A multidimensional discrete variable representation basis obtained by simultaneous diagonalization

Direct product basis functions are frequently used in quantum dynamics calculations, but they are poor in the sense that many such functions are required to converge a spectrum, compute a rate constant, etc. Much better, contracted, basis functions, that account for coupling between coordinates, can be obtained by diagonalizing reduced dimension Hamiltonians. If a direct product basis is used, it is advantageous to use discrete variable representation (DVR) basis functions because matrix representations of functions of coordinates are diagonal in the DVR. By diagonalizing matrices representing coordinates it is straightforward to obtain the DVR that corresponds to any direct product basis. Because contracted basis functions are eigenfunctions of reduced dimension Hamiltonians that include coupling terms they are not direct product functions. The advantages of contracted basis functions and the advantages of the DVR therefore appear to be mutually exclusive. A DVR that corresponds to contracted functions is unknown. In this paper we propose such a DVR. It spans the same space as a contracted basis, but in it matrix representations of coordinates are diagonal. The DVR basis functions are chosen to achieve maximal diagonality of coordinate matrices. We assess the accuracy of this DVR by applying it to model four-dimensional problems.     

Using simultaneous diagonalization and trace minimization to make an efficient and simple multidimensional basis for solving the vibrational Schrodinger equation

In this paper we improve the product simultaneous diagonalization (SD) basis method we previously proposed [J. Chem. Phys. 122, 134101 (2005)] and applied to solve the Schrodinger equation for the motion of nuclei on a potential surface. The improved method is tested using coupled complicated Hamiltonians with as many as 16 coordinates for which we can easily find numerically exact solutions. In a basis of sorted products of one-dimensional (1D) SD functions the Hamiltonian matrix is nearly diagonal. The localization of the 1D SD functions for coordinate qc depends on a parameter we denote alphac. In this paper we present a trace minimization scheme for choosing alphac to nearly block diagonalize the Hamiltonian matrix. Near-block diagonality makes it possible to truncate the matrix without degrading the accuracy of the lowest energy levels. We show that in the sorted product SD basis perturbation theory works extremely well. The trace minimization scheme is general and easy to implement.     

Finite basis representations with nondirect product basis functions having structure similar to that of spherical harmonics

The currently most efficient finite basis representation (FBR) method [Corey et al., in Numerical Grid Methods and Their Applications to Schrodinger Equation, NATO ASI Series C, edited by C. Cerjan (Kluwer Academic, New York, 1993), Vol. 412, p. 1; Bramley et al., J. Chem. Phys. 100, 6175 (1994)] designed specifically to deal with nondirect product bases of structures phi(n) (l)(s)f(l)(u), chi(m) (l)(t)phi(n) (l)(s)f(l)(u), etc., employs very special l-independent grids and results in a symmetric FBR. While highly efficient, this method is not general enough. For instance, it cannot deal with nondirect product bases of the above structure efficiently if the functions phi(n) (l)(s) [and/or chi(m) (l)(t)] are discrete variable representation (DVR) functions of the infinite type. The optimal-generalized FBR(DVR) method [V. Szalay, J. Chem. Phys. 105, 6940 (1996)] is designed to deal with general, i.e., direct and/or nondirect product, bases and grids. This robust method, however, is too general, and its direct application can result in inefficient computer codes [Czako et al., J. Chem. Phys. 122, 024101 (2005)]. It is shown here how the optimal-generalized FBR method can be simplified in the case of nondirect product bases of structures phi(n) (l)(s)f(l)(u), chi(m) (l)(t)phi(n) (l)(s)f(l)(u), etc. As a result the commonly used symmetric FBR is recovered and simplified nonsymmetric FBRs utilizing very special l-dependent grids are obtained. The nonsymmetric FBRs are more general than the symmetric FBR in that they can be employed efficiently even when the functions phi(n) (l)(s) [and/or chi(m) (l)(t)] are DVR functions of the infinite type. Arithmetic operation counts and a simple numerical example presented show unambiguously that setting up the Hamiltonian matrix requires significantly less computer time when using one of the proposed nonsymmetric FBRs than that in the symmetric FBR. Therefore, application of this nonsymmetric FBR is more efficient than that of the symmetric FBR when one wants to diagonalize the Hamiltonian matrix either by a direct or via a basis-set contraction method. Enormous decrease of computer time can be achieved, with respect to a direct application of the optimal-generalized FBR, by employing one of the simplified nonsymmetric FBRs as is demonstrated in noniterative calculations of the low-lying vibrational energy levels of the H3+ molecular ion. The arithmetic operation counts of the Hamiltonian matrix vector products and the properties of a recently developed diagonalization method [Andreozzi et al., J. Phys. A Math. Gen. 35, L61 (2002)] suggest that the nonsymmetric FBR applied along with this particular diagonalization method is suitable to large scale iterative calculations. Whether or not the nonsymmetric FBR is competitive with the symmetric FBR in large-scale iterative calculations still has to be investigated numerically.     

Dynamical pruning of static localized basis sets in time-dependent quantum dynamics

We investigate the viability of dynamical pruning of localized basis sets in time-dependent quantum wave packet methods. Basis functions that have a very small population at any given time are removed from the active set. The basis functions themselves are time independent, but the set of active functions changes in time. Two different types of localized basis functions are tested: discrete variable representation (DVR) functions, which are localized in position space, and phase-space localized (PSL) functions, which are localized in both position and momentum. The number of functions active at each point in time can be as much as an order of magnitude less for dynamical pruning than for static pruning, in reactive scattering calculations of H2 on the Pt(211) stepped surface. Scaling of the dynamically pruned PSL (DP-PSL) bases with dimension is considerably more favorable than for either the primitive (direct product) or DVR bases, and the DP-PSL basis set is predicted to be three orders of magnitude smaller than the primitive basis set in the current state-of-the-art six-dimensional reactive scattering calculations.     

Electronic coupling matrix elements from charge constrained density functional theory calculations using a plane wave basis set

We present a plane wave basis set implementation for the calculation of electronic coupling matrix elements of electron transfer reactions within the framework of constrained density functional theory (CDFT). Following the work of Wu and Van Voorhis [J. Chem. Phys. 125, 164105 (2006)], the diabatic wavefunctions are approximated by the Kohn-Sham determinants obtained from CDFT calculations, and the coupling matrix element calculated by an efficient integration scheme. Our results for intermolecular electron transfer in small systems agree very well with high-level ab initio calculations based on generalized Mulliken-Hush theory, and with previous local basis set CDFT calculations. The effect of thermal fluctuations on the coupling matrix element is demonstrated for intramolecular electron transfer in the tetrathiafulvalene-diquinone (Q-TTF-Q(-)) anion. Sampling the electronic coupling along density functional based molecular dynamics trajectories, we find that thermal fluctuations, in particular the slow bending motion of the molecule, can lead to changes in the instantaneous electron transfer rate by more than an order of magnitude. The thermal average, (<|H(ab)|(2)>)(1/2)=6.7 mH, is significantly higher than the value obtained for the minimum energy structure, |H(ab)|=3.8 mH. While CDFT in combination with generalized gradient approximation (GGA) functionals describes the intermolecular electron transfer in the studied systems well, exact exchange is required for Q-TTF-Q(-) in order to obtain coupling matrix elements in agreement with experiment (3.9 mH). The implementation presented opens up the possibility to compute electronic coupling matrix elements for extended systems where donor, acceptor, and the environment are treated at the quantum mechanical (QM) level.     

Treating singularities present in the Sutcliffe-Tennyson vibrational Hamiltonian in orthogonal internal coordinates

Two methods are developed, when solving the related time-independent Schrodinger equation (TISE), to cope with the singular terms of the vibrational kinetic energy operator of a triatomic molecule given in orthogonal internal coordinates. The first method provides a mathematically correct treatment of all singular terms. The vibrational eigenfunctions are approximated by linear combinations of functions of a three-dimensional nondirect-product basis, where basis functions are formed by coupling Bessel-DVR functions, where DVR stands for discrete variable representation, depending on distance-type coordinates and Legendre polynomials depending on angle bending. In the second method one of the singular terms related to a distance-type coordinate, deemed to be unimportant for spectroscopic applications, is given no special treatment. Here the basis set is obtained by taking the direct product of a one-dimensional DVR basis with a two-dimensional nondirect-product basis, the latter formed by coupling Bessel-DVR functions and Legendre polynomials. With the basis functions defined, matrix representations of the TISE are set up and solved numerically to obtain the vibrational energy levels of H3+. The numerical calculations show that the first method treating all singularities is computationally inefficient, while the second method treating properly only the singularities having physical importance is quite efficient.     

Full-dimensional quantum calculations of vibrational spectra of six-atom molecules. I. Theory and numerical results

Two quantum mechanical Hamiltonians have been derived in orthogonal polyspherical coordinates, which can be formed by Jacobi and/or Radau vectors etc., for the study of the vibrational spectra of six-atom molecules. The Hamiltonians are expressed in an explicit Hermitian form in the spatial representation. Their matrix representations are described in both full discrete variable representation (DVR) and mixed DVR/nondirect product finite basis representation (FBR) bases. The two-layer Lanczos iteration algorithm [H.-G. Yu, J. Chem. Phys. 117, 8190 (2002)] is employed to solve the eigenvalue problem of the system. A strategy regarding how to carry out the Hamiltonian-vector products for a high-dimensional problem is discussed. By exploiting the inversion symmetry of molecules, a unitary sequential 1D matrix-vector multiplication algorithm is proposed to perform the action of the Hamiltonian on the wavefunction in a symmetrically adapted DVR or FBR basis in the azimuthal angular variables. An application to the vibrational energy levels of the molecular hydrogen trimer (H2)3 in full dimension (12D) is presented. Results show that the rigid-H2 approximation can underestimate the binding energy of the trimer by 27%. Finally, it is demonstrated that the two-layer Lanczos algorithm is also capable of computing the eigenvectors of the system with minor effort.     

Use of a nondirect-product basis for treating singularities in triatomic rotational-vibrational calculations

A technique has been developed which in principle allows the determination of the full rotational-vibrational eigenspectrum of triatomic molecules by treating the important singularities present in the triatomic rotational-vibrational kinetic energy operator given in Jacobi coordinates and the R(1) embedding. The singular term related to the diatom-type coordinate, R(1), deemed to be unimportant for spectroscopic applications, is given no special attention. The work extends a previous [J. Chem. Phys., 2005, 122, 024101] vibration-only approach and employs a generalized finite basis representation (GFBR) resulting in a nonsymmetric Hamiltonian matrix [J. Chem. Phys., 2006, 124, 014110]. The basis set to be used is obtained by taking the direct product of a 1-D DVR basis, related to R(1), with a 5-D nondirect-product basis, the latter formed by coupling Bessel-DVR functions depending on the distance-type coordinate causing the singularity, associated Legendre polynomials depending on the Jacobi angle, and rotational functions depending on the three Euler angles. The robust implicitly restarted Arnoldi method within the ARPACK package is used for the determination of a number of eigenvalues of the nonsymmetric Hamiltonian matrix. The suitability of the proposed approach is shown by the determination of the rotational-vibrational energy levels of the ground electronic state of H(3)(+) somewhat above its barrier to linearity. Convergence of the eigenenergies is checked by an alternative approach, employing a Hamiltonian expressed in Radau coordinates, a standard direct-product basis, and no treatment of the singularities.     

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.     

Direct inversion of the iterative subspace with contracted planewave basis functions

Ways to reduce the computational cost of periodic electronic structure calculations by using basis functions corresponding to linear combinations of planewaves have been examined recently. These contracted planewave (CPW) basis functions correspond to Fourier series representations of atom‐centered basis functions, and thus provide access to some beneficial properties of planewave (PW) and localized basis functions. This study reports the development and assessment of a direct inversion of the iterative subspace (DIIS) method that employs unique properties of CPW basis functions to efficiently converge electronic wavefunctions. This method relies on access to a PW‐based representation of the electronic structure to provide a means of efficiently evaluating matrix–vector products involving the application of the Fock matrix to the occupied molecular orbitals. These matrix–vector products are transformed into a form permitting the use of direct diagonalization techniques and DIIS methods typically employed with atom‐centered basis sets. The abilities of this method are assessed through periodic Hartree–Fock calculations of a range of molecules and solid‐state systems. The results show that the method reported in this study is approximately five times faster than CPW‐based calculations in which the entire Fock matrix is calculated. This method is also found to be weakly dependent upon the size of the basis set, thus permitting the use of larger CPW basis sets to increase variational flexibility with a minor impact on computational performance. © 2018 Wiley Periodicals, Inc.     

Energy and energy gradient matrix elements with N-particle explicitly correlated complex Gaussian basis functions with L=1

In this work we consider explicitly correlated complex Gaussian basis functions for expanding the wave function of an N-particle system with the L=1 total orbital angular momentum. We derive analytical expressions for various matrix elements with these basis functions including the overlap, kinetic energy, and potential energy (Coulomb interaction) matrix elements, as well as matrix elements of other quantities. The derivatives of the overlap, kinetic, and potential energy integrals with respect to the Gaussian exponential parameters are also derived and used to calculate the energy gradient. All the derivations are performed using the formalism of the matrix differential calculus that facilitates a way of expressing the integrals in an elegant matrix form, which is convenient for the theoretical analysis and the computer implementation. The new method is tested in calculations of two systems: the lowest P state of the beryllium atom and the bound P state of the positronium molecule (with the negative parity). Both calculations yielded new, lowest-to-date, variational upper bounds, while the number of basis functions used was significantly smaller than in previous studies. It was possible to accomplish this due to the use of the analytic energy gradient in the minimization of the variational energy.     

Optimization of augmentation functions for correlated calculations of spin-spin coupling constants and related properties

A new hierarchy of augmented basis sets optimized for the calculation of molecular properties such as indirect spin-spin coupling constants is presented. Based on the Dunning hierarchy of cc-pVXZ (X = D, T, Q, and 5) basis sets augmentation functions with tight exponents have been optimized for coupled-cluster calculations of indirect spin-spin coupling constants. The optimal exponents for these tight functions have been obtained by optimizing the sum of the absolute values of all contributions to the coupling constant. On the basis of a series of test cases (CO, HF, N(2), F(2), H(2)O, NH(3), and CH(4)) we propose a set of tight s, p, and d functions to be added to the uncontracted Dunning basis sets, and, subsequently, to recontract. The resulting ccJ-pVXZ (X = D, T, Q, and 5) basis sets demonstrate excellent cost efficiency in benchmark calculations. These new basis sets should generally be applicable for the calculation of spin-spin coupling constants and other properties that have a strong dependence on powers of 1r or even contain a delta distribution for correlated ab initio methods.     

Calculating vibrational energies and wave functions of vinylidene using a contracted basis with a locally reorthogonalized coupled two-term Lanczos eigensolver

We use a contracted basis+Lanczos eigensolver approach to compute vinylidene-like vibrational states of the acetylene-vinylidene system. To overcome problems caused by loss of orthogonality of the Lanczos vectors we reorthogonalize Lanczos vector and use a coupled two-term approach. The calculations are done in CC-HH diatom-diatom Jacobi coordinates which make it easy to compute states one irreducible representation at a time. The most costly parts of the calculation are parallelized and scale well. We estimate that the vinylidene energies we compute are converged to approximately 1 cm(-1).     

A fully relativistic method for calculation of nuclear magnetic shielding tensors with a restricted magnetically balanced basis in the framework of the matrix Dirac-Kohn-Sham equation

A new relativistic four-component density functional approach for calculations of NMR shielding tensors has been developed and implemented. It is founded on the matrix formulation of the Dirac-Kohn-Sham (DKS) method. Initially, unperturbed equations are solved with the use of a restricted kinetically balanced basis set for the small component. The second-order coupled perturbed DKS method is then based on the use of restricted magnetically balanced basis sets for the small component. Benchmark relativistic calculations have been carried out for the (1)H and heavy-atom nuclear shielding tensors of the HX series (X=F,Cl,Br,I), where spin-orbit effects are known to be very pronounced. The restricted magnetically balanced basis set allows us to avoid additional approximations and/or strong basis set dependence which arises in some related approaches. The method provides an attractive alternative to existing approximate two-component methods with transformed Hamiltonians for relativistic calculations of chemical shifts and spin-spin coupling constants of heavy-atom systems. In particular, no picture-change effects arise in property calculations.     

Pseudodiagonalization-based wavefunction optimization with contracted planewave basis functions

Electronic structure calculations representing the molecular orbitals (MOs) with contracted planewave basis functions (CPWBFs) have been reported recently. CPWBFs are Fourier-series representations of atom-centered basis functions. The mathematical features of CPWBFs permit the construction of matrix–vector products, FC _o , involving the application of the Fock matrix, F , to the set of occupied MOs, C _o , without the explicit evaluation of F . This approach offers a theoretical speed-up of M/n over F -based methods, where M and n are the number of basis functions and occupied MOs, respectively. The present study reports methodological advances that permit FC _o -based optimization of wavefunction formed from CPWBFs. In particular, a technique is reported for optimizing wavefunctions by combining pseudodiagonalization techniques based on an exact representation of FC _o , approximate information regarding the virtual orbital energies, and direct inversion of the iterative subspace optimization schemes to guide the wavefunction to a converged solution. This method is found to speed-up wavefunction optimizations by factors of up to ~6 − 8 over F -based optimization methods while providing identical results. Further, the computational cost of this technique is relatively insensitive to basis set size, thus providing further benefits in calculations using large CPWBF basis sets. The results of density functional theory calculations show that this method permits the use of hybrid exchange-correlation (XC) functionals with a small increase in effort over analogous calculations using generalized gradient approximation XC functionals. © 2019 Wiley Periodicals, Inc.     

Accurate and highly efficient calculation of the highly excited pure OH stretching resonances of O(1D)HCl, using a combination of methods

Accurate calculation of the energies and widths of the resonances of HOCl--an important intermediate in the O(1D)HCl reactive system--poses a challenging benchmark for computational methods. The need for very large direct product basis sets, combined with an extremely high density of states, results in difficult convergence for iterative methods. A recent calculation of the highly excited OH stretch mode resonances using the filter diagonalization method, for example, required 462,000 basis functions, and 180,000 iterations. In contrast, using a combination of new methods, we are able to compute the same resonance states to higher accuracy with a basis less than half the size, using only a few hundred iterations-although the CPU cost per iteration is substantially greater. Similar performance enhancements are observed for calculations of the high-lying bound states, as reported in a previous paper [J. Theo. Comput. Chem. 2, 583 (2003)].     

Using a nondirect product discrete variable representation for angular coordinates to compute vibrational levels of polyatomic molecules

In this paper we test a nondirect product discrete variable representation (DVR) method for solving the bend vibration problem and compare it with well-established direct product DVR and finite basis representation approaches.     

Vibrational matrix elements of the quadrupole moment functions of H2, N2 and CO

The quadrupole moment functions (molecular quadrupole moment versus internuclear distance) have been determined by quantum mechanical calculations for H₂ (by Kolos and Wolniewicz), N₂ (by Wahl and Nesbet), and CO (by Nesbet). These functions are used with numerical vibrational wave functions to compute matrix elements which are useful for calculations of scattering cross sections, energy transfer rates and excitation probabilities, and infrared intensities of forbidden bands.     

Spherical particle in Poiseuille flow between planar walls

We study a spherical mesoparticle suspended in Newtonian fluid between plane-parallel walls with incident Poiseuille flow. Using a two-dimensional Fourier transform technique we obtain a symmetric analytic expression for the Green tensor for the Stokes equations describing the creeping flow in this geometry. From the matrix elements of the Green tensor with respect to a complete vector harmonic basis, we obtain the friction matrix for the sphere. The calculation of matrix elements of the Green tensor is done in large part analytically, reducing the evaluation of these elements to a one-dimensional numerical integration. The grand resistance and mobility matrices in Cartesian form are given in terms of 13 scalar friction and mobility functions which are expressed in terms of certain matrix elements calculated in the spherical basis. Numerical calculation of these functions is shown to converge well and to agree with earlier numerical calculations based on boundary collocation. For a channel width broad with respect to the particle radius, we show that an approximation defined by a superposition of single-wall functions is reasonably accurate, but that it has large errors for a narrow channel. In the two-wall geometry the friction and mobility functions describing translation-rotation coupling change sign as a function of position between the two walls. By Stokesian dynamics calculations for a polar particle subject to a torque arising from an external field, we show that the translation-rotation coupling induces sideways migration at right angles to the direction of fluid flow.     

Transformation of Bloch representation into atomic orbital representation and combined matrix element calculation technique for spatially bounded basis functions in crystals

A method is proposed for transforming the Hamiltonian from Bloch to atomic function representation. For spatially bounded functions, this is a rigorous method based on solution of a certain algebraic system of equations. Unlike the conventional procedure based on integration over the Brillouin zone, the new method requires knowledge of the matrix elements of the Bloch representation only at several points of the Brillouin zone. The number of these points is determined by the trimming radius for the spatially bounded functions and by the lattice constant. The method can be used for calculating matrix elements in a basis of atomic functions and for reducing computations in matrix element calculations of the Bloch representation for procedures using numerical integration.