The Quadrature Discretization Method (QDM) in the solution of the Schrödinger equation

The Quadrature Discretization Method (QDM) is employed in the solution of several onedimensional Schrödinger equations that have received considerable attention in the literature. The QDM is based on the discretization of the wave function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of nonclassical polynomials orthogonal with respect to a weight function. For a certain class of problems with potentials of the form that occur in supersymmetric quantum mechanics, the ground state wavefunction is known. In the present paper, the weight functions that are used are related to the ground state wavefunctions if known, or some approximate form. The eigenvalues and eigenfunctions of four different potential functions discussed extensively in the literature are calculated and the results are compared with published values.     

Potentials of an alternative scheme for ab initio polymer band structure calculations. Illustration on the chain of hydrogen atoms

An alternative scheme for ab initio polymer band structure calculations based on a Filon-type quadrature is proposed. This scheme avoids the explicit calculation and the storage of the “troublesome” Fourier transforms of the LCAO density matrix elements and is a first step towards a better control of the convergence of the different lattice sums appearing in the configuration space LCAO-SCF-CO method. The potential of the proposed technique is illustrated by a minimal basis set calculations on an infinite chain of H atoms.     

Atomic spectral methods for molecular electronic structure calculations

Theoretical methods are reported for ab initio calculations of the adiabatic (Born-Oppenheimer) electronic wave functions and potential energy surfaces of molecules and other atomic aggregates. An outer product of complete sets of atomic eigenstates familiar from perturbation-theoretical treatments of long-range interactions is employed as a representational basis without prior enforcement of aggregate wave function antisymmetry. The nature and attributes of this atomic spectral-product basis are indicated, completeness proofs for representation of antisymmetric states provided, convergence of Schrodinger eigenstates in the basis established, and strategies for computational implemention of the theory described. A diabaticlike Hamiltonian matrix representative is obtained, which is additive in atomic-energy and pairwise-atomic interaction-energy matrices, providing a basis for molecular calculations in terms of the (Coulombic) interactions of the atomic constituents. The spectral-product basis is shown to contain the totally antisymmetric irreducible representation of the symmetric group of aggregate electron coordinate permutations once and only once, but to also span other (non-Pauli) symmetric group representations known to contain unphysical discrete states and associated continua in which the physically significant Schrodinger eigenstates are generally embedded. These unphysical representations are avoided by isolating the physical block of the Hamiltonian matrix with a unitary transformation obtained from the metric matrix of the explicitly antisymmetrized spectral-product basis. A formal proof of convergence is given in the limit of spectral closure to wave functions and energy surfaces obtained employing conventional prior antisymmetrization, but determined without repeated calculations of Hamiltonian matrix elements as integrals over explicitly antisymmetric aggregate basis states. Computational implementations of the theory employ efficient recursive methods which avoid explicit construction the metric matrix and do not require storage of the full Hamiltonian matrix to isolate the antisymmetric subspace of the spectral-product representation. Calculations of the lowest-lying singlet and triplet electronic states of the covalent electron pair bond (H(2)) illustrate the various theorems devised and demonstrate the degree of convergence achieved to values obtained employing conventional prior antisymmetrization. Concluding remarks place the atomic spectral-product development in the context of currently employed approaches for ab initio construction of adiabatic electronic eigenfunctions and potential energy surfaces, provide comparisons with earlier related approaches, and indicate prospects for more general applications of the method.     

Optimal grids for generalized finite basis and discrete variable representations: definition and method of calculation

The method of optimal generalized finite basis and discrete variable representations (FBR and DVR) generalizes the standard, Gaussian quadrature grid-classical orthonormal polynomial basis-based FBR/DVR method to general sets of grid points and to general, nondirect product, and/or nonpolynomial bases. Here, it is shown how an optimal set of grid points can be obtained for an optimal generalized FBR/DVR calculation with a given truncated basis. Basis set optimized and potential optimized grids are defined. The optimized grids are shown to minimize a function of grid points derived by relating the optimal generalized FBR of a Hamiltonian operator to a non-Hermitian effective Hamiltonian matrix. Locating the global minimum of this function can be reduced to finding the zeros of a function in the case of one dimensional problems and to solving a system of D nonlinear equations repeatedly in the case of D>1 dimensional problems when there is an equal number of grid points and basis functions. Gaussian quadrature grids are shown to be basis optimized grids. It is demonstrated by a numerical example that an optimal generalized FBR/DVR calculation of the eigenvalues of a Hamiltonian operator with potential optimized grids can have orders of magnitude higher accuracy than a variational calculation employing the same truncated basis. Nevertheless, for numerical integration with the optimal generalized FBR quadrature rule basis optimized grids are the best among grids of the same number of points. The notions of Gaussian quadrature and Gaussian quadrature accuracy are extended to general, multivariable basis functions.     

Overlap integrals of B functions

Two different methods for the evaluation of overlap integrals of B functions with different scaling parameters are analyzed critically. The first method consists of an infinite series expansion in terms of overlap integrals with equal scaling parameters [14]. The second method consists of an integral representation for the overlap integral which has to be evaluated numerically. Bhattacharya and Dhabal [13] recommend the use of Gauss-Legendre quadrature for this purpose. However, we show that Gauss-Jacobi quadrature gives better results, in particular for larger quantum number. We also show that the convergence of the infinite series can be improved if suitable convergence accelerators are applied. Since an internal error analysis can be done quite easily in the case of an infinite series even if it is accelerated, whereas it is very costly in the case of Gauss quadratures, the infinite series is probably more efficient than the integral representation. Overlap integrals of all commonly occurring exponentially declining basis functions such as Slater-type functions, can be expressed by finite sums of overlap integrals of B functions, because these basis functions can be represented by linear combinations of B functions.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

Approximation of functions of Lipschitz class and solution of Fokker-Planck equation by two-dimensional Legendre wavelet operational matrix

In this paper, Lipschitz class of two-variables is considered. This is the genralization of well-known Lipschitz class of functions. A new estimator of functions belonging to generalized Lipschitz class has been obtained. Also, the solutions for the Fokker-Planck equations have been obtained for two different cases by two-dimensional Legendre wavelet operational matrix method. The approximated solutions of the time-and space-Fokker Planck equation have been compared with the exact solutions and the solutions obtained by homotopy perturbation method. The proposed scheme is simple, effective and suitable for the solution of Fokker-Planck equation.

     

System-specific discrete variable representations for path integral calculations with quasi-adiabatic propagators

Discrete variable representations (DVRs), constructed numerically from eigenstates of the one-dimensional adiabatic potential, provide the optimal quadrature for evaluating quasi-adiabatic propagator path integrals (QUAPI) for a system coupled to a harmonic bath. Calculations of partition functions and reaction rates for a multiple-minimum potential in a dissipative environment illustrate the convergence characteristics of this approach. The small number of quadrature points required, along with the rapid convergence of QUAPI methods, results in a powerful numerical scheme, complementary to Monte Carlo methods, for performing condensed phase dynamics calculations over the entire temperature range of interest in chemical physics.     

Pseudospectral methods of solution of the Schrödinger equation

Several different pseudospectral methods of solution of the Schrödinger equation are applied to the calculation of the eigenvalues of the Morse potential for I₂ and the Cahill–Parsegian potential for Ar₂ [Cahill, Parsegian, J. Chem. Phys. 121, 10839 (2004)]. The calculation of the eigenvalues for the Woods–Saxon potential are also considered. The convergence of the eigenvalues with a quadrature discretization method is found to be very fast owing to the judicious choice for the weight function, basis set and quadrature points. The weight function used is either related to the exact ground state wavefunction, if known, or an approximation to it from some reference potential. We compare several different pseudospectral methods.     

HOA (Heaviside Operational Ansatz) revisited: recent remarks on novel exact solution methodologies in wavefunction analysis

A viable methodology for the exact analytical solution of the multiparticle Schrodinger and Dirac equations has long been considered a holy grail of theoretical chemistry. Since a benchmark work by Torres-Vega and Frederick in the 1990s, the QPSR (Quantum Phase Space Representation) has been explored as an alternate method for solving various physical systems. Recently, the present author has developed an exact analytical symbolic solution scheme for broad classes of differential equations utilizing the HOA (Heaviside Operational Ansatz). An application of the scheme to chemical systems was initially presented in Journal of Mathematical Chemistry (Toward chemical applications of Heaviside Operational Ansatz: exact solution of radial Schrodinger equation for nonrelativistic N-particle system with pairwise 1/r(I) radial potential in quantum phase space. Journal of Mathematical Chemistry, 2009; 45(1):129–140). It is believed that the coupling of HOA with QPSR represents not only a fundamental breakthrough in theoretical physical chemistry, but it is promising as a basis for exact solution algorithms that would have tremendous impact on the capabilities of computational chemistry/physics. The novel methods allow the exact determination of the momentum [and configuration] space wavefunction from the QPSR wavefunction by way of a Fourier transform. In this note some remarks, examples and further directions, concerning HOA as a tool to solve and provide analytical insight into solutions of dynamical systems occurring in, but not limited to Mathematical Chemistry, are also posited.     

Irreducible correlation functions of the S matrix in the coordinate representation: application in calculating Lorentzian half-widths and shifts

By introducing the coordinate representation, the derivation of the perturbation expansion of the Liouville S matrix is formulated in terms of classically behaved autocorrelation functions. Because these functions are characterized by a pair of irreducible tensors, their number is limited to a few. They represent how the overlaps of the potential components change with a time displacement, and under normal conditions, their magnitudes decrease by several orders of magnitude when the displacement reaches several picoseconds. The correlation functions contain all dynamical information of the collision processes necessary in calculating half-widths and shifts and can be easily derived with high accuracy. Their well-behaved profiles, especially the rapid decrease of the magnitude, enables one to transform easily the dynamical information contained in them from the time domain to the frequency domain. More specifically, because these correlation functions are well time limited, their continuous Fourier transforms should be band limited. Then, the latter can be accurately replaced by discrete Fourier transforms and calculated with a standard fast Fourier transform method. Besides, one can easily calculate their Cauchy principal integrations and derive all functions necessary in calculating half-widths and shifts. A great advantage resulting from introducing the coordinate representation and choosing the correlation functions as the starting point is that one is able to calculate the half-widths and shifts with high accuracy, no matter how complicated the potential models are and no matter what kind of trajectories are chosen. In any case, the convergence of the calculated results is always guaranteed. As a result, with this new method, one can remove some uncertainties incorporated in the current width and shift studies. As a test, we present calculated Raman Q linewidths for the N2-N2 pair based on several trajectories, including the more accurate "exact" ones. Finally, by using this new method as a benchmark, we have carried out convergence checks for calculated values based on usual methods and have found that some results in the literature are not converged.     

Hermiticity of Hamiltonian Matrix using the Fourier Basis Sets in Bond-Bond-Angle and Radau Coordinates

In quantum calculations a transformed Hamiltonian is often used to avoid singularities in a certain basis set or to reduce computation time. We demonstrate for the Fourier basis set that the Hamiltonian can not be arbitrarily transformed. Otherwise, the Hamiltonian matrix becomes non-hermitian, which may lead to numerical problems. Methods for correctly constructing the Hamiltonian operators are discussed. Specific examples involving the Fourier basis functions for a triatomic molecular Hamiltonian (J=0) in bond-bond angle and Radau coordinates are presented. For illustration, absorption spectra are calculated for the OClO molecule using the time-dependent wavepacket method. Numerical results indicate that the non-hermiticity of the Hamiltonian matrix may also result from integration errors. The conclusion drawn here is generally useful for quantum calculation using basis expansion method using quadrature scheme.     

Perturbational approach to the electron correlation cusp applied to helium‐like atoms

A recently proposed perturbational approach to the electron correlation cusp problem 1 is tested in the context of three spherically symmetrical two‐electron systems: helium atom, hydride anion, and a solvable model system. The interelectronic interaction is partitioned into long‐ and short‐range components. The long‐range interaction, lacking the singularities responsible for the electron correlation cusp, is included in the reference Hamiltonian. Accelerated convergence of orbital‐based methods for this smooth reference Hamiltonian is shown by a detailed partial wave analysis. Contracted orbital basis sets constructed from atomic natural orbitals are shown to be significantly better for the new Hamiltonian than standard basis sets of the same size. The short‐range component becomes the perturbation. The low‐order perturbation equations are solved variationally using basis sets of correlated Gaussian geminals. Variational energies and low‐order perturbation wave functions for the model system are shown to be in excellent agreement with highly accurate numerical solutions for that system. Approximations of the reference wave functions, described by fewer basis functions, are tested for use in the perturbation equations and shown to provide significant computational advantages with tolerable loss of accuracy. Lower bounds for the radius of convergence of the resulting perturbation expansions are estimated. The proposed method is capable of achieving sub‐μHartree accuracy for all systems considered here. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

三电子原子的超球坐标分析和基态能态计算

用超球坐标表示多电子原子的薛定谔方程,从符合Pauli原理的全反称波函数出发,推导出相应的超球径耦合微分方程,其形式与双电子原子的相似。以九维超球谐为投影函数构造S₃群不可约表示的对称基,将超球谐-广义Laguerre展开法(HHGL)推广到Li原子,首次对其Schrdinger方程的精确求解进行了研究,基态能量的计算值与实验值接近。最后,还从波函数出发对提高计算的精度进行了讨论。     

Relativistic and nonrelativistic solutions for diatomic molecules in the presence of double ring-shaped Kratzer potential

The authors investigate solutions of the three dimensional Klein-Gordon and Schrodinger equations in the presence of a new exactly solvable potential of V(r,theta)=-2De(re/r-(1/2)(re2/r2))+b/r2 sin2 theta+a/r2 cos2 theta type, the so-called double ring-shaped Kratzer potential. For a diatomic molecule system in double ring-shaped Kratzer potential, the exact bound state energy eigenvalues and corresponding wave functions have been determined within the framework of the asymptotic iteration method. Bound state eigenfunction solutions used in applications related to molecular spectroscopy are obtained in terms of confluent hypergeometric function and Jacobi polynomial. This new formulation is tested by calculating the energies of rovibrational states of a number of diatomic molecules. Also, the author-prove that in the nonrelativistic limit c-->infinity, where c is the speed of light, solutions of the Klein-Gordon system converge to those of the Schrodinger system.     

An adaptive pseudospectral method for wave packet dynamics

We solve the time-dependent Schro?dinger equation for molecular dynamics using a pseudospectral method with global, exponentially decaying, Hagedorn basis functions. The approximation properties of the Hagedorn basis depend strongly on the scaling of the spatial coordinates. Using results from control theory we develop a time-dependent scaling which adaptively matches the basis to the wave packet. The method requires no knowledge of the Hessian of the potential. The viability of the method is demonstrated on a model for the photodissociation of IBr, using a Fourier basis in the bound state and Hagedorn bases in the dissociative states. Using the new approach to adapting the basis we are able to solve the problem with less than half the number of basis functions otherwise necessary. We also present calculations on a two-dimensional model of CO(2) where the new method considerably reduces the required number of basis functions compared to the Fourier pseudospectral method.     

A comparison of efficiency and accuracy of two-electron integrals calculation between two methods in multi-configuration time-dependent hartree fock frame

An approach to the evaluation of the two-electron repulsion integrals exactly in sine finite basis representation is proposed. The two-electron coulomb potential integrals are calculated respectively in sine finite basis representation by using two-fold Gaussian quadrature rules and in discrete variable representation by using the natural potential expansion of coulomb potential $r_{12} $ . The efficiency and accuracy of two methods to calculate the two-electron repulsion integrals are compared. Some demonstrative calculations indicate that both the two ways are effective methods to do two-electron integrals calculations in the multi-configuration time-dependent hartree fock (MCTDHF) frame. By using the method to calculate the two-electron integrals in sine FBR, the working equations of MCTDHF are propagated in imaginary time. The ground state energy of helium atom obtained in the imaginary propagation is close to the Full Configuration interaction energy calculated by Molpro.     

Application of efficient algorithm for solving six-dimensional molecular Ornstein-Zernike equation

In this article, we propose an efficient algorithm for solving six-dimensional molecular Ornstein-Zernike (MOZ) equation. In this algorithm, the modified direct inversion in iterative subspace, which is known as the fast convergent method for solving the integral equation theory of liquids, is adopted. This method is found to be effective for the convergence of the MOZ equation with a simple initial guess. For the accurate averaging of the correlation functions over the molecular orientations, we use the Lebedev-Laikov quadrature. The appropriate number of grid points for the quadrature is decided by the analysis of the dielectric constant. We also analyze the excess chemical potential of aqueous ions and compare the results of the MOZ with those of the reference interaction site model.     

Distributed Gaussian discrete variable representation

A discrete variable representation (DVR) made from distributed Gaussians g_n(x) = e, (n = ?∞, …, ∞) and its infinite grid limit is described. The infinite grid limit of the distributed Gaussian DVR (DGDVR) reduces to the sinc function DVR of Colbert and Miller in the limit c → 0. The numerical performance of both finite and infinite grid DGDVRs and the sinc function DVR is compared. If a small number of quadrature points are taken, the finite grid DGDVR performs much better than both infinite grid DGDVR and sinc function DVR. The infinite grid DVRs lose accuracy due to the truncation error. In contrast, the sinc function DVR is found to be superior to both finite and infinite grid DGDVRs if enough grid points are taken to eliminate the truncation error. In particular, the accuracy of DGDVRs does not get better than some limit when the distance between Gaussians d goes to zero with fixed c, whereas the accuracy of the sinc function DVR improves very quickly as d becomes smaller, and the results are exact in the limit d → 0. An analysis of the performance of distributed basis functions to represent a given function is presented in a recent publication. With this analysis, we explain why the sinc function DVR performs better than the infinite grid DGDVR. The analysis also traces the inability of Gaussians to yield exact results in the limit d → 0 to the incompleteness of this basis in this limit. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Discrete variable representations and sudden models in quantum scattering theory

An exact formalism in which the scattering problem may be described by sets of coupled equations labeled either by basis functions or quadrature points is presented. Use of each frame and the simply evaluated unitary transformation which connects them results in an efficient procedure for performing quantum scattering calculations. Two approximations are compared with the IOS.     

Discretization of unknown functions: A new method to solve partial differential equations with mixed boundary conditions

In this paper, we develop a new numerical technique to obtain an approximate solution of partial differential equations subject to mixed boundary conditions (MBCs). The approach has been applied to a class of differential equations which frequently arise in a large variety of problems such as heat conduction, potential theory, and diffusion‐controlled chemical reactions. In our approach, based on the discretization of unknown functions (DF), the solution is expressed as a series expansion and the determination of the series coefficients is reduced to the solution of a system of algebraic equations. The main advantages of the DF procedure are: (a) the smoothness of the function and of its first derivative in the different domains, whereas the other numerical methods generally show a highly oscillating behavior; (b) the fast convergence of the series expansion. This method has been applied to solve diffusion problems in different coordinate systems (trigonometric, cylindrical and spherical). The obtained results have been compared with the analytical solution (when available) as well as with other numerical methods commonly used to solve MBCs problems. This revised version was published online in July 2006 with corrections to the Cover Date.