Variational Perturbation Theory Research Articles

Mass renormalization and binding energies in quantum field theory

We compare the predictions of two methods of determining the amount of binding energy between two distinguishable fermions that interact with each other through force-intermediating bosons. Both measures try to quantify this binding energy by the downward shift of the fully interacting two-fermion ground state energy relative to the sum of the corresponding two single-particle ground state energies. The first method computes this energy difference directly from the standard quantum field theoretical Hamiltonian. The second method uses the mass renormalized form of this Hamiltonian. In order to have a concrete example for this comparison, we employ a simple Yukawa-like model system in one spatial dimension. We find that both approaches lead to identical predictions in the second and fourth order perturbation of the coupling constant, and they remain remarkably close even in the strong coupling domain where perturbation theory diverges. This illustrates that there are field theoretical systems for which rather accurate binding energies can be obtained even without the mass renormalization procedure.

Nonperturbative calculations in the framework of variational perturbation theory in QCD

We discuss applications of the method based on the variational perturbation theory to perform calculations down to the lowest energy scale. The variational series is different from the conventional perturbative expansion and can be used to go beyond the weak-coupling regime. We apply this method to investigate the Borel representation of the light Adler function constructed from the τ data and to determine the residual condensates. It is shown that within the method suggested the optimal values of these lower dimension condensates are close to zero.

Are Various σ-Hole Bonds Steered by the Same Mechanisms?

Representative Lewis acid-Lewis base complexes linked by tetrel, pnicogen, chalcogen, and halogen bonds have been studied within the quantum theory of atoms in molecules (QTAIM) approach and the hybrid variation-perturbation theory (HVPT) to analyze possible relationships between these σ-hole dimers. Results obtained at the MP2/aug-cc-pVTZ level indicate numerous correlations similar to hydrogen-bonded systems.

Nonequilibrium variational cluster perturbation theory: Quench dynamics of the quantum Ising model

We introduce a variational implementation of cluster perturbation theory (CPT) to address the dynamics of spin systems driven out of equilibrium. We benchmark the method with the quantum Ising model subject to a sudden quench of the transverse magnetic field across the transition or within a phase. We treat both the one-dimensional case, for which an exact solution is available, as well the two-dimensional one, for which has to resort to numerical results. Comparison with exact results shows that the approach provides a quite accurate description of the real-time dynamics up to a characteristic time scale $\tau$ that increses with the size of the cluster used for CPT. In addition, and not surprisingly $\tau$ is small for quenches across the equilibrium phase transition, but can be quite larger for quenches within the ordered or disordered phases.

Electron correlation within the relativistic no-pair approximation.

This paper addresses the definition of correlation energy within 4-component relativistic atomic and molecular calculations. In the nonrelativistic domain the correlation energy is defined as the difference between the exact eigenvalue of the electronic Hamiltonian and the Hartree-Fock energy. In practice, what is reported is the basis set correlation energy, where the "exact" value is provided by a full Configuration Interaction (CI) calculation with some specified one-particle basis. The extension of this definition to the relativistic domain is not straightforward since the corresponding electronic Hamiltonian, the Dirac-Coulomb Hamiltonian, has no bound solutions. Present-day relativistic calculations are carried out within the no-pair approximation, where the Dirac-Coulomb Hamiltonian is embedded by projectors eliminating the troublesome negative-energy solutions. Hartree-Fock calculations are carried out with the implicit use of such projectors and only positive-energy orbitals are retained at the correlated level, meaning that the Hartree-Fock projectors are frozen at the correlated level. We argue that the projection operators should be optimized also at the correlated level and that this is possible by full Multiconfigurational Self-Consistent Field (MCSCF) calculations, that is, MCSCF calculations using a no-pair full CI expansion, but including orbital relaxation from the negative-energy orbitals. We show by variational perturbation theory that the MCSCF correlation energy is a pure MP2-like correlation expression, whereas the corresponding CI correlation energy contains an additional relaxation term. We explore numerically our theoretical analysis by carrying out variational and perturbative calculations on the two-electron rare gas atoms with specially tailored basis sets. In particular, we show that the correlation energy obtained by the suggested MCSCF procedure is smaller than the no-pair full CI correlation energy, in accordance with the underlying minmax principle and our theoretical analysis. We also show that the relativistic correlation energy, obtained from no-pair full MCSCF calculations, scales at worst as X(-2) with respect to the cardinal number X of our correlation-consistent basis sets optimized for the two-electron atoms. This is better than the X(-1) scaling suggested by previous studies, but worse than the X(-3) scaling observed in the nonrelativistic domain. The well-known 1/Z- expansion in nonrelativistic atomic theory follows from coordinate scaling. We point out that coordinate scaling for consistency should be accompanied by velocity scaling. In the nonrelativistic domain this comes about automatically, whereas in the relativistic domain an explicit scaling of the speed of light is required. This in turn explains why the relativistic correlation energy to the lowest order is not independent of nuclear charge, in contrast to nonrelativistic theory.

Time-dependent perturbation theory in quantum mechanics and the renormalization group

Time-dependent perturbation theory in quantum mechanics is divergent at long times when the perturbation induces a resonance between two eigenstates of the unperturbed Hamiltonian. Divergences in perturbation theory are also common in quantum field theory and in critical phenomena. The renormalization group (RG) was designed to deal with these divergences. In the last two decades, this procedure has been extended to dynamical systems where the perturbation theory diverges in the long-time limit. In this article, we first review the connection between RG in the context of field theory and RG in the context of dynamical systems. We then show that the long-time divergence in the resonant situation in the time-dependent perturbation theory in quantum mechanics can be removed by using a RG-aided calculational scheme.

Two-species Bose–Einstein condensate in an optical lattice: analytical approximate formulae

Employing a general variational method and perturbation theory, we derived explicit solutions for the description of one-dimensional two species Bose–Einstein condensates confined by a harmonic trap potential in an optical lattice. We consider the system of two coupled Gross–Pitaevskii equations (GPE) and derive explicit expressions for the chemical potentials and wavefunctions in terms of the atom–atom interaction parameters and laser intensity. We have compared our results with the numerical solutions of the GPE and performed a quantitative analysis for the both considered methods. We underline the importance of the obtained explicit solutions to characterize the density profile or degree of miscibility of the two components.

Strongly interacting Bose–Fermi mixtures in one dimension

We study one-dimensional (1D) strongly interacting Bose–Fermi mixtures by both the exact Bethe-ansatz method and variational perturbation theory within the degenerate ground state subspace of the system in the infinitely repulsive limit. Based on the exact solution of the 1D Bose–Fermi gas with equal boson–boson and boson–fermion interaction strengths, we demonstrate that the ground state energy is degenerate for different Bose–Fermi configurations and the degeneracy is lifted when the interaction deviates the infinitely interacting limit. We then show that the ground properties in the strongly interacting regime can be well characterized by using the variational perturbation method within the degenerate ground state subspace, which can be applied to deal with more general cases with anisotropic interactions and in external traps. Our results indicate that the total ground-state density profile in the strongly repulsive regime behaves like the polarized non-interacting fermions, whereas the density distributions of bosons and fermions display different properties for different Bose–Fermi configurations and are sensitive to the anisotropy of interactions.

Elucidation of the binding mechanism of renin using a wide array of computational techniques and biological assays

We investigate the binding mechanism in renin complexes, involving three drugs (remikiren, zankiren and enalkiren) and one lead compound, which was selected after screening the ZINC database. For this purpose, we used ab initio methods (the effective fragment potential, the variational perturbation theory, the energy decomposition analysis, the atoms-in-molecules), docking, molecular dynamics, and the MM-PBSA method. A biological assay for the lead compound has been performed to validate the theoretical findings. Importantly, binding free energy calculations for the three drug complexes are within 3kcal/mol of the experimental values, thus further justifying our computational protocol, which has been validated through previous studies on 11 drug-protein systems. The main elements of the discovered mechanism are: (i) minor changes are induced to renin upon drug binding, (ii) the three drugs form an extensive network of hydrogen bonds with renin, whilst the lead compound presented diminished interactions, (iii) ligand binding in all complexes is driven by favorable van der Waals interactions and the nonpolar contribution to solvation, while the lead compound is associated with diminished van der Waals interactions compared to the drug-bound forms of renin, and (iv) the environment (H2O/Na+) has a small effect on the renin–remikiren interaction.

Universal short-range ab initio atom-atom potentials for interaction energy contributions with an optimal repulsion functional form.

The repulsion term in conventional force fields constitutes a major source of error. Assuming that this could originate from a too simple analytical functional form, we analyzed various analytical functions using ab initio exchange component values as a reference and obtained (α + β R (-1))exp(-γ R) as the optimal form to represent the repulsion term. Universal exchange, delocalization, and electrostatic penetration potentials approximating the corresponding interaction energy components defined within hybrid variation-perturbation theory (HVPT) were derived using as a reference a training set of 660 biomolecular complexes. The electrostatic multipole term was calculated using cumulative atomic multipole moments, whereas correlation contribution including dispersion term and first-order correlation correction was estimated from nonempirical D a s functions derived by Pernal et al. The resulting non-empirical atom-atom potentials (NEAAP) were tested for several urokinase-inhibitor complexes yielding improved docking results.

Excited states from range-separated density-functional perturbation theory

We explore the possibility of calculating electronic excited states by using perturbation theory along a range-separated adiabatic connection. Starting from the energies of a partially interacting Hamiltonian, a first-order correction is defined with two variants of perturbation theory: a straightforward perturbation theory and an extension of the Görling–Levy one that has the advantage of keeping the ground-state density constant at each order in the perturbation. Only the first, simpler, variant is tested here on the helium and beryllium atoms and on the hydrogen molecule. The first-order correction within this perturbation theory improves significantly the total ground- and excited-state energies of the different systems. However, the excitation energies mostly deteriorate with respect to the zeroth-order ones, which may be explained by the fact that the ionisation energy is no longer correct for all interaction strengths. The second (Görling–Levy) variant of the perturbation theory should improve these results but has not been tested yet along the range-separated adiabatic connection.

Hybrid variational–perturbation method for calculating ro-vibrational energy levels of polyatomic molecules

A procedure for calculation of rotational–vibrational states of medium-sized molecules is presented. It combines the advantages of variational calculations and perturbation theory. The vibrational problem is solved by diagonalising a Hamiltonian matrix, which is partitioned into two sub-blocks. The first, smaller sub-block includes matrix elements with the largest contribution to the energy levels targeted in the calculations. The second, larger sub-block comprises those basis states which have little effect on these energy levels. Numerical perturbation theory, implemented as a Jacobi rotation, is used to compute the contributions from the matrix elements of the second sub-block. Only the first sub-block needs to be stored in memory and diagonalised. Calculations of the vibrational–rotational energy levels also employ a partitioning of the Hamiltonian matrix into sub-blocks, each of which corresponds either to a single vibrational state or a set of resonating vibrational states, with all associated rotational levels. Physically, this partitioning is efficient when the Coriolis coupling between different vibrational states is small. Numerical perturbation theory is used to include the cross-contributions from different vibrational states. Separate individual sub-blocks are then diagonalised, replacing the diagonalisation of a large Hamiltonian matrix with a number of small matrix diagonalisations. Numerical examples show that the proposed hybrid variational–perturbation method greatly speeds up the variational procedure without significant loss of precision for both vibrational–rotational energy levels and transition intensities. The hybrid scheme can be used for accurate nuclear motion calculations on molecules with up to 15 atoms on currently available computers.

Pairing in a system of a few attractive fermions in a harmonic trap

We study a strongly attractive system of a few spin-(1/2) fermions confined in a one-dimensional harmonic trap, interacting via two-body contact potential. Performing exact diagonalization of the Hamiltonian we analyze the ground state and the thermal state of the system in terms of one- and two-particle reduced density matrices. We show how for strong attraction the correlated pairs emerge in the system. We find that the fraction of correlated pairs depends on temperature and we show that this dependence has universal properties analogous to the gap function known from the theory of superconductivity. In contrast to the standard approach based on the variational ansatz and/or perturbation theory, our predictions are exact and are valid also in a strong-attraction limit. Our findings contribute to the understanding of strongly correlated few-body systems and can be verified in current experiments on ultra-cold atoms.

Modeling of the Amorphous Phase of Poly-CO with He and N

ABSTRACTDensity functional theory (DFT) and variational density functional perturbation theory (DFPT) simulations of amorphous poly-CO structures were performed to understand the stability of the polymerized structure at low pressures and to study the mechanism of destruction of the extended network at low pressures. Charge population analyses accompanied the search of the “weakest link” in the covalently bonded network. IR and Raman spectra of amorphous p-CO, calculated at 15 and 5.02 GPa, show significant contributions of CO molecules, carbonyl groups fragments decorating chains, and lactones of amorphous p-CO structures. DFT simulations of formation of amorphous polymeric structures were also done with the addition (as a result of replacement of CO molecules) of N or He atoms to the crystalline delta phase of CO. For the CO-N mixtures, the concentration of N was varied in the range from 6.25 % to 50% with different distribution patterns of N atoms in the unit cell. For all studied CO-N concentrations, isotropic compression led to CO polymerization beginning at a pressure of 11 GPa; the N was incorporated in the random network in low concentration. In CO-He mixtures He atoms appear to facilitate complete formation of the random structure which is almost completely polymerized at a pressure of 18 GPa. He atoms also help stabilize the structure at low pressures.

The influence of absolute minimum (l1–δ1) type inversion on the ionization energy of the ground state of shallow donors in n-Ge single crystals

Based on the Ritz variational method and perturbation theory, ionization energy of shallow donors for the cases of L 1 and Δ 1 model of the conduction band of germanium single crystals was calculated. It was shown that the absolute minimum (L 1 –Δ 1 ) type inversion in n-Ge leads to a significant increase in ionization energy of shallow donors. Using the Ritz variational method allows more accurately describe the experimental results with respect to the calculation based on perturbation theory. Comparison of theoretical calculations with experimental data shows that the hydrogenlike impurity model is approximate and may be used only for Sb impurity in germanium. For impurities, such as, P and As chemical shift, that is personality of the ion field potential of each impurity, which is not Coulomb must be considered.

Study on lattice vibrational properties and Raman spectra of Bi2Te3 based on density-functional perturbation theory

We present a variational density-functional perturbation theory (DFPT) to investigate the lattice dynamics and vibrational properties of single crystal bismuth telluride material. The phonon dispersion curves and phonon density of states (DOS) of the material were obtained. The phonon dispersions are divided into two fields by a phonon gap. In the lower field, atomic vibrations of both Bi and Te contribute to the DOS. In the higher field, most contributions come from Te atoms. The calculated Born effective charges and dielectric constants reveal a great anisotropy in the crystal. The largest Born effective charge generates a significant dynamic charge transferring along the c axis. By DFPT calculation, the greatest LO—TO splitting takes place in the infrared phonon modes and reaches 1.7 THz in the Brillouin zone center. The Raman spectra and peaks corresponding to respective atomic vibration modes were found to be in good agreement with the experimental data.

Review of Feynman’s Path Integral in Quantum Statistics: from the Molecular Schrödinger Equation to Kleinert’s Variational Perturbation Theory

AbstractFeynman’s path integral reformulates the quantum Schrödinger differential equation to be an integral equation. It has been being widely used to compute internuclear quantum-statistical effects on many-body molecular systems. In this Review, the molecular Schrödinger equation will first be introduced, together with the Born-Oppenheimer approximation that decouples electronic and internuclear motions. Some effective semiclassical potentials, e.g., centroid potential, which are all formulated in terms of Feynman’s path integral, will be discussed and compared. These semiclassical potentials can be used to directly calculate the quantum canonical partition function without individual Schrödinger’s energy eigenvalues. As a result, path integrations are conventionally performed with Monte Carlo and molecular dynamics sampling techniques. To complement these techniques, we will examine how Kleinert’s variational perturbation (KP) theory can provide a complete theoretical foundation for developing non-sampling/non-stochastic methods to systematically calculate centroid potential. To enable the powerful KP theory to be practical for many-body molecular systems, we have proposed a new path-integral method: automated integration-free path-integral (AIF-PI) method. Due to the integration-free and computationally inexpensive characteristics of our AIF-PI method, we have used it to performab initiopath-integral calculations of kinetic isotope effects on proton-transfer and RNA-related phosphoryl-transfer chemical reactions. The computational procedure of using our AIF-PI method, along with the features of our new centroid path-integral theory at the minimum of the absolute-zero energy (AMAZE), are also highlighted in this review.

Physical Nature of Intermolecular Interactions in [BMIM][PF6] Ionic Liquid

The intermolecular interaction energy in a popular ionic liquid, [BMIM][PF6] is analyzed using the Hybrid Variation-Perturbation Theory approach. The analysis is performed on a sample of configurations from molecular dynamics simulation, instead of minimized structures. The interaction energy components are quantified, showing that the electrostatics is the dominating but not the only important term. It is found that two- and three-body electron delocalization components also contribute to the stabilization of the complexes; however, these interactions vanish beyond the first coordination sphere. The presented study shows a systematic way to obtain the amount of physically meaningful components of the interaction energy, which possibly could be related to macroscopic properties of ionic liquids (e.g., viscosity, melting point) or electron transfer in ionic liquids.

Phase transitions in three-dimensional bosonic systems in optical lattices

We formulate the collective quantum field theory for three-dimensional bosonic optical lattices and evaluate its consequences in a mean-field approximation to two collective fields, proposed by Cooper et al (2011 Phys. Rev. A 83 053622), and in a lowest-order variational perturbation theory. It is shown that the present mean-field approximation predicts some essential features of the experimentally observed dependence of the critical temperature on the coupling strength and a second-order quantum phase transition. In contrast to a recent prediction for atomic gases given by Cooper et al, our finding indicates no superfluid state with zero condensate fraction.

Variational perturbation theory and nonperturbative calculations in QCD

A nonperturbative approach based on the variational perturbation theory in quantum chromodynamics is developed. The variational series is different from the conventional perturbative expansion and can be used to go beyond the weak-coupling regime. The approach suggested takes into account the summation of threshold singularities and the involvement of nonperturbative light quark masses. Phenomenological applications of this approach to describe physical quantities connected with the hadronic τ-decay data: the Rτ ratio, the light-quark Adler function, and the smeared RΔ function are presented. The description of examined quantities includes an infrared region and, therefore, they cannot be directly calculated within the standard perturbation theory. It is shown that in spite of this fact the approach suggested gives a rather good result for these quantities down to the lowest energy scale.