Study of the convergence radius of the Rayleigh–Schrödinger perturbation series for the delta-function model of H

The convergence radius of the series expansion for the energy of H₂⁺ in the δ-function model (in terms of the perturbation parameter μ/λ, where μ is the charge of the perturbing nucleus and λ the fixed charge of the other nucleus) is investigated. A lower bound of this convergence radius (possibly equal to it) previously defined by Robinson [5] is studied analytically as a function of the internuclear distance R and computed numerically. The results differ strikingly from those previously obtained by Robinson who used a simplified but poorer lower bound: in contrast with this poorer bound, the one studied in the present paper is larger than for I every R, from which fact it may be concluded that, contrary to Robinson's previous result, the series expansion of the energy, in the δ-function model under consideration, still converges when μ = λ for every R.     

The problem of unphysical states in the theory of intermolecular interactions

It was shown by Claverie that the interactions between atoms and molecules make unphysical electronic solutions of the Schradinger equation accessible in perturbation calculations of intermolecular interactions, accessible in the sense that the perturbation expansion is likely to converge to an unphysical solution if it converges at all. This is a difficult problem because there are generally an infinite number of unphysical states with energies below that of the physical ground state. We have carried out configuration interaction calculations on LiH of both physical and unphysical states. They show that avoided crossings occur between physical and unphysical energy levels as the interaction between the two atoms is turned on, i.e. as the expansion parameter is increased from 0 to 1. The avoided crossing for the lowest energy state occurs for < 0.8, implying that the perturbation expansion will diverge for larger values of . The behavior of the energy levels as functions of . is shown to be understandable in terms of a two-state model. In the remainder of the paper, we concentrate on designing effective Hamiltonians which have physical solutions identical to those of the Schrödinger equation, but which have no unphysical states of lower energy than the physical ground state. We find that we must incorporate ideas from the Hirschfelder-Silbey perturbation theory, as modified by Polymeropoulos and Adams, to arrive finally at an effective Hamiltonian which promises to have the desired properties, namely, that all unphysical states be higher in energy than the physical bound states, that the first and higher order corrections to the energy vanish in the limitR = . that the leading terms of the asymptotic 1/R expansion of the energy be given correctly in second order, and that the overlap between the zeroth order wave function and the corresponding eigenfunction of the effective Hamiltonian be close to one.     

Unified treatment of chemical and van der Waals forces via symmetry-adapted perturbation expansion

We propose a symmetry-adapted perturbation theory (SAPT) expansion of the intermolecular interaction energy which in a finite order provides the correct values of the constants determining the asymptotics of the interaction energy (the van der Waals constants) and is convergent when the energy of the interacting system is submerged in the continuum of Pauli-forbidden states-the situation common when at least one of the monomers has more than two electrons. These desirable features are achieved by splitting the intermolecular electron-nucleus attraction terms of the Hamiltonian into regular (long-range) and singular (short-range) parts. In the perturbation theory development, the regular part is treated as in the conventional polarization theory, which guarantees the correct asymptotics of the interaction energy, while the singular part is weakened sufficiently by an application of permutational symmetry projectors so that a convergent perturbation series is obtained. The convergence is demonstrated numerically, for both the chemical and van der Waals minima, by performing high-order calculations of the interaction energy of the ground-state lithium and hydrogen atoms-the simplest system for which the physical ground state is submerged in the Pauli-forbidden continuum. The obtained expansion enables a systematic extension of SAPT calculations beyond second order with respect to the intermolecular interaction operator.     

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.     

Long-range interaction between 1s and 2s or 2p hydrogen atoms

Long-range interaction energy between two hydrogen atoms has been computed in the second order of the perturbation theory. All states of the system arising when one of the atoms is in the 1s and the other in the 2s or 2p state have been considered. The energy represented by a series expansion in inverse powers of the internuclear distance, R, has been computed up to the terms in R^?8. The results are believed to give reliable interaction energies for R > 15 a.u. Accurate interaction energy for two ground-state hydrogen atoms has also been obtained up to the terms in R^?10. Results for the B′ ¹∑ state are employed to discuss the experimental ground-state dissociation energy of H₂, D₂, and HD. For H₂ all values of the dissociation energy obtained from various experimental absorption limits, by using the computed potential energy curve to separate off the effect of rotation, are shown to be satisfactorily consistent. The resulting total energy of H₂ is, however, higher than the most accurate theoretical value.     

Asymptotic behavior of the primitive function of different “symmetry-adapted” perturbation schemes for the H ground state

The leading terms in the asymptotic 1/R expansion of the wave functions and energies of various “symmetry-adapted” perturbation schemes for intermolecular forces, as well as for the “polarization approximation” (PA ) are derived for the H ground state, both exactly (i.e., to infinite order in λ) and in the first two orders of the λ expansion. It is pointed out that only in the PA and the Hirschfelder–Silbey scheme is the formal primitive function Φ genuinely primitive, whereas Φ in the other schemes is asymmetric in a rather strange way. In this lack of genuine primitivity lies the reason why in these schemes the leading term of the 1/R expansion is only recovered to infinite order in λ and requires knowledge of the R^?4 term of the wave function, provided that one uses the “internal” energy expression. Four different energy expressions are compared that behave differently in the different schemes. The results obtained here are basis independent, but the implications of the basis completeness problem are discussed as well.     

A new parametrizable model of molecular electronic structure

A new electronic structure model is developed in which the ground state energy of a molecular system is given by a Hartree-Fock-like expression with parametrized one- and two-electron integrals over an extended (minimal + polarization) set of orthogonalized atom-centered basis functions, the variational equations being solved formally within the minimal basis but the effect of polarization functions being included in the spirit of second-order perturbation theory. It is designed to yield good dipole polarizabilities and improved intermolecular potentials with dispersion terms. The molecular integrals include up to three-center one-electron and two-center two-electron terms, all in simple analytical forms. A method to extract the effective one-electron Hamiltonian of nonlocal-exchange Kohn-Sham theory from the coupled-cluster one-electron density matrix is designed and used to get its matrix representation in a molecule-intrinsic minimal basis as an input to the parametrization procedure--making a direct link to the correlated wavefunction theory. The model has been trained for 15 elements (H, Li-F, Na-Cl, 720 parameters) on a set of 5581 molecules (including ions, transition states, and weakly bound complexes) whose first- and second-order properties were computed by the coupled-cluster theory as a reference, and a good agreement is seen. The model looks promising for the study of large molecular systems, it is believed to be an important step forward from the traditional semiempirical models towards higher accuracy at nearly as low a computational cost.     

Dynamic dipole polarizabilities of the ground and excited states of confined hydrogen atom computed by means of a mapped Fourier grid method

Accurate theoretical estimates of static and dynamic dipole polarizabilities are reported for the ground and excited states of hydrogen atom confined at the center of a spherical “box” with impenetrable walls using a novel theoretical algorithm that combines the variational–perturbation approach with an appropriately adapted mapped Fourier grid method and uniformly maintains the numerical accuracy. A variation of computed polarizabilities is observed as a function of the number of grid points. However, rapid convergence to their correct values, even far into the anomalous dispersion region, is achieved by an extrapolation procedure to the limit of an infinitely large number of grid points using a small number of the lowest‐order Padé approximants. It is shown that dipole polarizabilities strongly depend upon the electronic state and the radius of confinement. In particular, the static polarizability of 2s state changes sign under strong confinement. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Ab initio vibrational transition dipole moments and intensities of formaldehyde

Vibrational transition dipole moments and absorption band intensities for the ground state of formaldehyde, including the deuterated isotopic forms, are calculated. The analysis is based on ab initio SCF and CI potential energy and dipole moment surfaces. The formalism derives from second-order perturbation theory and involves the expansion of the dipole moment in terms of normal coordinates, as well as the incorporation of point group symmetry in the selection of the dipole moment components for the allowed transitions. Dipole moment expansion coefficients for the three molecule-fixed Cartesian coordinates of formaldehyde are calculated for internal and normal coordinate representations. Transition dipole moments and absorption band intensities of the fundamental, first overtone, combination, and second overtone transitions are reported. The calculated intensities and dipole moment derivatives are compared to experiment and discussed in the context of molecular orbital and bond polarization theory.     

Localized electron pair theory for the calculation of ground state energies of large molecules

The quantum mechanical energy is examined in which groups of one, two, three, and four localized electron pairs found within a molecule are separately computed. From these results, the interaction energies of the electron pairs taken one, two, three, and four at a time form the terms of a convergent molecular mechanics like expansion of the molecular ground state energy. This procedure can be used with any size consistent quantum mechanical method. The computational time for large molecules depends chiefly upon the order needed in the energy expansion to obtain sufficient convergence and not on the particular quantum mechanical method used. Preliminary results within the framework of a semiempirical CNDO/2 model Hamiltonian show at the Hartree–Fock and Møller–Plesset perturbation levels that relative energies converge to within a few tenths of a kcal/mol of the exact values at the four body level for molecules that have little delocalization. In strained ring and aromatic systems, convergence is however not nearly as rapid. Results can be improved somewhat by using larger interacting fragments containing two or more electron pairs over three or more atomic centers. © 1992 by John Wiley & Sons, Inc.     

Ab initio Hartree-Fock calculations of electronic wave functions for the c 3πu state of H2

Theoretical electronic wave functions, potential curves, and expectation values of some one-electron properties are given for the c³II_u state of the hydrogen molecule. The calculations are carried out by the matrix Hartree-Fock method and use a 2-center basis of Slater-type orbitals. A total energy of ?0.7292 a.u. is obtained in the best calculation. Our potential curve is reasonably consistent with that calculated by Browne, but we have examined the region of small internuclear distances (those at and below R_e for the ground state) more extensively than any previous calculation. At R ≦ 1.6 a.u. our calculated potential curve is in excellent agreement with experiment.     

A new single-center perturbation theory for the treatment of simple hydrides

The perturbation expansion of Hartree-Fock potential energy surfaces of simple hydrides in a single-center scale 1/Z-expansion is carried out using diagrammatic methods. Diagrams having the topological structure of third-order diagrams are summed to all orders using an integral equation technique. The resulting one-electron wavefunction is used to generate a resummed perturbation series correct through fourth order.     

Singularities of Møller-Plesset energy functions

The convergence behavior of M?ller-Plesset (MP) perturbation series is governed by the singularity structure of the energy, with the energy treated as a function of the perturbation parameter. Singularity locations, determined from quadratic approximant analysis of high-order series, are presented for a variety of atoms and small molecules. These results can be used as benchmarks for understanding the convergence of low-order methods such as MP4 and for developing and testing summation methods that model the singularity structure. The positions and types of singularities confirm previous qualitative predictions based on functional analysis of the Schrodinger equation.     

Convergence of multipole expanded intermolecular interaction energies for Gaussian-type-function and Slater-type-function basis sets

 It is shown that the multipole expansion of each order of the polarization series converges for large enough intermolecular distances when finite basis sets of Gaussian or Slater-type functions are used to approximate molecular response properties. Convergence of the multipole expansion for each order of the polarization series does not imply convergence of the polarization series itself. A corresponding convergence condition is extracted from the general perturbation theory in a finite-dimensional space and is applied to the H + H⁺ problem. Received: 29 September 1999 / Accepted: 22 May 2000 / Published online: 18 August 2000     

Configuration interaction benchmark for Be ground state

A set of 432 energy-optimized Slater-type radial orbitals together with spherical harmonics up to ? = 30 is used to approximate the corresponding full configuration interaction (CI) expansion for Be ground state. An analysis of radial and angular patterns of convergence for the energy yields a basis set incompleteness error of 8.7 μhartree of which 85% comes from radial basis truncations for ? ≤ 30. Select-divide-and-conquer CI (Bunge in J Chem Phys 125:014107, 2006; Bunge and Carbó-Dorca in J Chem Phys 125:014108, 2006) produces an energy upper bound 0.02(1) μhartree above the full CI limit. The energy upper bound E = ?14.6673473 corrected with these two truncation energy errors yields E = ?14.6673560 a.u. (Be) in fair agreement with the latest explicitly correlated Gaussian results of E = ?14.66735646 a.u. (Be). The new methods employed are discussed. It is acknowledged that at this level of accuracy traditional atomic CI has reached a point of diminishing returns. Modifications of conventional (orbital) CI to seek for significantly higher accuracy without altering a strict one-electron orbital formalism are proposed.     

Long- and intermediate-range interaction in three lowest sigma states of the HeH+ ion

Accurate variational energies have been calculated for three lowest sigma states of the HeH⁺ ion. This includes the ground state (5 ≤ R ≤ 9 a.u.) which dissociates into He + H⁺, as well as the A ¹Σ⁺ state (4 ≤ R ≤ 10) and the a ³Σ⁺ state (3 ≤ R ≤ 10) which both dissociate into He⁺ + H. The variational results are compared with those obtained using a perturbation theory expansion.     

Autoionization spectra of Li2 and the XΣg ground state of Li2: Experimental and theoretical investigations

Autoionizing Rydberg levels of Li₂ molecules in a supersonic molecular beam are populated by stepwise excitation with two tunable pulsed dye lasers. The observed autoionization spectra show severe perturbations. Based on calculations of quantum defects and a perturbation treatment of l-uncoupling a tentative assignment of Rydberg series up to n = 32 is proposed. The convergence limits of these series yield a value of IP = 41475 cm⁻¹ for the adiabatic ionization potential and a vibrational constant ω_e = 263 cm⁻¹ for the X²Σ⁺_g ground state of Li⁺₂. The experimental results are compared with ab initio calculations combined with a core polarization potential, which yield the potential curve. the dissociation energy, the quadrupole moment and the vibrational frequency for the X²Σ⁺_g ground state of Li⁺₂, in the excellent agreement with experimental findings.     

On the nature of the Møller-Plesset critical point

It has been suggested [F. H. Stillinger, J. Chem. Phys. 112, 9711 (2000)] that the convergence or divergence of M?ller-Plesset perturbation theory is determined by a critical point at a negative value of the perturbation parameter z at which an electron cluster dissociates from the nuclei. This conjecture is examined using configuration-interaction computations as a function of z and using a quadratic approximant analysis of the high-order perturbation series. Results are presented for the He, Ne, and Ar atoms and the hydrogen fluoride molecule. The original theoretical analysis used the true Hamiltonian without the approximation of a finite basis set. In practice, the singularity structure depends strongly on the choice of basis set. Standard basis sets cannot model dissociation to an electron cluster, but if the basis includes diffuse functions then it can model another critical point corresponding to complete dissociation of all the valence electrons. This point is farther from the origin of the z plane than is the critical point for the electron cluster, but it is still close enough to cause divergence of the perturbation series. For the hydrogen fluoride molecule a critical point is present even without diffuse functions. The basis functions centered on the H atom are far enough from the F atom to model the escape of electrons away from the fluorine end of the molecule. For the Ar atom a critical point for a one-electron ionization, which was not previously predicted, seems to be present at a positive value of the perturbation parameter. Implications of the existence of critical points for quantum-chemical applications are discussed.     

Convergence radii of the perturbation expansions for the ground-state energies of finite Hubbard models

The ground state energies of finite Hubbard molecules are calculated by numerically solving the Lieb–Wu equations for a complex Hubbard repulsion parameter U. From the positions of the singular points located in the complex plane, the radii of convergence of the perturbation expansions for the ground state energies are determined.