Electron correlation effects on the shape of the Kohn–Sham molecular orbital

Even systems in which strong electron correlation effects are present, such as the large near-degeneracy correlation in a dissociating electron pair bond exemplified by stretched H₂, are represented in the Kohn–Sham (KS) model of non-interacting electrons by a determinantal wavefunction built from the KS molecular orbitals. As a contribution to the discussion on the status and meaning of the KS orbitals we investigate, for the prototype system of H₂ at large bond distance, and also for a one-dimensional molecular model, how the electron correlation effects show up in the shape of the KS σ _g orbital. KS orbitals φ^HL and φ^FCI obtained from the correlated Heitler-London and full configuration interaction wavefunctions are compared to the orbital φ^LCAO, the traditional linear combination of atomic orbitals (LCAO) form of the (approximate) Hartree-Fock orbital. Electron correlation manifests itself in an essentially non-LCAO structure of the KS orbitals φ^HL and φ^FCI around the bond midpoint, which shows up particularly clearly in the Laplacian of the KS orbital. There are corresponding features in the kinetic energy density t _s of the KS system (a well around the bond midpoint) and in the one-electron KS potential v _s (a peak). The KS features are lacking in the Hartree-Fock orbital, in a minimal LCAO approximation as well as in the exact one. Received: 11 December 1996 / Accepted: 10 January 1997     

Nodal surfaces of the wave functions of the hydrogen molecule in the triplet state 3Σu +

For molecular hydrogen in the triplet state ³Σ_u ⁺, the nodal surfaces of the wave function corresponding to the minimum basis set of Slater orbitals in the Hartree—Fock approximation and those of the wave function used in calculations by the diffusion quantum Monte Carlo method were plotted and analyzed. Taking account of the condition for antisymmetrical wave function of the triplet state ³ S of He atom, the Hartree-Fock approximation in the minimum basis set of one-electron orbitals is inappropriate for a priori determination of the nodal surfaces of many-electron wave functions (MWF). An MWF quantum chemical method developed by the authors is outlined. The alternative nodal surfaces for H₂ (³Σ_u ⁺) a priori specified in this method are presented.     

The Potential Energies of Cohesion of a Crystalline Organic Enantiomer and the Racemate; on the Energy for the Liquid Racemate [1]

Summary. The cohesion potential energy of the crystal of one enantiomer of ethyl 3-cyano-3-(3,4-dimethyloxyphenyl)-2,2,4-trimethylpentanoate, −47.7 ± 0.1 kJ mol⁻¹ (0–90°C), was found out from the heat of sublimation (123.2 ± 5.1 kJ mol⁻¹, 78.6°C) and the kinetic energies for the gas phase and the crystal. It was found that the entropy function of Debye’s theory of solids mathematically agreed with the vibrational entropy of the gas (variationally obtained), allowing to disclose the vibrational energy using the Debye energy function (E _vib 835.0 kJ mol⁻¹ (78.6°C), E ⁰ included). E _kin for the crystal (771.1 kJ mol⁻¹ (78.6°C)) was obtained by Debye’s theory with the experimental heat capacity. The cohesion energy represented a moderate part of the sublimation energy. The cohesion energy of the racemic crystal, −44.2 kJ mol⁻¹, was obtained by the heat of formation of the crystal in the solid state (3.0 kJ mol⁻¹, 83.3°C) and E _kin for the crystal (by Debye’s theory). The decrease in cohesion on formation of the crystal accounted for the energy of formation. The change in potential energy on liquefaction of the racemate from the gas state was disclosed obtaining added-up E _{vib + rot} for the liquid in the way as to E _vib for the gas, the Debye entropy function being increasedly suited for the liquid (E _{vib + rot} 763.4 kJ mol⁻¹ (115.4°C)). Positive ΔE _pot, 13.0 kJ mol⁻¹, arised from the increase in electronic energy (Δ _l ν_mean − 154.3 cm⁻¹, by the dielectric nature of the liquid), added to the cohesion energy.     

The Potential Energies of Cohesion of a Crystalline Organic Enantiomer and the Racemate; on the Energy for the Liquid Racemate [1]

The cohesion potential energy of the crystal of one enantiomer of ethyl 3-cyano-3-(3,4-dimethyloxyphenyl)-2,2,4-trimethylpentanoate, −47.7 ± 0.1 kJ mol⁻¹ (0–90°C), was found out from the heat of sublimation (123.2 ± 5.1 kJ mol⁻¹, 78.6°C) and the kinetic energies for the gas phase and the crystal. It was found that the entropy function of Debye’s theory of solids mathematically agreed with the vibrational entropy of the gas (variationally obtained), allowing to disclose the vibrational energy using the Debye energy function (E _vib 835.0 kJ mol⁻¹ (78.6°C), E ⁰ included). E _kin for the crystal (771.1 kJ mol⁻¹ (78.6°C)) was obtained by Debye’s theory with the experimental heat capacity. The cohesion energy represented a moderate part of the sublimation energy. The cohesion energy of the racemic crystal, −44.2 kJ mol⁻¹, was obtained by the heat of formation of the crystal in the solid state (3.0 kJ mol⁻¹, 83.3°C) and E _kin for the crystal (by Debye’s theory). The decrease in cohesion on formation of the crystal accounted for the energy of formation. The change in potential energy on liquefaction of the racemate from the gas state was disclosed obtaining added-up E _{vib + rot} for the liquid in the way as to E _vib for the gas, the Debye entropy function being increasedly suited for the liquid (E _{vib + rot} 763.4 kJ mol⁻¹ (115.4°C)). Positive ΔE _pot, 13.0 kJ mol⁻¹, arised from the increase in electronic energy (Δ _l ν_mean − 154.3 cm⁻¹, by the dielectric nature of the liquid), added to the cohesion energy.     

Magnetic properties of Cu(II) bischelate complexes with 3-imidazoline nitroxides. 2. Anab initio analysis of exchange interaction mechanisms

The mechanism responsible for the emergence of ferromagnetic exchange interactions in bischelate complexes of Cu²⁺ with enaminoketone derivatives of 3-imidazoline nitroxide CuL₂ is studied by ab initio quantum chemical methods. The parameters J_cu-L and J_L-L’ of exchange interactions between the unpaired electrons of the paramagnetic centers (Cu²⁺ ion and N-O groups of nitroxyl ligands L and L’) of these complexes were calculated in terms of the full 3x3 configuration interaction between the singlet states constructed in a basis set of molecular orbitals of unpaired electrons. It is shown that for variations of the structure of the coordination polyhedron around the Cu²⁺ ion from square planar to tetrahedral the exchange interactions between the unpaired electrons of the paramagnetic centers is ferromagnetic J_Cu-L >J_L -L’>0, which agrees with the data of magnetic measurements. The principal mechanism of exchange interactions in CuL₂ complexes is the delocalization mechanism that is due to a minor transfer of spin density from the 3d-orbitals of Cu²⁺ to the Σ-orbitals of the N-O groups of L and L’ ligands. Translated fromZhurnal Strukturnoi Khimii, Vol. 38, No. 5, pp. 850–856, September-October, 1997.     

An energy decomposition scheme applicable to strongly interacting systems

An energy decomposition scheme useful for the analysis of the coupled types of interactions in strongly interacting systems is developed within the Hartree-Fock approximation. A dominant characteristic of the scheme is that it involves the interactions between vacant orbitals of component molecules, as can be justified from the third-order perturbation theory. On the basis ofab initio molecular orbital calculations, the utility of the scheme is illustrated for the BH₃-NH₃ complexation and the S_N2 reaction of CH₄ with H^–. It is found that the charge transfer from electron donor (i.e. NH₃ or H^–) to acceptor (i.e. BH₃ or CH₄) is strongly coupled with the polarization of the acceptor, to contribute appreciably to the stabilization of the entire system. A specific role of this coupling mode in the progress of reactions is discussed.     

Second-order Møller–Plesset perturbation theory with terms linear in the interelectronic coordinates and exact evaluation of three-electron integrals

 The second-order correlation energy of M?ller–Plesset perturbation theory is computed for the neon atom using a wave function that depends explicitly on the interelectronic coordinates (MP2-R12). The resolution-of-identity (RI) approximation, which is invoked in the standard formulation of MP2-R12 theory, is largely avoided by rigorously computing the necessary three-electron integrals. The basis-set limit for the second-order correlation energy is reached to within 0.1 mE _h. A comparison with the conventional RI-based MP2-R12 method shows that only three-electron integrals over s and p orbitals need to be computed exactly, indicating that the RI approximation can be safely used for integrals involving orbitals of higher angular momentum. Received: 9 May 2001 / Accepted: 31 October 2001 / Published online: 9 January 2002     

Dispersion interaction of conjugated molecules: One- and many-electron models

A scheme for calculating van der Waals constants C₆, C₈, and C₁₀ in terms of the full configuration interaction for π-shells is suggested. The required imaginary frequency dynamic polarizabilities are also investigated with one-electron schemes. The variational perturbation theory of the Hartree-Fock method considerably exaggerates the constant C_2l the conjugated variant of perturbation theory is preferable. The additive scheme for estimating the Σ-contribution allows one to calculate the energy of dispersion attraction in real conjugated systems, as demonstrated by semiempirical estimates of evaporation heat in butadiene, benzene, naphthalene, and phenylacetylene. Translated fromZhurnal Strukturnoi Khimii, Vol. 38, No. 6, pp. 1029–1037, November–December, 1997. Original article submitted December 14, 1996.     

The scaled one-electron Hamiltonian model for open-shelllcao-mo-scf calculations: further corrections to thesoeh energy

The wavefunction generated by the scaled one-electron Hamiltonian (soeh) model has been modified further by including all single excitations from the space of thesoeh function. A perturbation-variation ansatz has been invoked for obtaining the corresponding energy correction (ΔE ₂). [E(soeh)+ΔE ₂] is shown to reproduceE(Roothaan) very closely. It has been demonstrated that by making ΔE ₂ stationary with respect to change in μ results. The correctedsoeh [E(soeh)+ΔE ₂] is shown to be quite useful for the calculation of geometrical parameters of open-shell systems even though it lacks the ‘upper-boundedness’ exhibited byE(soeh).     

Electrochemical behavior and parameters of hydrofullerene C60H36

Electrochemistry of hydrofullerene C₆₀H₃₆ was studied by cyclic voltammetry in THF and CH₂Cl₂ in the −47–14 °C temperature range. Hydrofullerene undergoes reversible one-electron reduction to form a radical anion in THF (E ⁰=−3.18 V (Fc⁰/Fc⁺), Fc=ferrocene) and irreversible one-electron oxidation in CH₂Cl₂ (E _p ^a =1.22 V (Fc⁰/Fc⁺)). The reduction potential was used to estimate electron affinity of hydrofullerene as EA=−0.33 eV. It was suggested that C₆₀H₃₆ is an isomer withT-symmetry in which 12 double bonds form four isolated benzenoid rings located in vertices of an imaginary inscribed tetrahedron on the molecular surface. For hydrofullerene, the “electrochemical gap” is an analog of the energy gap (HOMO−LUMO), equal to (E ^Ox−E ^Red)=4.4 V, and indicates that C₆₀H₃₆ is a sufficiently “hard” molecule with a low reactivity in redox reactions. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 11, pp. 2083–2087, November, 1999.     

Combined Open Shell Hartree–Fock Theory of Atomic–Molecular and Nuclear Systems

In this study, the combined Hartree–Fock (HF) and Hartree–Fock–Roothaan equations are derived for multideterminantal single configuration states with any number of open shells of atoms, molecules and nuclei. It is shown that the postulated orbital-dependent energy and Fock operators are invariant to the unitary transformation of orbitals. This new methodology is based entirely on the spin-restricted HF theory. As an application of combined open shell theory of atomic–molecular and nuclear systems presented in this paper, we have solved Hartree–Fock–Roothaan equations for the ground state of electronic configuration C(1s ²2s ²2p ²) using Slater type orbitals as a basis.     

Synthesis, crystal structure, and reduction of bis(2-phenylindenyl)hafnium dichloride

Reduction of the bent-sandwich [·⁵-(Ph)Ind]₂HfCl₂ complex (1) (where (Ph) Ind is the 2-phenylindenyl anion) in a THF medium was studied by low-temperature cyclic voltammetry. Complex1 is stable in THF at a temperature lower than −50°C and undergoes reversible one-electron reduction to radical anion1 ^{. −}. Further one-electron reduction of1 ^{. −} to dianion1 ²⁻ is accompanied by the elimination of two Cl⁻ ions to form the bisindenyl sandwich [·⁵-(Ph)Ind]₂Hf complex (2). This complex can undergo reversible one-electron reduction to the corresponding radical anion2 ^{. −}, which is stable within the cyclic voltammetry time scale. AtT=−30°C in a THF solution, complex1 was reduced to a diamagnetic (apparently, binuclear) Hf^III complex, which was characterized by cyclic voltammetry. Synthesis and the crystal structure of complex1 are reported. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 12, pp. 2161–2165, December, 1997.     

Correlation effects to the expectation values of atomic systems

The Hartree-Fock and first natural spin determinants were compared as reference determinants for calculating various one-electron properties such as ρ(0), 〈½?〉, 〈r^?2〉,…, 〈r³〉, and r^?1₁₂〉. Calculations were made on various small atoms and their positive and negative ions. For nearly all the expectation values studied, the first natural spin orbital determinant gave consistently superior results. In particular, the Hartree-Fock functions gave markedly inferior results for some long range properties such as the magnentic susceptibilities of negative ions. The major correlation error in the expectation values is primarily an orbital effect which may be accounted for by including correlation terms in the one-particle Hamiltonian. Such approximate Brueckner or best overlap orbitals should reproduce most one-electron expectation values accurately.     

A conjecture for some partial differential operators on L 2( R n)

OnX =L ²(R ⁿ), letQ = (Q ₁,Q ₂,…,Q _n) andP = (P ₁,P ₂, …,P _n) be the operators given by (Q _jf) (x) =x _jf(x),P _j = - i∂/∂x _j. For anyC ^∞ functionh:R ⁿ →R putH ₀ =h(P) andH =H ₀ + (1 +Q ²)^-δ, where δ > 1/2. By the method of scattering theory we prove thatH _ac, the absolutely continuous part ofH is unitarily equivalent toH ₀ when (a)n = 1 and (b) forn ≥ 2, whenh is in a large class of polynomials. It is conjectured that the results are true for any polynomialh. We use the techniques of Enss’ method and the idea of bound states for momentum.     

Additive density functional correlation corrections to single particle theories

The correlation energy functional E_c of the Hartree-Fock density is investigated. It was previously established that E_c produces the exact ground-state energy when added to the Hartree-Fock energy. Except when certain degeneracies occur, it is here shown that E_c is bounded from below when the coordinates of the Hartree-Fock density are scaled uniformly by λ, as λ → ∞. Consequently, approximations to E_c should display this bounded property, which is a second-order perturbation energy. It is also shown that a corresponding result applies to that correlation energy functional, Ë_c, which is to be added to a completed exact exchange-only (in the OPM sense) density functional calculation. Scaling requirements are presented for each order in the perturbation expansion for Ë_c. For instance, the second-order term is dimensionless. © 1997 John Wiley & Sons, Inc.     

Glass transition and crystallization study of chalcogenide Se<Subscript>70</Subscript>Te<Subscript>15</Subscript>In<Subscript>15</Subscript> glass

Differential scanning calorimetry data at different heating rates (5, 10, 15 and 20 °C min⁻¹) of Se₇₀Te₁₅In₁₅ chalcogenide glass is reported and discussed. The crystallization mechanism is explained in terms of recent analyses developed for use under non-isothermal conditions. The value of Avrami exponent (n) indicates that the glassy Se₇₀Te₁₅In₁₅ alloy has three-dimensional growth. The average values of the activation energy for glass transition, E _g, and crystallization process, E _c, are (154.16 ± 4.1) kJ mol⁻¹ and (98.81 ± 18.1) kJ mol⁻¹, respectively. The ease of glass formation has also been studied. The reduced glass transition temperature (T _rg), Hruby’ parameter (K _gl) and fragility index (F _i) indicate that the prepared glass is obtained from a strong glass forming liquid.     

Spectrum of states in icosahedral structures of the electronic configuration gn (N = 1–7). 1. Invariant expansion for integrals and energies

The energy spectrum of the states that appear in structures of icosahedral (I,I_h symmetry with open electronic shells g^N (dim g = 4; N = 1–7) is reported. The energies are obtained in terms of integral invariants (reduced matrix elements of electron-electron interaction) H^k (g, g). The latter are analogs of the Slater-Condon parameters F^k(l,l) for atoms with the l^N electronic configuration. A similar representation is proposed for the integrals mm’≨’) of electron-electron interaction on the 4-fold degenerate g orbitals in the “standard” representation. The relation between the terms of the g^N(I,I_h) configuration and the parent states of the orthogonal group O⁺(4) is discussed. Translated fromZhurnal Strukturnoi Khimii, Vol. 38, No. 1, pp. 3–13, January–February, 1997.     

Conformational Properties of (Z,Z)-, (E,Z)-, and (E,E)-Cycloocta-1,4-dienes

Summary.  Ab initio calculations at the HF/6-31G^* level of theory for geometry optimization and the MP2/6-31G^*//HF/6-31G^* level for a single point total energy calculation are reported for (Z,Z)-, (E,Z)-, and (E,E)-cycloocta-1,4-dienes. The C ₂-symmetric twist-boat conformation of (Z,Z)-cycloocta-1,4-diene was calculated to be by 3.6 kJ·mol⁻¹ more stable than the C _S-symmetric boat-chair form; the calculated energy barrier for ring inversion of the twist-boat conformation via the C _S-symmetric boat-boat geometry is 19.1 kJ·mol⁻¹. Interconversion between twist-boat and boat-chair conformations takes place via a half-chair (C ₁) transition state which is 43.5 kJ·mol⁻¹ above the twist-boat form. The unsymmetrical twist-boat-chair conformation of (E,Z)-cycloocta-1,4-diene was calculated to be by 18.7 kJ·mol⁻¹ more stable than the unsymmetrical boat-chair form. The calculated energy barrier for the interconversion of twist-boat-chair and boat-chair is 69.5 kJ·mol⁻¹, whereas the barrier for swiveling of the trans-double bond through the bridge is 172.6 kJ·mol⁻¹. The C _S symmetric crown conformation of the parallel family of (E,E)-cycloocta-1,4-diene was calculated to be by 16.5 kJ·mol⁻¹ more stable than the C _S-symmetric boat-chair form. Interconversion of crown and boat-chair takes place via a chair (C _S) transition state which is 37.2 kJ·mol⁻¹ above the crown conformation. The axial- symmetrical twist geometry of the crossed family of (E,E)-cycloocta-1,4-diene is 5.9 kJ·mol⁻¹ less stable than the crown conformation. Corresponding author. E-mail: isayavar@yahoo.com Received March 25, 2002; accepted April 3, 2002     

Thermal behavior study and decomposition kinetics of salbutamol under isothermal and non-isothermal conditions

The thermal decomposition of salbutamol (β₂ — selective adrenoreceptor) was studied using differential scanning calorimetry (DSC) and thermogravimetry/derivative thermogravimetry (TG/DTG). It was observed that the commercial sample showed a different thermal profile than the standard sample caused by the presence of excipients. These compounds increase the thermal stability of the drug. Moreover, higher activation energy was calculated for the pharmaceutical sample, which was estimated by isothermal and non-isothermal methods for the first stage of the thermal decomposition process. For isothermal experiments the average values were E _act=130 kJ mol⁻¹ (for standard sample) and E _act=252 kJ mol⁻¹ (for pharmaceutical sample) in a dynamic nitrogen atmosphere (50 mL min⁻¹). For non-isothermal method, activation energy was obtained from the plot of log heating rates vs. 1/T in dynamic air atmosphere (50 mL min⁻¹). The calculated values were E _act=134 kJ mol⁻¹ (for standard sample) and E _act=139 kJ mol⁻¹ (for pharmaceutical sample).