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.     

Micellar perturbation of enone reduction

The reduction of several enones using BH₄^? in the presence of cationic surfactants has been investigated. These reductions are all shifted by the micelle towards the formation of conjugate reduction products.     

Iterative degenerate perturbation theory

A fairly general degenerate perturbation theory has been considered which involves two versions, i.e., the Rayleigh-Schrodinger and Brillouin-Wigner treatments. The simple recursive formulas have been found for effective Hamiltonians which allow one to split the degenerate energy levels in any orders of this theory.     

Open-shell generalized perturbation theory

Open shell generalized perturbation theory (GPT) is a systematic scheme for the improvement of a multi-configurational zeroth order wavefunction that can be chosen so the lowest order corresponds to a sum-of-the-pairs- type theory of electron correlation (with or without pair-pair couplings) in the spirit of the theories of Kelly, Sinano?lu, and Nesbet for the single configurational case. Applications axe presented to the generalized perturbative solution for individual states and for the simultaneous solution for many nondegenerate states. The (2n+ 1)-rule of GPT and the interchange theorems of double Girl' are illustrated. A brief discussion is included concerning the use of GPT in ab initio evaluations of the true parameters that are customarily, only modeled and then fit to experiment in semi-empirical theories of molecular electronic structure. The conceptual value of expressing the results of ab initio calculations in terms of effective hamiltonians is stressed and the relationship between open shell GPT and other theories of electron correlation is also considered.     

Fock space perturbation theory

Diagonal and non-diagonal operators in Fock space are defined. With a universal Fock space wave operator W the Fock space hamiltonian H can be transformed to a diagonal operator L containing all relevant information about eigenvalues of H for arbitrary particle number in a simply coded form. W and L are constructed by perturbation theory, even in a spinfree form, and illustrated diagrammatically.     

Nonorthogonal density-matrix perturbation theory

Recursive density-matrix perturbation theory [A.M.N. Niklasson and M. Challacombe, Phys. Rev. Lett. 92, 193001 (2004)] provides an efficient framework for the linear scaling computation of materials response properties [V. Weber, A.M.N. Niklasson, and M. Challacombe, Phys. Rev. Lett. 92, 193002 (2004)]. In this article, we generalize the density-matrix perturbation theory to include properties computed with a perturbation-dependent nonorthogonal basis. Such properties include analytic derivatives of the energy with respect to nuclear displacement, as well as magnetic response computed with a field-dependent basis. The theory is developed in the context of linear scaling purification methods, which are briefly reviewed.     

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.     

Auxiliary density perturbation theory

A new approach, named auxiliary density perturbation theory, for the calculation of second energy derivatives is presented. It is based on auxiliary density functional theory in which the Coulomb and exchange-correlation potentials are expressed by auxiliary function densities. Different to conventional coupled perturbed Kohn-Sham equations the perturbed density matrix is obtained noniteratively by solving an inhomogeneous equation system with the dimension of the auxiliary function set used to expand the auxiliary function density. A prototype implementation for the analytic calculation of molecular polarizabilities is presented. It is shown that the polarizabilities obtained with the newly developed auxiliary density perturbation approach match quantitative with the ones from standard density functional theory if augmented auxiliary function sets are used. The computational advantages of auxiliary density perturbation theory are discussed, too.     

Variable basis sets in perturbation theory: Numerical finite perturbation versus analytic approach

We consider the problem of determining variational, external-field-dependent corrections to nonoptimal zero-field nonlinear parameters. Both a direct analytic perturbation analysis and finite perturbation methods are described in a general way and in detail for the SCF approximation. The abstract theory is illustrated by reference to the results of several explicit calculations. Also, the sensitivity of the results to the choice of zero-field values is discussed.     

Parallelization of a multiconfigurational perturbation theory

In this work, we present a parallel approach to complete and restricted active space second‐order perturbation theory, (CASPT2/RASPT2). We also make an assessment of the performance characteristics of its particular implementation in the Molcas quantum chemistry programming package. Parallel scaling is limited by memory and I/O bandwidth instead of available cores. Significant time savings for calculations on large and complex systems can be achieved by increasing the number of processes on a single machine, as long as memory bandwidth allows, or by using multiple nodes with a fast, low‐latency interconnect. We found that parallel efficiency drops below 50% when using 8–16 cores on the shared‐memory architecture, or 16–32 nodes on the distributed‐memory architecture, depending on the calculation. This limits the scalability of the implementation to a moderate amount of processes. Nonetheless, calculations that took more than 3 days on a serial machine could be performed in less than 5 h on an InfiniBand cluster, where the individual nodes were not even capable of running the calculation because of memory and I/O requirements. This ensures the continuing study of larger molecular systems by means of CASPT2/RASPT2 through the use of the aggregated computational resources offered by distributed computing systems. © 2013 Wiley Periodicals, Inc.     

Relativistic perturbation theory of molecular structure

Summary The recently developed relativistic double perturbation theory is extended to handle relativistic changes of molecular structure more easily. This is achieved by simple coordinate scalings. Accurate higher order mixed perturbation energies for H ₂ ⁺ are calculated. The relativistic changes of bond energy,DE, of bond length,R _e , and especially of force constant,k, and of anharmonicity,a, are large, up to 100%·(Z/c)². The dominant contributions tok anda are due to the indirect change of the nonrelativistick anda connected with the relativistic change of bond length. Accordingly the relativistic changes obey Badger's and Gordy's rules (–RDEk).Dedicated to Prof. Klaus Ruedenberg in appreciation of his fundamental contributions to both formal theory and physical explanations in quantum chemistry     

Thermodynamic perturbation theory of simple liquids

It was proven that after averaging over the canonical Gibbs ensemble, the mean perturbation energy was singled out of the classical partition function before the expansion in a series of perturbation theory. Therefore, the term that formally coincides with first order perturbation theory in a decomposition of the Helmholtz free energy bears no relationship to perturbation theory. Then the proper series of the thermodynamic perturbation theory always starts with a second order infinitesimal. Therefore, the wellknown condition of applicability of the thermodynamic perturbation theory, “...the requirement that the perturbation energy per particle be small compared with T...” (L. D. Landau and E. M. Livshits, Statistical Physics, Vol. V, Pt. I), can be substantially weakened. The most important factor for applicability of thermodynamic perturbation theory is the value of many-particle correlations in an unperturbed system, but not the smallness of the perturbation potential.     

SCF perturbation calculations on metalloporphyrins

The influence of metal ion on the oxidation and ionisation potentials of metalloporphyrins is investigated by the simple electrostatic model using SCF perturbation theory. The zero order wavefunctions are obtained from PPP and CNDO/2 methods. The wide variations in redox potentials with metal and the relative insensitivity of the optical transitions with metal are very well accounted for by the perturbation approach.     

Bounds for Hartree-Fock perturbation theory

Upper and lower bounds for the second-order energy in both coupled and uncoupled Hartree-Fock perturbation theories are derived. Using these bounds inequalities are derived for the error in the geometric approximation.     

Relativistic perturbation theory of chemical properties

Summary Double perturbation theory is developed for the case where relativity is one perturbation and the other perturbation describes a chemically interesting observable such as molecular structure, force constant or polarizability. Relativity is treated according to Rutkowski's nonsingular perturbation approach. Expressions for four-component and two-component wave-functions and for the Hartree-Fock approximation are given. The method is applied analytically to the relativistic corrections of the electric polarizability of the H atom, and algebraically to the potential curve of the H ₂ ⁺ molecule. Second and third order double perturbation interchange relations are numerically verified. In the present formalism, terms up to third order are needed to qualitatively understand the relativistic corrections of chemical observables.