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     

An attempt to decompose the force constants for some diatomic molecules by the derivatives of the electronic kinetic energy

The force constants for several diatomic molecules were calculated by the derivatives of the electronic kinetic energy within the restricted Hartree–Fock formalism. The uniform scaling procedure was utilized in order to satisfy the virial theorem. The decomposition of the force constant was performed by partitioning the derivatives of the kinetic energy in several ways.     

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.     

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.     

A simple variational-numerical approach to the ro-vibrational spectrum of diatomic molecules. An application to CH+

We describe a simple way of obtaining numerically the manifold of energies for ro-vibrational transitions for a centrifugally distorted oscillator, starting from the potential energy of the non-rotating oscillator calculated by an accurate ab initio method. It is shown that the energies so obtained compare well with those obtained variationally. The species of astrophysical interest methylidyne ion, CH(+), has been selected as an example that allow us to show the computational efficiency of the method with respect to the variational one. It is applied for the determination of ro-vibrational levels up J=6, and the spectroscopic parameters corresponding to the ground electronic state X(1)Sigma(+). From the potential energy surface computed at the MRCI/cc-pV5Z level, the fundamental frequency, B(0) and D(0) are determined to be 2724.8, 13.85688 and -1.53322x10(-3)cm(-1), respectively. We provide also an estimation of anharmonic constants.     

Franck–Condon factors for diatomic molecules with anharmonic corrections

Some of the band systems of several astrophysically important molecules are calculated and compared with the results obtained by calculations based on realistic Klein–Dunham and Rydberg–Klein–Rees potential functions. The Morse potential is approximated by means of a fourth-order anharmonic oscillator model. In the second-quantized formalism, the anharmonic Hamiltonian is diagonalized by using the Bogoliubov–Tyablikov transformation. The diagonalization process gives a shift in the frequency associated with each normal mode of harmonic vibration of the molecules presented here. The Franck–Condon factors are estimated using this new frequency within the framework of a harmonic oscillator.     

Exact quantization rule to the Kratzer-type potentials: an application to the diatomic molecules

For arbitrary values of n and l quantum numbers, we present a simple exact analytical solution of the D-dimensional (D ≥ 2) hyperradial Schr?dinger equation with the Kratzer and the modified Kratzer potentials within the framework of the exact quantization rule (EQR) method. The exact bound state energy eigenvalues (E _nl ) are easily calculated from this EQR method. The corresponding normalized hyperradial wave functions (ψ _nl (r)) are also calculated. The exact energy eigenvalues for these Kratzer-type potentials are calculated numerically for a few typical LiH, CH, HCl, CO, NO, O₂, N₂ and I₂ diatomic molecules for various values of n and l quantum numbers. Numerical tests using the energy calculations for the inter dimensional degeneracy (D = 2 − 4) for I₂, LiH, HCl, O₂, NO and CO are also given. Our results obtained by EQR are in exact agreement with those obtained by other methods.     

Relativistic bond lengthening of UO 2 2+ and UO2

Summary Relativistic calculations on UO₂ [1] have shown that relativity leads to substantial bondlengthening in this compound, in contrast to the bond contraction found almost exclusively for other compounds. The bond lengthening isnot caused by the relativistic expansion of the 5f valence AO of U, which is the primary bond forming orbital on U in UO₂. The origin of the bond lengthening can be traced back to the semi-core resp. subvalence character of the U 6p AO. The valence character of 6p shows up in an increasing depopulation of the 6p upon bond shortening, and hence loss of mass-velocity stabilization. The core character of 6p shows up in large off-diagonal mass-velocity matrix elements 5p|h _MV|6p which are shown to have an overall bond lengthening effect. The larger expansion in UO₂ than in UO ₂ ²⁺ is due to destabilization of U levels in UO₂, caused by repulsion of the two additional 5f electrons.The present analysis corroborates the picture of relativistic bond length effects of Ref. [2].     

An application of the virial theorem to study AH2 structures II. Use of CNDO/2 wavefunctions

The simple form for the virial theorem for polyatomic molecules takes the form W=–t _i where t _i is an orbital kinetic energy. It is applied to study shapes of molecules of AH₂ type. Kinetic-energy-versus-angle diagrams are constructed with CNDO/2 wavefunctions that satisfy the virial theorem. The shapes of the molecules can be explained with the aid of the diagrams.     

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.     

Benchmark calculations with correlated molecular wave functions XII. Core correlation effects on the homonuclear diatomic molecules B2-F2

Using systematic sequences of the newly developed correlation consistent core-valence basis sets from cc-pCVDZ through cc-pCV6Z, the spectroscopic constants of the homonuclear diatomic molecules containing first row atoms, B–F, are calculated both with and without inclusion of 1s correlation. Internally contracted multireference configuration interaction (IC-MRCI) and singles and doubles coupled cluster (CCSD) theory with a perturbational estimate of connected triple excitations, CCSD(T), have been investigated. By exploiting the convergence of the correlation consistent basis sets, complete basis set (CBS) limits have been estimated for total energies, dissociation energies, equilibrium geometries, and harmonic frequencies. Based on the estimated CBS limits the effects of 1s correlation on D _e (kcal/mol), r _e (?), and ω _e (cm⁻¹) are: +1.1, −0.0070, +10 for B₂; +1.5, −0.0040, +13 for C₂; +0.9, −0.0020, +9 for N₂; +0.3, −0.0020, +6 for O₂; and −0.1, −0.0015, +1 for F₂. Received: 20 January 1997 / Accepted: 6 May 1997     

The inversion of spectroscopic data to the diatomic effective Hamiltonian containing the generalized potential energy function. Ground state of PbO

A method of inversion of spectroscopic data of a diatomic molecule to the effective Hamiltonian containing adiabatic and non-adiabatic corrections is reported. The method is based on the use of a previously suggested generalized potential energy function with correct long-range part. The potential energy function for the X¹Σ⁺ state of PbO is calculated by inversion of the infrared and microwave spectra.     

The second virial coefficient of 3-centerLennard-Jones molecules and its relation to 1- and 2- center molecules

The second virial coefficients of homonuclear three-centerLennard-Jones molecules are calculated with various parameters of the isosceles triangle connecting the three sites. A special effort is made to establish the reducedBoyle temperaturesT _B and the values of the second virial coefficients atT/T _B=0.3 for the sake of comparison with one- and two-centerLennard-Jones molecules. It is shown that it is possible to find parameter values of the interaction potential of one- and two-centerLennard-Jones molecules which give very similar values of second virial coefficients forT/T _B0.3, and the equivalence conditions are established. These conditions might not only give a basis for a microscopic scaling of state variables, but also some restrictions for the validity of the group contribution concept.Presented in part at the DFG-Colloquium at Paderborn, 19th April 1982, and at the 5th Conference on Mixtures of Nonelectrolytes and Intermolecular Interactions, April 18–22, 1983, at Halle (GDR).     

Binding/antibinding analyses for diatomic interactions H 2 + system

The region-functional concept of electron density has been quantitatively examined for 1s_g, 2p_u, 2p_u, and 3d_g states of H ₂ ⁺ system on the basis of Berlin diagram which divides the three-dimensional molecular space into binding and antibinding regions. The electronic charge, Hellmann-Feynman (H-F) force, and stabilization energy of the system are partitioned into the binding and antibinding contributions by the regional integrations.Dynamic behaviors of the electron density (i.e. electron-cloud preceding and following) during the interaction processes are also clarified using the centers of electron density and force density.Differences in attractive and repulsive, and - and -type interactions are discussed from the force and density point of view.     

Molecular orbital studies on small molecules using H2+-type elliptical basis orbitals. Application to H2+, H2, He2++ and H3+

H₂⁺-type elliptical orbitals are defined in Section 1. These orbitals, which in elliptical coordinates involve a factor (1 + ξ)^σ, are employed in variational calculations on the ground states of H₂⁺ and H₂ (Sections 2 and 3). Various choices of σ are explored for H₂⁺, while two choices are used for H₂ : the “boundary condition” (Equation 6) and the “cusp condition” (Equation 9) values. Variational energies are calculated and compared to the results of similar calculations. Section 3 concludes by employing the H₂⁺-type orbitals in LCETO-MO-SCF calculations on the ground states of H₂ and He₂⁺⁺. For both molecules a four-function basis set with two (nonlinear) variational parameters yields more than 99% of the Hartree-Fock limit. Section 4 deals with LCETO-MO-SCF calculations on triangular H₃⁺. Three four-function basis sets are used, and the best energy is -1.2306 a.u., which is in reasonable agreement with the Hartree-Fock limit, -1.2999 a.u. Our best basis set is a four-term two-center expansion of the wave function with only one nonlinear variational parameter. Section 5 concludes the paper with a summary of the methods used to evaluate the integrals which arise in SCF calculations in the H₂⁺-type elliptical orbital basis.     

CEPA calculations on open-shell molecules. II. Singlet-triplet energy splitting in π2 configurations of diatomic molecules

The CEPA-PNO method is used for calculating the energy difference ΔE _ST between the³∑⁻ and the¹Δ states of diatomic molecules in electronic π² configurations. An analysis of the contribution of electron correlation to ΔE _ST is performed in terms of physically understandable effects such as direct correlation, dynamic spin polarization, semiinternal and internal excitations. It is shown that these effects are of completely different importance for the molecules treated in this study: For C₂ the direct correlation between the two singly occupied π-orbitals is the dominant correlation contribution to ΔE _ST; for O₂, S₂, SO the internal excitation π _u ² → π _g ² is predominant, whereas for NH and PH there is a close competition between the direct correlation and the spin polarization of the underlying σ-orbitals. The basis set dependence of these effects is investigated, in particular for NH. Our final results reproduce experimental values of ΔE _ST within 0.05–0.10 eV.     

Application of the FAKE molecular-orbital method to diatomic molecules XY (X,Y = H,F, Cl,Br, I)

Theoretical Chemistry Accounts - The FAKE (fast, accurate kinetic energy) method of semiempirical molecular orbital calculation is applied to diatomic molecules XY (X, Y= H, F, Cl, Br, I). The...     

Zero virial reaction processes. Reactions of Si and Mg atoms with H2 and HF molecules

By adopting the bond angle as a reaction coordinate and imposing on appropriate conditions, we determine a reaction path named the zero virial path starting from the reactant and arriving at the product via the transition state. At every point on this path, the polyatomic virial theorem is reduced to the simple atom-like form, and within the LCAO framework the total energy including the nuclear repulsion is partitioned exactly into the one center (atomic) and two center (bond) terms through the negative kinetic energy. Using these advantages of the zero virial path, the processes of bond formation and fission are examined in detail for the reactions of Si and Mg atoms with H₂ and HF molecules.     

ChemInform Abstract: Molecular Structure and Thermal Stability of B2H4 and B2H+ 4 Species.

B₂H₄ is produced by reaction of B₂H₆ with fluorine atoms generated in a microwave discharge.     

A method for molecular correlation energy calculations. Application to the determination of dissociation energies of diatomic and polyatomic molecules

A simple and economical method for molecular correlation energy calculations is developed. In this method, the internal part of the correlation energy is calculated by means of a CI in a minimal basis set and the non-internal part (semi-internal and all-external) is evaluated using an original atoms-in-molecule method. It is successfully applied to the determination of dissociation energies of some diatomic (H₂, NH, C₂, CN, N₂, CO, NO, O₂, F₂) and polyatomic (H₂O, N₂O, CO₂, N₃H, CH₂N₂, CH₂CO, C₂N₂) molecules. The results are compared to those obtained using very elaborate variational methods.Aspirant du Fonds National Belge de la Recherche Scientifique.