A test of perturbation theory for determining anharmonic vibrational corrections to properties of diatomic molecules

The convergence behaviour of the Rayleigh-Schrödinger perturbation series for obtaining anharmonic vibrational corrections to properties of diatomic molecules is studied. Various procedures for ordering these corrections are examined. It is verified numerically that an asymptotically divergent energy series results when the complete anharmonic potential is taken as the first-order hamiltonian. By contrast, when each term in a Taylor series expansion of the energy about the equilibrium distance is identified with successively higher orders of perturbation, convergence problems are essentially obviated. Properties other than the energy are also investigated with regard to convergence.     

Newtonian Kinetic Theory and the Ergodic-Nonergodic Transition

In a recent work we have discussed how kinetic theory, the statistics of classical particles obeying Newtonian dynamics, can be formulated as a field theory. The field theory can be organized to produce a self-consistent perturbation theory expansion in an effective interaction potential. In the present work we use this development for investigating ergodic-nonergodic (ENE) transitions in dense fluids. The theory is developed in terms of a core problem spanned by the variables ρ, the number density, and B, a response density. We set up the perturbation theory expansion for studying the self-consistent model which gives rise to a ENE transition. Our main result is that the low-frequency dynamics near the ENE transition is the same for Smoluchowski and Newtonian dynamics. This is true despite the fact that term by term in a density expansion the results for the two dynamics are fundamentally different.     

Infrared behaviour of on shell form factors in anN=4 supersymmetric Yang-Mills field theory

We discuss the infrared and mass singular behaviour of some on shell form factors which are calculated up to the two loop level in anN=4 supersymmetric Yang-Mills field theory. One of the outcomes of our investigation is that the finite terms in the perturbation series of the form factor are determined by exponentiating the lowest order term. The same phenomeno has been discovered in some infrared and mass finite quantities.     

Dynamic polarizability,dispersion coefficient C6 and dispersion energy in the effective fragment potential method

The development of a fragment–fragment dispersion energy expression, for the general effective fragment potential (EFP2) method is presented. C₆ dispersion coefficients, expressed in terms of the dynamic polarizabilties over the imaginary frequency range (α(iν)), were calculated for a set of homo and hetero dimers. Using these coefficients the dispersion energy has been calculated. The dispersion energy is expressed using a simple London series expansion terminated after the n=6 term and implemented using distributed localized molecular orbitals (LMOs). The EFP2 dispersion energy is compared to symmetry adapted perturbation theory (SAPT) values. From this comparison, it is apparent that one needs to include higher order terms in the dispersion energy. Adding an estimated C₈ term to the C₆ energy greatly improves the agreement with the benchmark SAPT energies.     

Quasilinearization method: Nonperturbative approach to physical 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 Schrödinger 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.     

KAM Theory in Configuration Space and Cancellations in the Lindstedt Series

The KAM theorem for analytic quasi-integrable anisochronous Hamiltonian systems yields that the perturbation expansion (Lindstedt series) for any quasi-periodic solution with Diophantine frequency vector converges. If one studies the Lindstedt series by following a perturbation theory approach, one finds that convergence is ultimately related to the presence of cancellations between contributions of the same perturbation order. In turn, this is due to symmetries in the problem. Such symmetries are easily visualised in action-angle coordinates, where the KAM theorem is usually formulated by exploiting the analogy between Lindstedt series and perturbation expansions in quantum field theory and, in particular, the possibility of expressing the solutions in terms of tree graphs, which are the analogue of Feynman diagrams. If the unperturbed system is isochronous, Moser’s modifying terms theorem ensures that an analytic quasi-periodic solution with the same Diophantine frequency vector as the unperturbed Hamiltonian exists for the system obtained by adding a suitable constant (counterterm) to the vector field. Also in this case, one can follow the alternative approach of studying the perturbation expansion for both the solution and the counterterm, and again convergence of the two series is obtained as a consequence of deep cancellations between contributions of the same order. In this paper, we revisit Moser’s theorem, by studying the perturbation expansion one obtains by working in Cartesian coordinates. We investigate the symmetries giving rise to the cancellations which makes possible the convergence of the series. We find that the cancellation mechanism works in a completely different way in Cartesian coordinates, and the interpretation of the underlying symmetries in terms of tree graphs is much more subtle than in the case of action-angle coordinates.     

An analysis of the “tail” of the series in density for the Coulomb crystal energy

The crystal energy of the metallic hydrogen is found within the framework of the Wigner-Seitz method, in the form of the power series in the density parameter r_s. The series appears to be absolutely convergent within the radius r₀ = 4.2a_B. The correspondence between this series and the perturbation theory expansion is established for terms of 2nd, 3rd, and 4th order. The value of the “tail” is estimated from the sum of all higher order terms of the first series.     

On the validity of scaling theory for the anisotropic ising model

When scaling theory is applied to successive terms of a perturbation expansion for the anisotropic Ising model, a consistent gap index of 1.75 is predicted. We have recently derived a series expansion for this model and present here an analysis of the series which suggests that the scaling theory results are not obeyed.     

Formulation of the theory of turbulence in an incompressible fluid

The theory of turbulence in an incompressible fluid is formulated using methods similar to those of quantum field theory. A systematic perturbation theory is set up, and the terms in the perturbation series are shown to be in one to one correspondence with certain diagrams analogous to Feynman diagrams. From a study of the diagrams it is shown that the perturbation series can be rearranged and partially summed in such a way as to reduce the problem to the solution of three simultaneous integral equations for three functions, one of which is the second order velocity correlation function. The equations have the form of infinite power series integral equations, and the first few terms in the power series are derived from an analysis of the diagrams to sixth order. Truncation of the integral equations at the lowest nontrivial order yields Chandrasekhar's equation, and truncation at a higher order yields the equations discussed by Kraichnan.     

Accelerated higher order radiative perturbation computation: an application of GDOM

In a previous paper by the authors, the computational expressions for the higher order terms of the radiative perturbation series have been developed, and numerical results have been obtained. In this paper, new computational expressions are developed that use the analytic Green's function and the GDOM code, which were developed recently by Qin and Box. Our analysis and numerical computations indicate that the new scheme dramatically improves the efficiency of the computation (by reducing the CPU-time to less than 2% of that required by the previous scheme), and the huge demand on memory usage has been completely removed. This new scheme allows us to expand the high order perturbation computations to more general cases, for example, to include azimuth dependence, and to use the perturbation theory in more potential applications.     

氦原子1snd(n=4—11)组态下1D—3D谱项分裂值的计算

利用多体微扰理论(MBPT)计算了氦原子1snd(n=4—11)组态的１Ｄ—^３Ｄ谱项分裂值.基于两种不同的模型分别计算Rayleigh-Shrdinger微扰展开式中仅含束缚态的部分和包含连续态的部分.对于束缚态，较严格地通过自洽迭代求解Hartree方程构造零级近似波函数，并利用积分处理方法对无穷项求和中的余项给出了近似算法.而对于连续态波函数，则采用简化的氢原子势模型.按照Rayleigh-Shrdinger微扰展开方法，将Rydberg态的微扰论修正计算至三级.计算表明，二级和三级微扰对谱项分裂的贡献主要来自于束缚态求和部分.单态-三重态精细结构分裂的计算结果与两组实验结果基本符合. 关键词： 氦原子 Rydberg态 多体微扰 组态波函数 能级分裂     

Multireference second-order Brillouin–Wigner perturbation theory§

The multireference, state-specific, second-order, Brillouin–Wigner perturbation theory (MR-BWPT2) is presented. A posteriori corrections are made which, in the case of a single reference function, recover the well-known formula of second-order many-body perturbation theory, i.e. Møller–Plesset ‘MP2’ theory, and in the multireference case suppress terms which scale non-linearly with the number of electrons in the system. Prototype calculations are reported for the (H₂)₄ model in which the degree of quasi-degeneracy can be varied by changing a single geometric parameter. The calculated total energies obtained by a second-order Brillouin–Wigner treatment are compared with those supported by CAS-MP2 (complete active space self-consistent field followed by second-order Møller–Plesset-like treatment of dynamic correlation effects), by MR-BWCC (multireference Brillouin–Wigner coupled cluster expansion), and by full configuration interaction. MR-BWPT2 provides a theoretical apparatus comparable to the widely used MP2 theory, but which can be applied to quantum chemical problems requiring a multireference formalism.     

The ground state and the thermodynamics of an extended BCS model of superconductivity

The thermodynamics of an extended BCS model of superconductivity is investigated. A physical system is described by a Hamiltonian containing the BCS interaction and an attractive four-fermion interaction. The four-fermion potential is caused by attractions between Cooper pairs mediated by the phonon field. The weakness of this potential allows the use of perturbation theory. The perturbation expansion was restricted to the first order because in the ground state the second order terms are not larger than 0.5 percent of first order correction for parameters used for calculations. The BCS Hamiltonian is an unperturbed one. The ground state and the thermal properties are examined. As a result the jump in the specific heat is higher than that in the BCS case. Moreover, the squared critical field is larger than the corresponding one in the BCS theory. Additionally, we show connections with the Bogolyubov's mean field approach used earlier in order to investigate general physical consequences of the model.     

Extended perturbation formulae for Jahn-Teller molecules

The effect of the Jahn-Teller interaction on a symmetric top molecule with a threefold axis of symmetry in a ^2S + 1E state has been investigated by perturbation theory. Contributions up to sixth order are included. Explicit formulae for various quantities have been derived on the assumption that there is only one Jahn-Teller active mode of vibration; both linear and quadratic Jahn-Teller interactions are considered. The quantities concerned are (i) the vibronic energy levels, (ii) the orbital quenching factor d_t, (iii) the correction to the A-rotational constant, (iv) the correction to the spin-spin dipolar coupling term, and (v) the correction to the spin-rotation coupling constant ε_aa. Because the perturbation expansion converges slowly, the results are only applicable to molecules subject to a weak Jahn-Teller effect. There are several examples of this type of molecule which have been studied experimentally.     

Quantum-chemical study of the complexation of maleimide with benzene and water molecules

The density functional theory methods (B3LYP and PBE0) and the Möller-Plesset perturbation theory methods of second, third, and fourth orders (MP2, MP3, and MP4) are used to calculate the structure and energy of molecular complexes of benzene with maleimide and water, maleimide dimers in the ground and lowest excited triplet state, and the energies of interaction of the molecules in the complexes. The perturbation theory methods predict the existence of bound states for all the complexes studied. The largest binding energies are obtained for the benzene-maleimide complex, a result that explains why maleimide-based dispersants are used to disintegrate molecular associates of hydrocarbons in petroleum fractions.     

Improved ɛ expansion in the theory of turbulence: summation of nearest singularities by inclusion of an infrared irrelevant operator

A method is put forward to improve the ε expansion in the theory of developed d-dimensional turbulence on the basis of the renormalization of random forcing in the stochastic Navier-Stokes equation. This renormalization takes into account additional divergences, which appear as d →2. In the nth order of the perturbation theory with the extended renormalization the first n terms of the usual ε expansion are correctly reproduced as well as the first n terms of the Laurent expansion in the parameter d-2 of the terms of the rest of the usual ε expansion. The Kolmogorov constant and skewness factor calculated in the one-loop approximation of the improved perturbation theory are in reasonable agreement with their recommended experimental values.     

Studies of scattering theory using numerical methods

Abstract

Numerical simulations, using both exact and approximate methods, are used to study rough surface scattering in both the smd and large roughness regimes. This study is limited lo scattcring lrom rough one-dimensional surfaces that obey the Dirichlet boundary condition and have a Gaussian roughness spectrum. For surfdces with small roughness (kh?1, where k is the radiation wavenumber and h is the root-mean-square (RMS) Surface height), perturbation theory is known to be valid. However, it is shown numerically that when kh?1 and kl?6 (where I is the surface correlation length) the Kirchhoffapprorimation is valid except at low grazing angles, and one must sum the first three orders of perturbation theory obtain the correct result. For kh?1 and kl?1, first-order perturbation theory is accurate. In this region, the accuracy of the first two terms of the iterative series solution of the exact integral equation is examined; the first term a1 this series is the Kirchhoff approximation, It is shown numerically that lor very small kh these first two terms reduce to first-order perturbation theory. However, lor this reduction to occur, kh must be made smaller than necessdry lor first-order perturbation theory to be accurate. In the regime of large roughness (kh?1) backscattering enhancement occurs when the RMS slope is on the order of unity. Several investigators have recently shown that the second term of the iterative series solution (the double-scattering term) replicates the properties of backscattering enhancement reasonably well. However, the double-scattering term has a lundamental flaw: predictions lor the scattering cross section per unit length based on the double-scattering term increase as the surfdce length is increased. This is shown here with numerical simulations and with an approximate analytical result based on the high frequency limit. The physical significance of this finding is also discussed. The final topic is the use of the double-scattering approximation to study the mechanism for backscattering enhancement with the Dirichlet boundary condition. This mechanism is usually assumed to be interference between reciprocal scattering paths. When the interlerence between reciprocal scattering paths is removed, the enhancement is eliminated. This shows that interference between reciprocal paths is almost certainly the dominant mechanism for backscattering enhancement in the scattering regime studied.     

Spectral distribution methods in effective interaction theory

Spectral distribution methods are used to suggest alternative forms for effective interaction calculation. Using the configuration centroid operator as the unperturbed Hamiltonian is found to yield a residual interaction with minimal Euclidean norm. If the effective interaction is expanded in terms of orthogonal polynomials appropriate to the configuration density, the leading term is more meaningful than those in existing series. Using the same technique of orthogonal polynomial expansion, the calculation of diagonal matrix elements of higher order terms in the perturbation expansions reduces to an evaluation of traces which are more easily calculable. Convergence of both polynomial expansions is tied to the convergence of the spectral density to normal form which itself is governed by a strong principle, the central limit theorem.     

Thermodynamic properties of a hard-core Lennard-Jones fluid from computer simulation and the coupling parameter series expansion

ABSTRACT

The thermodynamic properties of a fluid with an interaction potential consisting in a hard-sphere core plus a Lennard-Jones tail have been obtained by Monte Carlo (MC) NVT simulation as a function of the density along several isotherms. In addition, the liquid–vapour coexistence has been determined by means of histogram-reweighting MC. These data have been used to analyse the performance of perturbation theory. To this end, the first three perturbation terms of the inverse temperature expansion of the Helmholtz free energy have been obtained by means of MC NVT simulations to test the convergence of the perturbation series and to compare with the predictions of the coupling parameter series expansion. Then, the predictions of the latter theory for the thermodynamic properties have been compared with the simulations, revealing the overall excellent performance of this perturbation theory for this model fluid, except in the vicinity of the critical point.