A change of variable for the perturbation parameter in rayleigh–Schrödinger perturbation theory

The Feenberg–Goldhammer change of scale, whereby the H₀ in perturbation theory is replaced by (1/μ)H₀ with μ a scaling parameter, is shown to be equivalent to a change of variable for the perturbation parameter. A more general change of variable is shown to lead to a perturbation series with perturbation energies E₃, …, E_2n+1 equal to zero. The resulting energy through (2n + 1)th order has the same form as that found from the Brillouin–Wigner series by different methods.     

Connections between perturbation theory and the unitary transformation methods of deriving effective Hamiltonians

The connections between open shell Brillouin–Wigner perturbation theory and the Van Vleck unitary transformation formalisms for generating effective Hamiltonians are explored. An explicit expression is obtained relating the generator ? of the unitary transformation e^i? with the amplitudes to be found from perturbation theory. The “renormalization effects” needed to produce the explicit “orthogonal-Hermitian” form of the effective Hamiltonian in perturbation theory are related directly to the generator of the unitary transformation. The conclusions reached previously by Jørgensen and Brandow regarding the identity of the effective Hamiltonians of the formalisms are explicitly verified for the case that the generator ? satisfied the Kemble condition. The procedure suggests how the powerful techniques of perturbation theory can be used within the unitary transformation framework to guarantee properly renormalized wave functions.     

Resolvent technique and Padé approximants in configuration interaction calculations

A perturbation approach based on resolvent technique and Padé approximants is proposed. The eigenvalue of interest is part of a solution of two nonlinear algebraic equations. The nonlinear equations are arrived at by considering two different expression of the expectation value of the resolvent of an outer projection of the Hamiltonian. The first expression is based on the spectral resolution of the resolvent, and the second one is obtained by a power series expansion analogous to that applied in the derivation of the energy expression in the Brillouin–Wigner perturbation theory. The truncated power series is extrapolated by Padé approximants of type II. The method is tested on a CI calculation of the energy of the lowest ¹Σ state of the B₂ molecule.     

An accurate calculation of the ground state energy of the helium atom and the hydrogen negative ion using hyperspherical coordinates

The method of hyperspherical harmonics using an adiabatic separation between the hyper-radial coordinate and the hyperspherical angles is applied to the calculation of the ground state of the helium atom and the hydrogen negative ion. For each system the simple adiabatic separation provides much of the electron correlation energy. The non-adiabatic coupling is included by means of a perturbation treatment using both the Rayleigh—Schrödinger and Brillouin—Wigner approaches. These provide rapid convergence to the exact energy of −2.90372 au.     

The Rayleigh—Schrödinger BK method applied to the lower electronic states of pyrrole

A second-order Rayleigh—Schrödinger B_K method (RS B_K) is described and applied to computing the lower electronic states of pyrrole. This method is more nearly size consistent than the previously used Brillouin—Wigner B_K (BW B_K) method. Pyrrole RS B_K excitation energies compare closely to BW B_K, second-order perturbation theory, and extrapolated CI energies. To provide insight into the origin of the bands in the experimental absorption spectrum, absorption coefficients and second moments are also reported.     

On the single-root approach within the framework of coupled-cluster theory in Fock space

A thorough formulation of Fock Space Brillouin–Wigner Coupled Cluster method is presented following previous developments [N.D.K. Petraco, Ľ. Horný, H.F. Schaefer, I. Hubač, J. Chem. Phys. 117 (2002) 9580]. The new method is designed to avoid the intruder states problem, and introduces the single-root solution feature which has not been considered yet within valence-universal methods. The explicit equations for the (0,1) sector of the Fock space are introduced.     

Electron correlation effects in the 4f14 shell

This paper serves a twofold purpose. First, Löwdin's inner projection in both nonperturbative and perturbative forms is applied to the quartic anharmonic oscillator. Inner projection with perturbation theory yields rational approximations to Brillouin–Wigner-type perturbation expansions. These lower bounds are compared with [N ? 1, N] Padé approximants to the Rayleigh–Schrödinger perturbation series for this problem. These Padés are also expressible as the even convergents, w_2N, of a Stieltjes-type continued fraction. The latter representation has certain advantages with respect to its Padé counterpart. Inner projection without perturbation theory provides significantly better results than the perturbative version. The application of inner projection techniques to a perturbed hydrogen atom is not straightforward. The usual problems associated with the continuum spectrum of hydrogen are present. By means of a nonunitary “tilting” transformation associated with the Lie group SO(4, 2), these problems may be bypassed. In the SO(4, 2)-reformulated eigenvalue problem, a reinterpretation of the basic variables, as developed by Silverstone and Moats, yields a new Hamiltonian that permits direct use of the inner projection method. This method has been applied to the ground state of the hydrogen atom in a magnetic field, using both four- and eight-dimensional basis manifolds. This represents the first application of inner projection to this problem.     

Bond length alternation in cyclic polyenes. V. Local finite-order perturbation theory approach

The finite-order many-body perturbation theory using the localized Wannier orbital basis is applied to the problem of bond length alternation in the Pariser–Parr–Pople model of cyclic polyenes C_N H_N, N = 4v + 2, which may be regarded as a simplified model of polyacetylene. Both the Møller–Plesset and the Epstein–Nesbet-type partitionings of the model Hamiltonian are employed. The localized orbital basis enables an efficient truncation of the perturbation theory summations over the intermediate states as well as an elimination of energetically unimportant diagrams, thus enabling one to obtain the fourth-order Møller–Plesset-perturbation energies with a relatively small computational effort even for large polyenes. The results obtained with the second-, third-, and fourth-order Møller–Plesset and with the third-order Epstein–Nesbet perturbation theories yield very similar bond length distortions (about 0.05 Å) and stabilization energies per site (about 0.04 eV) as obtained earlier with the RHF , one-parameter AMO , and delocalized orbital perturbation theories. The effects of truncation and diagram elimination in the fourth-order Møller–Plesset perturbation theory and the abnormal behavior of the second-order Epstein–Nesbet perturbation theory results in the localized Wannier basis near the instability threshold of the RHF solutions are discussed.     

Quantum hydrodynamic model of density functional theory

In this paper, we extend the method in Cai et al. (J Math Phys 53:103503, 2012) to derive a class of quantum hydrodynamic models for the density-functional theory (DFT). The most popular implement of DFT is the Kohn–Sham equation, which transforms a many-particle interacting system into a fictitious non-interacting one-particle system. The Kohn–Sham equation is a non-linear Schrödinger equation, and the corresponding Wigner equation can be derived as an alternative implementation of DFT. We derive quantum hydrodynamic models of the Wigner equation by moment closure following Cai et al. (J Math Phys 53:103503, 2012). The derived quantum hydrodynamic models are globally hyperbolic thus locally wellposed. The contribution of the Kohn–Sham potential is turned into a nonlinear source term of the hyperbolic moment system. This work provides a new possible way to solve DFT problems.     

Evidence for the γ‐relaxation in the light scattering spectra of poly (n‐hexyl methacrylate) near the glass transition

The full range of relaxation processes present in optically pure poly‐(n‐hexyl methacrylate) (PHMA) was studied using Rayleigh–Brillouin and photon correlation spectroscopy (PCS). Brillouin shifts, linewidths, and Landau–Placzek ratios (LPR) were measured over the temperature range from ?11 to 21 °C. The Brillouin splitting and linewidth were consistent with previous studies of PHMA, but the LPR was much lower, indicating that the scattered light primarily comes from intrinsic density fluctuations. Relaxation functions of the same PHMA sample were measured using PCS over the temperature range 0.5–52.5 °C. The average relaxation times calculated from a Williams–Watts fit follow a VFT temperature dependence, with the stretching parameter β decreasing with decreasing temperature. The distribution of relaxation times reveals a merging of the α and β‐relaxations over this temperature range, and the temperature dependent width confirms that there are at least two processes with separate temperature dependences. Furthermore, there appears a process at short times in the correlation function window at low temperatures. This upturn at the fastest relaxation times is attributed to the γ‐relaxation present in higher order methacrylate polymers. The effect of the γ‐relaxation is discussed in terms of the dynamic behavior over 12 decades in time. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 1504–1519, 2005     

Structural,vibrational and thermodynamics propertiesof Zn-based semiconductors

We have preformed first-principle calculations for the structural, vibrational and thermodynamic properties of the IIB–VIA Zn-based semiconductor compounds ZnX (X = O, S, Se, Te). The phonon dispersion curves along several high-symmetry lines at the Brillouin zone together with the corresponding phonon density of states are calculated using density-functional perturbation theory. The calculated phonon frequencies at the Γ, X, and L points of the Brillouin zone show good agreement with the experimental values and other calculations. The thermodynamics properties including the phonon contribution to the Helmholtz free energy ΔF, the phonon contribution to the internal energy ΔE, the entropy S, and the constant-volume specific heat C_V are determined within the harmonic approximation based on the calculated phonon dispersion relations. If 298 K is taken as a reference temperature, the difference values of H ? H₂₉₈ have been also calculated and compared with the available experimental data.     

Application of Wigner and Husimi intracule based electron correlation models to excited states

A new approach to the electron correlation problem based on phase space intracules derived from the Wigner distribution is applied to excited states. The computed electron correlation energy reduces the mean absolute error in the prediction of the excitation energies of 55 atomic excited states from 0.65 eV for unrestricted Hartree-Fock to 0.32 eV. This compares favorably to a mean absolute deviation of 0.52 eV for second order Moller-Plesset perturbation theory and 0.35 eV for the Lee-Yang-Parr functional. An analogous correlation model based on the Husimi distribution is developed. Predicted correlation energies and excitation energies from this model are significantly worse than for the Wigner intracule based model. Alternative correlation kernels may be more suitable for the Husimi intracule based approach.     

Near-degeneracy corrections for second-order perturbation theory: comparison of two approaches

 We compare two approximate perturbation schemes which were developed recently to deal with the (quasi)degeneracy problem in many-body perturbation theory. We conclude that although the two methods were introduced on quite different theoretical grounds, their performances are quite similar, and present an improvement over traditional perturbation theory. Both methods are cheap in computation time, but cannot compete in accuracy with more sophisticated schemes such as complete-active-space perturbation theory or dressed particle theories. Received: 1 August 2000 / Accepted: 2 August 2000 / Published online: 19 January 2001     

Density functional theory on phase space

Forty‐five years after the point de départ [Hohenberg and Kohn, Phys Rev, 1964, 136, B864] of density functional theory, its applications in chemistry and the study of electronic structures keep steadily growing. However, the precise form of the energy functional in terms of the electron density still eludes us—and possibly will do so forever [Schuch and Verstraete, Nat Phys, 2009, 5, 732]. In what follows we examine a formulation in the same spirit with phase space variables. The validity of Hohenberg–Kohn–Levy‐type theorems on phase space is recalled. We study the representability problem for reduced Wigner functions, and proceed to analyze properties of the new functional. Along the way, new results on states in the phase space formalism of quantum mechanics are established. Natural Wigner orbital theory is developed in depth, with the final aim of constructing accurate correlation‐exchange functionals on phase space. A new proof of the overbinding property of the Müller functional is given. This exact theory supplies its home at long last to that illustrious ancestor, the Thomas–Fermi model. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Phonon-induced surface charge density oscillations in quantum wells: a first-principles study of the (2 × 2)-K overlayer on Be(0001)

Density functional perturbation theory has been applied to study the surface vibrations of (2 × 2)-K monolayer on the Be(0001) surface. We present the full phonon dispersion curves along the high symmetry directions of the surface Brillouin zone (SBZ) together with the layer-projected phonon density of states and the phonon-induced surface charge density oscillations at Γ and M for the alkali SV and L modes. Surprisingly, at the M point, the L-phonon displacements produce a more pronounced perturbation on the surface charge density than the SV-phonon displacements. These results apparently solve the long-standing question regarding helium atom scattering (HAS) experiments performed on the similar system (2 × 2)-K on graphite, where the alkali SV phonon mode is not observed. Moreover, this result confirms the previous finding that HAS from free-electron metal surfaces probes directly the phonon-induced charge density oscillations and the related electron-phonon interaction.     

Superconductivity origin of PdTe and pressure effect: Insights from first-principles investigation

The phonon spectra and electron-phonon interaction properties of hexagonal superconductor PdTe are studied systematically for the first time by density functional perturbation theory (DFPT). We present phonon dispersion with non-negative frequency in the whole Brillouin zone and reveal its three-dimensional character and strong vibrational coupling from both electronic and lattice dynamic viewpoint. First-principles calculation of logarithmically averaged frequency, Debye temperature, electron-phonon coupling constant and transition temperature Tc agree well with experimental values. It is definitely believed superconductivity of PdTe originates from isotropical nonlocal electron-phonon interaction. Moreover, Unlike FeTe, the transition temperature of PdTe decreases with increasing of pressure.     

Different forms of perturbation theory for the calculation of the correlation energy

Different forms of perturbation theory for the calculation of correlation energy in both closed-and open-shell systems are discussed. For closed-shell systems, Epstein–Nesbet perturbation theory is compared with Møller–Plesset (MP ) perturbation theory based on canonical Hartree–Fock orbitals and with MP theory based on internally consistent SCF orbitals. The traditional MP theory gives superior results despite its use of an inferior zeroth-order Hamiltonian. This behavior is rationalized in terms of the larger denominators present in the traditional MP theory. These conclusions are used to support the restricted open-shell perturbation methods proposed recently by Murray and Davidson, and these new methods are compared with spin-restricted Epstein–Nesbet theory and the unrestricted MP (UMP ) approach. © 1992 John Wiley & Sons, Inc.     

PERTURB: A special-purpose algebraic manipulation program for classical perturbation theory

Classical perturbation theory provides a particularly promising route to EBK quantization of nonseparable systems. However, the number of terms generated when implementing perturbation theory for systems with more than two degrees of freedom can prove too large for general purpose symbolic manipulators to handle. We describe PERTURB, a specialized algebraic manipulation program written in C for quantization of multidimensional systems. A review of operator based classical perturbation theory is given, and the relationship between this type of perturbation theory and quantum mechanical Van Vleck perturbation theory discussed. The relative performance of the Dragt–Finn and Lie transform algorithms is assessed.     

First-order properties and the Hellmann–Feynman theorem in the case of a limited CI wave function

Two distinct approaches to the calculation of first-order properties with a limited CI wave function are discussed. One is based on the Hellmann–Feynman theorem and the other on the direct evaluation of the total energy derivative at zero perturbation. Corrections to the Hellmann–Feynman expectation value are given for the CI wave function consisting of a single determinant reference state and all single and double replacements of this. These corrections are the extended Brillouin matrix elements and involve interactions between the zeroth-order wave function and triply substituted configurations. The usefulness of these matrix elements for the generation of MC SCF orbitals and for the calculation of cluster corrections to the wave function is briefly discussed. The formulas for the Brillouin matrix elements expressed in terms of one- and two-electron integrals have been automatically generated using the syntax of the algebraic program SCHOONSCHIP.     

Dominant channels of vibronic transitions in molecules with several identical modes

Weak-coupling radiationless transitions (internal conversion or inter system crossing) are studied assuming separability and symmetry over N identical modes. Franck–Condon factors control the branching ratios between exciting just one of the equivalent modes, or equally distributing the available energy. The dominant process can be predicted by an exact quantum mechanical solution if the wavefunctions are known (Gaussian initial distributions and accepting Morse or Poeschl-Teller oscillators, for example); or more generally by a Wigner phase space surface-jumping analysis based on a classical limit of the Wigner function, using only the donor distribution and the acceptor potential surface.