New isospectral generalized potentials

In quantum chemistry, supersymmetry, shape invariance and intertwining techniques are used to determine the class of potentials that are solvable as well as to find their isospectral and generalized partners. To do that, it is necessary to have the corresponding Witten superpotential defined by W(x)=/ where is a particular wavefunction of the Hamiltonian under study. In this work, we propose an alternative way to express the Witten superpotential in terms of reciprocal wavefunctions. Thus, when this new definition of W(x) is used as an ansatz in the Riccati equation involved, one is led to a potential identical to that resulting from the use of the standard Darboux transform, which means that it is possibly the generalization of it. Moreover, the generalization of the new Witten superpotential gives rise to a new generalized isospectral potential other than that obtained from the generalized Darboux transform. As an example of an application of the proposed approach, we found the new generalized isospectral potentials that correspond to the one-dimensional free particle, harmonic oscillator and Morse potential models. Also, owing to the fact that the proposed method is general our proposal can be used straightforwardly to obtain new, exactly solvable potentials as well as to find their isospectral generalized partners which can be used advantageously in the modeling of important quantum chemical applications.From the Proceedings of the 28th Congreso de Químicos Teóricos de Expresíon Latina (QUITEL 2002) Acknowledgement.This work was supported by CONACYT-Mexico, under scientific project No. 32762-E.     

The generalized Hulthén potential

A generalization of the Hulthén potential is presented on the basis of an approach that uses the factorization of a general Hamiltonian by means of a specific model of operational equations with the structure ∼β(r)∓(d/dr). To achieve this goal, the treatment of the V _Hs(r) standard Hulthén potential for bound s states is carried out by proposing a particular β_p(r) ansatz to identify V _Hs(r) by means of a particular Riccati-type relationship. Once the identification has been achieved, the generalized Hulthén potential is obtained straightforwardly with the aid of a general Riccati formula. As expected, the Hamiltonian of the generalized Hulthén potential is isospectral when compared with the corresponding standard Hamiltonian. Moreover, according to the Darboux transform there exists a modified Hulthén potential which is also isospectral. We show that the latter is just a particular case of the generalized Hulthén interaction model. Received: 14 September 1999 / Accepted: 3 February 2000 / Published online: 19 April 2000     

A simple and effective technique to variationally interpret the structure of SUSY partners of mirror image potentials

Precise supersymmetric partner potentials can be generated for exactly solvable problems of the stationary Schrödinger equation. Construction of isospectral potential is not always possible for exactly solvable systems. This is a restriction, as most problems are not exactly solvable. Employment of mirror-image property can help to construct an exact isospectral partner of that potential. These potentials have chemical relevance to enantiomers. In this paper, we present a formulation as modelling to explore the form of SUSY pair of these potentials. Through polynomial fit, we correlate all possible basic SUSY partners and optimise it to best fit polynomial to present a typical energy value of N = 50.     

Energy-gap opening and quenching in graphene under periodic external potentials

We investigated the effects of periodic external potentials on properties of charge carriers in graphene using both the first-principles method based on density functional theory (DFT) and a theoretical approach based on a generalized effective spinor Hamiltonian. DFT calculations were done in a modified Kohn-Sham procedure that includes the effects of the periodic external potential. Unexpected energy band gap opening and quenching were predicted for the graphene superlattice with two symmetrical sublattices and those with two unsymmetrical sublattices, respectively. Theoretical analysis based on the spinor Hamiltonian showed that the correlations between pseudospins of Dirac fermions in graphene and the applied external potential, and the potential-induced intervalley scattering, play important roles in energy-gap opening and quenching.     

Generalized relativistic effective core potential: Theoretical grounds

The generalized relativistic effective core potential (GRECP) method is analyzed from theoretical and computational points of view. The Hamiltonian in the frozen‐core approximation is compared with the Hamiltonian containing the GRECP operator. It is demonstrated that the GRECP operator can be derived from rather natural physical grounds and the procedure of the GRECP generation can be justified theoretically. The accuracy of the RECP approximations in the simulation of the interactions and densities in the valence and outer‐core regions is analyzed. The reliability of the simulation of the interaction with the inner‐core electrons removed from the calculations with the GRECP is also studied. The importance of additional nonlocal terms both with the potentials for the outer‐core pseudospinors and with the potentials depending on the occupation numbers of the outermost core shells in the expression for the GRECP operator is demonstrated in calculations on the Ag, Ba, Hg, Tl, and U atoms. The difference between the outer core and valence potentials was investigated. It is shown that in the valence region the two‐component pseudospinors coincide with the large components of four‐component spinors in calculations for the same configuration states with a very high accuracy. Problems of Gaussian approximation caused by rather singular shapes of the potentials are considered. To attain a required high accuracy of approximation of the numerical potentials by Gaussians, serious additional efforts were undertaken. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 359–401, 1999     

Stress and heat flux for arbitrary multibody potentials: a unified framework

A two-step unified framework for the evaluation of continuum field expressions from molecular simulations for arbitrary interatomic potentials is presented. First, pointwise continuum fields are obtained using a generalization of the Irving-Kirkwood procedure to arbitrary multibody potentials. Two ambiguities associated with the original Irving-Kirkwood procedure (which was limited to pair potential interactions) are addressed in its generalization. The first ambiguity is due to the nonuniqueness of the decomposition of the force on an atom as a sum of central forces, which is a result of the nonuniqueness of the potential energy representation in terms of distances between the particles. This is in turn related to the shape space of the system. The second ambiguity is due to the nonuniqueness of the energy decomposition between particles. The latter can be completely avoided through an alternate derivation for the energy balance. It is found that the expressions for the specific internal energy and the heat flux obtained through the alternate derivation are quite different from the original Irving-Kirkwood procedure and appear to be more physically reasonable. Next, in the second step of the unified framework, spatial averaging is applied to the pointwise field to obtain the corresponding macroscopic quantities. These lead to expressions suitable for computation in molecular dynamics simulations. It is shown that the important commonly-used microscopic definitions for the stress tensor and heat flux vector are recovered in this process as special cases (generalized to arbitrary multibody potentials). Several numerical experiments are conducted to compare the new expression for the specific internal energy with the original one.     

Efficiency of nested Markov chain Monte Carlo for polarizable potentials and perturbed Hamiltonians

Nested Markov chain Monte Carlo is a rigorous way to enhance sampling of a given energy landscape using an auxiliary, approximate potential energy surface. Its practical efficiency mainly depends on how cheap and how different are the auxiliary potential with respect to the reference system. In this article, a combined efficiency index is proposed and assessed for two important families of energy surfaces. As illustrated for water clusters, many‐body polarizable potentials can be approximated by simplifying the polarization contribution and keeping only the two‐body terms. In small systems, neglecting polarization entirely is also acceptable. When the reference potential energy is obtained from diagonalization of a quantum mechanical Hamiltonian, a first‐order perturbation scheme can be used to estimate the energy difference occuring on a Monte Carlo move. Our results indicate that this perturbation approximation performs well provided that the number of steps between successive diagonalization is adjusted beforehand. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem 110:2342–2346, 2010     

On the use of effective core potentials in the calculation of magnetic properties, such as magnetizabilites and magnetic shieldings

State-of-the art effective core potentials (ECPs) that replace electrons of inner atomic cores involve non-local potentials. If such an effective core potential is added to the Hamiltonian of a system in a magnetic field, the resulting Hamiltonian is not gauge invariant. This means, magnetic properties such as magnetisabilities and magnetic shieldings (or magnetic susceptibilities and nuclear magnetic resonance chemical shifts) calculated with different gauge origins are different even for exact solutions of the Schro?dinger equation. It is possible to restore gauge invariance of the Hamiltonian by adding magnetic field dependent terms arising from the effective core potential. Numerical calculations on atomic and diatomic model systems (potassium mono-cation and potassium dimer) clearly demonstrate that the standard effective core potential Hamiltonian violates gauge invariance, and this affects the calculation of magnetisabilities more strongly than the calculation of magnetic shieldings. The modified magnetic field dependent effective core potential Hamiltonian is gauge invariant, and therefore it is the correct starting point for distributed gauge origin methods. The formalism for gauge including atomic orbitals (GIAO) and individual gauge for localized orbitals methods is worked out. ECP GIAO results for the potassium dimer are presented. The new method performs much better than a previous ECP GIAO implementation that did not account for the non-locality of the potential. For magnetic shieldings, deviations are clearly seen, but they amount to few ppm only. For magnetisabilities, our new ECP GIAO implementation is a major improvement, as demonstrated by the comparison of all-electron and ECP results.     

Calculation of free-energy differences and potentials of mean force by a multi-energy gap method

A method is proposed to significantly accelerate the convergence of free-energy calculations. It introduces a bias factor in Monte Carlo simulations or, equivalently, a bias force in molecular dynamics simulations. The bias factor targets the energy gap, i.e., the difference in energy function between two states, and is therefore specifically designed for calculating free-energy differences. The goal is to make the probability density of the energy gap as uniform as possible, thus allowing for its accurate determination. An iterative procedure, based on simulations at higher temperatures, is devised to obtain the bias factor. The same method naturally extends to the calculation of potentials of mean force. The generalized coordinate, for which the potential of mean force is to be calculated, now plays the role of the energy gap. Applications to model systems confirm the expected increase in accuracy of calculated free-energy differences and potentials of mean force.     

Bohr Sommerfeld quantisation and molecular potentials

We combine, within the Bohr Sommerfeld quantization rule, a systematic perturbation with asymptotic analysis of the action integral for potentials which support a finite number of bound states with E < 0 to obtain an interpolation formula for the energy eigenvalues. We find interpolation formulae for the Morse potential as well as potentials of the form \({V=V_0 \left[ {\left( {\frac{a}{x}} \right)^{2k}-\left( {\frac{a}{x}}\right)^{k}} \right]}\). For k = 6 i.e. the well known Lennard Jones potential this yields results within 1 per cent of the highly accurate numerical values. For the Morse potential this procedure yields the exact answer. We find that the result for the Morse potential which approaches zero exponentially is the \({k\rightarrow\infty}\) limit of the Lennard Jones class of potentials.     

The application of direct potential fitting to the X1sigma+ ground electronic states of LiCl, TlCl, RbF and CsF.

A procedure for directly fitting the potential energy curve of a diatomic molecule has been applied to the X1sigma+ ground states of LiCl, TlCl, RbF and CsF. Extensive, high-precision infrared and pure-rotational data sets for all isotopomers of the aforementioned molecules have been employed in direct least-squares fits of a radially dependent Hamiltonian operator. The Born-Oppenheimer potentials are represented by a modified Lennard-Jones function that is shown to be flexible and provide the proper behavior in the long-range region of the potential. While the potential fits of LiCl and TlCl required the inclusion of Born-Oppenheimer breakdown functions, no such functions were necessary for either RbF or CsF.     

Improvement to the hypervirial theorem and Kramer's rule for the calculation of central potential matrix elements

Ameliorated recurrence relations for the calculation of matrix elements of central potential wavefunctions are presented. These were obtained by using the hypervirial theorem with V(r) three-dimensional potentials and f (r) arbitrary functions. This procedure leads to a generalization of the usual l = l′ diagonal three-dimensional and l == 0 one-dimensional hypervirial relations of first and second class. The use of this kind of generalization to the calculation of r^k integrals, allows one to obtain all off-diagonal recursion formulas for any V(r). Besides, for hydrogenic wavefunctions one gets to equations that reduce to the usual Kramer's rule as a particular diagonal case. The proposed approach can be straightforwardly extended to obtain recurrence relations for the calculation of two center integrals.     

Anisotropic effective core potentials

We present a convenient method for accounting for the anisotropy of partially filled d shells that are incorporated in effective core potentials by including the leading anisotropy term of the d-type electron density, which is the quadrupole moment, as an electrostatic potential energy operator in the model Hamiltonian. We present sample calculations on the cobalt hydride and dicobalt systems. We find the quadrupole anisotropy to have a very large effect in the distance regime of CoH. In the dicobalt system, which has a relatively long internuclear distance, the quadrupole anisotropy shifts the equilibrium bond length by nearly 0.02 bohr.     

Axially symmetric potentials in the oscillator representation

The Wick-ordering method called the Oscillator Representation in the nonrelativistic Schrödinger equation is proposed to calculate the energy spectrum for axially symmetric potentials allowing the existence of a bound state. In particular, the method is applied to calculate the energy spectrum of (2s) states of a hydrogen atom in a uniform magnetic field of an arbitrary strength. In the perturbation (external field) approximation, the energy spectrum of the so-called quadratic and spherical quadratic Zeeman problem and the problem of a hydrogen atom in a generalized van der Waals potential is calculated analytically. The results of the zeroth approximation of oscillator representation are in good agreement with the exact values     

Generalized relativistic effective core potentials for superheavy elements

The generalized relativistic effective core potentials (GRECPs) for calculations of electronic structure and physical-chemical properties of compounds containing superheavy elements (Z ≥ 104) are presented. Features of accounting for the finite nuclear size effects which are unusually large for superheavy elements are discussed in details. Accuracy of the GRECPs is analyzed in atomic calculations compared to all-electron studies with the Dirac-Coulomb-Breit Hamiltonian. Applications of the GRECP method in molecular and cluster calculations are surveyed.     

Information theoretic spreading measures of orthogonal functions

We calculate information theoretic spreading measures of orthogonal functions associated with solutions of quantum mechanical isospectral potentials. In particular, Shannon, Renyi and Fisher lengths have been evaluated for potentials isospectral to the linear harmonic oscillator and the symmetric Rosen-Morse potentials. We have also compared the behaviour of different lengths for the orthogonal functions and the associated orthogonal polynomials.     

Calculating particle pair potentials from fluid‐state pair correlations: Iterative ornstein–zernike inversion

An iterative Monte Carlo inversion method for the calculation of particle pair potentials from given particle pair correlations is proposed in this article. The new method, which is best referred to as Iterative Ornstein–Zernike Inversion, represents a generalization and an improvement of the established Iterative Boltzmann Inversion technique (Reith, Pütz and Müller‐Plathe, J. Comput. Chem. 2003, 24, 1624). Our modification of Iterative Boltzmann Inversion consists of replacing the potential of mean force as an approximant for the pair potential with another, generally more accurate approximant that is based on a trial bridge function in the Ornstein–Zernike integral equation formalism. As an input, the new method requires the particle pair correlations both in real space and in the Fourier conjugate wavenumber space. An accelerated iteration method is included in the discussion, by which the required number of iterations can be greatly reduced below that of the simple Picard iteration that underlies most common implementations of Iterative Boltzmann Inversion. Comprehensive tests with various pair potentials show that the new method generally surpasses the Iterative Boltzmann Inversion method in terms of reliability of the numerical solution for the particle pair potential. © 2018 Wiley Periodicals, Inc.     

Analytical density-dependent representation of Hartree—Fock atomic potentials

A procedure to represent atomic electron charge densities [L. Fernandez Pacios, J. Phys. Chem., 95 , 10653 (1991); J. Phys. Chem., 96 , 7294 (1992)] is here generalized to obtain simple analytical functions for potential energy contributions. Based upon suitable functions to describe atomic electron densities in a physically meaningful form, the procedure is developed to define density-dependent analytical expressions for the electrostatic (classical) and exchange (quantum) potentials by means of proper approximate functionals. Calculations of correlation energies by using various density-functional approaches are also performed. The whole scheme is used to represent Hartree–Fock limit atomic wave functions by Clementi–Roetti. This way, a set of analytically simple, nonbasis set-dependent functions are defined with the aim to be further implemented in energy decomposition schemes for molecular interactions studies using atomic instead of electronic building blocks. © 1993 John Wiley & Sons, Inc.     

Deriving effective mesoscale potentials from atomistic simulations

We demonstrate how an iterative method for potential inversion from distribution functions developed for simple liquid systems can be generalized to polymer systems. It uses the differences in the potentials of mean force between the distribution functions generated from a guessed potential and the true distribution functions to improve the effective potential successively. The optimization algorithm is very powerful: convergence is reached for every trial function in few iterations. As an extensive test case we coarse-grained an atomistic all-atom model of polyisoprene (PI) using a 13:1 reduction of the degrees of freedom. This procedure was performed for PI solutions as well as for a PI melt. Comparisons of the obtained force fields are drawn. They prove that it is not possible to use a single force field for different concentration regimes.