Dicretization to avoid singularities in vibration–rotation Hamiltonians: A bisector embedding for AB2 triatomics

A previously proposed [Sutcliffe and Tennyson, Int. J. Quantum Chem. 29 , 183 (1991)] body-fixed Hamiltonian is applied to AB₂ systems in Radau coordinates with the x-axis embedded along the bisector of the angle. It is shown that by using a discrete variable representation for the angular coordinate it is possible to avoid singular regions of the Hamiltonian. A two-step variational procedure is used to obtain rotationally excited states of the system. The results of test calculations H₂S and D₂S with J = 0, 1, 5, and 10 are discussed along with computer-usage characteristics.     

The centrifugally decoupled exponential distorted wave (CDEDW) approximation for the calculation of rotationally inelastic molecular collision cross se

A new approximate method is presented for the rapid calculation of rotationally inelastic molecular collision cross sections. The method is called the centrifugally decoupled exponential distorted wave (CDEDW) approximation and involves the combination of two well known approximations. The first approximation is the neglect of the off-diagonal coupling terms which arise from the orbital angular momentum operator in the coupled differential equations in the body-fixed axis system. The second approximation is to treat the remaining coupling terms, which arise from the interaction potential, using a unitary perturbation approximation. The CDEDW method is applied to the calculation of total and partial rotationally inelastic cross sections in the ArN₂ system, and detailed comparisons are made with exact and several other types of approximate calculations. Agreement with exact calculations is good and often comparable with the coupled states and p-helicity decoupled approximations. The CDEDW method requires a similar amount of computational effort to the infinite order sudden (IOS) approximation, and we show that for the present system the CDEDW method gives more reliable results.     

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.     

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.     

Exact quantum mechanical vibration-rotation kinetic energy operator of four-particle system in various coordinates

The vibration-rotational kinetic energy operators of four-particle system in various coordinates are derived using a new and simple angular momentum method. The operators are respectively suitable for studying the systems described by scattering coordinate, valence coordinate, Radau coordinate, Radau/Jacobi and Jacobi/valence hybrid coordinates and so on. Certain properties of these operators and their possible applications are discussed.     

The isotopic effects in the IR-multiphoton excitation of molecules with c_(2v) symmetry

Quantum dynamical calculations for the IR-multiphoton excitation of H_2O,D_2O andT_2O are presented using a generalized Hamiltonian for XY_2 molecules.The energy spectraobtained from this Hamiltonian are in good agreement with those of the experiments.Thelong time average of the transition probabilities and the isotopic effects are discussed in detail,     

A simple method for derivation of the rovibrational Hamiltonian: application to orthogonal radau coordinate systems as special cases

The molecular Hamiltonian of polyatomic molecules has been obtained. A general choice of internal coordinates depending on external parameters was considered. The rovibrational Hamiltonian for this set of coordinate system was derived in general terms as a function of the external parameters a and b. This procedure is also applicable to various kinds of internal coordinates in a straightforward way. The rovibrational Hamiltonian of triatomic molecules is considered as an application of this general formulation. In addition, orthogonal Radau coordinates are considered as cases of this new approach     

Evaluation of Hylleraas-CI atomic integrals by integration over the coordinates of one electron. I. Three-electron integrals

A method to evaluate the nonrelativistic electron-repulsion, nuclear attraction and kinetic energy three-electron integrals over Slater orbitals appearing in Hylleraas-CI (Hy-CI) electron structure calculations on atoms is shown. It consists on the direct integration over the interelectronic coordinate r _ij and the sucessive integration over the coordinates of one of the electrons. All the integrals are expressed as linear combinations of basic two-electron integrals. These last are solved in terms of auxiliary two-electron integrals which are easy to compute and have high accuracy. The use of auxiliary three-electron ones is avoided, with great saving of storage memory. Therefore this method can be used for Hy-CI calculations on atoms with number of electrons N ≥ 5. It has been possible to calculate the kinetic energy also in terms of basic two-electron integrals by using the Hamiltonian in Hylleraas coordinates, for this purpose some mathematical aspects like derivatives of the spherical harmonics with respect to the polar angles and recursion relations are treated and some new relations are given.     

Quantum solution of coupled harmonic oscillator systems beyond normal coordinates

Normal coordinates can be defined as orthogonal linear combinations of coordinates that remove the second order couplings in coupled harmonic oscillator systems. In this paper we go further and explore the possibility of using linear although non-orthogonal coordinate transformations to get the quantum solution of coupled systems. The idea is to use as non-orthogonal linear coordinates those which allow us to express the second-order Hamiltonian matrix in a block diagonal form. To illustrate the viability of this treatment, we first apply it to a system of two bilinearly coupled harmonic oscillators which admits analytical exact solutions. The method provides in this case, as an extra mathematical result, the analytical expressions for the eigenvalues of a certain type of symmetrical tridiagonal matrices. Second, we carry out a numerical application to the Barbanis coupled oscillators system, which contains a third order coupling term and cannot be solved in closed form. We demonstrate that the non-orthogonal coordinates used, named oblique coordinates, are much more efficient than normal coordinates to determine the energy levels and eigenfunctions of this system variationally.     

A quartic force field coordinate substitution scheme using hyperbolic sine coordinates

Quartic force fields (QFF) are currently the most cost‐effective method for the approximation of potential energy surfaces for the calculation of anharmonic vibrational energies. It is known, although, that its performance can be less than satisfactory due to limitations related to slow convergence of the series. In this article, we present a coordinate substitution scheme using a combination of Morse and sinh coordinates, well adapted for its use with cartesian normal coordinates. We derive expressions for analytical integrals for use in VSCF and VCI calculations and show that the simultaneous substitution of symmetric and antisymmetric normal coordinates by Morse and sinh coordinates, respectively, significantly improves the vibrational transition frequencies for these modes in a well‐balanced fashion. The accuracy of this substitution scheme is demonstrated by comparing one and two‐dimensional sections of substituted and unsubstituted QFF with ab initio potential energy grids, as well as with vibrational energy calculations using as test cases two well‐studied benchmark molecules: water and formaldehyde. We conclude that the coordinate substitution scheme presented constitutes a very attractive alternative to simple QFFs in the context of cartesian normal coordinates.     

Variational calculation of vibrational energies of triatomic molecules using SCF optimized modes

A self-consistent field optimization of the vibrational coordinates for nonlinear triatomic molecules is presented. The optimal coordinates are obtained by making a three-dimensional rotational transformation of the normal modes and determining the rotation angles as those for which the SCF energy is stationary. The utility of the optimized coordinates in full variational calculations of vibrational energies is studied for the molecules of H₂O, O₃, H₂D⁺, H₂T⁺, and D₂T⁺. For H₂O and O₃, the optimization procedure leads to the local mode representation. It is shown that the use of the optimal coordinates in variational calculations allows a large reduction of the dimension of the Hamiltonian matrix to be diagonalized in order to reach convergence.     

On the long way from the Coulombic Hamiltonian to the intermolecular energy surfaces: Concluding remarks at the tromsø conference,June 20–22, 1989

After a brief review of the main results given at the conference, the general properties of the Coulombic Hamiltonian for a system of electrons moving in a framework of moving atomic nuclei—considered as point charges—are discussed. Since this Hamiltonian is invariant under translations and rotations, the total momentum and the total angular momentum are constants of motion, which means that it is possible to separate the motion of the center of mass and the rotation of the system as a whole. Even if these separations are simple in principle, they lead to a mixing of the electronic and nuclear coordinates that complicates the transformed Hamiltonian. The general features of this Hamiltonian are discussed both in pure quantum mechanics and general quantum theory dealing with wave functions Ψ respective density matrices ρ or system operators T. The principles of the latter are derived from five simple axioms, and it is shown that pure quantum mechanics is a special case of the general theory and that the analogy between these two approaches is essential for the “economy of thinking.” It is indicated that the general theory of the shape and topology of the energy surface 〈H〉 = TrHΓ and its critical points, as a function of the system operator Γ involving both electronic and nuclear coordinates, is a very difficult mathematical problem and that calculation of this surface even for simple molecular systems represents a formidable computational problem, which has to be solved in order to be able to understand the nature of chemical reactions from first principles.     

On the interweaving of partial cross sections of different parity

Partial cross sections of definite parity, calculated for electronic-rotational energy transfer in the F + H₂ collision system, interweave with increasing total angular momentum J. An explanation, in terms of diabatic curve crossings induced by the centrifugal potential in the body-fixed coordinate system, predicts the interweaving to occur only in systems having half-integer J.     

Implementation of the IMOMM methodology for performing combined QM/MM molecular dynamics simulations and frequency calculations

A technique for implementing the integrated molecular orbital and molecular mechanics (IMOMM) methodology developed by Maseras and Morokuma that is used to perform combined quantum mechanics/molecular mechanics (QM/MM) molecular dynamics simulations, frequency calculations and simulations of macromolecules including explicit solvent is presented. Although the IMOMM methodology is generalized to any coordinate system, the implementation first described by Maseras and Morokuma requires that the QM and MM gradients be transformed into internal coordinates before they are added together. This coordinate transformation can be cumbersome for macromolecular systems and can become ill-defined during the course of a molecular dynamics simulation. We describe an implementation of the IMOMM method in which the QM and MM gradients are combined in the cartesian coordinate system, thereby avoiding potential problems associated with using the internal coordinate system. The implementation can be used to perform combined QM/MM molecular dynamics simulations and frequency calculations within the IMOMM framework. Finally, we have examined the applicability of thermochemical data derived from IMOMM framework. Finally, we have examined the applicability of thermochemical data derived from IMOMM frequency calculations. Received: 11 May 1998 / Accepted: 14 August 1998 / Published online: 16 November 1998     

三原子分子振转激发态的理论研究

三原子分子振转激发态的理论研究谢代前,鄢国森,田安民（四川大学化学系,成都,６１００６４）关键词振转激发态,三原子分子,变分法能级较高的振转激发态通常包含大振幅运动,其波函数分布于很广的势能面区域内,传统的正则模理论已不适合于解决这类问题．近年来,Ｈ...     

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.     

Coordinate transformation methods to calculate state-to-state reaction probabilities with wave packet treatments

A procedure for the transformation from reactant to product Jacobi coordinates is proposed, which is designed for the extraction of state-to-state reaction probabilities using a time-dependent method in a body-fixed frame. The method consists of several steps which involve a negligible extra computational time as compared with the propagation. Several intermediate coordinates are used, in which the efficiency depends on the masses of the atoms involved in the reaction. A detailed study of the relative efficiency of using reactant and product Jacobi coordinates is presented for several systems, and simple arguments are found depending on the masses of the atoms involved in the reaction. It is found that the proposed method is, in general, more efficient than the use of product Jacobi coordinates, specially for nonzero total angular momentum. State-to-state reaction probabilities are obtained for Li+FH-->LiF+H and F+HO-->FH+O collisions for several total angular momenta.