of long polyenes.     

Comparison of high-order many-body perturbation theory and configuration interaction for H2O

Diagrammatic many-body perturbation theory, coupled with a recursive computational procedure, is employed to obtain the correlation energy of H₂O within a 39-STO basis set by evaluating all double-excitation diagrams through twelfth order without any approximations. This provides, for the first time, the complete double-excitation diagrams contributions to the correlation energy, which is ?0.28826 hartree, compared with a correlation energy of ?0.27402 hartree obtained from a configuration interaction calculation which includes all double excitations. The difference of 0.0142 hartree includes the “size consistency” correction to the all-double-excitations CI energy, due to the “pathological” unliked-diagram terms remaining in that result, but also involves certain fourth- and higher-order rearrangement diagrams. Contrary to conventional belief, the unshifted, or Møller-Plesset partitioning of the hamiltonian provides a much more rapid convergence of the perturbation series that does the shifted, or Epstein-Nesbet partitioning. In both cases. Padé approximants enhance the convergence of the series considerably. A simple variation-perturbation scheme based on the first-order MBPT wavefunction is sufficient to provide 97.5% of the all-doubles CI correlation energy.     

A PCILO Study of Conformation and Internal Rotation in Mono-substituted Benzenes

The effect of Kékulé representation and hybrid function of O-atoms in the PCILO-CNDO framework of conformation and internal rotation in mono-sub stituted benzenes Ph-X (X?NH₂, OH, OCH₃, CH₃, CHO, NO₂) is studied. Three variational criteria for the choice of the appropriate third-order energy, proposed to symmetrize the PCILO results, are critically examined in relation with the height of rotational barrier in these molecules. The study shows that, in all cases, the most stable conformation is qualitatively correct predicted by the PCILO method. Since the barrier to internal rotation in the studied aromatic systems arises predominantly from delocalization effect, it is proposed to employ the arithmetic mean of the third-order energy of the two Kékulé structures. In molecules, in which the third-order energy between the two Kékulé structures is larger than 2 kcal/mol, however, the lower third-order energy representation alone seems to be appropriate. In phenol and anisole the spa-hybridization type of the O-atoms offers better values of rotational barrier, whereas in the sp³-type the delocalization is overestimated in the planar conformation.     

Quadratic Padé approximants and the intruder state problem of multireference perturbation methods

Simple and quadratic Padé resummation methods are applied to high‐order series from multireference many‐body perturbation theory (MR‐MBPT) calculations using various partitioning schemes (Møller–Plesset, Epstein–Nesbet, and forced degeneracy) to determine their efficacy in resumming slowly convergent or divergent series. The calculations are performed for the ground and low‐lying excited states of (i) CH₂, (ii) BeH₂ at three geometries, and (iii) Be, for which full configuration interaction (CI) calculations are available for comparison. The 49 perturbation series that are analyzed include those with oscillatory and monotonic divergence and convergence, including divergences that arise from either frontdoor or backdoor intruder states. Both the simple and quadratic Padé approximations are found to speed the convergence of slowly convergent or divergent series. However, the quadratic Padé method generally outperforms the simple Padé resummation. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Diagrammatic perturbation theory: Potential curves for the ground state of the carbon monoxide molecule

Diagrammatic many-body perturbation theory is used to calculate the potential energy function for the X¹ σ+ state of the CO molecule near the equilibrium nuclear configuration. Spectroscopic constants are derived from a number of curves which are obtained from calculations taken through third order in the energy. By forming [2/1] Padé approximants to the constants we obtain: r_e = 1.125 Å (1.128 Å), B_e = 1.943 cm^?1 (1.9312 cm^?1), a^B_e = 0.0156 cm^?1 (0.0175 cm^?1), W_e = 2247 cm^?1 (2170 cm^?1), W_eX_e = 12.16 cm^?1 (13.29 cm^?1), where the experimental values are given in parenthesis.     

Many-body perturbation theory,coupled-pair many-electron theory,and the importance of quadruple excitations for the correlation problem

Many-body (diagrammatic) perturbation theory (MBPT ), coupled-pair many-electron theory (CPMET ), and configuration interaction (CI ) are investigated with particular emphasis on the importance of quadruple excitations in correlation theories. These different methods are used to obtain single, double, and quadruple excitation contributions to the correlation energy for a series of molecules including CO₂, HCN, N₂, CO, BH₃, and NH₃. It is demonstrated that the sum of double and quadruple excitation diagrams through fourth-order perturbation theory is usually quite close to the CPMET result for these molecules at equilibrium geometries. The superior reliability of the CPMET model as a function of internuclear separation is illustrated by studying the ¹∑ potential curve of Be₂. This molecule violates the assumption common to nondegenerate perturbation theory that only a single reference function is important and this causes improper behavior of the potential curve as a function of R. This is resolved once the quadruple excitation terms are fully included by CPMET .     

Electrostatic energy of polygonal charge distributions

An asymptotic series for the electrostatic energy E₁(N){\mathcal{E}_1(N)} of an N-gonal charge distribution, i.e., a set of unit charges occupying vertices of a regular N-gon with a unit circumradius, is derived. Application of Padé approximants to truncations of this expansion produces compact approximate formulae capable of estimating E₁(N){\mathcal{E}_1(N)} with great accuracy. A closed-form expression for the energy of electrostatic interaction of two polygonal charge distributions is obtained from the respective Fourier series. The availability of this expression allows for a rapid calculation of the relevant energy with computational effort independent of the numbers of particles involved.     

On a new partitioning of the Hamiltonian in many-body calculation of pair-correlation energies in closed-shell systems

We introduce here a new partitioning of the Hamiltonian in calculating pair-correlation energies using many-body perturbation theory, by which we are able to eliminate the off-diagonal particle–hole (p–h) ladders exactly to all orders in the perturbation expansion. In this formulation, the particle states turn out to be different for each distinct pair of hole states in the correlation energy calculation. We have also included the contributions of the diagonal particle–particle (p–p) and hole–hole ladders exactly to all orders. The effect of the off-diagonal p–p ladders has been estimated for each pair by computing the third-, foruth- and fifth-order energies. For highly symmetric systems the present partitioning yields in general symmetry-broken orbitals. Here one may use an average kind of partitioning for all the partners of the degenerate sets, which restores the symmetry and at the same time ensures cancellation of the p–h ladders exactly at the lowest order and approximately at the higher orders. Results are presented for a selection of 6π-electron conjugated systems. The correlation energy for each pair is in excellent agreement with that obtained from a partial CI calculation involving all double excitations from this pair. The advantages of implementing the present scheme in larger systems has been discussed.     

Ab initio molecular fragment calculations with pseudopotentials: Model peptide studies

The energetics of rotation about the N? C′(ω); N? C^α(φ), and C^α? C′(?) bonds of the peptide unit have been investigated in the pseudo-FSGO fragment scheme on model compounds formamide and N-methylacetamide. The results indicated that the position of the minimum in ω is in the near vicinity of 0°, i.e., the planar arrangement of the peptide unit. The minimum in φ (C′? N? C^α? H) has been found to be 180° and in ψ(H? C^α? C′? N) to be 60°, in good agreement with PCILO and Gaussian-70 results.     

Fourth-order diagrammatic MB–RSPT calculations of the correlation energy: N2, CO,F2 and the reaction energy of the process ½F2 + ½H2 = HF

The correlation energies obtained by the fourth-order diagrammatic perturbation theory were analyzed for three diatomic molecules: N₂, CO, and F₂. The results were compared with correlation energies obtained previously for the ten-electron hydrides HF, H₂O, and NH₃. The relative importance of contributions which arise from the double excitations, from the quadruple excitations, as well as from the renormalization term was investigated. It is shown that for the diatomic molecules under study these contributions are considerably larger than for the ten-electron hydrides not only in absolute value but also in percentage: they represent about 3, 3, and 5%, respectively, of the valence shell correlation energy obtained by the perturbation theory up to the fourth order. A careful analysis of the fourth-order correlation effects is also presented for the reaction energy of the process ½H₂ + ½F₂ = HF.     

Intermolecular and intramolecular interactions calculated with ab initio perturbative configuration interaction method using strongly localized orbitals

We have applied the ab initio formulation of the perturbative configuration interaction using localized orbitals (PCILO ) method up to third order to calculate intermolecular and intramolecular interaction energies going beyond the ab initio Hartree–Fock calculation. For the rotational barrier in ethane our results agree well with the experimental value and the cis- and even the trans-barriers in HOOH are at least qualitatively reproduced with the aid of the STO -3G basis set. In the case of the water dimer we obtain an equilibrium intermolecular distance and interaction energy which are confirmed by other calculations. We can further conclude from our studies that one has to include higher orders in the perturbation expansion as the system becomes more complicated. It is especially the last aspect which hinders the application of the ab initio PCILO to estimate the major part of the electron correlation energy for large molecules.     

Theoretical study of the relative stabilities of H+(X)2 and H+(X)3 conformers and their clustering energies: X  CO and N2

Third-order Møller–Plesset perturbation theory (MP 3) with a 6-31G** basis set was applied to study the relative stabilities of H⁺(X)₂ conformations (X ? CO and N₂) and their clustering energies. The effect of both basis set extensions and electron correlation is not negligible on the relative stabilities of the H⁺(CO)₂ clusters. The most stable conformation of H⁺(CO)₂ is found to be a C_∞v structure in which a carbon atom of CO bonds to the proton of H⁺(CO), whereas that of H⁺(N₂)₂ is a symmetry D_∞h structure. The second lowest energy conformations of H⁺(CO)₂ and H⁺(N₂)₂ lie within 2 kcal/mol above the energies of the most stable structures. Clustering energies computed using MP 3 method with the 6-31G** basis set are in good agreement with the experimental findings of Hiraoka, Saluja, and Kebarle. The low-lying singlet conformations of H⁺(X)₃ (X ? CO and N₂) have been studied by the use of the Hartree–Fock MO method with the 6-31G** basis set and second-order Møller–Plesset perturbation theory with a 4-31G basis set. The most stable structure is a T-shaped structure in which a carbon atom of CO (or a nitrogen atom of N₂) attacks the proton of the most stable conformation of H⁺(X)₂ clusters.     

Thomas–Fermi theory from a perturbative treatment of the hydrogenic limit energy density functional: A possible link to local density functional theory

A perturbation expansion which connects the hydrogenic limit energy density functional to the Thomas–Fermi functional is discussed. This perturbation series, where the Coulomb energy density functional is treated as the perturbation to the hydrogenic limit functional, is, in fact, the q = (N/Z) expansion of Thomas–Fermi theory. A truncated form of the first-order correction to the functional provides further insight into the model which treats the ground state energy as a local functional of the electron density.