Molecular electronic virial theorem

The partitioning of the molecular electronic energy into true one-electron quantities defined by a molecular electronic virial theorem (MEVT) is studied for a number of molecules. Since the theorem is derived for exact wavefunctions, its applicability to various ab initio wavefunctions at difierent levels of accuracy is examined. The average percentage deviations of the theorem for near Hartree-Fock, double zeta, STO-6G and STO-3G type wave functions are 0.4, 1.7, 2.3 and 3.3, respectively.     

Relativistic virial theorem for atoms

A relativistic virial theorem is derived for atoms in a general manner. The virial ratio consists of the usual V/T term and a correction term W/T, where T, V, and W are the kinetic energy, the potential energy, and correction terms, respectively. Explicit forms of W are presented for four specific nuclear potential models. Numerical calculations for a uniform nuclear charge model show that the magnitude of the correction term W/T increases with increasing atomic numbers and that it modifies the ratio V/T considerably for atoms with large atomic numbers in particular. Received: 21 November 2000 / Accepted: 8 January 2001 / Published online: 3 April 2001     

Spin virial theorem in the density functional theory

The spin virial theorem is derived in the density functional theory. The theorem establishes a relation between the differences of spin-up and -down kinetic and potential energies. The theorem is useful for checking the accuracy of spin orbitals. As an illustration, the example of the Xα method is studied. © 1994 John Wiley & Sons, Inc.     

Confined systems and the modified virial theorem from semiclassical considerations

We show that two simple semiclassical strategies, one based on the Wilson–Sommerfeld rule and the other on the uncertainty principle, yield the exact modified form of the virial theorem for confined systems. An alternative, easier quantum mechanical route to arrive at this result is also sketched. Pilot calculations on confined oscillators reveal decisive trends. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

A modification of Koopmans' theorem by imposition of the virial theorem in molecules

Imposition of the virial theorem on Koopmans' theorem permits the introduction of some relaxation effect in the electronic cloud of atomic (less than 5%) or molecular (less than 1.3% for the systems studied) systems and a partitioning of the ionization energy. The method is applied in some diatomic hydrides. It is observed that the imposition of the virial theorem improves the ionization of the innermost molecular orbitals significantly, while the improvement is negligible for the outermost orbitals. The ionization energy is divided among three different terms that elucidate some aspects of the nature of the ionization process.     

Extremely localized nonorthogonal orbitals by the pairing theorem

Using the concepts of L?wdin pairing theorem, a method is developed to calculate extremely localized, but nonorthogonal, sets of molecular orbitals and their strictly localized counterparts. The method is very suitable to study to what extent a given model of bonding in a given molecule can be considered adequate from the point of view of the actual LCAO-MO (Hartree Fock or DFT) wave function and is expected to be useful for doing local approximations of electron correlation.     

Some comments about tensor virial theorem and linear transformations

The tensor virial theorem is discussed in relation to different transformation matrices which are compatible with it.     

The virial theorem as an entail for Koopmans' theorem in atomic calculations

The frozen approximation in the Koopmans' theorem suggests that the virial theorem is not obeyed. By imposing the virial theorem to Koopmans' theorem, we observe that the ionization potential of an atomic orbital is directly related to the respective kinetic energy and that the virial theorem introduces some reorganization effect on the electronic cloud. The quantity of reorganization introduced is not hazard, depending on the type of atom as well as the atomic orbital.     

定域分子轨道对称性

提出了利用定域分子轨道重心确定分子轨道的对称性。确定了线型的、非线型的、环状的分子定域分子轨道对称性。结果表明非线型分子和其中双中心键具有相同对称性,但含孤对或三中心键的分子则存在一些差别。还讨论了定域分子轨道的特征。     

A general procedure for obtaining wave functions obeying the virial theorem

The virial theorem for molecules is shown to have two different forms, one employing the energy gradient the other involving the Hellmann–Feynman force. While the former VT can be fulfilled by a uniform scaling of the basis set, the latter cannot be satisfied in certain basis sets, and can give unrealistic results in others. The scaling procedure is applied to molecules at nonstationary points on the potential energy surface and it is found that energy components can change substantially, especially at short bondlengths, while the change in total energy is small. The effects on molecular properties are also small.     

Molecular orbitals of polynuclear complexes

The theory of the orbital structure of various complexes and complicated molecules has been further developed, and the scope of application of the orbital model has been extended. Formulas have been suggested for determining the molecular shells of all types from the sets of atom packing numbers.     

Eigensolutions,virial theorem and molecular study of nonrelativistic Krazer-Fues potential

A Kratzer-Fues potential is coupled with its self to form a double Kratzer-Fues potential. The solutions of the radial Schrödinger equation with the combined potential are obtained via two different methodologies. Using the supersymmetric approach, the energy equation for a non-approximated centrifugal term is obtained. The energy equation for the approximated centrifugal term is obtained using the parametric Nikiforov-Uvarov method. Numerical results are computed for five molecules using MATLAB software program. The computed numerical values are compared with the experimental values and the ground state energy obtained from Herzberg's energy level equation. The study revealed that the results of the approximated centrifugal term are better than the results of the non-approximated centrifugal term for four molecules.     

The virial theorem for non-coulombic interactions and scaled approximate wave functions

The virial theorem for various not necessarily homogeneous potential functions V(r) is derived. It is proved in a transparent way that approximate wave functions at optimal scale satisfy the general virial theorem for any radial potential V(r). The proposed method for improving a variational wave function which by the scaling procedure obeys the virial theorem is tested in the case of some simple screened coulomb potentials.

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Zusammenfassung Das Virialtheorem für verschiedene, nicht unbedingt homogene Potentialfunktionen V(r) wird abgeleitet. Es wird gezeigt, da\ eine NÄherungsfunktion mit optimalem Skalenparameter auch das allgemeine Virialtheorem für beliebige Radialpotentiale V(r) erfüllt. Die Methode der Streckung des Grundgebietes (scaling) dient zur Verbesserung von NÄherungsfunktionen, was am Beispiel einiger abgeschirmter Coulomb-Potentiale demonstriert wird.

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Résumé Le théorème du viriel pour quelques fonctions de potentiel V(r) non nécessairement homogènes est dérivé du théorème général. Il est prouvé d'une manière intuitive que des fonctions d'onde approchées, à l'échelle optimum, satisfont le théorème du viriel général pour tous les potentiels radiaux V(r). La méthode proposée pour améliorer une fonction d'onde variationnelle, qui obéit au théorème du viriel, est essayée dans le cas de quelques simples potentiels coulombiens écrantés.

     

Obtaining self–consistent wave functions which satisfy the virial theorem

The virial theorem has played an important role in applying quantum mechanics to chemical problems. It has served as one criterion of a satisfactory wave function and its consequences on chemical bonding, molecular structure, and substituent effects have been analyzed extensively. A common method of gaining compliance with the virial theorem is to introduce a “scale” factor which adjusts all distances by a factor η. Optimizing the scale factor through the variational principle produces a wave function satisfying the virial theorem. In the present paper it is shown that when this “scaling” procedure is applied to self-consistent wave functions, the virial theorem can be satisfied, but self-consistency is lost. Scaling generally has a small effect on the total energy, but the effects on the energy components (T, V_ne, V_ee, V_nn) can be two to three orders of magnitude larger and in the range of tens to hundreds of kcal. Consequently, for applications where the energy components are useful, it is highly desirable to obtain wave functions which satisfy the virial theorem and are self-consistent. In the present paper, a simple, inexpensive extrapolation technique is reported which requires one integral evaluation and two SCF cycles to achieve convergence. Applications to atoms and small molecules are reported.     

Evanescent high pressure during hypersonic cluster-surface impact characterized by the virial theorem

Matter under extreme conditions can be generated by a collision of a hypersonic cluster with a surface. The ultra-high-pressure interlude lasts only briefly from the impact until the cluster shatters. We discuss the theoretical characterization of the pressure using the virial theorem and develop a constrained molecular-dynamics procedure to compute it. The simulations show that for rare-gas clusters the pressures reach the megabar range. The contribution to the pressure from momentum transfer is comparable in magnitude and is of the same sign as that ("the internal pressure") due to repulsive interatomic forces. The scaling of the pressure with the reduced mechanical variables is derived and validated with reference to the simulations.     

Application of the virial theorem to the study of molecular electronic wave functions

With aid of the virial theorem formulated for the energy differences of two electronic states some theorems on the wave functions of diatomic molecules have been proven. It is shown how proper Rydberg states can be distinguished from other electronic states with a diffuse outer orbital by virtue of the virial theorem and that a singlet-triplet pair of excited states cannot have the same equilibrium geometry and identical orbitals simultaneously. Furthermore if the two states have the same dissociation limit a theorem on the differences of the kinetic and the potential energy can be derived which allows an understanding of the shape of the electronic wave functions. As an application the wave functions and the ordering of the lowest states of H ₂ ⁺ and H₂ have been discussed.This work is part of the project Nr. SR 2.159.74 of the Schweizerischer Nationalfonds.     

Molecular properties from coupled-cluster Brueckner orbitals

We investigate to which extent a single determinant made up from orbitals obtained by a Brueckner coupled-cluster doubles calculation is able to reproduce correlated one-electron properties. It is shown that dipole and quadrupole moments and radial expectation values compare quite well with BCCD finite-field results for a test selection of nine molecules enclosing HF, H₂O, NH₃, CO, N₂, NO⁺, HCN, CuH and CH₃OH and three rare-gas atoms He, Ne and Ar. Furthermore, we find that even second-order properties such as dipole and quadrupole polarizabilities are reproduced fairly well when determined as first derivatives of the corresponding Brueckner orbital expectation values.