Scaling,renormalisation and accuracy of perturbation calculations

For certain eigenvalue problems which lead to divergent Rayleigh—Schrödinger perturbation series, energy estimates of acceptable accuracy may be obtained easily in first order by optimising a variational scale factor. Some simple calculations on the quartic anharmonic oscillator and on the quadratic Zeeman effect show errors which are generally quite small over a very wide range of perturbation parameter values.     

The semiclassical and the quasiclassical partition function of a quartic anharmonic oscillator

After a substitution a known Laplace-type integral is used to derive quantum corrections to the classical partition function of a quartic anharmonic oscillator in the framework of the Wigner—Kirkwood perturbation expansion. By straightforward calculations results are given in a closed form allowing the analytical formulation of the thermodynamic functions H, E, S, C_υ. The numerical results agree for arbitrary anharmonicity and for high and intermediate temperatures with the numerical partition function calculated from the Hioe—Montroll eigenvalues. Furthermore, the same integral type is used for the analytical calculation of a “quasiclassical” partition function and of “quasiclassical” moments. In the trace formulation of the partition function all commutators are neglected. The harmonic oscillator density matrix is applied to the evaluation of the truncated trace expressions. The “quasiclassical” partition function is an exact upper bound and lies always below the classical partition function.     

Time-dependent perturbation theory for vibrational energy relaxation and dephasing in peptides and proteins

Without invoking the Markov approximation, we derive formulas for vibrational energy relaxation (VER) and dephasing for an anharmonic system oscillator using a time-dependent perturbation theory. The system-bath Hamiltonian contains more than the third order coupling terms since we take a normal mode picture as a zeroth order approximation. When we invoke the Markov approximation, our theory reduces to the Maradudin-Fein formula which is used to describe the VER properties of glass and proteins. When the system anharmonicity and the renormalization effect due to the environment vanishes, our formulas reduce to those derived by and Mikami and Okazaki [J. Chem. Phys. 121, 10052 (2004)] invoking the path-integral influence functional method with the second order cumulant expansion. We apply our formulas to VER of the amide I mode of a small amino-acid like molecule, N-methylacetamide, in heavy water.     

Embedding scaling relations in Padé approximants: Detours to tame divergent perturbation series

A divergent perturbation series is known to yield very unreliable results for observables even at moderate coupling strengths. One of the most popular techniques in handling such series is to express them as rational functions, but it is often faithful only for small coupling. We outline here how one can gain considerable advantages in the large‐coupling regime by properly embedding known asymptotic scaling relations for selected observables during construction of the aforesaid Padé approximants. Three new bypass routes are explored in this context. The first approach involves a weighted geometric mean of two neighboring PA. The second idea is to consider series for specific ratios of observables. The third strategy is to express observables as functionals of the total energy in the form of series expansions. Symanzik's scaling relation, and the virial and Hellmann–Feynman theorems, are used at appropriate places to aid each of the strategies. Pilot calculations on the ground‐state perturbation series of certain observables for the quartic anharmonic oscillator problem reveal readily the benefit and novelty. © 2012 Wiley Periodicals, Inc.     

Accurate solutions for the spiked oscillators

The one‐dimensional harmonic oscillator potential plus a term of the form λ / x^α is known as the spiked oscillator potential; it constitutes a very interesting system because of its difficulties to accept perturbative and variational solutions for certain regimes of the α parameter. By the use of a numerical method, we obtain accurate energy eigenvalues and eigenfunctions for a wide range of λ values and a few α values. The accuracy of the present results is by much higher than previously published results. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

An alternative series solution to the isotropic quartic oscillator inN dimensions

The series solution of theN-dimensional isotropic quartic oscillator weighted by an appropriate function which exhibits the correct asymptotic behavior of the wave function is presented. The numerical performance of the solution in Hill's determinant picture is excellent, and yields the energy spectrum of the system to any desired accuracy for the full range of the coupling constant. Furthermore, it converges to the well-known exact solution of the unperturbed harmonic oscillator wave function, when the anharmonic interaction vanishes.     

Perturbative treatment of anharmonic vibrational effects on bond distances: An extended langevin dynamics method

In this work, we first review the perturbative treatment of an oscillator with cubic anharmonicity. It is shown that there is a quantum‐classical correspondence in terms of mean displacement, mean‐squared displacement, and the corresponding variance in the first‐order perturbation theory, provided that the amplitude of the classical oscillator is fixed at the zeroth‐order energy of quantum mechanics . This correspondence condition is realized by proposing the extended Langevin dynamics (XLD), where the key is to construct a proper driving force. It is assumed that the driving force adopts a simple harmonic form with its amplitude chosen according to , while the driving frequency chosen as the harmonic frequency. The latter can be improved by using the natural frequency of the system in response to the potential if its anharmonicity is strong. By comparing to the accurate numeric results from discrete variable representation calculations for a set of diatomic species, it is shown that the present method is able to capture the large part of anharmonicity, being competitive with the wave function‐based vibrational second‐order perturbation theory, for the whole frequency range from ～4400 cm^?1 (H₂) to ～160 cm^?1 (Na₂). XLD shows a substantial improvement over the classical molecular dynamics which ceases to work for hard mode when zero‐point energy effects are significant. © 2013 Wiley Periodicals, Inc.     

Large-Order perturbation theory

The original motivation for studying the asymptotic behavior of the coefficients of perturbation series came from quantum field theory. An overview is given of some of the attempts to understand quantum field theory beyond finite-order perturbation series. At least in the case of the Thirring model and probably in general, the full content of a relativistic quantum field theory cannot be recovered from its perturbation series. This difficulty, however, does not occur in quantum mechanics, and the anharmonic oscillator is used to illustrate the methods used in large-order perturbation theory. Two completely different methods are discussed, the first one using the WKB approximation, and a second one involving the statistical analysis of Feynman diagrams. The first one is well developed and gives detailed information about the desired asymptotic behavior, while the second one is still in its infancy and gives instead information about the distribution of vertices of the Feynman diagrams.     

Asymptotic series for the quantum quartic anharmonic oscillator

We propose a numerical method to find solutions of the one-dimensional Schrödinger equation when the potential is symmetric and can be expanded in a polynomial form. We used a non-perturbative method, in which we include explicitly the correct asymptotic behavior of the wave function computed by the WKB method. The numerical convergence is very fast and allows to compute the energy eigenvalues and eigenfunctions simultaneously. The method is applied to the quartic anharmonic oscillator with one and two wells, we compute the energy eigenvalues for the ground state and for the first six excited states, the results obtained are in agreement with those reported previously in the literature.     

Accurate estimates of asymptotic indices via fractional calculus

We devise a three-parameter random search strategy to obtain accurate estimates of the large-coupling amplitude and exponent of an observable from its divergent Taylor expansion, known to some desired order. The endeavor exploits the power of fractional calculus, aided by an auxiliary series and subsequent construction of Padé approximants. Pilot calculations on the ground-state energy perturbation series of the octic anharmonic oscillator reveal the spectacular performance.     

The calculation of vibrational energy levels of polyatomic molecules including anharmonic effect using contact transformation perturbation method

Using contact transformation perturbation method based on the Taylor expansion of the potential energy function in terms of dimensionless normal coordinates up to sixth‐order, the vibrational energy levels in terms of force constants are derived. The contact transformation theory has been applied to simplify the calculation of perturbation effects. To calculate the second‐order vibrational energy correction, the third and fourth‐order terms of potential function have been placed in the first‐order perturbation Hamiltonian and the second‐order Hamiltonian contains hexatic ones. We present expressions which give relations between the fourth‐ and sixth‐order terms in dimensionless normal coordinates of the potential and the anharmonicity coefficients. For illustration, a set of vibrational energies levels of SO₂, and H₂O molecules including anharmonic effects has been calculated. © 2013 Wiley Periodicals, Inc.     

Renormalized perturbation theory by the moment method for degenerate states: Anharmonic oscillators

We apply renormalized perturbation theory by the moment method to an anharmonic oscillator in two dimensions with a perturbation that couples unperturbed degenerate states. The method leads to simple recurrence relations for the perturbation corrections to the energy and moments of the eigenfunction. We calculate accurate energy eigenvalues, illustrate the general features of the method, and comment on the application of the approach to other quantum mechanical models. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 261–272, 1998     

Orbital-dependent correlation energy in density-functional theory based on a second-order perturbation approach: success and failure

We have developed a second-order perturbation theory (PT) energy functional within density-functional theory (DFT). Based on PT with the Kohn-Sham (KS) determinant as a reference, this new ab initio exchange-correlation functional includes an exact exchange (EXX) energy in the first order and a correlation energy including all single and double excitations from the KS reference in the second order. The explicit dependence of the exchange and correlation energy on the KS orbitals in the functional fits well into our direct minimization approach for the optimized effective potential, which is a very efficient method to perform fully self-consistent calculations for any orbital-dependent functionals. To investigate the quality of the correlation functional, we have applied the method to selected atoms and molecules. For two-electron atoms and small molecules described with small basis sets, this new method provides excellent results, improving both second-order Moller-Plesset expression and any conventional DFT results significantly. For larger systems, however, it performs poorly, converging to very low unphysical total energies. The failure of PT based energy functionals is analyzed, and its origin is traced back to near degeneracy problems due to the orbital- and eigenvalue-dependent algebraic structure of the correlation functional. The failure emerges in the self-consistent approach but not in perturbative post-EXX calculations, emphasizing the crucial importance of self-consistency in testing new orbital-dependent energy functionals.     

势阱中粒子能级与波函数微扰计算的代数递推公式

利用超位力定理(HVT)和Hellmann-Feynman 定理(HFT),导出了由有精确解的势阱的能级值用微扰法直接计算一维势阱的各级近似能级的普遍代数公式,并导出了由能级近似值计算定态波函数近似表达式的代数公式.给出了代数公式具体应用的几个典型一维势阱实例.此法可推广到二维势阱与三维势阱的情形.     

The problem of unphysical states in the theory of intermolecular interactions

It was shown by Claverie that the interactions between atoms and molecules make unphysical electronic solutions of the Schradinger equation accessible in perturbation calculations of intermolecular interactions, accessible in the sense that the perturbation expansion is likely to converge to an unphysical solution if it converges at all. This is a difficult problem because there are generally an infinite number of unphysical states with energies below that of the physical ground state. We have carried out configuration interaction calculations on LiH of both physical and unphysical states. They show that avoided crossings occur between physical and unphysical energy levels as the interaction between the two atoms is turned on, i.e. as the expansion parameter is increased from 0 to 1. The avoided crossing for the lowest energy state occurs for < 0.8, implying that the perturbation expansion will diverge for larger values of . The behavior of the energy levels as functions of . is shown to be understandable in terms of a two-state model. In the remainder of the paper, we concentrate on designing effective Hamiltonians which have physical solutions identical to those of the Schrödinger equation, but which have no unphysical states of lower energy than the physical ground state. We find that we must incorporate ideas from the Hirschfelder-Silbey perturbation theory, as modified by Polymeropoulos and Adams, to arrive finally at an effective Hamiltonian which promises to have the desired properties, namely, that all unphysical states be higher in energy than the physical bound states, that the first and higher order corrections to the energy vanish in the limitR = . that the leading terms of the asymptotic 1/R expansion of the energy be given correctly in second order, and that the overlap between the zeroth order wave function and the corresponding eigenfunction of the effective Hamiltonian be close to one.     

The effects of static quartic anharmonicity on the quantum dynamics of a linear oscillator with time-dependent harmonic frequency: Perturbative analysis and numerical calculations

The effects of quartic anharmonicity on the quantum dynamics of a linear oscillator with time-dependent force constant (K) or harmonic frequency (ω) are studied both perturbatively and numerically by the time-dependent Fourier grid Hamiltonian method. In the absence of anharmonicity, the ground-state population decreases and the population of an accessible excited state (k = 2, 4, 6…) increases with time. However, when anharmonicity is introduced, both the ground- and excited-state populations show typical oscillations. For weak coupling, the population of an accessible excited state at a certain instant of time (short) turns out to be a parabolic function of the anharmonic coupling constant (λ), when all other parameters of the system are kept fixed. This parabolic nature of the excited-state population vs. the λ profile is independent of the specific form of the time dependence of the force constant, K_t. However, it depends upon the rate at which K_t relaxes. For small anharmonic coupling strength and short time scales, the numerical results corroborate expectations based on the first-order time-dependent perturbative analysis, using a suitably repartitioned Hamiltonian that makes H₀ time-independent. Some of the possible experimental implications of our observations are analyzed, especially in relation to intensity oscillations observed in some charge-transfer spectra in systems in which the dephasing rates are comparable with the time scale of the electron transfer. © 1995 John Wiley & Sons, Inc.     

Anharmonicity of a molecular oscillator

The phenomenon of anharmonicity has been proved to be an effect of coupling between the change of nuclear positions in molecular vibrations ( Q ) and the electronic degrees of freedom as represented by the chemical potential (μ) at constant number of electrons (N). The coupling parameters have well‐founded meaning in the conceptual density functional theory (DFT), first approximations to their numerical values have recently become available. The effect of coupling between normal vibrational modes also appears to be the direct consequence of the electron‐nuclear coupling. To show the pure anharmonic effect, calculations for a collection of diatomic molecules have been presented. The anharmonicity, described in the present work as d³E/d Q ³ ≠ 0, has been proved to be the intrinsic property of every oscillating molecular system. A small anharmonic contribution exists even for the “strong harmonic” oscillator, when for the force constant k both a = (?k/? Q )_N = 0 and λ = (?k/?N) _Q = 0. The latter derivative of the force constant appears to be primary factor determining the anharmonic property of a molecule. An estimate of its values has been provided from the experimental data on the anharmonicity of diatomic molecules. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Analytical probability distribution function for a one-dimensional anharmonic oscillator in thermal equilibrium

A simple closed formula is proposed for the probability distribution function of a one-dimensional anharmonic oscillator in thermal equilibrium. It is shown that this formula yields a plausible approximation for the distribution function over wide ranges of the parameter θ = hcωelkT and the degree of anharmonicity. A more general formula with extended limits of applicability is also considered.     

On the variational solution of the coupled breathing rotation-vibration of a spherical top molecule

On the basis of the solutions of the isotropic 3-D harmonic oscillator, we show how to evaluate the matrix elements for the coupled rotation-vibration of the totally symmetric breathing mode of a rotating spherical top molecule, whereby the anharmonic potential energy is expanded in a power series of r. It is shown also how to obtain the ro-vibrational spectrum when used in conjunction with the variational method to obtain eigenvalues.