Coupled perturbation theory within the antisymmetrized product of separated geminals (APSG) framework

A coupled perturbation theory for the antisymmetrized product of separated geminals (APSG ) approximation is derived. The variational principle for the APSG wave function in an external oscillating field is employed and a set of equations of the form analogous to the normal RPA is obtained. At this level the reduced resolvent in the form of a spectral expansion is written and it is used for the evaluation of the second-order properties.     

Solid phase thermodynamic perturbation theory: test and application to multiple solid phases

A simple procedure for the determination of hard sphere (HS) solid phase radial distribution function (rdf) is proposed, which, thanks to its physical foundation, allows for extension to other crystal structures besides the fcc structure. The validity of the procedure is confirmed by comparing (1) the predicted HS solid phase rdf's with corresponding simulation data and (2) the predicted non-HS solid phase Helmholtz free energy by the present solid phase first-order thermodynamic perturbation theory (TPT) whose numerical implementation depends on the HS solid phase rdf's as input, with the corresponding predictions also by the first-order TPT but the required HS solid phase rdf is given by an "exact" empirical simulation-fitted formula. The present solid phase first-order TPT predicts isostructural fcc-fcc transition of a hard core attractive Yukawa fluid, in very satisfactory agreement with the corresponding simulation data and is far more accurate than a recent thermodynamically consistent density functional perturbation theory. The present solid phase first-order TPT is employed to investigate multiple solid phases. It is found that a short-ranged potential, even if it is continuous and differentiable or is superimposed over a long-ranged potential, is sufficient to induce the multiple solid phases. When the potential range is short enough, not only isostructural fcc-fcc transition but also isostructural bcc-bcc transition, simple cubic (sc)-sc transition, or even fcc-bcc, fcc-sc, and bcc-sc transitions can be induced. Even triple point involving three solid phases becomes possible. The multiple solid phases can be stable or metastable depending on the potential parameters.     

Extended multi-configuration quasi-degenerate perturbation theory: the new approach to multi-state multi-reference perturbation theory

The distinctive desirable features, both mathematically and physically meaningful, for all partially contracted multi-state multi-reference perturbation theories (MS-MR-PT) are explicitly formulated. The original approach to MS-MR-PT theory, called extended multi-configuration quasi-degenerate perturbation theory (XMCQDPT), having most, if not all, of the desirable properties is introduced. The new method is applied at the second order of perturbation theory (XMCQDPT2) to the 1(1)A(')-2(1)A(') conical intersection in allene molecule, the avoided crossing in LiF molecule, and the 1(1)A(1) to 2(1)A(1) electronic transition in cis-1,3-butadiene. The new theory has several advantages compared to those of well-established approaches, such as second order multi-configuration quasi-degenerate perturbation theory and multi-state-second order complete active space perturbation theory. The analysis of the prevalent approaches to the MS-MR-PT theory performed within the framework of the XMCQDPT theory unveils the origin of their common inherent problems. We describe the efficient implementation strategy that makes XMCQDPT2 an especially useful general-purpose tool in the high-level modeling of small to large molecular systems.     

Sensitivity analysis of state-specific multireference perturbation theory

State-specific multireference perturbation theory (SS-MRPT) developed by Mukherjee et al. [Int. J. Mol. Sci. 3, 733 (2002)] is examined focusing on the dependence of the perturbed energy on the initial model space coefficients. It has been observed earlier, that non-physical kinks may appear on the potential energy surface obtained by SS-MRPT while related coupled-cluster methods may face convergence difficulties. Though exclusion or damping of the division by small coefficients may alleviate the problem, it is demonstrated here that the effect does not originate in an ill-defined division. It is shown that non-negligible model space coefficients may also be linked with the problem. Sensitivity analysis is suggested as a tool for detecting the coefficient responsible. By monitoring the singular values of sensitivity matrices, orders of magnitude increase is found in the largest value, in the vicinity of the problematic geometry point on the potential energy surface. The drastic increase of coefficient sensitivities is found to be linked with a degeneracy of the target root of the effective Hamiltonian. The nature of the one-electron orbitals has a profound influence on the picture: a rotation among active orbitals may screen or worsen the effect.     

An application of perturbation theory ideas in configuration interaction calculations

Configuration interaction calculations of electronic wave functions for atoms and molecules have generally been limited to relatively small basis sets because of the exponential increase in the number of configurations as basis functions are added. While higher than quadruply excited configurations are of negligible importance in CI wave functions, it is shown that the effect of triple and quadruple excitation configurations can be substantially included even when the matrix elements between such configurations are neglected, leaving only their diagonal elements and the elements connecting them with the single and double excitations. This approximation is seen to be formally practically equivalent to a first-order perturbation expression for the wave function (second-order for the energy) based on an optimum linear combination of the zero, single, and double excitation configurations as the zero-order function. If suitable procedures are used, the amount of computational effort involved in such a calculation is roughly proportional to the fourth power of the number of basis functions employed, thus preventing the CI stage of the calculation from increasing in magnitude much faster than the stages involving the calculation and manipulation of the elementary integrals.     

The application of strictly localized geminals to the description of chemical bonds

A theory is developed in which closed-shell molecules are viewed as systems of weakly interacting chemical bonds. Composite-particle creation operators obtained by an appropriate quasiparticle transformation are used to create the wave function of two-electron bonds. These quasiparticles are bosons, since they are composed of two electrons, but the total many-electron wave function is properly antisymmetric. The internal structure of the quasi-Bose-particles is affected by inductive interbond interactions. Delocalization and dispersion interactions between different bonds are neglected, thus the approach corresponds to a first-order many-body PT (Perturbation Theory) with a correlated, but fully localized, reference state. The whole formalism is developed ab initio. The nonorthogonality problem is handled by a biorthogonal formulation. To illustrate the effectiveness of the model, numerical calculations are reported.     

Elimination of translational and rotational motions in nuclear orbital plus molecular orbital theory: application of Moller-Plesset perturbation theory

The translation- and rotation-free nuclear orbital plus molecular orbital (TRF-NOMO) theory was developed to determine the nonadiabatic nuclear and electronic wave functions. This study presents a formulation of TRF-NOMO second-order Moller-Plesset (MP2) perturbation and Epstein-Nesbet (EN) theory with the use of the TRF Hamiltonian. Numerical assessment of the TRF-NOMO/MP2 and EN is performed for several molecules. We confirm the importance of the elimination of translational and rotational motions in the many-body calculations.     

External coupled-cluster perturbation theory: description and application to weakly interaction dimers. Corrections to the random phase approximation

The formalism for developing perturbation theory by using an arbitrary fixed (external) set of amplitudes as an initial approximation is presented in a compact form: external coupled-cluster perturbation theory (xCCPT). Nonperturbative approaches also fit into the formalism. As an illustration, the weakly interacting dimers Ne(2) and Ar(2) have been studied in the various ring-coupled-cluster doubles (CCD) approximations; ring, direct-ring, antisymmetrized ring, and antisymmetrized direct ring, and a second-order correction in the xCCPT approach is added. The direct approaches include the summation of just Coulomb terms with the intention of selectively summing the largest terms in the perturbation first. "Coulomb attenuation" is effected by taking the random phase approximation to define such amplitudes, whose results are then improved upon using perturbation theory. Interaction energies at the ring-CCD level are poor but the xCCPT correction employed predicts binding energies which are only a few percent from the coupled-cluster single double (triple) values for the direct ring-CCD variants. Using the MP2 amplitudes which neglect exchange, the initial Coulomb-only term, leads to very accurate Ne(2) and Ar(2) potentials. However, to accurately compute the Na(2) potential required a different initial wavefunction, and hence perturbation. The potential energy surfaces of Ne(2) and Ar(2) are much too shallow using linear coupled-cluster doubles. Using xCCPT(2) with these amplitudes as the initial wavefunction led to slightly worse results. These observations suggest that an optimal external set of amplitudes exists which minimizes perturbational effects and hence improve the predictability of methods.     

An application of second-order n-electron valence state perturbation theory to the calculation of excited states

n–electron valence state perturbation theory (NEVPT) is a form of multireference perturbation theory where all the zero-order wave functions are of multireference nature, being generated as eigenfunctions of a two–electron model Hamiltonian. The absence of intruder states makes NEVPT an interesting choice for the calculation of electronically excited states. Test calculations have been performed on several valence and Rydberg transitions for the formaldehyde and acetone molecules; the results are in good accordance with the best calculations and with the existing experimental data.Contribution to the Jacopo Tomasi Honorary Issue     

Multiphoton ionization by intense fields: Corrections to lowest order perturbation theory,with an application to photoionization of H

We consider the perturbation expansion (in powers of the coupling strength of the electron interaction with the radiation field) of the standard time-independent expression for the multiphoton ionization amplitude. We show that corrections beyond lowest nonvanishing order contain divergent diagrams. These divergences cancel in the velocity gauge but not in the length gauge. We discuss the mathematical and physical nature of these divergences, and we present an application to the photoionization of hydrogen.     

Degenerate symmetry-adapted perturbation theory of weak interactions between closed- and open-shell monomers: application to Rydberg states of helium hydride

Symmetry-adapted perturbation theory is extended to the (quasi) degenerate, open-shell case. The new formalism is tested in calculations of the interaction energies for a helium atom in the ground state interacting with an excited hydrogen atom. It is shown that the method gives satisfactory results if the coupling with higher Rydberg states of the dimer is small, as is the case for the A²Σ⁺,B²Π,E²Π,3²Π, and 1²Δ states of HeH. For the C²Σ⁺ state convergence of the method is very slow, but it can be improved by including the n=3 states in the model space. Received: 3 June 1998 / Accepted: 9 September 1998 / Published online: 7 December 1998     

Application of SCF perturbation theory to molecular calculations

A method for solving Roothaan's molecular orbital equations by means of SCF perturbation theory is presented. An estimate of the accuracy of the third order expansion is made for the CNDO/2 approximation from a comparison of the results from direct calculations. It is found that the third order theory is sufficiently accurate for quantitative studies.     

Discrete perturbation theory applied to Lennard-Jones and Yukawa potentials

Discrete perturbation theory (DPT) is a powerful tool to study systems interacting with potentials that are continuous but can be approximated by a piecewise continuous function composed of horizontal segments. The main goal of this work is to analyze the effect of several variables to improve the representation of continuous potentials in order to take advantage of DPT. The main DPT parameters chosen for the purpose are the starting location and size of the horizontal segments used to divide the full range of the potential and its maximum reach. We also studied the effect of having each segment aligned to the left, to the right, or centered on the continuous function. The properties selected to asses the success of this strategy are the orthobaric densities and their corresponding critical points. Critical parameters and orthobaric densities were evaluated by DPT for each of an ample set of variables and compared with their values calculated via discontinuous molecular dynamics. The best sets of DPT parameters are chosen so as to give equations of state that represent accurately the Lennard-Jones and Yukawa fluids.     

A perturbation analysis of rate theory of self-regulating genes and signaling networks

A thorough kinetic analysis of the rate theory for stochastic self-regulating gene networks is presented. The chemical master equation kinetic model in terms of a coupled birth-death process is deconstructed into several simpler kinetic modules. We formulate and improve upon the rate theory of self-regulating genes in terms of perturbation theory. We propose a simple five-state scheme as a faithful caricature that elucidates the full kinetics including the "resonance phenomenon" discovered by Walczak et al. [Proc. Natl. Acad. Sci. U.S.A. 102, 18926 (2005)]. The same analysis can be readily applied to other biochemical networks such as phosphorylation signaling with fluctuating kinase activity. Generalization of the present approach can be included in multiple time-scale numerical computations for large biochemical networks.     

Diagrammatic analysis of adiabatic and time-independent perturbation theory for degenerate energy levels

Time-ordered folded diagrams are used to represent the effective hamiltonian in the adiabatic formalism. Resummation of the diagrams is shown to give a term-by-term correspondence with time-independent perturbation theory.