Perturbation Methods

A systematic QED treatment of electron correlation is presented for ions along the lithium isoelectronic sequence. We start with the zeroth-order approximation that accounts for a part of the electron-electron interaction by a local model potential introduced into the Dirac equation. The residual electron correlation is treated by perturbation theory. Rigorous QED evaluation is presented for the first two terms of the perturbative expansion; the third-order contribution is calculated within the many-body perturbation theory. We report accurate numerical results for the electronic-structure part of the ionization potential for the n=2 states of all Li-like ions up to uranium.

A many-body version of the symmetry-adapted perturbation theory is developed for a direct calculation of intermolecular potentials as a sum of the electrostatic, exchange, induction, and dispersion contributions. Since no multipole expansion is used, the obtained interaction energy components are properly dampened at short distance by the charge-overlap (penetration) effects. The influence of the intramonomer correlation is accounted for by the perturbation expansion in terms of the Mo/ller–Plesset type fluctuation potentials WA and WB for the individual molecules. For the electrostatic and for the dispersion energy, the terms of the zeroth, first, and second order in WA+WB are considered. In this way, the leading three-particle correlation contribution to the dispersion energy is taken into account. As a test of our method, we have performed calculations of the interaction energy for the water and hydrogen fluoride dimers. Both the geometry and the basis set dependence of the interaction energy components have been investigated. For a comparison, we have also computed the supermolecular interaction energies through the full fourth order of the many-body perturbation theory. On the basis of our results, we predict the association energy for (H2O)2 equal to −4.7±0.2 kcal/mol in relatively poor agreement with the experimental value of −5.4±0.7 kcal/mol, but still within the experimental error bars. For (HF)2, the predicted association energy is −4.2±0.2 kcal/mol, while the experimental value (corrected by a theoretical zero-point energy) is −4.9±0.1 kcal/mol.

The detailed spectroscope information about highly excited molecules and radicals such us as <i>H</i>⁺₃, <i>H</i>₂, <i>HI</i>, <i>H</i>₂<i>O</i>, <i>CH</i>₂ is needed for a number of applications in the field of laser physics, astrophysics and chemistry. Studies of highly excited molecular vibration-rotation states face several problems connected with slowly convergence or even divergences of perturbation expansions. The physical reason for a perturbation expansion divergence is the large amplitude motion and strong vibration-rotation coupling. In this case one needs to use the special method of series summation. There were a number of papers devoted to this problem: papers 1-10 in the reference list are only example of studies on this topic. The present report is aimed at the application of GET method (Generalized Euler Transformation) to the diatomic molecule. Energy levels of a diatomic molecule is usually represented as Dunham series on rotational <i>J(J+1)</i> and vibrational <i>(V+1/2)</i> quantum numbers (within the perturbation approach). However, perturbation theory is not applicable for highly excited vibration-rotation states because the perturbation expansion in this case becomes divergent. As a consequence one need to use special method for the series summation. The Generalized Euler Transformation (GET) is known to be efficient method for summing of slowly convergent series, it was already used for solving of several quantum problems Refs.13 and 14. In this report the results of Euler transformation of diatomic molecule Dunham series are presented. It is shown that Dunham power series can be represented of functional series that is equivalent to its partial summation. It is also shown that transformed series has the butter convergent properties, than the initial series.

Chapter 2 is a tutorial on computing spin-forbidden reactions. Most chemists solving problems with quantum chemical tools typically work on a single potential energy surface but there are many chemical transformations where two or more potential energy surfaces need to be included to describe properly the event that is taking place. How one goes about treating changes in spin multiplicity in this situation is a daunting task. Described for the novice is the importance of the minimum energy crossing point (MEXP) and a rationalization of how spin–orbit coupling provides a mechanism for spin-forbidden reactions. An explanation of crossing probabilities, the Fermi golden rule and the Landau–Zener semiclassical approximation are given. Methodologies for obtaining spin–orbit matrix elements are presented including, among others, the Klein–Gordon equation, the Dirac equation, the Foldy–Wouthuysen transformation and the Breit–Pauli Hamiltonian. With this background the novice is taken through a tutorial that explains how to locate the MEXP. Programs available for modeling spin-forbidden reactions are provided and examples of such calculations on diatomic and polyatomic molecules are given. The chapter covers: Overview of Reactions Requiring Two States Spin-Forbidden Reaction, Intersystem Crossing Spin–Orbit Coupling as a Mechanism for Spin-Forbidden Reactions General Considerations Atomic Spin–Orbit Coupling Molecular Spin–Orbit Coupling Crossing Probability Fermi Golden Rule Landau–Zener Semiclassical Approximation Methodology for Obtaining Spin–Orbit Matrix Elements Electron Spin in Non-Relativistic Quantum Mechanics Klein–Gordon Equation Dirac Equation Foldy–Wouthuysen Transformation Breit–Pauli Hamiltonian Zeff Method Effective Core Potential-Based Method Model Core Potential-Based Method Douglas–Kroll Transformation Potential Energy Surfaces Minimum Energy Crossing Point Location Available Programs for Modeling Spin-Forbidden Reactions Applications to Spin-Forbidden Reactions Diatomic Molecules Polyatomic Molecules Molecular Properties Dynamical Aspects Other Reactions Biological Chemistry

We discuss an alternative method of performing coupling constant perturbation expansions in nonrelativistic quantum mechanics. This method, called logarithmic perturbation theory, yields new expressions for any-order corrections En to an unperturbed bound-state energy which do not involve cumbersome sums over intermediate unperturbed states. In one dimension, these corrections En can be evaluated using a simple explicit form containing a small number of integrals. In more than one dimension the approach is systematic but computations require the solution of well-defined partial differential equations. For order n=2 this equation is identical to that appearing in the method of Dalgarno and Lewis. Numerous illustrative examples are presented.

In the first part, a review is made of the development of the many‐body perturbation theory for the last half century. The development of methods for quantum electrodynamics (QED) calculations, which have occurred essentially in parallel, is also briefly reviewed. In the second part, an effort of unifying the two is discussed. The covariant‐evolution‐operator method, developed primarily for QED calculations by the Gothenburg group, can be combined with atomic/molecular many‐body perturbation theory (MBPT) and for the first time making it possible to combine full QED with electron correlation of arbitrary order. The basic idea is to extend the standard procedure of MBPT to a time‐ or energy‐dependent formalism, which makes it possible to include QED perturbations in a rigorous fashion. The procedure has been implemented and as a first test applied to helium like ions. The method can also be applied to dynamical processes. © 2014 Wiley Periodicals, Inc.

Many-body Green's function perturbation theories, such as the GW and Bethe-Salpeter formalisms, are starting to be routinely applied to study charged and neutral electronic excitations in molecular organic systems relevant to applications in photovoltaics, photochemistry or biology. In parallel, density functional theory and its time-dependent extensions significantly progressed along the line of range-separated hybrid functionals within the generalized Kohn-Sham formalism designed to provide correct excitation energies. We give an overview and compare these approaches with examples drawn from the study of gas phase organic systems such as fullerenes, porphyrins, bacteriochlorophylls or nucleobases molecules. The perspectives and challenges that many-body perturbation theory is facing, such as the role of self-consistency, the calculation of forces and potential energy surfaces in the excited states, or the development of embedding techniques specific to the GW and Bethe-Salpeter equation formalisms, are outlined.

The preference for `reductive explanations', i.e., explanations of the behaviour of a system at one `basic' level of sub-systems, seems to be related, at least in the physical sciences, to the success of a formal technique –- perturbation theory –- for extracting insight into the workings of a system from a supposedly exact but intractable mathematical description of the system. This preference for a style of explanation, however, can be justified only in the case of `regular' perturbation problems in which the zeroth-order term in the perturbation expansion (characterizing the `basic' level) is the uniform limit of the exact solution as the perturbation parameter goes to zero. For the much more frequent case of `singular' perturbation problems, various techniques have been developed which all introduce a hierarchy of levels or scales into the solutions. These levels describe processes or sub-systems operating simultaneously at different time or spatial scales. No single level, no reductive explanation in the above sense will provide an adequate explanation of the system behaviour. Explanations involving multiple levels should be recognized as far more common even in supposedly reductionist disciplines like physics.

The main factors affecting the accuracy and computational cost of the Second-order Möller-Plesset perturbation theory (MP2) calculation of (77) Se NMR chemical shifts (methods and basis sets, relativistic corrections, and solvent effects) are addressed with a special emphasis on relativistic effects. For the latter, paramagnetic contribution (390-466 ppm) dominates over diamagnetic term (192-198 ppm) resulting in a total shielding relativistic correction of about 230-260 ppm (some 15% of the total values of selenium absolute shielding constants). Diamagnetic term is practically constant, while paramagnetic contribution spans over 70-80 ppm. In the (77) Se NMR chemical shifts scale, relativistic corrections are about 20-30 ppm (some 5% of the total values of selenium chemical shifts). Solvent effects evaluated within the polarizable continuum solvation model are of the same order of magnitude as relativistic corrections (about 5%). For the practical calculations of (77) Se NMR chemical shifts of the medium-sized organoselenium compounds, the most efficient computational protocols employing relativistic Dyall's basis sets and taking into account relativistic and solvent corrections are suggested. The best result is characterized by a mean absolute error of 17 ppm for the span of (77) Se NMR chemical shifts reaching 2500 ppm resulting in a mean absolute percentage error of 0.7%.

We develop an alternative approach to time independent perturbation theory in non-relativistic quantum mechanics. The method developed has the advantage to provide in one operation the correction to the energy and to the wave function, additionally we can analyze the time evolution of the system. To verify our results, we apply our method to the harmonic oscillator perturbed by a quadratic potential. An alternative form of the Dyson series, in matrix form instead of integral form, is also obtained.

Singular perturbation theory for quantum mechanics is considered in a framework generalizing the spectral concentration theory. Under very general conditions, asymptotic estimations on the Rayleigh–Schrödinger expansions of the perturbed spectral projections are obtained. As a consequence almost invariant subspaces of exponential order are constructed. The results cover practically all singular perturbations considered in nonrelativistic quantum mechanics. In the magnetic field case, under the condition that the magnetic field does not increase at infinity, a gauge invariant perturbation theory leading to convergent series with field-dependent coefficients is developed.

In nonrelativistic quantum mechanics, we require that the free and interacting Hamiltonians be related by a unitary transformation, as has been done by other authors. We then derive a stationary perturbation theory for the radial Schrödinger equation for scattering from a spherically symmetric potential. A resulting advantage over the more commonly used Green function method is that the expression for the interacting state vector is normalized to each order in the coupling constant, unlike, in general, the result of the Green function method. Other authors have applied the unitary transformation concept to time-dependent perturbation theory to give unitarity of the time evolution operator to each order in perturbation theory, with results that show improvement over the standard perturbation theory. In this paper, general formulas are obtained for the phase shifts at the first and second order in the coupling constant. We test the method on a simple system with a known exact solution and find complete agreement between our first- and second-order contributions to the s-wave phase shifts and the corresponding expansion to the second order of the exact solution.

Aim of this paper is trying to show the possible significance, and usefulness, of various non-selfadjoint operators for suitable Observables in non-relativistic and relativistic quantum mechanics, and in quantum electrodynamics: More specifically, this work starts dealing with: (i) the hermitian (but not selfadjoint) Time operator in non-relativistic quantum mechanics and in quantum electrodynamics; with (ii) idem, the introduction of Time and Space operators; and with (iii) the problem of the four-position and fourmomentum operators, each one with its hermitian and anti-hermitian parts, for relativistic spin-zero particles. Afterwards, other physical applications of non-selfadjoint (and even non-hermitian) operators are briefly discussed. We briefly mention how non-hermitian operators can indeed be used in physics [as it was done, elsewhere, for describing Unstable States]; and some considerations are added on the cases of the nuclear optical potential, of quantum dissipation, and in particular of an approach to the measurement problem in QM in terms of a chronon. This paper is largely based on work developed, along the years, in collaboration with V.S.Olkhovsky, and, in smaller parts, with P.Smrz, with R.H.A.Farias, and with S.P.Maydanyuk.

Wilson, Jankowski, and Paldus have recently applied nondegenerate many‐body perturbation theory (MBPT) to simple models, in which the degree of quasidegeneracy could be varied continuously, and concluded that the nondegenerate theory was applicable even near degeneracy. The error in their results changes, however, considerably with geometry, leading to an incorrect potential surface. An extension of their calculations shows convergence even at exact degeneracy (square planar H4). It is shown here that the apparently good convergence is due to the suppression of the large (infinite at exact degeneracy) component of the perturbation energy in low order by the way the Hamiltonian is partitioned. This component will, however, resurface at higher orders, leading to slow convergence or even divergence. The low‐order sum of the perturbation series is not very meaningful, depends strongly on details of the zero‐order Hamiltonian, and yields, in general, incorrect potential surfaces. Multireference MBPT eliminates these problems.

The general properties of the quasilinearization method (QLM), particularly its fast quadratic convergence, monotonicity, and numerical stability, are analyzed and illustrated on different physical problems. The method approaches the solution of a nonlinear differential equation by approximating the nonlinear terms by a sequence of linear ones and is not based on the existence of a small parameter. It is shown that QLM gives excellent results when applied to different nonlinear differential equations in physics, such as Blasius, Lane-Emden, and Thomas-Fermi equations, as well as in computation of ground and excited bound-state energies and wave functions in quantum mechanics (where it can be applied by casting the Schrodinger equation in the nonlinear Riccati form) for a variety of potentials most of which are not treatable with the help of perturbation theory. The convergence of the QLM expansion of both energies and wave functions for all states is very fast and the first few iterations already yield extremely precise results. The QLM approximations, unlike the asymptotic series in perturbation theory and 1/N expansions, are not divergent at higher orders. The method sums many orders of perturbation theory as well as of the WKB expansion. It provides final and accurate answers for large and infinite values of the coupling constants and is able to handle even supersingular potentials for which each term of the perturbation series is infinite and the perturbation expansion does not exist.