Scalable implementation of analytic gradients for second-order Z-averaged perturbation theory using the distributed data interface

The analytic gradient expression for second-order Z-averaged perturbation theory is revised and its parallel implementation is described in detail. The distributed data interface is used to access molecular-orbital integral arrays stored in distributed memory. The algorithm is designed to maximize the use of local data and reduce communication costs. The iterative solution and the preconditioner used to induce the convergence of the coupled-perturbed Hartree-Fock equations are presented. Several illustrative timing examples are discussed.     

A note on the convergence of multiconfigurational many-body perturbation theory

The convergence of multiconfigurational many-body perturbation theory (MC MBPT ) is discussed in connection with the intruder state. Its convergence properties are first examined with a fictitious three-level system employing a Hermitian version of MC MBPT , which permits a general model space. It is then applied to the H₂—H₂ and N₂ systems. The results suggest that a more extensive model space is likely to embrace new intruder states and the space extension be executed with due caution.     

On perturbation graph theory

A perturbation theoretical formalism is developed which enables the calculation of the (topological) resonance energy of arbitrary heteroconjugated π-electron systems. The previous method of Herndon and Párkányi is thus generalized.     

Accuracy and convergence of higher order distorted wave perturbation theory

A distorted wave perturbation theory which allows for a different distortion potential for each channel is investigated. The method is closely related to a perturbative solution of close coupled equations. Improvement in first order is found. However, higher order terms are not significantly improved over other computationally more convenient approaches. For purposes of comparison, numerical results are given for the oscillator atom collinear collision. The convergence of series solutions is studied as a function of scattering parameters. The accuracy of several approximation schemes is also investigated.     

On the convergence of the Møller-Plesset perturbation series

Møller-Plesset perturbation energies, up to as much as 48th order, have been calculated for H₂O (RHF and UHF framework) and NH₂ (UHF framework) atC _2v geometries (r _e , 1.5r _e and 2r _e ). Atr _e , the RHF and UHF series rapidly converge, but at2r _e , the RHF series converges erratically with an energy at 43rd order within 10^–7 hartree of the exact value whereas the UHF series converges smoothly, but very slowly, and for H₂O has an error in excess of 10^–5 hartree at 48th order. The significance of these results is discussed.     

On the performance of the open-shell perturbation theory

A few open-shell molecules are taken as examples in order to examine the performance of the open-shell perturbation theory for electron correlation(J Chem Theory Comput,2009,5:931–936).The convergence of the perturbation series is shown to be stable for the doublet state of NH2 at both the equilibrium and stretched geometries.The equilibrium bond lengths,and harmonic and anharmonic vibrational frequencies are calculated for NO(X2),OH(X2),CH(X2)and NH(X2)with different second-order perturbation theories at t...     

On the choice of unperturbed hamiltonians and the convergence of perturbation series

Perturbation theory for excited states is often only applicable in a restricted way due to the intruder-state problem. For the ¹Σ_g⁺ spectra of the hydrogen molecule four different decompositions of the Hamiltonian H into H₀ + V are examined. It is shown that the intruder-state problem heavily depends on the choice of H₀ and that with a scaled Hartree-Fock potential as a starting point, it is possible to calculate the first five states perturbatively.     

On the relation between Peierls' perturbation theory for projected states and exchange perturbation theory

The derivations of a rather general class of exchange perturbation theories (or symmetry-adapted perturbation theories) and Peierls' perturbation theory for projected states are discussed. While the approaches are different, the equivalence of the resulting formalisms is established.     

On a formal treatment of the Rayleigh-Schrödinger perturbation theory

The formal treatment of the Rayleigh-Schrödinger perturbation theory based on a first-order iteration procedure as described in a previous publication is discussed with reference to the properties of the terms of a Taylor series. The formalism is generalized to allow for multiple perturbation.     

Stationary perturbation theory

Summary After a short recapitulation of the basic concepts of stationary perturbation theory, this is applied to a many-electron Hamiltonian, with or without an external field, given in a Fock space formulation in terms of a finite basis, the exact eigenfunctions of which are the full-CI wave functions. The Lie algebra _c ⁿ of the variational group corresponding to this problem is presented. It has an important subalgebra _c ⁽¹⁾ of one-particle transformations. Hartree-Fock and coupled Hartree-Fock (also uncoupled Hartree-Fock) as well as MC-SCF and coupled MC-SCF are outlined in this framework. Many-body perturbation theory and Møller-Plesset perturbation theory are derived from the same kind of stationarity condition and a new non-perturbative iteration construction of the full-CI wave function is proposed, the first Newton-Raphson iteration cycle of which is CEPA-0. For the treatment of electron correlation for properties two variants of Møller-Plesset theory referred to as coupled (CMP) and uncoupled (UCMP) are defined, neither of which is fully satisfactory. While CMP satisfies a Brillouin condition, which implies that first order correlation corrections to first- and second-order properties vanish, it does not satisfy a Hellmann-Feynman theorem, i.e. a first order property isnot the expectation value of the operator associated with the property. Conversely UCMP satisfies a Hellmann-Feynman theorem but no Brillouin theorem. The incompatibility of the two theorems is related to an unbalanced treatment of one-particle- and higher excitations in MP theory. CMP, which is based on coupled Hartree-Fock as uncorrelated reference, appears to have slight advantages over UCMP, but neither variant looks very promising for the evaluation of 2nd order correlation corrections to 2nd-order properties. Then four variants of the perturbation theory of properties with a nonperturbative treatment of electron correlation on CEPA-0 level (but extendable to a higher level) are discussed. While those variants which are the direct counterpart of UCMP and CMP must be discarded, the perturbative CEPA-0 derived from a perturbative treatment on full-CI level appears to satisfy all important criteria, in particular it satisfies a Brillouin-Brueckner condition and a Hellmann-Feynman theorem. A simplified version, the coupled Brillouin-Brueckner CEPA-0 appears to have essentially the same qualities. It is important to replace the Brillouin condition of MP theory by the Brillouin-Brueckner condition in non-perturbative approaches, especially if one is interested in properties.     

Large-Order perturbation theory

The original motivation for studying the asymptotic behavior of the coefficients of perturbation series came from quantum field theory. An overview is given of some of the attempts to understand quantum field theory beyond finite-order perturbation series. At least in the case of the Thirring model and probably in general, the full content of a relativistic quantum field theory cannot be recovered from its perturbation series. This difficulty, however, does not occur in quantum mechanics, and the anharmonic oscillator is used to illustrate the methods used in large-order perturbation theory. Two completely different methods are discussed, the first one using the WKB approximation, and a second one involving the statistical analysis of Feynman diagrams. The first one is well developed and gives detailed information about the desired asymptotic behavior, while the second one is still in its infancy and gives instead information about the distribution of vertices of the Feynman diagrams.     

On the use of perturbation theory corrections to numerically selected MRDCI calculations

If numerical configuration selection procedures are used in MRDCI calculations, the full MRDCI energy may be estimated by adding energy corrections obtained by perturbation theory. Accurate results may then be obtained by including all selected CF s in the zero-order function instead of only the reference CF s.     

On the use of spin graphs for spin adapting many-body perturbation theory

In connection with spin adaptation in many-body perturbation theory, this paper reexamines the use of spin graphs in view of a Hugenholtz-like representation where both the orbital and the spin parts coexist. Together with the idea of essentially distinct diagrams, this representation leads to an economic handling of spin adaptation. As a side issue, the appropriate generalization of the Epstein–Nesbet partitioning for spin-adapted perturbation theory is obtained. Compact formulas up to fourth order of the ground-state energy, and up to third order for excitation energies and ionization potentials are given.     

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.