Variation-perturbation Approach Research Articles

A novel variational perturbation approach for formulating both linear and nonlinear acoustic model equations

A novel approach, based on variational principles, for obtaining linear acoustic model equations directly from an arbitrary Lagrangian is presented, followed by extension to the case of weakly nonlinear acoustics. The efficacy and ease of application of the methodology adopted is demonstrated for two different Lagrangians: one describing classical potential flow, the other relativistic flow including viscous dissipation. It is shown, following appropriate simplification, that corresponding classical, well-known wave equations available in the open literature, derived in the recognised standard manner, are recovered.

Multiple normalized solutions for a quasi-linear Schrödinger equation via dual approach

In this paper, we construct multiple normalized solutions of the following from quasi-linear Schrödinger equation: -\Delta u-\Delta(|u|^{2})u-\mu u=|u|^{p-2}u, \quad\text{in } \mathbb{R}^N, subject to a mass-subcritical constraint. In order to overcome non-smoothness of the associated variational formulation we make use of the dual approach. The constructed solutions possess energies being clustered at $0$ level which makes it difficult to use existing methods for non-smooth variational problems such as the variational perturbation approach.

A Variational Perturbative Approach to Planning in Graph-Based Markov Decision Processes

Coordinating multiple interacting agents to achieve a common goal is a difficult task with huge applicability. This problem remains hard to solve, even when limiting interactions to be mediated via a static interaction-graph. We present a novel approximate solution method for multi-agent Markov decision problems on graphs, based on variational perturbation theory. We adopt the strategy of planning via inference, which has been explored in various prior works. We employ a non-trivial extension of a novel high-order variational method that allows for approximate inference in large networks and has been shown to surpass the accuracy of existing variational methods. To compare our method to two state-of-the-art methods for multi-agent planning on graphs, we apply the method different standard GMDP problems. We show that in cases, where the goal is encoded as a non-local cost function, our method performs well, while state-of-the-art methods approach the performance of random guess. In a final experiment, we demonstrate that our method brings significant improvement for synchronization tasks.

High-beta analytic equilibria in circular, elliptical, and D-shaped large aspect ratio axisymmetric configurations with poloidal and toroidal flows

The Grad-Shafranov-Bernoulli system of equations is a single fluid magnetohydrodynamical description of axisymmetric equilibria with mass flows. Using a variational perturbative approach [E. Hameiri, Phys. Plasmas 20, 024504 (2013)], analytic approximations for high-beta equilibria in circular, elliptical, and D-shaped cross sections in the high aspect ratio approximation are found, which include finite toroidal and poloidal flows. Assuming a polynomial dependence of the free functions on the poloidal flux, the equilibrium problem is reduced to an inhomogeneous Helmholtz partial differential equation (PDE) subject to homogeneous Dirichlet conditions. An application of the Green's function method leads to a closed form for the circular solution and to a series solution in terms of Mathieu functions for the elliptical case, which is valid for arbitrary elongations. To extend the elliptical solution to a D-shaped domain, a boundary perturbation in terms of the triangularity is used. A comparison with the code FLOW [L. Guazzotto et al., Phys. Plasmas 11(2), 604–614 (2004)] is presented for relevant scenarios.

Nuclear induces effects and mass correlations in low and multiply charged helium-like ions

The ground-state electron energies, the mass correction and mass polarization of low and multiply charged helium-like ions are analytically and numerically calculated. Approximately 3500 different kinds of ions with charge Z = 2 ÷ 118 are considered. The two-electron Schrodinger equation was solved using a discrete variational-perturbation approach developed by the authors and based on explicitly correlated wave functions. This approach takes into account the motion of the nucleus and yields accurate values for the electron characteristics. The results are presented with and without the inclusion of the mass polarization in the minimization procedure. The relative importance of mass correlations and relativistic effects in the formation of the electron energy characteristics of the helium-like ions are studied for different values of Z. The role of the inclusion of the mass polarization in the minimization procedure as an instrument to present and take into account the effects induced by the nuclear properties, structure and characteristics has been shown.

Vibrational Signature of a Single Water Molecule Adsorbed on Pt(111): Toward a Reliable Anharmonic Description.

In this study, we present a thorough benchmarking of our direct anharmonic vibrational variation-perturbation approach for adsorbed molecules on surfaces. We then use our method to describe the vibrational structure of a water molecule adsorbed on a Pt(111) surface and compare our results with the available experimental data. By using an explicitly correlated hybrid method to describe the molecule-surface interaction, we improve on the initial periodic PBE/DZP potential energy landscape and obtain vibrational frequencies that are of near-experimental accuracy. We introduce an implementation of anharmonic z-polarized IR intensity calculation and explain the absence of antisymmetric O-H stretch in the experimental data for the adsorbed water molecule, while the symmetric O-H stretch is predicted to be visible.

Existence and symmetry of positive ground states for a doubly critical Schrödinger system

We study the following doubly critical Schrödinger system: \[ { − Δ u − λ 1 | x | 2 u = u 2 ∗ − 1 + ν α u α − 1 v β , x ∈ R N , − Δ v − λ 2 | x | 2 v = v 2 ∗ − 1 + ν β u α v β − 1 , x ∈ R N , u , v ∈ D 1 , 2 ( R N ) , u , v > 0 in R N ∖ { 0 } , \begin {cases}-\Delta u -\frac {\lambda _1}{|x|^2}u=u^{2^\ast -1}+ \nu \alpha u^{\alpha -1}v^\beta , \quad x\in \mathbb {R}^N,\\ -\Delta v -\frac {\lambda _2}{|x|^2}v=v^{2^\ast -1} + \nu \beta u^{\alpha }v^{\beta -1}, \quad x\in \mathbb {R}^N,\\ u,\, v\in D^{1, 2}(\mathbb {R}^N),\quad u,\, v>0\,\,\hbox {in $\mathbb {R}^N\setminus \{0\}$},\end {cases} \] where N ≥ 3 N\ge 3 , λ 1 , λ 2 ∈ ( 0 , ( N − 2 ) 2 4 ) \lambda _1, \lambda _2\in (0, \frac {(N-2)^2}{4}) , 2 ∗ = 2 N N − 2 2^\ast =\frac {2N}{N-2} and α > 1 , β > 1 \alpha >1, \beta >1 satisfying α + β = 2 ∗ \alpha +\beta =2^\ast . This problem is related to coupled nonlinear Schrödinger equations with critical exponent for Bose-Einstein condensate. For different ranges of N N , α \alpha , β \beta and ν > 0 \nu >0 , we obtain positive ground state solutions via some quite different variational methods, which are all radially symmetric. It turns out that the least energy level depends heavily on the relations among α , β \alpha , \,\beta and 2 2 . Besides, for sufficiently small ν > 0 \nu >0 , positive solutions are also obtained via a variational perturbation approach. Note that the Palais-Smale condition cannot hold for any positive energy level, which makes the study via variational methods rather complicated.

Long-range interactions from the self-broadened profile of Zn (41P1–41S0) and Cd (51P1–51S0) lines: correction of the interaction potential parameters

The self-broadened absorption profile of Zn (213.8 nm) and Cd (228.8 nm) lines are measured and carefully analysed. The interaction potentials for the electronic states 0+u and 1u dissociating into (41P1 + 41S0) and (51P1 + 51S0) for Zn2 and Cd2, respectively, are derived in the long-range region of (11–25) Å. It is found that although the resonance dipole–dipole interactions, Cm3/R3, dominate in this region, the higher order van der Waals terms, Cm6/R6 are also not negligible. The coefficients Cm3 and the differences ΔCm6 between Cm6 for each of the 0+u and 1u excited states and Cg6 for the ground state of both molecules are determined. For the energy in eV and R in Å it is found that C03 = −23.74 ± 0.40, ΔC06 = −483 ± 30 and C13 = 11.87 ± 0.20, ΔC16 = −227 ± 20 for Zn2, as well as C03 = −24.54 ± 0.10, ΔC06 = −280 ± 15 and C13 = 12.27 ± 0.05, ΔC16 = −215 ± 10 for Cd2. A problem of the Cg6 coefficient for the considered systems is discussed accepting finally the following values: −178±10 and −291±15 eVÅ6 for Zn2 and Cd2, respectively. Thus, for the excited states involved, one obtains C06 = −661 ± 40 and C16 = −405 ± 30 for Zn2 as well as C06 = −571 ± 30 and C16 = −506 ± 25 for Cd2. The C6m coefficients have never been determined experimentally in any of these cases and only for Zn2 were they calculated by Ellingsen et al (2001 J. Phys. B: At. Mol. Opt. Phys. 34 2313) by means of a time-dependent gauge-invariant method—a variation-perturbation approach. Our experimental results for this case agree very well with theoretical predictions.

Characteristics of X-H···π Interactions: Ab Initio and QTAIM Studies

The MP2/6-311++G(d,p) calculations were performed on several hydrogen-bonded systems. Different complexes were taken into account to analyze various types of hydrogen bonds, possessing different types of proton donors and proton acceptors as well as characterized by the broad range of the interaction energy. The Quantum Theory of Atoms in Molecules is applied. The results of the hybrid variational-perturbational approach are discussed. The unique properties of hydrogen bonds, where π-electrons act as the proton acceptor (X-H···π), are analyzed, and these interactions are compared with the other types of hydrogen bonds, mainly with C-H···Y interactions. It is shown that for X-H···π systems the ellipticity at the bond critical point of the proton···acceptor interaction is much greater than for the other types of hydrogen bonds. However, both X-H···π and C-H···Y interactions are characterized by the dominance of the dispersive energy.

An Integrated Protocol for the Accurate Calculation of Magnetic Interactions in Organic Magnets.

A new, fast, and efficient computational protocol for the accurate calculation of singlet-triplet magnetic splittings in organic diradicals is tested and validated. This procedure essentially consists of three steps: the adoption of modified virtual orbitals (MVO) and a mixed variational-perturbational approach (CSPA) are now combined with a third method that exploits the reduction of the configurational space dimensions achieved by fragmentation/localization criteria. This innovative approach is successfully tested on four different substituted m-phenylene bis(tert-butyl) nitroxides, which show paramagnetic behavior, by computing singlet-triplet energy gaps and comparing them with their experimental counterparts.

Vibrational spectra for P 4O 6 and P 4O 10 systems: Theoretical study from DFT quartic potential and mixed perturbation-variation method

This Letter presents the first anharmonic vibrational results obtained for the tetraphosphorus hexoxide and the tetraphosphorus decaoxide from PBE0/DFT calculations using a 6-311+G(2d) basis set. The quartic force fields are obtained from a linear regression procedure taking into consideration energies and analytical gradients and hessians. The vibrational energies levels are computed using a mixed variation-perturbation approach (VCI-P) in which the vibrational configuration interaction (VCI) space is guessed by a preliminary second order perturbative (P) treatment. Our theoretical results are compared with the existing Raman and IR data for a new vibrational analysis.

Dynamic dipole polarizabilities of the ground and excited states of confined hydrogen atom computed by means of a mapped Fourier grid method

AbstractAccurate theoretical estimates of static and dynamic dipole polarizabilities are reported for the ground and excited states of hydrogen atom confined at the center of a spherical “box” with impenetrable walls using a novel theoretical algorithm that combines the variational–perturbation approach with an appropriately adapted mapped Fourier grid method and uniformly maintains the numerical accuracy. A variation of computed polarizabilities is observed as a function of the number of grid points. However, rapid convergence to their correct values, even far into the anomalous dispersion region, is achieved by an extrapolation procedure to the limit of an infinitely large number of grid points using a small number of the lowest‐order Padé approximants. It is shown that dipole polarizabilities strongly depend upon the electronic state and the radius of confinement. In particular, the static polarizability of 2s state changes sign under strong confinement. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

New Relativistic ANO Basis Sets for Transition Metal Atoms

New basis sets of the atomic natural orbital (ANO) type have been developed for the first, second, and third row transition metal atoms. The ANOs have been obtained from the average density matrix of the ground and lowest excited states of the atom, the positive and negative ions, and the atom in an electric field. Scalar relativistic effects are included through the use of a Douglas-Kroll-Hess Hamiltonian. Multiconfigurational wave functions have been used with dynamic correlation included using second order perturbation theory (CASSCF/CASPT2). The basis sets are applied in calculations of ionization energies, electron affinities, and excitation energies for all atoms and polarizabilities for spherically symmetric atoms. These calculations include spin-orbit coupling using a variation-perturbation approach. Computed ionization energies have an accuracy better than 0.2 eV in most cases. The accuracy of computed electron affinities is the same except in cases where the experimental values are smaller than 0.5 eV. Accurate results are obtained for the polarizabilities of atoms with spherical symmetry. Multiplet levels are presented for some of the third row transition metals.

New relativistic ANO basis sets for actinide atoms

New basis sets of the atomic natural orbital (ANO) type have been developed for the actinide atoms Ac–Cm. The ANOs have been obtained from the average density matrix of the ground and lowest excited states of the atom, the positive ions, and the atom in a electric field. Scalar relativistic effects are included through the use of a Douglas–Kroll–Hess Hamiltonian. Multiconfigurational wave functions have been used with dynamic correlation included using second order perturbation theory (CASSCF/CASPT2). The basis sets are applied in calculations of ionization energies and some excitation energies. Computed ionization energies have an accuracy better than 0.2 eV in most cases. The lowest multiplet levels have been computed. These calculations include spin–orbit coupling using a variation-perturbation approach. The atomic polarizability of the spherically symmetric americium atom has been computed to be 116 au 3.

Sine-Gordon expectation values of exponential fields with variational perturbation theory

In this Letter, expectation values of exponential fields in the 2-dimensional Euclidean sine-Gordon field theory are calculated with variational perturbation approach up to the second order. Our numerical analysis indicates that for not large values of the exponential-field parameter a, our results agree very well with the exact formula conjectured by Lukyanov and Zamolodchikov [Nucl. Phys. B 493 (1997) 571].

Variational perturbation approach to the Coulomb electron gas

The efficiency of the variational perturbation theory [Phys. Rev. C {\bf 62}, 045503 (2000)] formulated recently for many-particle systems is examined by calculating the ground state correlation energy of the 3D electron gas with the Coulomb interaction. The perturbation beyond a variational result can be carried out systematically by the modified Wick's theorem which defines a contraction rule about the renormalized perturbation. Utilizing the theorem, variational ring diagrams of the electron gas are summed up. As a result, the correlation energy is found to be much closer to the result of the Green's function Monte Carlo calculation than that of the conventional ring approximation is.

CCSD(T) Determination of the Vibrational Structure in the Ã2Σ+ ← X̃2Π Spectrum of CCO-

An ab-inito quartic force field of CCO− in its first excited state Ã2∑+ is calculated from an aug-cc-VTZ basis set at CCSD(T) level of theory in order to compute the vibrational energy levels in the range 600−4800 cm-1. A variational-perturbational approach is used to diagonalize the most important vibrational configurations constructed from harmonic vibrational wave functions. The result agree well with the most reliable experimental values studied in a neon matrix and predict as yet unobserved overtones bands in the vibrational structure of the Ã2∑+ ← X̃2Π spectrum of CCO-.

A simple polynomial variational-perturbative approach to the evaluation of dynamic multipole polarizabilities for a ground-state hydrogen atom

Small well-shaped basis sets have been used to perform highly accurate variational-perturbative evaluations of dynamic (frequency-dependent) multipole polarizabilities for a hydrogen atom in its ground state. Their structure has been suggested by the ab initio polynomial power expansion we have recently developed to get correspondingly exact results within a finite range covering the region of low frequencies. The frequency-dependent coefficients of the optimized linear combinations of the excited pseudostate orbitals were obtained by diagonalizing the representative matrix of the excitation operator from the ground state. High levels of accuracy have been observed over a set of frequencies now covering the complete real and imaginary ranges.

A modified variation-perturbation approach to zero-point vibrational motion

We present a detailed investigation of the perturbation approach for calculating zero-point vibrational contributions to molecular properties. It is demonstrated that if the sum of the potential energy and the zero-point vibrational energy is regarded as an effective potential energy, the leading contribution to the first-order wave function vanishes in the perturbation approach. Two different perturbation approaches have been investigated numerically by calculations of some magnetic properties for a few diatomic molecules and the results obtained have been compared to the exact numerical results. It is shown that only a few terms need to be included in a perturbation expansion to obtain an accuracy in the calculated rovibrational contribution better than the quality normally obtained for the potential and property surfaces in electronic structure calculations.

A Variational Perturbative Approach to the Impurity Magnetopolarons in Parabolic Quantum Wires

We use a variational perturbative method for the calculation of the energy levels of impurity magnetopolarons in parabolic quantum wires. The method is reasonable for all electron-phonon coupling α, all Rydberg constant R , all parabolic frequency ω and all cyclotron frequency ω c . The ground state and first excited state energies are computed by the method and their dependence on α, R , ω and ω c is discussed.