Potential Energy Operator Research Articles

Volterra second-order integro-differential equations unresolved for the higher derivative. The case of semibounded operator coefficients

The theorems of existence and uniqueness of strong solutions to Volterra second-order integrodifferential equations in a Hilbert space are proved in the case where the energy pumping to a system occurs (the energy dissipation operator is bounded from below), and the system can be unstable (the potential energy operator is bounded from below as well).

Quantum Dynamics of Carbon Dioxide Encapsulated in the Cages of the sI Clathrate Hydrate: Structural Guest Distributions and Cage Occupation

We perform quantum multiconfiguration time-dependent Hartree calculations for the two cages of the sI CO2 clathrate hydrate, and we report, for first time, results on the translational, rotational, and vibrational states. The potential surface is constructed by the semiempirical SPC/E water model and pairwise additive interactions between the CO2 molecule and all water molecules forming each cage, while a spectroscopic-determined potential is used for the CO2 monomer. The exact kinetic energy operator is derived for the linear CO2 in the cage, and the potential energy operator is adapted to combined mode schemes, chosen adequately in order to obtain the desired product representation. The rotational ability of the CO2 is hindered in the small cavities (512), while in the large cage (51262) translational and rotational degrees of freedom are highly coupled. By analyzing the corresponding T-R eigenstates, we established a connection with experimental X-ray measurements on the orientation of the CO2 molecule in the two cages. Further, vibrational excitations of the fundamental symmetric and antisymmetric stretch modes, their overtones, and combination bands are computed from 7D quantum calculations. We found significant frequency shifts, which are comparable to those observed in the double-peak profile of the experimentally recorded infrared spectra, for the CO2 molecule trapped in the small and large cages of the sI clathrate hydrate. Such good agreement, with the data from recent spectroscopic studies of carbon dioxide clathrate hydrate at low temperatures, indicates a single occupancy of the sI structure cages, allowing an assessment of the theoretical approaches employed.

Quasi-Sturm Functions in Problems of the Continuous Spectrum

equation. In all of the enumerated approaches the Coulomb interaction enters into the unperturbed part of the Hamiltonian in the construction of the two-particle basis functions. Here, the Coulomb Green’s function was obtained in [6, 7] in terms of hypergeometric functions. In turn, the short-range potential energy operator is represented here in a finite subspace of L 2 basis functions. As a result of truncation of the basis space, the characteristics of scattering on the model potential undergo oscillations as the number of basis functions used to represent the potential is increased [8]. Thus, already in the solution of the two-particle scattering problem within the framework of the J-matrix method, additional measures are needed to hasten convergence. However, the basis functions here are found numerically, where the problem of generating the basis presents a problem almost more complex that the original scattering problem. In the present work, we propose basis functions that we call quasi-Sturm. These QS functions are formally solutions of the Schrodinger equation, whose right-hand side contains the Laguerre L 2 -functions. Thus, in contrast to the Sturm functions, the QS functions with prescribed asymptotic behavior can be obtained (with the help of the corresponding Green’s function) in closed analytical form. As our calculations show, the use of QS functions are as efficient in the solution of the two-particle scattering problem as the expansion over generalized Sturm functions. This work is organized as follows. In Section 1 we recall the necessary elements of the J-matrix method and the method of generalized Sturm functions. Section 2 gives a definition of QS functions with different asymptotic behavior

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On the basis of both experiment and theory, accurate rotational–vibrational line positions and energy levels, with associated critically reviewed labels and uncertainties, are reported for the ground electronic state of the H3+ molecular ion. An improved MARVEL algorithm is used to determine the validated experimental levels and their self-consistent uncertainties from a set of 1610 measured transitions and associated uncertainties, coming from 26 sources. 1410 transitions have been validated for ortho-H3+ and para-H3+, 78 belong to floating components of the spectroscopic network (SN) investigated and thus left unvalidated, while 122 measured transitions had to be excluded from the MARVEL analysis for one reason or another. The spectral range covered by the experiments is 7–16 506 cm–1 . Altogether 13 vibrational band origins are reported, the highest J value, where J stands for the rotational quantum number, for which energy levels are validated is 12. The MARVEL energy levels are checked against ones determined from accurate variational nuclear motion computations employing the best available adiabatic ab initio potential energy surface and exact kinetic energy operators. The number of validated and thus recommended experimental-quality rovibrational energy levels is 652, of which 259 belong to ortho-H3+ and 393 to para-H3+. There are 105 further energy levels within floating components of the SN. The variational computations have been performed both without and with a simple nonadiabatic correction, whereby nonadiabaticity is modeled by the use of a non-nuclear vibrational mass. The lists of validated lines and levels for H3+ are deposited in the Supporting Information to this paper.

Quantum Mechanical Characterization of the He4ICl Weakly Bound Complex

Vibrational calculations are performed for the 12-dimensional He4ICl van der Waals complex using the multiconfiguration time-dependent Hartree (MCTDH) method. The potential energy surface of the cluster is represented as a sum of the triatomic He-ICl ab initio parametrized terms plus the He-He interactions. The topology of the potential presents higher anisotropy compared to the one with a homonuclear dopant, and this is clearly reflected in the structure and energetics of the low-lying conformers of the system. In order to take advantage of the MCTDH method, natural potential fits are employed for the potential energy operator, and also, a mode combination scheme is introduced in order to speed up the computations. Zero-point energy, binding energies, and vibrationally averaged structures of different isomers of the He4ICl cluster are obtained. The present results predict that the (3,1,0) structure, involving three He atoms in the near T-shaped and one He atom in the linear configurations, to be the most stable one in accord with recent experimental findings. Comparisons with previous theoretical and experimental data are presented, and the stability of the high-order conformers is discussed in connection with the multiple minima (global and local) of the underlying potential surface.

Vibrational dynamics of the ${\rm H}_5^+$H5+ and its isotopologues from multiconfiguration time-dependent Hartree calculations

Full-dimensional multiconfiguration time-dependent Hartree (MCTDH) computations are reported for the vibrational states of the H(5)(+) and its H(4)D(+), H(3)D(2)(+), H(2)D(3)(+), HD(4)(+), D(5)(+) isotopologues employing two recent analytical potential energy surfaces of Xie et al. [J. Chem. Phys. 122, 224307 (2005)] and Aguado et al. [J. Chem. Phys. 133, 024306 (2010)]. The potential energy operators are constructed using the n-mode representation adapted to a four-combined mode cluster expansion, including up to seven-dimensional grids, chosen adequately to take advantage in representing the MCTDH wavefunction. An error analysis is performed to quantify the convergence of the potential expansion to reproduce the reference surfaces at the energies of interest. An extensive analysis of the vibrational ground state properties of these isotopes and comparison with the reference diffusion Monte Carlo results by Acioli et al. [J. Chem. Phys. 128, 104318 (2008)] are presented. It is found that these systems are highly delocalized, interconverting between equivalent minima through rotation and internal proton transfer motions even at their vibrational ground state. Isotopic substitution affects the zero-point energy and structure, showing preference in the arrangements of the H and D within the mixed clusters, and the most stable conformers of each isotopomer are the ones with the H in the central position. Vibrational excited states are also computed and by comparing the energies and structures predicted from the two surfaces, the effect of the potential topology on them is discussed.

Rovibrational interactions in linear triatomic molecules: a theoretical study in curvilinear vibrational coordinates

Variational solution of the rovibrational problem in curvilinear vibrational coordinates has been implemented and used to investigate the nuclear motions in several linear triatomic molecules, like HCN, OCS, and HCP. The dependence of the rovibrational energy levels on the rotational quantum numbers and the l-doubling has been studied. Two approximations to the rovibrational Hamiltonian have been examined, depending on the level of truncation of the potential energy operator. It turns out that the truncation after the fifth order in the potential is sufficient to produce vibrational energies of high accuracy. An interesting feature of the present formulation of the problem in terms of the curvilinear vibrational coordinates is the explanation of the l-doubling of the rovibrational levels, which in this picture is interpreted as the result of the inequivalency of the average rotational constants in mutually perpendicular planes, rather than as the effect of the Coriolis-type interactions between the vibrational and rotational motions. The present theoretical results are compared with the available experimental data from high-resolution spectroscopy, as well as with other ab initio calculations.

Kinetic energy principle and neoclassical toroidal torque in tokamaks

It is shown that when tokamaks are perturbed, the kinetic energy principle is closely related to the neoclassical toroidal torque by the action invariance of particles. Especially when tokamaks are perturbed from scalar pressure equilibria, the imaginary part of the potential energy in the kinetic energy principle is equivalent to the toroidal torque by the neoclassical toroidal viscosity. A unified description therefore should be made for both physics. It is also shown in this case that the potential energy operator can be self-adjoint and thus the stability calculation can be simplified by minimizing the potential energy.

Matrix-free application of Hamiltonian operators in Coifman wavelet bases

A means of evaluating the action of Hamiltonian operators on functions expanded in orthogonal compact support wavelet bases is developed, avoiding the direct construction and storage of operator matrices that complicate extension to coupled multidimensional quantum applications. Application of a potential energy operator is accomplished by simple multiplication of the two sets of expansion coefficients without any convolution. The errors of this coefficient product approximation are quantified and lead to use of particular generalized coiflet bases, derived here, that maximize the number of moment conditions satisfied by the scaling function. This is at the expense of the number of vanishing moments of the wavelet function (approximation order), which appears to be a disadvantage but is shown surmountable. In particular, application of the kinetic energy operator, which is accomplished through the use of one-dimensional (1D) [or at most two-dimensional (2D)] differentiation filters, then degrades in accuracy if the standard choice is made. However, it is determined that use of high-order finite-difference filters yields strongly reduced absolute errors. Eigensolvers that ordinarily use only matrix-vector multiplications, such as the Lanczos algorithm, can then be used with this more efficient procedure. Applications are made to anharmonic vibrational problems: a 1D Morse oscillator, a 2D model of proton transfer, and three-dimensional vibrations of nitrosyl chloride on a global potential energy surface.

A new electron-methanol molecule pseudopotential and its application for the solvated electron in methanol

A new electron-methanol molecule pseudopotential is developed and tested in the present paper. The formal development of the potential is based on quantum mechanical calculations on the electron-methanol molecule model in the static exchange approximation. The computational model includes a steep confining potential that keeps the otherwise unbound excess electron in the vicinity of the methanol molecule. Using the Phillips-Kleinman theorem we introduce a smooth pseudowave function of the excess electron with the exact eigenenergy and correct asymptotic behavior. The nonlocal potential energy operator of the model Hamiltonian is then replaced to a local potential that reproduces the ground-state properties of the excess electron satisfactorily. The pseudopotential is then optimized in an analytically simple functional form to fit this approximate local potential in conjunction with the point charges and the geometry of a classical, all-site methanol-methanol interaction potential. Of the adjustable parameters, the parameters for the carbon and the methyl hydrogen atoms are optimized, while those for the oxygen and the hydroxyl hydrogen are taken from a previous electron-water molecule pseudopotential. A polarization term is added to the potential a posteriori. The polarization parameters are chosen to reproduce the experimental position of the optical absorption spectrum of an excess electron in mixed quantum-classical molecular dynamics simulations. The energetic, structural and spectroscopic properties of the solvated electron in a methanol bath are simulated at 300 K and compared with previous solvated electron simulations and available experimental data.

New perspectives on the fundamental theorem of density functional theory

AbstractThe fundamental theorem of time‐independent/time‐dependent density functional theory due to Hohenberg–Kohn (HK)/Runge–Gross (RG) proves the bijectivity between the density ρ(r)/ρ(rt) and the Hamiltonian /(t) to within a constant C/function C(t), and wave function Ψ/Ψ (t). The theorems are each proved for scalar external potential energy operators. By a unitary or equivalently a gauge transformation that preserves the density, we generalize the realm of validity of each theorem to Hamiltonians, which additionally include the momentum operator and a curl‐free vector potential energy operator defined in terms of a gauge function α (R)/α (Rt). The original HK/RG theorems then each constitute a special case of this generalization. Thereby, a fourfold hierarchy of such theorems is established. As a consequence of the generalization, the wave function Ψ/Ψ (t) is shown to be a functional of both the density ρ(r)/ρ(rt), which is a gauge‐invariant property, and a gauge function α(R)/α(Rt). The functional dependence on the gauge function ensures that as required by quantum mechanics, the wave function written as a functional is gauge variant. The hierarchy and the dependence of the wave function functional on the gauge function thus enhance the significance of the phase factor in density functional theory in a manner similar to that of quantum mechanics. Various additional perspectives on the theorem are arrived at. These understandings also address past critiques of time‐dependent theory. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

Vibrational energy levels with arbitrary potentials using the Eckart-Watson Hamiltonians and the discrete variable representation

An effective and general algorithm is suggested for variational vibrational calculations of N-atomic molecules using orthogonal, rectilinear internal coordinates. The protocol has three essential parts. First, it advocates the use of the Eckart-Watson Hamiltonians of nonlinear or linear reference configuration. Second, with the help of an exact expression of curvilinear internal coordinates (e.g., valence coordinates) in terms of orthogonal, rectilinear internal coordinates (e.g., normal coordinates), any high-accuracy potential or force field expressed in curvilinear internal coordinates can be used in the calculations. Third, the matrix representation of the appropriate Eckart-Watson Hamiltonian is constructed in a discrete variable representation, in which the matrix of the potential energy operator is always diagonal, whatever complicated form the potential function assumes, and the matrix of the kinetic energy operator is a sparse matrix of special structure. Details of the suggested algorithm as well as results obtained for linear and nonlinear test cases including H(2)O, H(3) (+), CO(2), HCNHNC, and CH(4) are presented.

DETERMINATION OF BINDING ENERGY OF THE GROUND STATE FOR ONE-DIMENSIONAL HELIUM ATOM IN HYPERSPHERICAL APPROACH

Ground state binding energy for one-dimensional helium atom has been obtained in hyperspherical adiabatic approach. Logarithmic divergences in matrix elements of the potential energy operator were removed using the theory of residue, which led to the appearance of their complex values. An additional region of attraction was found for small values of radial variable. The width of this region depends on the dimension of basis, on which the adiabatic potentials are determined. The present results are compared with the calculations obtained in other approaches.

The intramolecular tunnelling of hydrogen†

This paper reports an approach to the analysis of the local anharmonic states of motion and tunnelling of hydrogen illustrated with the molecule malonaldehyde. A potential energy function is used accurately to fit the local potential energy surface. The basis fitting functions in the potential resemble the Morse basis function, , but are intrinsically Gaussian: . After rearrangement to isolate the Gaussian operators, the new form of the potential allows the direct use of mathematically friendly (Gaussian) manipulations that are needed to carry out the quantum mechanical analysis of the anharmonic features, namely tunnelling. The quantal analysis uses Cartesian harmonic oscillator basis functions together with the Talmi–Moshinsky–Smirnov methods to evaluate matrix elements of the potential energy operator. The longer term objective of this kind of work is to develop practical methods of hybrid analysis with which to treat the local atom dynamics associated with a reactive species while the rest of the dynamics are modelled as simple harmonic oscillators. The goal is eventually to have a computational theory of proton/hydrogen or small molecular group transfer reactions in condensed-phase systems. This paper addresses only the detailed local dynamical analysis. †Dedicated to Pavel Rosmus on the occasion of the celebration of his scienti.c achievements, in Prague.

A computationally efficient exact pseudopotential method. II. Application to the molecular pseudopotential of an excess electron interacting with tetrahydrofuran (THF)

In the preceding paper, we presented an analytic reformulation of the Phillips-Kleinman (PK) pseudopotential theory. In the PK theory, the number of explicitly treated electronic degrees of freedom in a multielectron problem is reduced by forcing the wave functions of the few electrons of interest (the valence electrons) to be orthogonal to those of the remaining electrons (the core electrons); this results in a new Schrodinger equation for the valence electrons in which the effects of the core electrons are treated implicitly via an extra term known as the pseudopotential. Although this pseudopotential must be evaluated iteratively, our reformulation of the theory allows the exact pseudopotential to be found without ever having to evaluate the potential energy operator, providing enormous computational savings. In this paper, we present a detailed computational procedure for implementing our reformulation of the PK theory, and we illustrate our procedure on the largest system for which an exact pseudopotential has been calculated, that of an excess electron interacting with a tetrahyrdrofuran (THF) molecule. We discuss the numerical stability of several approaches to the iterative solution for the pseudopotential, and find that once the core wave functions are available, the full e(-)-THF pseudopotential can be calculated in less than 3 s on a relatively modest single processor. We also comment on how the choice of basis set affects the calculated pseudopotential, and provide a prescription for correcting unphysical behavior that arises at long distances if a localized Gaussian basis set is used. Finally, we discuss the effective e(-)-THF potential in detail, and present a multisite analytic fit of the potential that is suitable for use in molecular simulation.

The Negative and Positive Charged Two-dimensional Three-body Systems in Adiabatic Hyperspherical Approach

Binding energies of ground 1S-states for two-dimensional three-body systems (e−e+e−, e−p+e−, He and Li+), which were received using the hyperspherical adiabatic approach in Born-Oppenheimer approximation, are presented. For description of electron correlation we use K-harmonics, which contain ground angular momentum equal to 0, 2, 4 and 6. We used expansion of potential energy operator in series on multipoles including up to forty terms for description of angular correlations. The values of binding energies are four times bigger than in three-dimensional cases with accuracy of about one-two percents. The convergence of energy values according to expansion parameters for channel function is shown. Avoided-crossing regions (R ≈ 0.2−5 a.u.) between adiabatic potential U2(R) and U3(R) were also found.

The collision-induced transitions, M=He, Ar, N2, CF4

The dependences of the I 2 ( E0 g + ,v E ,J E → M D0 u + ,v D ,J D ) , M=He, Ar, N 2 ,CF 4 , transition rate constants on the vibrational v E , v D , rotational J E , J D quantum numbers, energy gaps, as well as their correlations with Franck–Condon factors of the initial and final levels have been studied. Intramolecular transitions induced by the C 3 / R 3 term of multipole series of potential energy operator instantaneous dipole (I 2 (E–D) optical transition) – instantaneous dipole (vibrational transitions in CF 4 molecule) have been found out and discussed. The collision-induced intramolecular transitions observed in this and our previous works [Chem. Phys. 242 (1999) 263, 263 (2001) 459 and Chem. Phys. 277 (2002) 179] have been classified.

A General Variational Algorithm to Calculate Vibrational Energy Levels of Tetraatomic Molecules

A general variational method for calculating vibrational energy levels of tetraatomic molecules is presented. The quantum mechanical Hamiltonian of the system is expressed in a set of coordinates defined by three orthogonalized vectors in the body-fixed frame without any dynamical approximation. The eigenvalue problem is solved by a Lanczos iterative diagonalization algorithm, which requires the evaluation of the action of the Hamiltonian operator on a vector. The Lanczos recursion is carried out in a mixed grid/basis set, i.e., a direct product discrete variable representation (DVR) for the radial coordinates and a nondirect product finite basis representation (FBR) for the angular coordinates. The action of the potential energy operator on a vector is accomplished via a pseudo-spectral transform method. Six types of orthogonal coordinates are implemented in this algorithm, which is capable of describing most four-atom systems with small and/or large amplitude vibrational motions. Its application to the molecules H 2CO, NH 3, and HOOH and the van der Waals cluster He 2Cl 2 is discussed.

The energy as a functional of the charge density and the charge-density susceptibility: A simple, exact, nonlocal expression for the electronic energy of a molecule

Simple, new expressions relate the electronic potential energy 〈V〉 and the total electronic energy E of a molecule to its averaged electron density 〈ρe(r)〉, the nonlocal charge-density susceptibility χe(r,r′;iω), the nuclear positions {RN}, and the nuclear charges {ZN}. The expressions derived in this work are exact nonrelativistically, within the Born–Oppenheimer approximation. The results give a nonlocal form for the electronic energy in density functional theory. The virial theorem for a system with Coulomb forces is used to derive the expectation value of the kinetic energy in terms of the expectation values of the potential energy and the derivatives of the potential energy operator with respect to nuclear coordinates; gradient expansions of the kinetic energy functional are not needed. Exchange and correlation effects on 〈V〉 and E are determined by an integral of the charge-density susceptibility χe(r,r′;iω), over imaginary frequencies. The results for 〈V〉 and E are first derived by use of the fluctuation-dissipation theorem and the symmetry properties of the charge-density susceptibility with respect to a change in the sign of ω. Identical results are derived by integration of χe(r,r′;iω) over imaginary frequencies and use of the closure relation.

Full-dimensional quantum calculation of the vibrational energy levels of hydrogen peroxide (HOOH)

Vibrational levels of HOOH up to 6000 cm −1 are accurately determined on a newly developed ab initio potential using the Lanczos algorithm. The six-dimensional Hamiltonian is discretized in terms of a direct-product potential optimized discrete variable representation (DVR) for the radial coordinates and a non-direct product finite basis representation (FBR) comprised of uncoupled spherical harmonics for the angular coordinates. The action of the potential energy operator is calculated on a direct-product grid via a factorized FBR-DVR transformation. All energy levels below 3000 cm −1 are assigned and compared with existing theoretical and experimental data.