Relativistic virial theorem for diatomic molecules. Application to H 2 +

Summary The virial theorem for a molecule in the relativistic clamped-nuclei approximation is derived. The individual energy contributionsA (momentum energy),B (mass energy),T=A+B (kinetic energy) andV (potential energy) are expressed in terms ofE, E/R (derivate w.r.t. the nuclear coordinates) and the relativistic correction E/² (derivative w.r.t. Sommerfield's fine-structure constant ). IfE and E/R are known as functions of , then all individual energy terms are also known as functions of . As an example, numerical results for H ₂ ⁺ are presented. The relativistic and nonrelativistic potential energy curves and the paradoxical behavior of their different contributions are analyzed and interpreted in both the largeR and shortR ranges.Dedicated to Professor W. Kutzelnigg on the occasion of his 60th birthday     

Use of the virial theorem to calculate the molecular force constants of diatomic hydrides

The possibility of using the virial theorem to calculate the coefficients of the potential function for diatomic hydrides on the basis of the Fock-Roothaan wave function in the closed electron shell approximation has been demonstrated in relation to the LiH, BH, CH, NH, and OH molecules.     

Vibrational excitation and dissociation of a diatomic molecule in the diffusion approximation

A numeric solution is given for the excitation of vibration and dissociation in the diffusion approximation for an O₂-Ar (oxygen-argon) mixture. The time of vibrational relaxation is calculated, and also the rate constant for dissociation. The diffusion approximation describes the processes satisfactorily for a gas whose particles differ little in mass.I am indebted to B. V. Kuksenko, A. I. Osipov, and M. N. Safaryan for discussions and valuable advice.     

Dissociation of a diatomic molecule induced by discontinuous reversals of a static electric field

A diatomic molecule modeled by a Morse oscillator is shown to undergo dissociation if the external electric field, in which the molecule is placed, undergoes discontinuous reversals. A threshold frequency of reversal is found to exist. The classical behavior is compared with its quantum counterpart. The mechanism of dissociation is analyzed. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Application of the virial theorem to the calculation of anharmonic force constants for the water molecule

On the basis of the virial theorem, relationships have been found between the anharmonic force constants which make it possible to decrease the number of unknowns in the equations obtained from the general quantum-mechanical theory of the vibrations and rotations of molecules. The complete set of anharmonic force constants for the H₂O molecule have been used to calculate all the anharmonic constants X_SS' for the D₂O molecule.     

Use of the virial theorem to calculate molecular force constants. Harmonic force constants of the water molecule

The quantum mechanical virial theorem is used to calculate the harmonic part of the potential function for the water molecule on the basis of wave functions obtained in the self-consistent field approximation.     

The virial theorem scaling model for estimating the charge sensitivities of hydrogens in molecules

A simple variational model of hydrogens in molecules is presented, using the virial theorem (Fock) scaling of the wave function to account for the orbital relaxation due to a change in the number of electrons. The resulting interpolative formulas for the energy allow realistic predictions of the spin-nonpolarized or spin-polarized hardness (electron repulsion) parameters and other sensitivities of the hydrogen systems in molecules, including the realistic bare nuclei limit (N → 0) data.     

Quantum control of vibrational excitations in a heteronuclear diatomic molecule

Optimal control theory is applied to obtain infrared laser pulses for selective vibrational excitation in a heteronuclear diatomic molecule. The problem of finding the optimized field is phrased as a maximization of a cost functional which depends on the laser field. A time dependent Gaussian factor is introduced in the field prior to evaluation of the cost functional for better field shape. Conjugate gradient method^21,24 is used for optimization of constructed cost functional. At each instant of time, the optimal electric field is calculated and used for the subsequent quantum dynamics, within the dipole approximation. The results are obtained using both Morse potential as well as potential energy obtained using ab initio calculations.     

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.     

Differential virial theorem in relation to a sum rule for the exchange-correlation force in density-functional theory

Holas and March [Phys. Rev. A. 51, 2040 (1995)] gave a formally exact theory for the exchange-correlation (xc) force F(xc)(r)= -inverted Deltaupsilon(xc)(r) associated with the xc potential upsilon(xc)(r) of the density-functional theory in terms of low-order density matrices. This is shown in the present study to lead, rather directly, to the determination of a sum rule nF(xc)=0 relating the xc force with the ground-state density n(r). Some connection is also made with an earlier result relating to the external potential by Levy and Perdew [Phys. Rev. A. 32, 2010 (1985)] and with the quite recent study of Joubert [J. Chem. Phys. 119, 1916 (2003)] relating to the separation of the exchange and correlation contributions.     

Vibrational energy relaxation of a diatomic molecule in a room-temperature ionic liquid

Vibrational energy relaxation (VER) dynamics of a diatomic solute in ionic liquid 1-ethyl-3-methylimidazolium hexafluorophosphate (EMI(+)PF(6) (-)) are studied via equilibrium and nonequilibrium molecular dynamics simulations. The time scale for VER is found to decrease markedly with the increasing solute dipole moment, consonant with many previous studies in polar solvents. A detailed analysis of nonequilibrium results shows that for a dipolar solute, dissipation of an excess solute vibrational energy occurs almost exclusively via the Lennard-Jones interactions between the solute and solvent, while an oscillatory energy exchange between the two is mainly controlled by their electrostatic interactions. Regardless of the anharmonicity of the solute vibrational potential, VER becomes accelerated as the initial vibrational energy increases. This is attributed primarily to the enhancement in variations of the solvent force on the solute bond, induced by large-amplitude solute vibrations. One interesting finding is that if a time variable scaled with the initial excitation energy is employed, dissipation dynamics of the excess vibrational energy of the dipolar solute tend to show a universal behavior irrespective of its initial vibrational state. Comparison with water and acetonitrile shows that overall characteristics of VER in EMI(+)PF(6) (-) are similar to those in acetonitrile, while relaxation in water is much faster than the two. It is also found that the Landau-Teller theory predictions for VER time scale obtained via equilibrium simulations of the solvent force autocorrelation function are in reasonable agreement with the nonequilibrium results.     

Ro-vibrating energy states of a diatomic molecule in an empirical potential

An approximate analytical solution of the Schrödinger equation is obtained to represent the rotational–vibrational (ro-vibrating) motion of a diatomic molecule. The ro-vibrating energy states arise from a systematical solution of the Schrödinger equation for an empirical potential (EP) V _±(r) = D _e {1 ? (?/δ)[coth (ηr)]^±1/1 ? (?/δ)}² are determined by means of a mathematical method so-called the Nikiforov–Uvarov (NU). The effect of the potential parameters on the ro-vibrating energy states is discussed in several values of the vibrational and rotational quantum numbers. Moreover, the validity of the method is tested with previous models called the semiclassical (SC) procedure and the quantum mechanical (QM) method. The obtained results are applied to the molecules H₂ and Ar₂.     

Collinear collision of an atom with a homonuclear diatomic molecule

The problem of collinear scattering of an atom from a homonuclear diatomic molecule is formulated in terms of a first-order nonlinear matrix differential equation for the variable coefficient of reflection. For a homonuclear molecule when the target Hamiltonian is invariant under the parity transformation, only transitions between even states or odd states are possible. This selection rule reduces the number of open or closed channels that contribute to the reflection and transmission coefficients. But for numerical calculation, under the conditions of the problem, one can approximate the target Hamiltonian by the Hamiltonian of a displaced harmonic oscillator. In this approximation, the reflectional symmetry of the Hamiltonian is not preserved and transitions between any two levels of the target are possible. To simplify the problem further, the interaction between the projectile and the target is assumed to be a sum of two Gaussian terms. For this combination of the potentials the many-channel interaction can be expressed analytically. By fitting the Lennard–Jones potential with a sum of two Gaussian potentials and solving the matrix differential equation, transition probabilities are obtained for the He? H₂ collision. The numerical results are compared with the results found by Secrest and Johnson, and by Clark and Dickinson.     

Electronic effect on the rotational energy of a diatomic molecule

The rotational hamiltonian for a diatomic molecule has been rederived from the total classical hamiltonian. This procedure directly introduces the effect of electronic motion which is ordinarily neglected in zero-order approximation. Kronig's rotational hamiltonian is discussed and shown to be an approximation of our findings. Our general result is then specialized to ¹Σ states, and the theory tested by calculating the observed fractional discrepancy between the experimentally determined H³⁵Cl energy level constant Y₀₂ and its predicted value from Dunham's theory. When all corrections are summed, the results are in good agreement with experiment.     

Extension of the quantum virial and Hellmann–Feynman theorems to the quantum statistical averages

In the present article, we extended the quantum virial and Hellmann–Feynman theorems to the quantum statistical averages, that is, to the thermal states. We obtained some new formulas which make possible expressing the thermodynamical observables of the system as functions of the moments of coordinates, as we see in a short example relating to the pseudoharmonical oscillator. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 159–165, 1998     

On the variational solution of morphed molecular potential in a diatomic molecule

The general one-dimensional potential energy function, including centrifugal distortion, for a diatomic molecule is morphed with a series of Morse-like functions for each of the rotational quantum numbers $J$ . For each of the morphed potential, explicit formulae for the matrix elements of the complete energy matrix, on the basis of the solutions of the one-dimensional harmonic oscillator, are given and these may be used in connection with the variational procedure to solve the corresponding vibrational Schrödinger equation. From the set of vibrational levels $\{\text{ E}_\mathrm{vJ}\}, J = 0, 1, 2,{\ldots }$ the ro-vibrational transitions can be deduced.     

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.     

Branching ratio for preionization and predissociation of a superexcited state of a diatomic molecule

The branching ratio for preionization and predissociation is studied qualitatively on the basis of a previously proposed simplisitc classical probabilistic model of the superexcited states of molecules. The isotope effect is also discussed.