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.     

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     

On the quantum chemical origin for the nonvalidity of Koopmans' theorem in transitionmetal compounds

The quantum chemical origin for the nonvalidity of Koopmans' theorem in transitionmetal compounds of the 3d series is analyzed by means of the Green's function formalism applied in the framework of a semiempirical INDO Hamiltonian. In the case of ferrocene (1), cyclobutadiene iron tricarbonyl (2) and irontetracarbonyl dihydride (3) the self-energy part of a geometric approximation has been partitioned into relaxation and correlation (pair removal, pair relaxation) increments. The breakdown of Koopmans' theorem for strongly localized MOs with large Fe 3d amplitudes is predominantly the result of electronic relaxation lowering the calculated ionization potentials. On the other hand the variation of the pair correlation energy in the cationic hole-state is by no means negligible and acts into the opposite direction as the relaxation increment. These significant pair relaxation contributions explain the wellknown failtures of the ΔSCF approach in combination with large scaleab initio bases. The loss of ground state pair correlation in the outer valence region is small in comparison to relaxation and pair relaxation. The magnitude of the aforementioned reorganization increments has been studied as a function of the localization properties of the MOs and as a function of the one-electron energies of the available particle- and hole-states. The computational findings derived with the INDO model are compared with recentab initio studies.     

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     

Long‐range corrected functionals satisfy Koopmans' theorem: Calculation of correlation and relaxation energies

In this article, we show that the long‐range‐corrected (LC) density functionals LC‐BOP and LCgau‐BOP reproduce frontier orbital energies and highest‐occupied molecular orbital (HOMO)—lowest‐unoccupied molecular orbital (LUMO) gaps better than other density functionals. The negative of HOMO and LUMO energies are compared with the vertical ionization potentials (IPs) and electron affinities, respectively, using CCSD(T)/6‐311++G(3df,3pd) for 113 molecules, and we found LC functionals to satisfy Koopmans' theorem. We also report that the frontier orbital energies and the HOMO‐LUMO gaps of LC‐BOP and LCgau‐BOP are better than those of recently proposed ωM05‐D (Lin et al., J. Chem. Phys. 2012, 136 , 154109). We express the exact IP in terms of orbital relaxation, and correlation energies and hence calculate the relaxation and correlation energies for the same set of molecules. It is found that the LC functionals, in general, includes more relaxation effect than Hartree–Fock and more correlation effect than the other density functionals without LC scheme. Finally, we scan μ parameter in LC scheme from 0.1 to 0.6 bohr^?1 for the above test set molecules with LC‐BOP functional and found our parameter value, 0.47 bohr^?1, is usefully applicable to our tested systems. © 2013 Wiley Periodicals, Inc.     

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.     

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.     

Molecular states of atoms. I. Orbital energy parameters

Atomic valence state energies are analyzed to obtain values of orbital energy parameters that may be used in semiempirical molecular orbital calculations. Difficulty in defining the interaction between orbitals with non-integer electron populations is systematically avoided by distinguishing between a valence state and a molecular state of an atom, only the latter state having non-integer spin paired orbital occupancy. Application of the virial theorem to the molecular state enables a value for the orbital kinetic energy to be obtained from the valence state orbital energy parameters once an arbitrary configuration is defined as reference. The orbitals then are eigenfunctions of the atomic Fock operator for that reference molecular state and, with their energy parameters, may be employed as a fixed basis set for molecular orbital calculations.     

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.     

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.     

The question of the completeness of the natural orbitals with nonzero occupation numbers for atoms and molecules

Summary The completeness of the natural orbitals with nonzero occupation numbers is examined for several model Hamiltonians and for the helium atom. It is demonstrated that whether the occupied natural orbitals form complete sets depends on the nature of the electron-electron interaction in the model Hamiltonian. Discrepancies in the extant proofs of the exactness and inexactness of the extended Koopmans' theorem are resolved.     

Interatomic interactions in momentum space. Momentum density and kinetic energy in chemical bonding

On the basis of the virial theorem for a uniform scaling process of a polyatomic system, the total energy and its gradient are quantitatively related with the behavior of the electron density in momentum space through the kinetic energy of the system. For attractive and repulsive interactions, the behavior of the momentum density distribution and its effect on the stabilization energy and the interatomic force are examined. Some guiding principles are deduced for their interrelation. The results are used to clarify the role of kinetic energy in chemical bonding. Possible energy partitioning in this approach is also mentioned.     

Presentation of a new index for estimation of aromaticity in halo‐ and cyanobenzenes: The role of potential energy in aromaticity

A linear correlation has been obtained between average values of Hamiltonian kinetic energy ( ) and potential energy ( ) calculated at the bond critical points using atoms in molecules method. This relation was used to introduce a new index ( ) for estimation of aromaticity in halo‐ and cyanobenzenes. Potential energy has different terms such as attraction between nuclei and electrons, also repulsion of electrons which affect the inertia and mobility of electrons, respectively. Therefore, contribution of potential energy in this relation must be controlled. Contribution of potential energy in aromaticity has been managed using a fitting parameter. This parameter was obtained by fitting the aromaticity stabilization energy data with values of aromaticity calculated by index for halo‐ and cyanobenzenes. The contribution of potential energy in index is complete when molecule is nonaromatic and is negligible when molecule is antiaromatic. Indeed, molecule is aromatic when contribution of potential energy in index lies between above limits. © 2013 Wiley Periodicals, Inc.     

Variational calculations for the energy levels of confined two-electron atomic systems

An accurate variational calculation has been performed for the ground-state-energy values of confined two-electron isoelectronic series from He to Ar¹⁶⁺. The confinement is obtained by embedding the ion in an overall charge neutral environment like that of a plasma. The confinement potential is chosen as that of a screened Coulomb potential between charges, obtained from a Debye model. The wave function is expanded in terms of product basis sets involving interparticle coordinates. The energy levels are found to be less bound with an increase of the screening parameter and ultimately become unstable. One- and two-particle moments have been calculated for the first time under such screening. The study is expected to throw new light on the behavior of the energy levels of foreign atoms embedded in an overall neutral environment which can be treated like a plasma. Received: 1 May 2002 / Accepted: 4 September 2002 / Published online: 6 November 2002 Acknowledgements. P.K.M. thanks the Max Planck Institute for Astrophysics, Garching, for financial support of his research visit to the institute where part of the work was performed. He also thanks the Council of Scientific and Industrial Research, Government of India, for research grant no. (03)/(0888)/99/EMR II. Correspondence to: P.K. Mukherjee e-mail: sppkm@mahendra.iacs.res.in     

Study of the kinetic energy densities of electrons as applied to quantum dots in a magnetic field

There are three expressions for the kinetic energy density t( r ) expressed in terms of its quantal source, the single-particle density matrix: t _A( r ) , the integrand of the kinetic energy expectation value; t _B( r ) , the trace of the kinetic energy tensor; t _C( r ) , a virial form in terms of the ‘classical’ kinetic field. These kinetic energy densities are studied by application to ‘artificial atoms‘ or quantum dots in a magnetic field in a ground and excited singlet state. A comparison with the densities for natural atoms and molecules in their ground state is made. The near nucleus structure of these densities for natural atoms is explained. We suggest that in theoretical frameworks which employ the kinetic energy density such as molecular fragmentation, density functional theory, and information-entropic theories, one use all three expressions on application to quantum dots, and the virial expression for natural atoms and molecules. New physics could thereby be gleaned.     

On the suitability of the virial equation for modeling the solubility of solids in supercritical fluids

Five model systems, the van der Waals fluid, the Soave-Redlich-Kwong fluid, the Peng-Robinson fluid, the hard-sphere fluid, and the square-well fluid, are used to examine the performance of the truncated virial expansion in describing the fugacity of a solute at infinite dilution in a solvent. It is demonstrated that the virial fugacity results deteriorate at significantly lower densities as the solute becomes larger. This has consequences for attempts to describe the solubility of solids in supercritical fluids, where the virial expansion, truncated after the third virial coefficient, has been considered as a modeling option. The results of this work suggest that, for the densities and solute-to-solvent size ratios commonly encountered in supercritical extraction, the truncated virial expansion should not be expected to describe correctly the solute fugacity, and therefore any success it has in fitting solubility data should be viewed with caution.     

Kinetic energy density for orbital‐free density functional calculations by axiomatic approach

An axiomatic approach is herein used to determine the physically acceptable forms for general D‐dimensional kinetic energy density functionals (KEDF). The resulted expansion captures most of the known forms of one‐point KEDFs. By statistically training the KEDF forms on a model problem of noninteracting kinetic energy in 1D (six terms only), the mean relative accuracy for 1000 randomly generated potentials is found to be better than the standard KEDF by several orders of magnitudes. The accuracy improves with the number of occupied states and was found to be better than for a system with four occupied states. Furthermore, we show that free fitting of the coefficients associated with known KEDFs approaches the exactly analytic values. The presented approach can open a new route to search for physically acceptable kinetic energy density functionals and provide an essential step toward more accurate large‐scale orbital free density functional theory calculations.     

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.     

The microarc as an emission source for atomic spectrometry

The potential of an atmospheric-pressure microarc as a primary emission source for multi-element determinations is described. Emission from Na, Li, K, In, Cd, Sn, Ba, W, and Cu was observed. Several of these elements were detected in subnanogram quantities. The feasibility of using a microarc as a detector for the gas chromatography of an organotin compound is demonstrated.     

Quantum electrodynamic effects in atomic structure

Summary In high-Z atoms, quantum electrodynamic (QED) corrections are an important component in the theoretical prediction of atomic energy levels. The main QED effects in electronic atoms are the one-electron self-energy and vacuum-polarization corrections which are well known. At the next level of precision, estimates of the effect of electron interactions on the self energy and higher-order effects in two exchanged photon corrections are necessary. These corrections can be evaluated within the framework of QED in the bound interaction picture. For high-Z few-electron atoms, this approach provides a rapidly converging series in 1/Z for the corrections, which is the generalization of the well-known relativistic 1/Z expansion methods. This paper describes recent work on the effect of electron interactions on the self energy. The QED effects are particularly important for the theory for lithiumlike uranium where an accurate measurement of the Lamb shift has been made, as well as for numerous other cases where systematic differences appear between theory that does not include these QED effects and experiment.