Variational Perturbation Theory Research Articles

A conformally coupled massive scalar field in the de Sitter expanding universe with the mass term treated as a perturbation

We study conformally coupled massive scalar field theory with the mass term treated as a perturbation in the expanding half of de Sitter spacetime as a toy model for comparing various perturbative formalisms. We point out that the in–out perturbation theory breaks down due to infrared divergences coming from the infinite future. We then clarify the relation between the free-theory vacuum and the true vacuum using the Bogolubov transformation and show that the discrepancy between the free-theory out-vacuum and the true vacuum causes apparent pair creation of free-theory particles and makes the in–out two-point function differ from the true two-point function. We also identify the infinite Bogolubov coefficients as the cause of infrared divergences. We then examine two alternative approaches: the Euclidean and in–in formalisms. We verify that there are no infrared divergences in perturbation theory with the mass term treated as a perturbation in either of these approaches and that the two-point function of the massive scalar field is reproduced correctly by these methods.

Is second-order perturbation theory sufficient to treat second-order properties?

If low-order perturbation theory is used to approximate physical quantities, the linear series has in general limited validity. In particular, the second order contribution, when examined in the light of variation-perturbation theory, has the form ϵe2γ, with γ usually well approximated by (1 - ϵe3/ϵe2)−1. Applications of this technique to dispersion forces, polarizabilities, and susceptibilities is rewarding.

Phase transition of trapped interacting Bose gases

We review some recent theoretical work on the phase transition of interacting Bose gases in the presence of external trapping potentials. A general framework for the study of such questions is presented which is based on the application of perturbative momentum-shell renormalization group methods to the trapped gas in the uncondensed phase. After giving an overview of this approach, we first establish its validity by comparing to previous results for homogeneous and harmonically trapped gases. Using this theoretical framework, we then examine various aspects of how external potentials influence the physics of condensation. (i) By studying the case of general power-law potentials and complemented by arguments from variational perturbation theory, it is quantitatively worked out how a growing inhomogeneity of the trapping potential diminishes nonperturbative effects at the transition. (ii) It is shown how by superimposing a weak periodic potentials on the homogeneous system, the characteristic nonperturbative momentum scale of critical interacting Bose gases can be probed. (iii) For a gas in a random potential, it is studied how condensation is affected by the combined influence of disorder effects and particle interactions.

Equation of state and interaction potential of helium under high temperatures and high densities

Based on the thermodynamics statistic method, the improved variational perturbation theory and the modified quantum mechanics correction model have been used to calculate the equation of state of liquid helium at pressure from 0.7 to 108 GPa. The calculation results are in good agreement with the experimental data. The EXP-6 potential (α = 13.1) can more accurately describe the interaction of helium atoms than other potentials in the scheme. Finally, a comparison is shown between our interatomic potentials and other potentials.

Quantum phase diagram of bosons in optical lattices

We work out two different analytical methods for calculating the boundary of the Mott-insulator--superfluid (MI-SF) quantum phase transition for scalar bosons in cubic optical lattices of arbitrary dimension at zero temperature which improve upon the seminal mean-field result. The first one is a variational method, which is inspired by variational perturbation theory, whereas the second one is based on the field-theoretic concept of effective potential. Within both analytical approaches we achieve a considerable improvement of the location of the MI-SF quantum phase transition for the first Mott lobe in excellent agreement with recent numerical results from quantum Monte Carlo simulations in two and three dimensions. Thus, our analytical results for the whole quantum phase diagram can be regarded as being essentially exact for all practical purposes.

Analytic Equations of State for the Generalized Lennard-Jones Fluid Based on the Ross Variational Perturbation Theory and the Percus-Yevick Radial Distribution Function of Hard Spheres

Analytic expressions for the equation of state and thermodynamic properties have been derived for a generalized Lennard-Jones fluid, based on the Ross variational perturbation theory and the analytic Percus-Yevick (PY) expression for the radial distribution function of hard spheres. It is shown that the variational procedure is absolutely convergent and the calculations are convenient and fast. Numerical calculations have been made within wide temperature and density ranges. The results show that the precision is equivalent to the non-analytic modified Weeks-Chandler- Anderson (mWCA) and complete analytic mean spherical approximation (MSA) theories as compared with computer simulation results. It is concluded that the analytic theory can be applied to research practical fluids within wide temperature and density ranges.

A Systematic Approach for Computing Zero-Point Energy, Quantum Partition Function, and Tunneling Effect Based on Kleinert's Variational Perturbation Theory.

In this paper, we describe an automated integration-free path-integral (AIF-PI) method, based on Kleinert's variational perturbation (KP) theory, to treat internuclear quantum-statistical effects in molecular systems. We have developed an analytical method to obtain the centroid potential as a function of the variational parameter in the KP theory, which avoids numerical difficulties in path-integral Monte Carlo or molecular dynamics simulations, especially at the limit of zero-temperature. Consequently, the variational calculations using the KP theory can be efficiently carried out beyond the first order, i.e., the Giachetti-Tognetti-Feynman-Kleinert variational approach, for realistic chemical applications. By making use of the approximation of independent instantaneous normal modes (INM), the AIF-PI method can readily be applied to many-body systems. Previously, we have shown that in the INM approximation, the AIF-PI method is accurate for computing the quantum partition function of a water molecule (3 degrees of freedom) and the quantum correction factor for the collinear H(3) reaction rate (2 degrees of freedom). In this work, the accuracy and properties of the KP theory are further investigated by using the first three order perturbations on an asymmetric double-well potential, the bond vibrations of H(2), HF, and HCl represented by the Morse potential, and a proton-transfer barrier modeled by the Eckart potential. The zero-point energy, quantum partition function, and tunneling factor for these systems have been determined and are found to be in excellent agreement with the exact quantum results. Using our new analytical results at the zero-temperature limit, we show that the minimum value of the computed centroid potential in the KP theory is in excellent agreement with the ground state energy (zero-point energy) and the position of the centroid potential minimum is the expectation value of particle position in wave mechanics. The fast convergent property of the KP theory is further examined in comparison with results from the traditional Rayleigh-Ritz variational approach and Rayleigh-Schrödinger perturbation theory in wave mechanics. The present method can be used for thermodynamic and quantum dynamic calculations, including to systematically determine the exact value of zero-point energy and to study kinetic isotope effects for chemical reactions in solution and in enzymes.

Hard-sphere perturbation theory for a model of liquid Ga

Investigating thermodynamic properties of a model for liquid Ga, we have extended the application of the hard-sphere (HS) perturbation theory to an interatomic pair potential that possesses a soft repulsive core and a long-range oscillatory part. The model is interesting for displaying a discontinuous jump on the main-peak position of the radial distribution function at some critical density. At densities less than this critical value, the effective HS diameter of the model, estimated by the variational HS perturbation theory, has a substantial reduction with increasing density. Thus, the density dependence of the packing fraction of the HS reference fluid has an anomalous behavior, with a negative slope, within a density region below the critical density. By adding a correction term originally proposed by Mon to remedy the inherent deficiency of the HS perturbation theory, the extended Mansoori-Canfield/Rasaiah-Stell theory [J. Chem. Phys. 120, 4844 (2004)] very accurately predicts the Helmholtz free energy and entropy of the model, including an excess entropy anomaly. Almost occurring in the same density region, the excess entropy anomaly is found to be associated with the anomalous packing faction of the HS fluid.

Self-similar factor approximants for evolution equations and boundary-value problems

The method of self-similar factor approximants is shown to be very convenient for solving different evolution equations and boundary-value problems typical of physical applications. The method is general and simple, being a straightforward two-step procedure. First, the solution to an equation is represented as an asymptotic series in powers of a variable. Second, the series are summed by means of the self-similar factor approximants. The obtained expressions provide highly accurate approximate solutions to the considered equations. In some cases, it is even possible to reconstruct exact solutions for the whole region of variables, starting from asymptotic series for small variables. This can become possible even when the solution is a transcendental function. The method is shown to be more simple and accurate than different variants of perturbation theory with respect to small parameters, being applicable even when these parameters are large. The generality and accuracy of the method are illustrated by a number of evolution equations as well as boundary value problems.

Anharmonic infrared and Raman spectra in Car–Parrinello molecular dynamics simulations

The infrared and Raman spectra of naphthalene crystal with inclusion of anharmonic effects have been calculated by adopting the generalized variational density functional perturbation theory in the framework of Car-Parrinello molecular dynamics simulations. The computational approach has been generalized for cells of arbitrary shape. The intermolecular interactions have been analyzed with and without the van der Waals corrections, showing the importance of such interactions in the naphthalene crystal to reproduce the structural, dynamical, and spectroscopic properties.

Stability of the homogeneous Bose-Einstein condensate at large gas parameter

The properties of the uniform Bose gas are studied within the optimized variational perturbation theory (Gaussian approximation) in a self-consistent way. It is shown that the atomic Bose-Einstein condensate with a repulsive interaction becomes unstable when the gas parameter $\ensuremath{\gamma}=\ensuremath{\rho}{a}^{3}$ exceeds a critical value ${\ensuremath{\gamma}}_{crit}\ensuremath{\approx}0.01$. The quantum corrections beyond the Bogoliubov-Popov approximation to the energy density, chemical potential, and pressure in powers of $\sqrt{\ensuremath{\gamma}}$ expansions are presented.

Convexity of the self-energy functional in the variational cluster approximation

In the variational cluster approximation (VCA) (or variational cluster perturbation theory), widely used to study the Hubbard model, a fundamental problem that renders variational solutions difficult in practice is its known lack of convexity at stationary points, i.e., the physical solutions can be saddle points rather than extrema of the self-energy functional. Here, we suggest two different approaches to construct a convex functional $\ensuremath{\Omega}[\ensuremath{\Sigma}]$. In the first approach, one can show analytically that in the approximation where the irreducible particle-hole vertex depends only on center of mass coordinates, the functional is convex away from phase transitions in the corresponding channel. Numerical tests on a tractable version of that functional show that convexity can be a nuisance when looking for instabilities both in the pairing and particle-hole channels. Therefore, an alternative phenomenological functional is proposed. Convexity is explicitly enforced only with respect to a restricted set of variables, such as the cluster chemical potential that is known to be otherwise problematic. Numerical tests show that our functional is convex at the physical solutions of VCA and allows second-order phase transitions in the pairing channel as well. This opens the way to the use of more efficient algorithms to find solutions of the VCA equations.

Analytical equations of state for multi-Yukawa fluids based on the Ross variational perturbation theory and the Percus-Yevick radial distribution function of hard spheres

Analytical expressions for equation of state and thermodynamic properties have been derived for the multi-Yukawa fluids, based on the Ross variational perturbation theory and the analytical Percus–Yevick (PY) expression for the radial distribution function of hard spheres. It is shown that the variational procedure is absolutely convergent and the calculations are convenient and fast. By using the parameters fitted to the Lennard–Jones 12–6 potential from the literature, numerical calculations have been made within wide temperature and density ranges. Comparison with computer simulations shows that the precision of the analytical Ross theory based on the Yukawa-type potential is equivalent to the non-analytical Ross theory based on the Lennard–Jones 12–6 potential. It is concluded that the analytical theory based on the Yukawa-type potential can be applied to research practical fluids within wide temperature and density ranges.

An automated integration-free path-integral method based on Kleinert’s variational perturbation theory

Based on Kleinert's variational perturbation (KP) theory [Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3rd ed. (World Scientific, Singapore, 2004)], we present an analytic path-integral approach for computing the effective centroid potential. The approach enables the KP theory to be applied to any realistic systems beyond the first-order perturbation (i.e., the original Feynman-Kleinert [Phys. Rev. A 34, 5080 (1986)] variational method). Accurate values are obtained for several systems in which exact quantum results are known. Furthermore, the computed kinetic isotope effects for a series of proton transfer reactions, in which the potential energy surfaces are evaluated by density-functional theory, are in good accordance with experiments. We hope that our method could be used by non-path-integral experts or experimentalists as a "black box" for any given system.

Correlation and prediction of osmotic pressures for three aqueous protein–electrolyte solutions based on the Ross perturbation theory and soft-core three-Yukawa potential

A simple analytic equation of state (EOS) for the three-Yukawa potential is derived based on the Ross variational perturbation theory. The model is used to describe the non-idealities of the aqueous protein–salt solutions. The three-Yukawa potential can describe the double-layer repulsive charge–charge interactions, the attractive dispersion (or the van der Waals) interaction and the hard-sphere repulsion. A soft-core Yukawa repulsion is introduced to replace the hard-core repulsion used in previous hard-core two-Yukawa potential. It is assumed to be able to describe the deformation of protein molecules with heavy weight under pressure. The EOS has been applied to the charged bovine serum albumin (BSA), lysozyme and α-chymotrypsin saline solutions with the various pH values and concentrations, the model parameters are determined by correlation of experimental data. The correlation and prediction results of osmotic pressure are compared with experimental data and other theories in literature. The comparisons show that the three-Yukawa model proposed in this work is reasonable, and the soft-core modification is important for the protein solutions with heavier molecular weight, and is not evident for the protein solutions with lighter molecular weight.

Effective potential and resummation procedure to multidimensional complex cubic potentials for weak and strong coupling

The method for the recursive calculation of the effective potential is applied successfully in case of weak-coupling limit (g→0) to a multidimensional complex cubic potential. In strong-coupling limit (g→∞), the result is resumed using the variational perturbation theory (VPT). It is found that the convergence of VPT results approaches those expected.

Analytic equation of state for exponential-six fluid based on the Ross variational perturbation theory and the Percus–Yevick radial distribution function of hard spheres

The analytic expressions for equation of state and thermodynamic properties have been derived for the exp-6 fluids, by using the Ross variational perturbation theory and the analytic Percus–Yevick (PY) expression of radial distribution function of hard spheres. It is shown that the variational procedure is absolutely convergent and the calculations are fast. The comparison of the numerical results with the computer simulations shows that the precision of the analytic Ross theory is equivalent to the non-analytic modified Weeks–Chandler–Anderson (mWCA) theory, and is slightly better than the complicated optimized reference hypernetted chain (RHNC) theory.

Bond variations in the complexes [CuCl6]4−

An extended version of the angular overlap model developed from the fourth-order variational perturbation theory was used to explain why sd mixing for [CuCl6]4− and other pseudooctahedral complexes [CuL6]n− with the split ground state 2E is responsible for strong softening of the vibrational E mode, which results in lengthening and even cleavage of the bonds between the metal atom and the axial ligands.

Magnetopolaron effects on the donor states in InP

In this work, we investigate the magnetopolaron effects on the transition energies between the intradonor levels 1s–2p± placed in InP bulk semiconductor. In spite of the InP Frölich electron–longitudinal optical phonon coupling constant (α=0.12) be greater than the GaAs (α=0.07), we employ in the calculation the same theoretical tools successfully applied for GaAs structures, a mix between variational method and perturbation theory. A good agreement between our theoretical results and the available experimental results were observed mainly in the region below the resonance, for magnetic field B⩽25T.

Intermolecular Interactions: Physical Picture, Computational Methods and Model Potentials

Preface. 1 Background Knowledge. 1.1 The Subject and its Specificity. 1.2 A Brief Historical Survey. 1.3 The Concept of Interatomic Potential and Adiabatic Approximation. 1.4 General Classification of Intermolecular Interactions. References. 2 Types of Intermolecular Interactions: Qualitative Picture. 2.1 Direct Electrostatic Interactions. 2.2 Resonance Interaction. 2.3 Polarization Interactions. 2.4 Exchange Interaction. 2.5 Retardation Effects in Long-Range Interactions and the Influence of Temperature. 2.6 Relativistic (Magnetic) Interactions. 2.7 Interaction Between Macroscopic Bodies. References. 3 Calculation of Intermolecular Interactions. 3.1 Large Distances. 3.2 Intermediate and Short Distances. References. 4 Nonadditivity of Intermolecular Interactions. 4.1 Physical Nature of Nonadditivity and the Definition of Many-Body Forces. 4.2 Manifestations of Nonadditive Effects. 4.3 Perturbation Theory and Many-Body Decomposition. 4.4 Many-Body Effects in Atomic Clusters. 4.5 Atom-Atom Potential Scheme and Nonadditivity. References. 5 Model Potentials. 5.1 Semiempirical Model Potentials. 5.2 Determination of Parameters in Model Potentials. 5.3 Reconstructing Potentials on the Basis of Experimental Data. 5.4 Global Optimization Methods. References. Appendix 1: Fundamental Physical Constants and Conversion Table of Physical Units. Appendix 2: Some Necessary Mathematical Apparatus. A2.1 Vector and Tensor Calculus. A2.1.1 Definition of vector the addition law. A2.1.2 Scalar and vector products triple scalar product. A2.1.3 Determinants. A2.1.4 Vector analysis gradient, divergence and curl. A2.1.5 Vector spaces and matrices. A2.1.6 Tensors. A2.2 Group Theory. A2.2.1 Properties of group operations. A2.2.2 Representations of groups. A2.2.3 The permutation group. A2.2.4 The linear groups. The three-dimensional rotation group. A2.2.5 Point groups. A2.2.6 Irreducible tensor operators. Spherical tensors. References. Appendix 3: Methods of Quantum-Mechanical Calculations of Many-Electron Systems. A3.1 Adiabatic Approximation. A3.2 Variational Methods. A3.2.1 Self-consistent field method. A3.2.2 Methods taking into account the electron correlation. A3.2.2.1 r12-dependent wave functions. A3.2.2.2 Configuration interaction. A3.2.2.3 Coupled cluster method. A3.2.2.4 Density functional theory approach. A3.3 Perturbation Theory. A3.3.1 Rayleigh-Schr odinger perturbation theory. A3.3.2 Moller-Plesset perturbation theory. A3.3.3 Operator formalism and the Brillouin-Wigner perturbation theory. A3.3.4 Variational perturbation theory. A3.3.5 Asymptotic expansions Pade approximants. References. Index.