Nuclear Attraction Energy Research Articles

Coulomb, exchange, kinetic, nuclear attraction and ionization energies of two-electron quantum dot with finite confinement potential

We have calculated Coulomb, exchange and configuration-average ionization energies for the ground and excited configurations in helium atom confined by a spherical cavity of finite barrier height. In addition, we have carried out kinetic energies, nuclear attraction energies, confining potential energies and probabilities of finding the electrons that occupy different subshells. In the strong and intermediate confinement regions, the results show that the barrier height of potential well and dot size have a strong influence on the energies and probabilities. The kinetic energies of electrons in 1s and nl orbitals increase as the height of the potential barrier increases, but in nuclear attraction energies the opposite behavior occurs. It is found that Coulomb and exchange energies rapidly decrease after the critical confinement radii, in which the electron in nl orbital escapes out of the well. The configuration-ionization confinement radius depends strongly on the potential barrier height.

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.

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.

Pseudo-Jahn−Teller Distortion from Planarity in Heterocyclic Seven- and Eight-Membered Ring Systems with Eight π Electrons

To elucidate the nature of a pseudo-Jahn−Teller (JT) distortion from planarity, ab initio MCSCF calculations with the 6-31G(d,p) basis sets were carried out in the ground states of seven- and eight-membered unsaturated heterocycles with eight π electrons. The monocyclic systems examined were found to show a variety of structural changes. Azepine and oxepin undergo the pseudo-JT distortions from a planar C2v to a boat Cs conformation. In 1,5-diazocine, the pseudo-JT distortion takes place in two stages, the initial step being from a symmetric planar D2h to a skew C2h structure and the subsequent step from the skew C2h to a tub C2 structure. The 1,3,5,7-tetrazocine molecule undergoes pseudo-JT distortions from a symmetric planar D4h to skew C4h and crown-like D2d structures through the respective in-plane and out-of-plane nuclear deformations. Moreover, the C4h and D2d structures are distorted into the same tub-shaped S4 conformation. An inspection of the energy components comprised in the total energy reveals that the stability of nonplanar structures arises from a lowering of the electron−nuclear attraction energy. The energy variation is accounted for in terms of an electrostatic interaction and the proximity among the nuclei and electron clouds owing to a folding of the molecular skeleton. It is shown that the theoretical structural characteristics for azepine, oxepin, 1,5-diazocine, and 1,3,5,7-tetrazocine are in good agreement with available experimental facts.

The electron–electron interaction in the Douglas–Kroll–Hess approach to the Dirac–Kohn–Sham problem

Incorporation of relativistic effects into all-electron density functional (DF) calculations via scalar relativistic corrections to the kinetic energy and the nuclear attraction energy has become rather common. On the other hand, a self-consistent treatment of the spin–orbit interaction and relativistic corrections to the electron–electron interaction (or both simultaneously) requires substantially more computational effort. We present an extension of the Douglas–Kroll approach to the Dirac–Kohn–Sham problem that accounts for relativistic corrections to the Hartree potential and permits a self-consistent treatment of spin–orbit interaction. To construct computationally efficient approximations, we exploit the electron charge density fitting scheme with an auxiliary basis set. These approximate schemes introduce effects of the relativistic transformation of the Hartree part of the electron–electron interaction, but leave the (smaller) exchange-correlation contributions untransformed. These approximations were implemented in the parallel DF program PARAGAUSS. Quantitative effects of the new relativistic DF procedures were illustrated for the spin-orbit splittings of the Kohn–Sham levels in the Hg atom and the g-tensor shifts of NO2 where we show the value of new scheme. We also studied how properties of the diatomic molecules TlH, PbO, Pb2, and Bi2 change due to the improved treatment of relativistic effects.

A fuzzy ARTMAP based on quantitative structure-property relationships (QSPRs) for predicting aqueous solubility of organic compounds.

Quantitative structure-property relationships (QSPRs) for estimating aqueous solubility of organic compounds at 25 degrees C were developed based on a fuzzy ARTMAP and a back-propagation neural networks using a heterogeneous set of 515 organic compounds. A set of molecular descriptors, developed from PM3 semiempirical MO-theory and topological descriptors (first-, second-, third-, and fourth-order molecular connectivity indices), were used as input parameters to the neural networks. Quantum chemical input descriptors included average polarizability, dipole moment, resonance energy, exchange energy, electron-nuclear attraction energy, and nuclear-nuclear (core-core) repulsion energy. The fuzzy ARTMAP/QSPR correlated aqueous solubility (S, mol/L) for a range of -11.62 to 4.31 logS with average absolute errors of 0.02 and 0.14 logS units for the overall and validation data sets, respectively. The optimal 11-13-1 back-propagation/QSPR model was less accurate, for the same solubility range, and exhibited larger average absolute errors of 0.29 and 0.28 logS units for the overall and validation sets, respectively. The fuzzy ARTMAP-based QSPR approach was shown to be superior to other back-propagation and multiple linear regression/QSPR models for aqueous solubility of organic compounds.

Ab Initio MCSCF Study on Eight π-Electron Heterocyclic Conjugated Systems: Energy Component Analysis of the Pseudo-Jahn−Teller Distortion from Planarity

To clarify the origin of nonplanarity in eight π-electron heterocyclic conjugated systems, an energy component analysis was carried out for the ground states by using ab initio MCSCF method with 6-31++G(d,p) basis set. Inspection of the energy components comprised in the total energy reveals that the type of pseudo-JT stabilization is classified into two groups, one in which the stability of nonplanar structure arises from a lowering of the interelectronic and internuclear repulsion energies and the other in which the stability results from a lowering of the electron−nuclear attraction energy. This sharp distinction in energy changes is accounted for in terms of an expansion or contraction of the molecular skeleton and a proximity among the nuclei and the electron clouds owing to a folding of the six-membered ring. Besides, it is shown that the theoretical structural characteristics for 1,2-dihydro- and 1,4-dihydro-1,2,4,5-tetrazines are in good agreement with available experimental facts.

Kinetic and potential energy of isoelectronic atomic ions from density functional theory compared with exact values

The performance of the Fock exchange and the two most common Becke exchange functionals, B88 and B3, in combination with Perdew and Wang's correlation functional (PW91) has been studied for highly charged atomic ions. The study focuses on the accuracy of the components of total energy: kinetic and potential energy. It is found that Becke's exchange functionals introduce a significant error in the kinetic and nuclear attraction energy for most ions studied, and the use of these exchange functionals might lead to inaccurate momentum distributions when their Kohn-Sham orbitals are transformed to momentum space.

Stationary direct perturbation theory of relativistic corrections.

The stationary variant of direct perturbation theory of relativistic effects is presented. In this variant neither the unperturbed (nonrelativistic) equation nor the equations for the relativistic corrections are solved exactly, but each of them is replaced by the condition that a certain functional becomes stationary. Let ${\mathrm{\ensuremath{\psi}}}_{0}$=(${\mathrm{\ensuremath{\Phi}}}_{0}$,${\mathrm{\ensuremath{\chi}}}_{0}$) be the four-component spinor with modified metric in the nonrelativistic limit and \ensuremath{\psi}${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{2}$=(${\mathrm{\ensuremath{\Phi}}}_{2}$,${\mathrm{\ensuremath{\chi}}}_{2}$) the leading relativistic correction of O(${\mathit{c}}^{\mathrm{\ensuremath{-}}2}$), then one can define functionals ${\mathit{F}}_{0}$(${\mathrm{\ensuremath{\Phi}}}_{0}$,${\mathrm{\ensuremath{\chi}}}_{0}$) and ${\mathit{F}}_{4}$(${\mathrm{\ensuremath{\Phi}}}_{2}$,${\mathrm{\ensuremath{\chi}}}_{2}$) called respectively the L\'evy-Leblond and the Rutkowski-Hylleraas functional, such that stationarity of ${\mathit{F}}_{0}$ with respect to variation of ${\mathrm{\ensuremath{\Phi}}}_{0}$ and ${\mathrm{\ensuremath{\chi}}}_{0}$ determines ${\mathrm{\ensuremath{\Phi}}}_{0}$ and ${\mathrm{\ensuremath{\chi}}}_{0}$, and stationarity of ${\mathit{F}}_{4}$ with respect to variation of ${\mathrm{\ensuremath{\Phi}}}_{2}$ and ${\mathrm{\ensuremath{\chi}}}_{2}$ determines ${\mathrm{\ensuremath{\Phi}}}_{2}$ and ${\mathrm{\ensuremath{\chi}}}_{2}$. The unperturbed (i.e., nonrelativistic) energy ${\mathit{E}}_{0}$ as well as the leading relativistic correction ${\mathit{c}}^{\mathrm{\ensuremath{-}}2}$${\mathit{E}}_{2}$ are expressible through ${\mathrm{\ensuremath{\Phi}}}_{0}$ and ${\mathrm{\ensuremath{\chi}}}_{0}$ while for the next higher corrections ${\mathit{c}}^{\mathrm{\ensuremath{-}}4}$${\mathit{E}}_{4}$ and ${\mathit{c}}^{\mathrm{\ensuremath{-}}6}$${\mathit{E}}_{6}$, ${\mathrm{\ensuremath{\Phi}}}_{2}$ and ${\mathrm{\ensuremath{\chi}}}_{2}$ are also needed. Either of the two functionals ${\mathit{F}}_{0}$ and ${\mathit{F}}_{4}$ can be decomposed into two contributions, the error of one of which is \ensuremath{\ge}0 while that of the other is \ensuremath{\le}0. An upper-bound property is obtained if the error of the second part vanishes. A strict variation perturbation theory requires that the approximate \ensuremath{\Phi}${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{2}$ and \ensuremath{\chi}${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{2}$ reproduce the behavior of the exact ${\mathrm{\ensuremath{\Phi}}}_{2}$ and ${\mathrm{\ensuremath{\chi}}}_{2}$ near a nucleus, which implies terms in ln r. If one regularizes \ensuremath{\Phi}${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{2}$ one must also regularize \ensuremath{\chi}${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{2}$; otherwise ${\mathit{E}}_{6}$ diverges. If one regularizes both ${\mathrm{\ensuremath{\Phi}}}_{2}$ and ${\mathrm{\ensuremath{\chi}}}_{2}$ in the sense of a kinetic balance, one gets regular results for ${\mathit{E}}_{4}$ and ${\mathit{E}}_{6}$, but one loses the strict upper-bound property. The Breit-Pauli expression for ${\mathit{E}}_{2}$ is shown to be correct only if the nonrelativistic wave equation has been solved exactly. Otherwise there is an extra term. Finally the question as to which extent some of the singularities in the perturbation theory of relativistic effects might be artifacts due to the unphysical assumption of a point nucleus is discussed. It is shown, however, that these singularities are not removed if one uses realistic extended nuclei. For all atoms, the critical radius ${\mathit{r}}_{\mathit{c}}$ inside of which the nuclear attraction energy is larger than the rest energy of the electron is larger than the extension of the nucleus. \textcopyright{} 1996 The American Physical Society.

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.

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.

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.

A 1/Z-expansion analysis of singlet-triplet differences in the lowest1,3Πstates of H2, HeH+, and LiH++

Energy differences, and differences in kinetic energies, nuclear attraction energies and interelectronic repulsion energies are given for the lowest 1,3 Pi states of H2, HeH+, and LiH++, and an analysis of the results in terms of 1/Z-expansions is presented. If Z is used to denote the charge of the nucleus of higher atomic number, then it is found that at low Z the difference in interelectronic repulsion is dominated by higher-order terms, and that the often-quoted statement that the singlet-triplet energy difference is twice an exchange integral is true only for high values of the nuclear charge Z.

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.

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.

Quantum-Mechanical Study of Some Three-Center Two-Electron Systems. II. A Natural-Orbital, Electron-Population, and Energy Analysis

Configuration-interaction wavefunctions for a series of pseudomolecular ions of the form ZHZ+2Z−1, where Z = 1.0 (0.4) 2.2, were reformulated in terms of natural orbitals. Consequently, changes in the electron density could be investigated, as a function of Z and the bond angle ZĤZ, by means of the population analysis of Mulliken. Contour diagrams were also determined for the charge density evaluated in the plane of the molecule. The total energy for each system was analyzed in terms of the kinetic energy, nuclear attraction energy, and the electron and nuclear repulsion energies. The results are presented graphically as a function of the bond angle θ for each value of Z. For Z = 1.0, i.e., H3+, the occupation numbers for the natural orbitals revealed the united atom character of this system, and the contour diagrams indicated that, in the equilibrium configuration, the “bonds” were directed from each nucleus towards the centroid of the positive charges. As θ was increased beyond 60°, it was found that the “central” proton within H3+ began to dominate the system. For Z = 2.2, the ions approach most closely to their separated systems of minimum energy as θ tends to 180°. The contour diagrams and the electron-population analysis also show that the ZHZ+2Z−1 ions change from a three-center to, basically, a two-center system and a strongly perturbing proton as the values for Z and θ increase and decrease, respectively. The θ dependence of the kinetic energy is perhaps the most interesting of the energy curves. By associating a decrease in kinetic energy with an increase in spatial freedom for the electrons, it was possible to understand the θ variations of all the energy components for each value of Z. The present examination has allowed us to observe, in detail, changes which occur in the electron density and energy components for a series of three-center two-electron systems when θ and Z are allowed to vary.