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.     

改进的三维分离变量表象方法计算振动光谱

The modified discrete variable representation for three-dimension (DVR3D) method was applied to the determination of the vibrational energy levels of the fundamental electronic state of H2S and H2O. The Hamiltonian was expressed in Jacobi coordinates and developed on a DVR basis for each internal coordinate. The angular coordinate used a DVR based on Legendre polynomials and the radial coordinates utilized a DVR based on sine basis functions. Successive diagonalization and truncation technique was used to reduce the size of the final Hamiltonian matrix to be diagonalized. Calculations were presented for H2S and H2O to demonstrate the accuracy of these algorithms.     

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.     

Gegenbauer polynomials in a theoretical study of the vibrational structure of the Ca+–H2 system

This work presents an application of Gegenbauer polynomials in vibrational calculations. We illustrated that by example calculations of vibrational structure of the Ca⁺–H₂ exciplex, in the state correlated with 3D calcium ion state. For this case Gegenbauer polynomials are used for formation of a basis set for a bending mode. We showed that this basis set leads to a faster convergence of results than a basis set formed from Legendre polynomials. Additionally we compared vibrational structure obtained in this manner with results of discrete variable representation-distributed Gaussian basis (DVR–DGB) method.     

Discrete variable approaches to tetratomic molecules: part I: DVR(6) and DVR(3) + DGB methods

We present two discrete variable representation (DVR) based methods for the determination of the vibrational energy levels of tetratomic molecules. Both methods are designed for orthogonal internal coordinates in a body-fixed reference frame and make use of the DVR of three angular variables. The angular DVRs allow the construction of a fixed-angle three-mode Hamiltonian for the stretching vibrations. For each of the angular triples, the stretching eigenvalue problems are solved by employing 3D radial DVRs in the DVR(6) approach and real three-dimensional distributed Gaussian functions in the DVR(3) + DGB method. The angular degrees of freedom are taken sequentially into account in conjunction with a contraction scheme resulting from several diagonalization/truncation steps. Vibrationally averaged geometries, expectation values of rotational constants, and several adiabatic projection schemes developed in this work for tetratomic molecules are used to characterize the vibrational levels calculated by the DVR(6) and DVR(3) + DGB.     

On the use of optimal internal vibrational coordinates for symmetrical bent triatomic molecules

The use of generalized internal coordinates for the variational calculation of excited vibrational states of symmetrical bent triatomic molecules is considered with applications to the SO2, O3, NO2, and H2O molecules. These coordinates depend on two external parameters which can be properly optimized. We propose a simple analytical method to determine the optimal internal coordinates for this kind of molecules based on the minimization with respect to the external parameters of the zero-point energy, assuming only quadratic terms in the Hamiltonian and no quadratic coupling between the optimal coordinates. The optimal values of the parameters thus obtained are shown to agree quite well with those that minimize the sum of a number of unconverged energies of the lowest vibrational states, computed variationally using a small basis function set. The unconverged variational calculation uses a basis set consisting of the eigenfunctions of the uncoupled anharmonic internal coordinate Hamiltonian. Variational calculations of the excited vibrational states for the four molecules considered carried out with an increasing number of basis functions, also evidence the excellent convergence properties of the optimal internal coordinates versus those provided by other normal and local coordinate systems.     

Discrete variable representation method applied to the determination of J = 0 vibrational bound states of NO2

The discrete variable representation method (DVR ) is applied to the calculation of the J = 0 vibrational energy levels of the ground electronic state of nitrogen dioxide, a molecule which shows a large amplitude bending vibration. The Hamiltonian is expressed in Johnson hyperspherical coordinates and developed on a DVR basis set for each coordinate. A successive diagonalization–truncation method is applied which gives accurate values for the energy levels up to ? 7000 cm^?1. © 1995 John Wiley & Sons, Inc.     

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.     

Finite vibrational displacements in the Eckart—Sayvetz basis

Curvilinear internal coordinates are considered in terms of cartesian displacements in a molecule-fixed basis determined by the Eckart-Sayvetz conditions. The latter are interpreted as a set of restrictions on the metrics of the space and define cartesian displacements of “pure” vibrational character expanded to any order in terms of internal coordinates. Explicit expressions for expansion coefficients are given as a function of contravariant components of the metric tensor taken from existing table. A compact notation is proposed for anharmonic force constants, expansion coefficients of redundancies and coupling terms of the rotation—vibration hamiltonian.     

Discrete variable representation method applied to the determination of rotation-vibration bound states of NO2

The discrete variable representation method is applied to the determination of the rotation-vibration energy levels of the fundamental electronic state of NO₂. The Hamiltonian is expressed in Johnson hyperspherical coordinates and developed on a DVR basis for each internal coordinate, while parity-adapted linear combinations of Wigner functions are used to describe the rotational motion. The diagonalization of the Hamiltonian matrix is performed using the Lanczos algorithm for large symmetric and Hermitian matrices. Results for rovibrational states up to J = 11 for the first five vibrational energy levels are presented. © 1997 John Wiley & Sons, Inc.     

A finite B-spline basis set for accurate diatomic molecule calculations

A finite basis set particularly adapted for solving the Hartree-Fock equation for diatomic molecules in prolate spheroidal coordinates has been constructed. These basis functions have been devised as products of B-splines times associated Legendre polynomials. Due to the large number of B-splines, the resulting set of eigenfunctions is amply distributed over excited states. This gives the possibility of using these basis sets to calculate sums over excited states, appearing in various orders of perturbation theory. As an illustration, the second-order corrections to the ground-state energy of some atoms and diatomic molecules with closed electron shells have been calculated.     

Molecular dynamic structure and dimerization of polyacetylene

The vibrational self-consistent-field approximation is used to calculate excited vibrational energy levels of the water molecule in hyperspherical coordinates. The calculations are made for a global realistic Sorbie–Murrell-type potential surface for which exactum quantum variational results are known for comparison. The coupled SCF equations are solved using the discrete variable representation (DVR ) method, which allows computation of the coupled multidimensional integrals in a very simple and efficient way. The results are in good agreement with exactum quantum calculations and are more accurate than SCF energy eigenvalues obtained using normal mode coordinates.     

The unimolecular dissociation of electronic state-selected methyl iodide cations

Within the context of a recently proposed model, the angle-bending Schrödinger equation for linear molecules is cast in the form of the Legendre equation with an added potential. This is solved by a complete set expansion in terms of associated Legendre functions. The resulting solutions are incorporated into the existing model. Computation of the lower vibrational excitations of HCN is carried out.     

Potential distribution in the solution interface of a scanning tunneling microscope

The linearized Poisson—Boltzmann equation is solved in the region between a sphere and a plane, which is modelling the electrolyte solution interface between the tip and the substrate in a scanning tunneling microscope. A series expansion in modified Bessel functions and Legendre polynomials, which are solutions to the linearized Poisson—Boltzmann equation, is used to fit the boundary conditions. Another numerical method of finite difference is also used with the domain transformed into bispherical coordinates. Results for cases of different potential values on the boundary surfaces and different distances of the sphere from the plane are presented.     

First-principle calculation of reduced masses in vibrational analysis using generalized internal coordinates: some crucial aspects and examples

In this paper we present and analyze the most essential aspects of reduced masses along generalized internal coordinates. The definition of reduced masses in the internal coordinate formalism is established through the Wilson G-matrix concept and includes sophisticated relations between internal and Cartesian coordinates. Moreover, reduced masses in internal coordinates are, in general, no longer constant but coordinate-dependent. Based on the approach presented earlier [Stare, J.; Balint-Kurti, G. G. J. Phys. Chem. A 2003, 107, 7204-7214] and on our experience with reduced masses discussed in this paper, we have developed a robust program for the calculation of Wilson G-matrix elements and their functional coordinate dependence. The approach is based on the first principles and can be used in virtually any (internal) coordinate set. Since the program allows for projection of any kind of nuclear motion on the selected internal coordinates, the method is particularly suitable for ab initio or DFT potential energy functions calculated by partial geometry optimization. Moreover, reduced masses obtained by this program can be used as a decision tool for selecting the most appropriate internal coordinates for the considered vibrational problem and for the inclusion or omission of the kinetic coupling terms in the vibrational Hamiltonian.     

Reduced dimension discrete variable representation study of cis-trans isomerization in the S1 state of C2H2

Isomerization between the cis and trans conformers of the S(1) state of acetylene is studied using a reduced dimension discrete variable representation (DVR) calculation. Existing DVR techniques are combined with a high accuracy potential energy surface and a kinetic energy operator derived from FG theory to yield an effective but simple Hamiltonian for treating large amplitude motions. The spectroscopic signatures of the S(1) isomerization are discussed, with emphasis on the vibrational aspects. The presence of a low barrier to isomerization causes distortion of the trans vibrational level structure and the appearance of nominally electronically forbidden A? (1)A(2)←X? (1)Σ(g)(+) transitions to vibrational levels of the cis conformer. Both of these effects are modeled in agreement with experimental results, and the underlying mechanisms of tunneling and state mixing are elucidated by use of the calculated vibrational wavefunctions.     

Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

We present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

     

How to choose one-dimensional basis functions so that a very efficient multidimensional basis may be extracted from a direct product of the one-dimensional functions: energy levels of coupled systems with as many as 16 coordinates

In this paper we propose a scheme for choosing basis functions for quantum dynamics calculations. Direct product bases are frequently used. The number of direct product functions required to converge a spectrum, compute a rate constant, etc., is so large that direct product calculations are impossible for molecules or reacting systems with more than four atoms. It is common to extract a smaller working basis from a huge direct product basis by removing some of the product functions. We advocate a build and prune strategy of this type. The one-dimensional (1D) functions from which we build the direct product basis are chosen to satisfy two conditions: (1) they nearly diagonalize the full Hamiltonian matrix; (2) they minimize off-diagonal matrix elements that couple basis functions with diagonal elements close to those of the energy levels we wish to compute. By imposing these conditions we increase the number of product functions that can be removed from the multidimensional basis without degrading the accuracy of computed energy levels. Two basic types of 1D basis functions are in common use: eigenfunctions of 1D Hamiltonians and discrete variable representation (DVR) functions. Both have advantages and disadvantages. The 1D functions we propose are intermediate between the 1D eigenfunction functions and the DVR functions. If the coupling is very weak, they are very nearly 1D eigenfunction functions. As the strength of the coupling is increased they resemble more closely DVR functions. We assess the usefulness of our basis by applying it to model 6D, 8D, and 16D Hamiltonians with various coupling strengths. We find approximately linear scaling.