Electron-nuclear Attraction Energy Research Articles

The response electron–electron repulsion energy and energy component analysis in CC/MBPT methods

Molecular properties are computed as responses to perturbations (energy derivatives) in coupled-cluster (CC)/many-body perturbation theory (MBPT) models. Here, the CC/MBPT energy derivative with respect to a general two-electron (2-e) perturbation is assembled from gradient theory for 2-e property evaluation, including the electron repulsion energy. The correlation energy (∆E) is shown to be the sum of response kinetic (∆T), electron–nuclear attraction (∆V), and electron repulsion (∆V ee ) energies. Thus, evaluation of total V ee for energy component analysis is simple: For total energy (E), total 1-e responses T and V, and nuclear–nuclear repulsion energy (V NN ), V ee = E − V NN − T − V is the true 2-e response value. Component energy analysis is illustrated in an assessment of steric repulsion in ethane’s rotational barrier. Earlier SCF-based results (Bader et al. in J Am Chem Soc 112:6530, 1990) are corroborated: The higher-energy eclipsed geometry is favored versus staggered in the two repulsion energies (V NN and V ee ), while decisively disfavored in electron–nuclear attraction energy (V). Our best quality calculations (CCSD/cc-pVQZ) attain practical Virial Theorem compliance (i.e., agreement among the kinetic energy, potential energy, and total energy representations) in assigning 2.70 ± 0.06 to the barrier height; −195.80 kcal/mol is assigned to the drop in “steric” repulsion upon going to the eclipsed geometry. Steric repulsion is not responsible for any fraction of the ~3 kcal/mol barrier.

Kinetic and electron-electron energies for convex sums of ground state densities with degeneracies and fractional electron number.

Properties of exact density functionals provide useful constraints for the development of new approximate functionals. This paper focuses on convex sums of ground-level densities. It is observed that the electronic kinetic energy of a convex sum of degenerate ground-level densities is equal to the convex sum of the kinetic energies of the individual degenerate densities. (The same type of relationship holds also for the electron-electron repulsion energy.) This extends a known property of the Levy-Valone Ensemble Constrained-Search and the Lieb Legendre-Transform refomulations of the Hohenberg-Kohn functional to the individual components of the functional. Moreover, we observe that the kinetic and electron-repulsion results also apply to densities with fractional electron number (even if there are no degeneracies), and we close with an analogous point-wise property involving the external potential. Examples where different degenerate states have different kinetic energy and electron-nuclear attraction energy are given; consequently, individual components of the ground state electronic energy can change abruptly when the molecular geometry changes. These discontinuities are predicted to be ubiquitous at conical intersections, complicating the development of universally applicable density-functional approximations.

Ab initio MCSCF study on several azide molecules: energy component analysis of the pseudo-Jahn–Teller effect

An exploration of the most stable geometrical structures for the azide molecules R–N3 (R = H, BeH, BH2, CH3, C6H5) has been carried out by using the ab initio multi-configuration self-consistent field (MCSCF) method together with the aug-cc-pVTZ basis sets. The unknown hydridoberyllium azide (R = BeH) is predicted to suffer no pseudo-Jahn–Teller (JT) distortion and, hence, to assume a linear structure of C∞v symmetry. In contrast, each of the others is found to adopt a trans bent Cs structure with regard to an interior NN bond. To elucidate the nature of the pseudo-JT distortions, energy component analyses have been carried out at all of the stationary points for the relevant molecules. The results show that pseudo-JT stabilizations are classified into two groups, one in which the stability of the bent structure about the azido group arises from lowering of the interelectronic and internuclear repulsion energies and the other in which the stability results from lowering of the electron-nuclear attraction energy. These energy behaviors are accounted for in terms of an elongation or a contraction of the heavy-atom skeleton and a charge migration from some part of the molecule to the other ones. Moreover, it is shown that the theoretical structural characteristics for the known molecules are in good agreement with available experimental facts.

ChemInform Abstract: Energy Component Analysis of the Jahn-Teller Effect in the Methane Radical Cation.

Extensive configuration‐interaction calculations with double‐zeta plus polarization and near triple‐zeta plus polarization basis sets are used to analyze the Jahn–Teller (JT) effect in the methane radical cation. Energy component analysis shows that the Jahn–Teller effect leads to a decrease in the expectation value of the electron‐nuclear attraction energy, an increase in the expectation value of the interelectronic repulsion energy, and an increase in the internuclear repulsion energy. These observations are consistent with a contraction of the electron cloud. The dominant factor in the −0.0550 hartree Jahn–Teller distortion (Td→C2v) in CH+4 is the −0.5262 hartree change in the electron–nuclear attraction energy. The differences in all energy components are large in relation to the JT distortion. Interelectronic repulsion plays a dominant role in determining the relative energies of the possible JT distorted structures, but electron correlation effects are relatively unimportant.

Properties and applications of the average interparticle distance

The first and second moment operators are used to define the origin invariant shape and size of a molecule or functional group, as well as expressions for the distance between two electrons and the distance between an electron and a nucleus. The measure of molecular size correlates quite well with an existing theoretical measure of molecular volume calculated from isodensity contours. Also, the measure of size is effective in predicting steric effects of substituents which have been measured experimentally. The electron-electron and electron-nuclear distances are related to components of the Hartree-Fock energy. The average distance between two-electrons can model the Coulomb energy quite well, especially in the case of localized molecular orbitals. The average distance between an electron and a nucleus is closely related to the electron-nuclear attraction energy of a molecule.

Using PM3 Hamiltonian, factor analysis and regression analysis in developing quantitative structure-property relationships for photohydrolysis quantum yields of substituted aromatic halides

Based on the descriptors computed by the PM3 hamiltonian and the application of factor analysis and multiple regression analysis, several Quantitative Structure-Property Relationship (QSPR) equations for disappearance quantum yields of 45 substituted halogen compounds were obtained. After the exclusion of the outliers, the remaining two QSPR equations can be used to predict disappearance quantum yields for compounds with similar structure. By stepwise variable selection regression analysis, two significant factors relating to the disappearance quantum yields of the compounds were identified: the energy of the lowest unoccupied molecular orbital and the electron-nuclear attraction energy of the two-center term for carbon-halogen bonds. By factor analysis, other factors that may relate to the quantum yields of the compounds were obtained. The quantum chemical descriptors used in the study can be determined by calculation from any given structural formula. So the method used in this study permits near a priori prediction of environmental properties for compounds for which the empirical parameters are not readily available.

Ab initio and density functional study of the Jahn-Teller distortion in the silane radical cation

An ab initio and density functional study is made of the Jahn-Teller distortion of the silane radical cation using Dunning's cc-pVTZ basis set. For this cation, the D 2d, C 3v, C 2v and C s structures are fully optimized and analyzed using an energy component analysis. The distortion towards the lowest energy geometry results in an increase in kinetic energy and a decrease in the electron-nuclear attraction energy, electron-electron repulsion energy and nuclear-nuclear repulsion. The exact exchange density functional methods B3LYP and B3PW91 turn out to be the methods best suited for the study of geometries, energies and vibrational spectra of these cations.

Electronic structures and bonding in Rydberg molecules: NH4, H3O and related molecules

The electronic structures of the Rydberg and related molecules belonging to the series [Formula: see text], NH4, and [Formula: see text] and H3O+, H3O, and H3O− have been investigated by analysis of the electron densities generated by the Hartree–Fock method (HF), the quadratic configuration interaction method including single and double excitations (QCISD), the coupled-cluster method including double excitations (CCD), and the density functional method with the Becke–Perdew gradient correction functionals (DFT-BP). Adding an electron to the stable cation core ([Formula: see text] and H3O+) results in an increase in the bond length, a larger increase in the distance of the bond critical point (BCP) from the nitrogen atom than from the hydrogen atom, a decrease in the electron density at the bond critical point, and an increase of the Laplacian of the electron density at the BCP. These trends are consistent with a decrease in the electronegativity of the central atom and a weakening of the bond as electrons are added to diffuse Rydberg orbitals. There is little effect, however, when a second electron is added to form the double Rydberg molecules ([Formula: see text] and H3O−). For NH4, the preferred geometry corresponds to a high atomic energy associated with the hydrogen basin, and to a low kinetic energy and a high potential energy in the basin. The high atomic energy associated with the hydrogen basin is attributed to the dominant contribution of the electron–nuclear attraction energy. Similar effects are observed in the Jahn–Teller distorted molecules: [Formula: see text] and [Formula: see text]. The DFT-BP method yields results in qualitative agreement with the results of the QCISD and CCD methods; however, it underestimates the bond strengths of the Rydberg molecules relative to high-level ab initio methods.

Energy Component Analysis of the Pseudo Jahn-Teller Effect in Pentalene and Heptalene

Abstract An energy component analysis of the pseudo Jahn-Teller effect from D2h to C2h within the PPP-type HF approximation reveals that the energetic stability for heptalene arises essentially from a lowering of the electron-nuclear attraction energy, while that for pentalene arises entirely from a decrease in the interelectronic repulsion energy.

The first-order Jahn–Teller distortion and the symmetry of the electron density in the BH+3 radical

Self-consistent field (SCF) and complete active space multiconfiguration (CAS-MCSCF) calculations with extended basis sets are used to show that the electron–nuclear attraction energy is the dominant contributor to the total energy lowering associated with the Jahn–Teller distortion of BH+3. Furthermore the Jahn–Teller distortion raises the interelectronic and internuclear repulsion energies. These observations are consistent with a contraction of the molecular framework and electron cloud. In the context of the relationship between the electron distribution and the Jahn–Teller distortion direction, it is observed that the electron density associated with the canonical highest occupied molecular orbital (HOMO) of the D3h constrained BH3 radical has C2v symmetry, which corresponds to the preferred point group of the Jahn–Teller distortion in BH+3. The effect of electron correlation on the electron density distribution is shown to be inconsequential for the prediction of the direction of the Jahn–Teller distortion.

Singlet-triplet energy component differences in homonuclear diatomics: A multi-reference configuration interaction study of Hund's rule

Multireference configuration interaction calculations for the low-lying electronic states of C 2 (a 3Π u and A 1Π u) and O 2 (X 3Σ g −, a 1Δ g) over a range of internuclear separations reveal that the singlet-triplet electron-electron repulsion energy differences tend to have the same sign as the singlet-triplet kinetic energy differences, while the corresponding electron-nuclear attraction energy differences tend to have the opposite sign at a given geometry. Adiabatic energy component differences for C 2 and O 2 show that the electron-electron repulsion in the triplet state is greater than in the singlet state, while there is more electron-nuclear attraction in the triplet than in the singlet state. The latter observations are consistent with the results for atomic states arising from the same configuration, but are contrary to the traditional explanation of Hund's rule.

Energy component analysis of the Jahn–Teller effect in the methane radical cation

Extensive configuration-interaction calculations with double-zeta plus polarization and near triple-zeta plus polarization basis sets are used to analyze the Jahn–Teller (JT) effect in the methane radical cation. Energy component analysis shows that the Jahn–Teller effect leads to a decrease in the expectation value of the electron-nuclear attraction energy, an increase in the expectation value of the interelectronic repulsion energy, and an increase in the internuclear repulsion energy. These observations are consistent with a contraction of the electron cloud. The dominant factor in the −0.0550 hartree Jahn–Teller distortion (Td→C2v) in CH+4 is the −0.5262 hartree change in the electron–nuclear attraction energy. The differences in all energy components are large in relation to the JT distortion. Interelectronic repulsion plays a dominant role in determining the relative energies of the possible JT distorted structures, but electron correlation effects are relatively unimportant.

Application of potential constants: Empirical determination of molecular energy components for diatomic molecules—II

Abstract The harmonic and anharmonic potential (force) constants which are usually available from a normal coordinate analysis for a diatomic molecule are applied to determine the molecular energy components such as the electronic kinetic energy, the total electrostatic potential energy, the electron—nuclear attraction energy, the electron—electron repulsion energy, the nuclear—nuclear repulsion energy, and the Hartree—Fock eigen-value sum. The method developed here is based on an inhomogeneous linear third-order differential equation derived from the quantum mechanical virial theorem, and a homogeneity hypothesis of the molecular energy with respect to the atomic number. To confirm the utility of the method, the calculated molecular energy components of diatomic molecules are compared with available Hartree—Fock data. It is concluded from this data that the present method is simple and useful for evaluating the molecular energy components of various diatomic molecules.

Reduction of derivatives and simple applications of the Legendre transformed density functional theory

A reduction of derivatives in the Legendre transformed density functional theory (DFT) of the single-component (atomic or fixed-nuclei molecular) systems is described. It is shown that a given derivative can be expressed by an identity in terms of a set of basic derivatives. The basic partial functional derivatives for the general case of N electrons in the external field v(r) are: α(r)=[∂ρ(r)/∂N]/ρ(r), β(r, r′)=−[∂ρ(r)/∂v(r′)]N/ρ(r), and γ(r, r′)=−[∂ρ(r)/∂v(r′)]μ/ρ(r); ρ is the density and μ denotes the chemical potential. For atoms, an additional alternative set of four integral derivatives, involving only global (referring to a system as a whole) parameters, is introduced: α̃=−[∂vne/∂N]Z/vne, β̃=−[∂vne /∂Z]N/vne, γ̃=[∂N/∂Z]μ/N, and χ̃=−[∂vne/∂Z]μ/vne; Z is the nuclear atomic number and vne is the electron-nuclear attraction energy per unit nuclear charge. The reduction procedure makes free use of the DFT ‘‘thermodynamic’’ diagram equations, Maxwell relations, and mathematical manipulations on functional derivatives. Several representative applications are indicated dealing with the system responses to slight changes in one of the system parameters subject to different constraints.

Hund's rules and the interpretation and common misinterpretation of energy differences

With a Z expansion formalism which is exact in non-relativistic quantum mechanics it has been shown that for multiplets of neutral atoms and of many positive ions the state of the highest energy has the lowest expectation value for the interelectronic repulsion energy. This reversed order of the repulsion energy occurs for cases which obey Hund's first rule as well as for cases which obey the second of Hund's rules. It can be shown that the energy differences are in all cases dominated by the difference in electron nuclear attraction energy and not by the difference in electron repulsion energy. The lowest 1Π u and 3Π u states of the H2 molecule have similar features. At many internuclear distances, including the equilibrium ones, the 1Π u state has the highest energy but the lower kinetic energy and electron repulsion energy but again the higher electron nuclear attraction energy. These results contradict clearly the usual theories for energy differences between spectroscopic states in atoms and molecules.

Singlet-triplet differences in the lowest Πustates of H2

A study of the energy differences between the 1 Pi u and 3 Pi u states of H2 has been undertaken, and differences in kinetic energies, electron-nuclear attraction energies, and interelectronic repulsion energies are also given for a range of internuclear separations 0.5 au<or=R<or=5.0 au. Wavefunctions for both states were calculated as part of the study, using a basis set of twenty Slater-type orbitals to do a self-consistent-field calculation followed by configuration interaction. Optimized exponents were used for the most important orbitals at each internuclear separation. It has been found that the interelectronic repulsion energy of the triplet is greater than that of the singlet, and that the triplet lies lower in energy because of its greater electron-nuclear attraction energy.

Hund's rule and theZexpansion for the energy, electron repulsion and electron nuclear attraction

A Z expansion is derived for the expectation values of the interelectronic repulsion energy and the electron-nuclear attraction energy in atomic states. The formalism, which is based on a non-relativistic hamiltonian, is applied to excited states in iso-electronic sequences. It appears that there are many cases where the state of the highest multiplicity has the larger expectation value for the interelectronic repulsion energy although it has a lower total energy.

Comments on the Analysis of Atomic Correlation Energies

Analysis of correlation energies is advocated in terms of average values of electronic kinetic energy, electron–nuclear attraction energy, and electron–electron repulsion energy, rather than pair energies. Exploratory calculations on the two-electron and three-electron isoelectronic sequences are reported.