Automatic computer procedure for generating exact and analytical kinetic energy operators based on the polyspherical approach

We develop a new general code to automatically derive exact analytical kinetic energy operators in terms of polyspherical coordinates. Computer procedures based on symbolic calculations are implemented. Sets of orthogonal or non-orthogonal vectors are used to parametrize the molecular systems in space. For each set of vectors, and whatever the size of the system, the exact analytical kinetic energy operator (including the overall rotation and the Coriolis coupling) can be derived by the program. The correctness of the implementation is tested for different sets of vectors and for several systems of various sizes.     

Kinetic energy operators in linearized internal coordinates

It is customary to describe molecular vibrations using as exact kinetic energy operators and as accurate potentials as possible. It has become a standard approach to express Hamiltonians in curvilinear internal displacement coordinates, because they offer a simple and physical picture of vibrational motions, including large amplitude changes in the shape. In the older normal mode model of molecular vibrations, the nuclei are thought to vibrate infinitesimally about the reference configuration, and the shape of the molecule is described using linearized approximations of the true geometrically defined internal displacement coordinates. It is natural to ask how the two approaches are related. In this work, I present a general yet practical way to obtain curvilinear displacement coordinates as closed function of their linearized counterparts, and vice versa. In contrast to the conventional power series approach, the body-frame dependency is explicitly taken into account, and the relations are valid for any value of the coordinates. The present approach also allows one to obtain easily exact kinetic energy operators in linearized shape coordinates.     

The use of interparticle coordinates in electronic energy calculations for atoms and molecules

The use of interparticle coordinates in atomic and molecular calculations is reviewed and emphasized. For states where the wave function is dependent only upon interparticle coordinates the Hamiltonian is a relatively simple expression. If the wave function is a linear combination of products of functions of single interparticle coordinates of the exponential power form the formulas for local energy are particularly simple.

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Zusammenfassung Es wird über Teilchenabständc als Koordinaten in Atom- und Molekülproblemen referiert und auf ihren Wert hingewiesen. In Fällen, wo die Wellenfunktion nur von Teilchenabständen abhängt, wird der Hamilton-Operator verhältnismäßig einfach. Ist die Wellenfunktion eine Linearkombination von Produkten von Potenz-Exponential-Funktionen einzelner Teilchenabstände, so werden die Ausdrücke für die lokale Energie besonders einfach.

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Résumé L'auteur fait une revue et souligne l'importance de l'emploi des distances relatives des particules comme coordonnées pour les calculs atomiques et moléculaires. Pour des états dont les fonctions d'onde ne dépendent que de ces distances le l'Hamiltonien devient relativement simple. Si la fonction d'onde s'écrit comme une combinaison linéaire de produits de fonctions exponentielles (à puissances) ne dépendant que d'une seule coordonnée interparticulaire, les formules pour l'énergie locale se simplifient particulièrement.

     

A new method for the derivation of exact vibration-rotational kinetic energy operator in internal coordinates

An efficient angular momentum method is presented and used to derive analytic expressions for the vibration-rotational kinetic energy operator of polyatomic molecules.The vibration-rotational kinetic energy operator is expressed in terms of the total angular momentum operator J,the angular momentum operator J and the momentum operator p conjugate to Z in the molecule-fixed frame Not only the method of derivation is simpler than that in the previous work,but also the expressions ot the kinetic energy operators arc more compact.Particularly,the operator is easily applied to different vibrational or rovibrational problems of the polyatomic molecules by variations of matrix elements Gn of a mass-dependent constant symmetric matrix     

A new exact quantum mechanical rovibrational kinetic energy operator for penta-atomic systems in internal coordinates

The concrete molecule-fixed (MF) kinetic energy operator for penta-atomic molecules is expressed in terms of the parameterδ, the matrix element G_(?), and angular momentum operator (?). The applications of the operator are also discussed. Finally, a general compact form of kinetic energy operator suitable for calculating the rovibrational spectra of polyatomie molecules is presented.     

Perimetric scale-shape coordinates for triatomic molecules

Perimetric nuclear coordinates of a triatomic molecule treat all three nuclei equivalently and are not subject to the triangle conditions. Through an appropriate orthogonal transformation they can be separated into one scale coordinate, viz., the circumference, and two shape coordinates, which are determined by the angles. The parameter space of the shape coordinates is an equilateral triangle. The basic formulas are given and the relationship between points in coordinate space and molecular shapes are elucidated. Received: 10 January 1996 / Accepted: 2 January 1997     

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.     

Hierarchical expansion of the kinetic energy operator in curvilinear coordinates for the vibrational self-consistent field method

A new hierarchical expansion of the kinetic energy operator in curvilinear coordinates is presented and modified vibrational self-consistent field (VSCF) equations are derived including all kinematic effects within the mean field approximation. The new concept for the kinetic energy operator is based on many-body expansions for all G matrix elements and its determinant. As a test application VSCF computations were performed on the H(2)O(2) molecule using an analytic potential (PCPSDE) and different hierarchical approximations for the kinetic energy operator. The results indicate that coordinate-dependent reduced masses account for the largest part of the kinetic energy. Neither kinematic couplings nor derivatives of the G matrix nor its determinant had significant effects on the VSCF energies. Only the zero-point value of the pseudopotential yields an offset to absolute energies which, however, is irrelevant for spectroscopic problems.     

Automatic generation of potential energy and property surfaces of polyatomic molecules in normal coordinates

A procedure for the automatic construction of Born-Oppenheimer (BO) potential energy and molecular property surfaces in rectilinear normal coordinates is presented and its suitability and accuracy when combined with vibrational structure calculations are assessed. The procedure relies on a hierarchical n-mode representation of the BO potential energy or molecular property surface, where the n-mode term of the sequence of potentials/molecular properties includes only the couplings between n or less vibrational degrees of freedom. Each n-mode cut of the energy/molecular property surface is first evaluated in a grid of points with ab initio electronic structure methods. The ab initio data are then spline interpolated and a subsequent polynomial fitting provides an analytical semiglobal representation for use in vibrational structure programs. The implementation of the procedure is outlined and the accuracy of the method is tested on water and difluoromethane. Strategies for improving the proposed algorithm are also discussed.     

Dissociative electron attachment to polyatomic molecules: Ion kinetic energy measurements

Dissociative electron attachment to SO₂, NO₂, NF₃ and H₂O₂ is studied in terms of the kinetic energies of the dominant fragment ions. The O^? data from SO₂ show that the two major resonances at 4.6 and 7.2 eV respectively have the same dissociation limit. Similarly, the resonances at 1.8 and 3.5 eV in the O^? channel in NO₂ appear to have same dissociation limit of NO (X ²Π) + O^?, while the resonance at 8.5 eV appears to dissociate to give NO (a ⁴Π_i) along with O^?. We find considerable internal excitation of the neutral fragments in all these cases along with that of NF₃, whereas the negative ion resonance in H₂O₂ appears to fragment almost like a diatomic system with very little internal excitation of the OH and OH^? fragments.     

A general treatment of vibration-rotation coordinates for triatomic molecules

An exact, within the Born–Oppenheimer approximation, body-fixed Hamiltonian for the nuclear motions of a triatomic system is presented. This Hamiltonian is expressed in terms of two arbitrarily defined internal distances and the angle between them. The body-fixed axis system is related to these coordinates in a general fashion. Problems with singularities and the domain of the Hamiltonian are discussed using specific examples of axis embedding. A number of commonly used coordinate systems including Jacobi, bond-length-bond-angle, and Radau coordinates are special cases of this Hamiltonian. Sample calculations on the H₂S molecule are presented using all these and other coordinate systems. The possibility of using this Hamiltonian for reactive scattering calculations is also discussed.     

Real‐space grid representation of momentum and kinetic energy operators for electronic structure calculations

We show that the central finite difference formula for the first and the second derivative of a function can be derived, in the context of quantum mechanics, as matrix elements of the momentum and kinetic energy operators on discrete coordinate eigenkets defined on a uniform grid. Starting from the discretization of integrals involving canonical commutations, simple closed‐form expressions of the matrix elements are obtained. A detailed analysis of the convergence toward the continuum limit with respect to both the grid spacing and the derivative approximation order is presented. It is shown that the convergence from below of the eigenvalues in electronic structure calculations is an intrinsic feature of the finite difference method. © 2018 Wiley Periodicals, Inc.     

Kinetic energy functional in hyperspherical coordinates

Hyperspherical coordinates and a generator coordinate representation are employed to find a simple form of the kinetic energy for a general three-particle problem. An expression is developed for the determination of adiabatic hyperangular states in a local potential using the finite element method.     

The hierarchical expansion of the kinetic energy operator in curvilinear coordinates extended to the vibrational configuration interaction method

The hierarchical expansion of the kinetic energy operator in curvilinear coordinates presented earlier for the vibrational self-consistent field technique is extended to the vibrational configuration interaction (VCI) method. The high accuracy of the modified VCI method is demonstrated by computing first excitation energies of the H(2)O(2) molecule using an analytic potential (PCPSDE) and showing convergence to accurate results from full dimensional discrete variable representation calculations.     

The sum rule for dipolar absorptions and rotational kinetic energy of wate and some dipolar molecules in condensed phases

Another form of the sum rule for dipolar absorptions has been derived by means of quantum statistics. The difference between this and usually used form results from a quantum effect on the molecular rotational motion. By the joint use of the two forms, average rotational kinetic energies of water molec in the liquid and solid phases and some dipolar molecules in solutions have been estimated. It has been shown that the average rotational kinetic energ larger than the value expected from the classical equipartition rule, with an increase in the hindering potential for the rotational motion of the mole The dipole moments of water molecules in liquid and solid water have been estimated. These are considerably smaller than the gas-phase value.