ALBERT

Notes: A method has been developed to analyzed the bond and current correlation structures of a molecular many-electron wave function. It is shown that the second order density matrix contains information about the bond and current correlations in its off-diagonal components with respect to the indices of orbital basis functions. We break down the off-diagonal correlation functions into five kinds: charge, spin scalar, spin quadrupole, charge spin, and spin polar correlation functions. For a real wave function, the four correlation functions, except for the spin polar one, have only symmetric-symmetric and antisymmetric-antisymmetric components. The former components give site-bond and bond-bond correlations of charges and spins, while the latter components give current-current correlations of charges and spins. The spin polar correlation function has only symmetric-antisymmetric components that give site-current and bond-current correlations of spins. The five off-diagonal correlation functions are expressed in terms of the off-diagonal components of the second order density matrix. The linked off-diagonal correlation functions are defined in that they give dynamical bond and current correlations. The method is applied to the analyses of the bond and current correlations in the low lying exact eigenstates of the PPP Hamiltonian of benzene.

Notes: This work amalgamates some basic elements defined in the first paper of this series and in related papers with the theory of coupling coefficients for an arbitrary group with the view of generating the Clebsch-Gordan coefficients and V symbols for point symmetry groups. The connection between Clebsch-Gordan coefficients and V symbols is established for an arbitrary group in a form that reduces to the one known for the chain SU(2) ⊃ U(1). The Clebsch-Gordan coefficients and V symbols of any point symmetry group G are shown to be obtainable from Clebsch-Gordan coefficients and \documentclass{article}\pagestyle{empty}\begin{document}$ {\bar f} $\end{document} symbols of the chain SU(2) ⊃ G through the resolving of a system of nonlinear equations.

Notes: The nature of the correlation function G(1,2) appearing in the definition of the reduced first order density operator γ′(1,2) = ρ(1)1/2ρ(2)1/2G(1,2) is analyzed. It is shown that when G(1,2) is expanded in terms of plane waves in the context of a single-determinant approximation to the wave function, the correction to the Weizsacker term in the kinetic energy density expression is the Thomas-Fermi term.

Notes: In the framework of our theoretical approach of the structure and reactivity of chemical intermediates we have been led to reexamine the concept of stability which is widely used by the experimentalists often without specifying its true meaning. In this work we propose a more general definition of the concept of stabilization energy, namely, \documentclass{article}\pagestyle{empty}\begin{document}$$ SE = \Delta H_a - \Sigma N_{AB} E_{AB} $$\end{document} where ΔHa is the heat of atomization of the species under consideration and the EAB's are standard bond energy terms derived from the heats of atomization of reference compounds. Using experimental heats of formation or semiempirical ones deduced from theoretical heats of reaction of appropriate isodesmic processes, we have calculated the stabilization energies of various types of chemical species: saturated, unsaturated and conjugated molecules, free radicals, carbocations, and carbanions. The results obtained can be rationalized in terms of steric hindrance, angular strain, polar interactions, electron delocalization, and substituent effects. Moreover, we have shown that heats of hydrogenation and bond dissociation energies do not provide accurate information on the thermodynamic stabilization of unsaturated compounds and free radicals, respectively. Among other applications the concept of stabilization energy allowed us to propose a detailed classification of free radicals and to rationalize their reactivity. Considering the particular case of radical recombination reactions we have been able to deduce interesting equations showing the relations between the concept of stabilization energy and other quantities commonly used in chemical physics, namely, the bond cleavage enthalpy [BDE(C—C) if one considers alkane thermolysis], the thermodynamic stability measured by the free enthalpy change of a given reaction, and the kinetic stabilization related to the activation energy of a chosen chemical process \documentclass{article}\pagestyle{empty}\begin{document}$$ \begin{array}{*{20}c} {{\rm BDE}(R - R) = E(R - R) - 2{\rm SE(R}^ \cdot {\rm)} + {\rm SE(R} - R{\rm),}} \\ {\Delta G^0_r 2{\rm SE(}R^ \cdot {\rm)} - {\rm SE(R} - R{\rm)} - E(R - R) - T\Delta S^0,} \\ {E_a (r) = \alpha [ - {\rm BDE(}R - R{\rm)}] + \beta ({\hbox{Evans - Polanyi relation}}),}\\ {E_a (r) = a[2{\rm SE(R}^ \cdot )} - {\rm SE(}R - R{\rm)] + }b{\rm .} \end{array} $$\end{document} This analysis allowed us to give a new interpretation of the adjectives transient, persistent, and stable introduced by Griller and Ingold and to show that the persistence of a radical may be due to other factors than steric ones. In conclusion, the concept of stabilization energy appears to be a good tool for rationalizing the static and dynamic properties of chemical species.

Notes: Two-dimensional fully numerical solutions of the Hartree-Fock problem are reported for the singlet ground states of H-, He, H2, and HeH+. The H2 energy at R = 1.4 a.u. is -1.13362957 a.u.

Notes: In this paper, some operator endomorphisms which give rise to conditions for N representability of pth order density matrices are presented.

Notes: A simple analysis of the bonding between the d orbitals in binuclear complexes of chromium indicate that the usual multiple bond picture used to describe these systems is incomplete. This analysis is fairly general and depends only on the symmetry of these complexes and the rather weak coupling between the 3d orbitals on the two metal centers. A series of INDO calculations on Cr2Cl8-4 and Cr2(CH3)8-4 are reported, and suggest that the dominant description of the bonding is one of two Cr atoms antiferromagnetically coupled. Although this description properly accounts for the eclipsed versus staggered conformations found for these systems, the calculated bond length is too long. An analysis of the components of the wave function suggests that the role of multiple bonded structures (configurations) may be small but is important in its influence in shifting the very flat potential energy surfaces to shorter Cr—Cr distances.

Notes: The interaction energies of proton with ethylene and substituted ethylenes in a positive bridged ion have been calculated by ab initio MO method with STO-3G basis set, and the energies were further decomposed according to decomposition scheme proposed by Kitaura and Morokuma [Int. J. Quantum Chem. 10, 325 (1976)].

Notes: Correlation corrections to coupled Hartree-Fock (CHF) static dipole polarizabilities and dipole moment of the HF molecule are calculated using third-order Rayleigh-Schrödinger perturbation theory with Møller-Plesset partitioning (RSMP) in a finite-field procedure. Computations are also made for the dipole moment and polarizability derivatives at the equilibrium internuclear separation. Three different basis sets using contracted Gaussian orbitals augmented by field-induced diffuse polarization functions have been employed to investigate the nature of variation of the properties under study, and the importance of field-induced diffuse polarization functions is discussed. [2/1] Padé approximants are used to accelerate the convergence of the properties. The correlated dipole moment and polarizability values are in excellent agreement with the existing theoretical and experimental values. The dipole moment derivative is in perfect agreement with the existing correlated value. However, the polarizability derivatives at SCF and correlated levels differ appreciably from the existing SCF values and represent as improvement.

Notes: The observed proton hyperfine coupling constants of radicals from resorcinol and 2-carboxylic, 5-carboxylic, and 5-methyl resorcinols in acidic and alkaline solution were reproduced by means of the INDO (intermediate neglect of differential overlap) with molecular geometry adjusting method. The bond lengths of C—O were longer in acidic solution than in alkaline solution and the total energies were lower in acidic solution than in alkaline solution. The assignments of the coupling constants of the radicals from resorcinols were |A2| 〉 |A5| in alkaline oxidation and |A2| 〈 |A5| in acidic oxidation.

Notes: In the present work, the problem of the determination of the potential energy surface for nonrigid molecules is examined in the case of the double rotation of the methyl groups in acetone. From the symmetry adapted functional form for the potential, the minimum number of configurations to be calculated is deduced in order to have a reliable surface. With this consideration in mind, the potential energy surface of acetone is determined in some Hartree-Fock semiempirical (CNDO/2) and ab initio procedures with different standard basis sets. In addition, ab initio calculations are performed using different sets of floating Gaussian functions in order to introduce some polarization effects in the wave function. Finally, the influence of the electronic correlation effects in the barrier height, and the role of the possible relaxation of the structure during the rotation is discussed.

Notes: Cross sections for energy transfer into many-body systems can be expressed in terms of time-correlation functions (TCFS) of transition operators. A semiclassical version is presented by treating internal motions as quantized and relative motions as classical. The time evolution of internal motions can be calculated in the Heisenberg picture and avoids expansions in target states. The decoupling of fast and slow internal motions is treated and applied to vibrational-rotational decoupling in polyatomic molecules. Results are presented for Li+-CO2 collisions.

Notes: Water chain intervention in the hydrolysis of methyl fluoride and in some tautomeric 1,3 proton transfer reactions has been studied. The results show that in solution the process are neither unimolecular nor bimolecular as in the gas phase, but are multimolecular, where the geometric parameters of the solvent intervene in the reaction coordinate.

Notes: The computational considerations involved in calculating ordinary and rotatory intensities and electronic excitation energies in the random phase approximation (RPA) are examined. We employ a localized orbital formulation in order to analyze the results in terms of local and charge-transfer excitations. Occupied orbitals are localized by the Foster-Boys procedure. The virtual space is transformed into a localized “valence” set that maximizes dipole strengths with the occupied counterparts, and a delocalized remainder. The two-electron integral transformation is performed with an efficient algorithm, based on Diercksen's, that generates only the particle-hole-type integrals required in the RPA. The lowest solutions of the RPA equations are obtained iteratively using a modification of the Davidson-Liu simultaneous vector expansion method. This allows the inclusion of the entire set of particle-hole states supported by a basis set of up to 102 orbitals. Calculations at this level give better excitation energies and intensities than SDCI methods, at substantial savings in computational effort. Comparative timings, computed results and analysis in terms of localized orbitals are given for planar and distorted ethylene using extended atomic orbital bases including diffuse functions. The results for planar ethylene are in excellent agreement with experiment.

Notes: Nonlinear equations are important for the description of dissipative processes and soliton-related phenomena. One of the principal aims of the theory of symmetry breaking in quantum mechanics is to provide a systematic method for introducing interactions which reduce the symmetry of a system. Such a method is applied here to compute the general form of the symmetry-breaking interaction F(x,t, Ψ, Ψ*) for the time-dependent Schrödinger equation in one spatial dimension \documentclass{article}\pagestyle{empty}\begin{document}$$ \left[{\partial _{xx} + 2i\partial_t - 2g_2(t)x^2 - 2g_1(t)x - 2g_0 (t)} \right]\psi (x,t) = F(x,t,\psi,\psi *). $$\end{document} For F = 0, it has been shown that the Lie algebra of space-time invariances of this equation is S1 = s1 (2,IR)□w1, the Schrödinger algebra. Following the method of Boyer, Sharp, and Winternitz, all conjugacy classes of subalgebras of S1 are given. For each subalgebra, the most general form of the interaction term F(x,t, Ψ, ψ*) is constructed. The potential F then reduces the symmetry from S1 to the considered subalgebra. Furthermore, the one-dimensional subalgebras of S1 obtained above partition S1 into orbits of operators. To each orbit there corresponds a coordinate system in which the above equation separates variables. A partial resolution of the solutions of the above equation has been obtained by exploiting this relation. The algebraic approach to symmetry breaking yields a rich variety of interaction terms F, which appear to generalize the nonlinear Schrödinger-Langevin-Kostin equation for nonconservative systems and the so-called nonlinear Schrödinger equation. A criterion in terms of the existence or nonexistence of Bäcklund transformations is conjectured in order to distinguish between dissipative and soliton equations.

Notes: Calculating the electrostatic potential around the Ser-His-Asp catalytic triad in serine proteinases the very important substituent effect of the buried aspartate is revealed. It is found that the strong Coulomb field of this distant but charged side chain considerably enhances the nucleophilicity of the active serine hydroxyl group. The interpretation of the vital importance of aspartate may replace the “charge-relay” hypothesis which seems to be disproved in the light of recent experiments.

Notes: Effective solutions for two difficulties which may be present in MCSCF calculation are discussed: (i) We show how the large configuration state function state expansion case may be handled simply and effectively without the introduction of extraneous projection operators or Lagrange multipliers; (ii) we present a simplified two-electron integral transformation procedure which significantly reduces the operation count (and hence computational efficiency is increased) for second order and particularly for third order MCSCF procedures. The procedures we introduce use some freedom available in the orthogonal complement Cl space and the virtual orbital space to simplify MCSCF calculations.

Notes: A modified scheme for SCF interaction energy decomposition has been proposed where the nonphysical basis set superposition error (BSSE) has been corrected by means of the counterpoise method. A new procedure to separate the exchange and induction energy terms free of nonphysical BSSE has been tested in the case of the H2O dimer. The first order BSSE appears to be non-negligible for strong hydrogen bonded complexes. In addition the scheme allows separation of the long-controversial charge-transfer contribution within the induction term, which has been considerably overestimated in previous studies.

Notes: The carbene reaction 1CH2 + C2H6 was studied with the semiempirical method SINDO 1. The pertinent transition states and products were obtained by geometry optimization on a CI surface. We find the insertion reaction forming C3H8 greatly favored compared to the double insertion leading to CH4 + C2H4. The result is qualitatively the same for the reaction of the two possible intermediate radicals CH3 and C2H5. The geometries and energies of all transition states are presented.

Notes: Exact computer-generated formulas can be produced for each term of an infinite series that gives the value of three-center Coulomb integrals over Slater-type orbitals of the s-type with equal screening constants. As a specific example, the Coulomb energy of an equilateral triangular arrangement of all 1s orbitals is calculated using seven terms of an infinite expansion to obtain an answer comparable to earlier work in elliptic coordinates. Generalization to all three-center cases of this implementation of the Löwdin α-function method is straightforward.

Notes: Hartree-Bogoliubov-Valatin (HBV) theory may be implemented with Lipkin Hamiltonians to obtain self-consistent BCS wave functions which describe bond formation and dissociation. These wave functions are in turn vacuua for Nambu's representation of Feynman-Dyson-Goldstone diagrammatic perturbation theory, and hence provide suitable references for the many-body treatment of correlation. Exact SCF solutions of the HBV equations are equivalent to special even-replacement MC-SCF solutions. The latter are similar to generalized valence bond theory, and require one Fock operator for each one-particle shell. The commutative coupling case of HBV theory is realized when the number-conserving renormalized one-body and number-nonconserving pairing operators commute. In this case, a set of orbital equations which involves a single Fock operator may be solved. Since this could prove to be a significant simplification for large systems, the commutative coupling and exact solutions are compared here for the fragmentation of H2 and F2. Results suggest that commutative coupling orbitals will be useful for the aforementioned many-body theory.

Notes: A MNDO study of molecular geometries, enthalpies, formation, atomization, bond separation, and hydrogenation of a series of unsaturated boron-nitrogen compounds, linear with two and three members and cyclic with three, four, five, and six members are presented. For all these molecules, MNDO calculations are in excellent agreement with available ab initio calculations or experimental data. The high rotational barrier in aminoborane H2BNH2, 30.6 kcal/mol, and the length of bond in iminoborane HBNH, 1.183 Å, imply strong double and triple BN bond character in these two molecules. In the odd membered heterocycles, examination of the molecular geometries and energies of equilibrium states shows that in all cases, the stability of the compounds grows with the number of boron atoms and decreases with that of the nitrogen atoms. Moreover, compared study of the two BN-fulvenes with their homologous hydrocarbon shows that only BN (BB)-fulvene has a polyenic structure similar to that of fulvene.

Notes: The linear, quadratic, and cubic response of a multiconfigurational Hartree-Fock state to a time independent one-electron perturbation has been derived. A comparison between the exact response functions as obtained from Rayleigh-Schrödinger perturbation theory and the multiconfigurational Hartree-Fock response functions allows a identification of matrix elements of the perturbation operator between the ground and excited states and between excited states. We discuss some ambiguities which result from such an identification.

Notes: Monte Carlo simulations in the isothermal-isobaric ensemble for the α phase of solid N2 have been carried out with two different pair potentials obtained from ab initio quantum chemical calculations. Comparison is made with data obtained from empirical potentials as well as with experimental data, and in general, the agreement with experiment is found to be good. It is also found that the differences between experiment and theory can largely be attributed to correlation effects, not considered in the quantum mechanical calculations of the pair potential.

Notes: The AGP wave function is used to describe the ground state potential energy curves of Li2 and CH+. Moderate size basis sets are used. The computational aspects of the AGP optimization are discussed and the nature of the AGP wave function versus internuclear separation is examined.

Notes: Doped semiconductors with shallow impurities provide an ideal situation to study a collection of hydrogenlike atoms under controllable experimental conditions. We present the results of recent cluster calculation on the electronic properties of such systems.

Notes: Experimental base stacking and hydrogen bonding enthalpies, as determined from double stranded DNA melting studies, are compared with theoretical calculations of interactions between the respective DNA interconstituent components. Comparisons are made in relation to the differences in magnitudes, sequence dependent spreads, and trends. It is shown that the use of an effective dielectric constant reconciles the differences between the experimental and theoretical values. Suggestions are made for calibration of potential functions suitable for nucleic acid conformational analysis.

Notes: An overview is given here of band structure calculations on the fourth and fifth group transition metal monocarbides, mononitrides, and monoxides, published since the review article by Calais [J.-L. Calais, Adv. Phys. 26, 847 (1977)]. Furthermore, the relations of three categories of experimental properties, which allow insight into the electronic structure of the above mentioned compounds, and the results of band structure calculations are discussed. Theoretical predictions are compared with experimental findings. The considered experimental properties are valence band photoemission spectra, valence band x-ray emission spectra, and optical properties.

Notes: Conformational energy calculations at the PCILO level of approximation were performed on β(1 → 3) linked disaccharide consisting of N-acetylgalactosamine and galactose of the Forssman antigen to examine the side group conformations and their influence on the mutual orientations of the two pyranosyl rings. Two low energy regions for the glycosidic bond conformation have been located in the grid search using classical potential functions. The PCILO energy minimizations were then carried out in each of these regions. The preferred orientations of the nonreducing pyranosyl ring relative to the reducing ring were found to be in agreement with the available x-ray results. Moreover, the orientations of groups attached to the anomeric carbons were in good accordance with the requirements of the exo-anomeric effect.

Notes: The molecular electrostatic potential of the triple helix poly(dT)·tpoly(dA)·poly(dT) is calculated, and the results are examined in relation to those obtained for its component double and single helical parts. For the double helix presenting the standard Watson-Crick hydrogen bonds, the deepest potentials are formed on the side of the major groove, a situation similar to that observed in the A-DNA duplex. For the double helix presenting Hoogsteen-type hydrogen bonds the deepest potentials lie in the major groove, on the side of the pyrimidine strand. In the triple helix the deepest potentials are located in the major groove in a narrow zone over the thymine bases of the Watson-Crick pair.

Notes: Interacting stereo-irregular chains of hydrogen atoms, which simulate the topological structure of many conducting polymers, are generated by a computer and solved numerically with the unrestricted Hartree-Fock method with a modified spin polarized potential. The electron localization is investigated, and a mechanism for the interchain tunneling is discovered. Local antiferromagnetic ordering is derived which may explain the AF behavior observed in some conducting polymers.

Notes: The glycan moiety of the bacterial peptidoglycan consists of alternatingly β(1 → 4) linked disaccharides of N-acetylglucosamine (NAG) and its 3-O-D lactic derivative, N-acetyl β-D-muramic acid (NAM). PCILO conformational energy calculations have been carried out for NAG-NAM and NAM-NAG disaccharides to see whether or not the glycan strands possess a chitinlike structure as suggested by earlier workers. In agrement with recent experimental findings, the present results also suggest that the chitinlike structure is energetically disallowed. Furthermore, the bulky N-acetyl substituents at C2 positions of the two sugar molecules are found to be relatively less important in stabilizing mutual orientations of the two pyranosyl rings.

Notes: The MO theory of structural polyhedrons is one of the basic subjects closely related to the bonding theory of complex compounds and clusters. A series of valuable papers on this subject have been contributed by Hoffman, Wade, King, and Lauher et al. A method called “group overlap method” for constructing the symmetry orbitals and for calculating the group overlap integrals is proposed. It is hoped that the group overlap method may be used for the qualitative and quantitative MO study of structural polyhedrons.

Notes: Quantum chemical ab initio MODPOT/VRDDO calculations have been carried out on the following aminonitrobenzenes for which crystal structures had been determined experimentally: 4-nitroaniline; N,N-dimethyl-p-nitroaniline; 2,4,6-trinitroaniline; 1,3-diamino-2,4,6-trinitrobenzene (DATB - Form I); 1,3,5-triamino-2,4,6-trinitrobenzene (TATB); 2,3,4,6-tetranitroaniline; N-methyl-N,2,4,6-tetranitroaniline (Tetryl); and N-(β,β,β-trifluoroethyl)-N,2,4,6-tetranitroaniline. These quantum chemical calculations were performed on the molecules in their conformations as found in their crystal structures. The calculations were carried out with our own ab initio programs which also incorporate as options several desirable features for calculations on large molecules: ab initio effective core model potentials (MODPOT) which enable calculations of valence electrons only explicitly, yet accurately, and a charge conserving integral prescreening evaluation (which we named VRDDO-variable retention of diatomic differential overlap) especially effective for spatially extended molecules. Aminonitrobenzenes are especially interesting since there are inherent intramolecular ring distortions and deviations from planarity and intramolecular hydrogen bonds as well as intermolecular hydrogen bonds causing further deviations from planarity. The theoretical indices resulting from the quantum chemical calculations are relevant to a number of properties and behavioral characteristics of these molecules, both intramolecular and intermolecular. The charges on the atoms [from the gross atomic populations (GAP's)] are needed for calculation of the atomic multipole-atomic multipole electrostatic contributions (a dominant factor) to the intermolecular interaction energies. These electrostatic interaction energies are part of the input necessary for calculations on the crystal packing and densities of these molecules. These GAP's are also of value in interpreting the experimental photoelectron and ESCA spectra of these molecules. The total overlap populations (TOP's) between atoms are related to the inherent bond strengths and can serve as a quantitative replacement for the old empirical bond length-bond order-bond energy relationship still used by explosives chemists to identify the “target bonds” (the weakest bonds). The TOP's are of considerable value in predicting and tracing initiation and subsequent steps of explosive phenomena. The molecular orbital energies of the lowest unoccupied orbitals are of interest since nitroexplosives have been implicated in testicular toxicity and the initial metabolic activation appears to proceed through a one-electron reduction of the nitroexplosive.

Notes: An analysis of the leading effects of a propagator approach to the calculation of resonance energies is presented. By comparison with ΔSCF results, a simple approximation formula has been developed which contains the main features of the VN-1 potential option and the diagonal Tamm-Dancoff approximation. This result is expected to be a useful guide for further refinements of the many-body aspects of the theory.

Notes: The electron density and the molecular electrostatic potential of the β-carbolines are studied using ab initio STO-3G wave functions. The analysis was done from the point of view of a previous model built with monoamine oxidase substrates and irreversible inhibitors. The results confirm the usefulness of the model and make it possible to propose new precision to the molecular electrostatic potential patterns needed to have monoamine oxidase inhibitory activity.

Notes: The transition energy and geometry of the lowest excited (nπ*) singlet and triplet states of CO, CS, HNO, H2CO, HFCO, and F2CO molecules are calculated by CNDO/S and CNDO/2-VN-1 potential methods, and the results are compared with those of experimental and ab initio theoretical studies, wherever available. In the calculation of the vertical transition energy, the performance of the CNDO/S method is seen to be generally more satisfactory than that of the CNDO/2-VN-1 potential method, while the reverse is true for the excited-state geometry. The CNDO/S method as such fails to describe the geometry of the excited state, but a combined version (CNDO/S-2) of CNDO/S and CNDO/2, as well as the CNDO/2-VN-1 potential method is fairly successful in this regard.

Notes: On the basis of a high-quality LCAO MO SCF calculation, covalency versus ionicity of a Co-F bond in the CoF6(n-) complexes, where n = 4, 3, and 2, is discussed. The overlap and gross atomic populations, delocalization of certain MOS, and the charge densities in the bond region as well as around F's all indicate that the covalency increases as n decreases or the valency of Co increases in these complexes.

Notes: In an n level quantum system there is a relation between each density operator and an element of su (n) Lie algebra. This relation is also established between Cartan's subalgebras and the complete sets of compatible observables. A scalar product is then defined in this algebra in order to introduce orthonormal bases and to simplify many calculations about expectation values of observables. Therefore simple general rules were established which show how to determine (completely or partly), from linearly independent observables, the density operator of the system.

Notes: The elastic forward scattering of high-energy electrons from molecules has been studied in the second Born approximation. An integral transformation has been adopted to evaluate the second Born integrals analytically without explicit use of molecular wave functions. In the high-energy limit, the differential cross section for the forward scattering is expressed in terms of electric dipole and quadrupole moments, the second moment of charge distribution with respect to the molecular center, and transition dipole moments. All these quantities are shown to be computable from molecular electron densities in the ground state.

Notes: We examine the short-range behavior of the spherically averaged Hartree-Fock exchange charge density by performing a simple Taylor expansion. On the basis of this expansion, a theoretical model is constructed that generates gradient correction terms to the local density approximation for the exchange energy of an inhomogeneous electron gas. In particular, we derive the Xαβ exchange energy functional and a theoretical value for the parameter β. Our value for β agrees well with previous empirical estimates, and with empirical calculations in the present work.

Notes: By the means of gauge invariance of the continuity equation Fick's law of diffusion can be extended to also comprehend the chemical reaction. In the case of first-order reactions a complete set of eigenfunctions is obtained. These solutions provide a tool for pattern recognition in biochemical and biological problems (e.g., formation of chromosome banding). The transport equation, including reactions of second order, exhibits soliton solutions describing the propagation of a kinetic process in a medium (molecular chain, fluids, etc.). A relationship to the soliton solutions of the nonlinear Schrödinger equation and Korteweg-de Vries equation of hydrodynamics is also indicated. The propagation of a reactive process (transition state) occurs in many problems of molecular biology. The Brownian motion of ions undergoing a reaction of first order in a constant magnetic field is also exactly solved.

Notes: In the study of the ground-state binding energy of atoms, relatively more attention has been paid to the gradient expansion of the kinetic energy term of the Thomas-Fermi-Dirac model than to the gradient expansion of the exchange term of this model. Recently Shih and Shih, Murphy, and Wang have focused on this problem and calculated the first gradient expansion correction to the Dirac exchange term by making use of electron densities constructed from Hartree-Fock wave functions. In previous work, aimed to introduce the shell structure of atoms via a variational procedure using the energy density functional formalism, electron densities have been obtained for the Na atom within the Thomas-Fermi-Dirac model with and without the Weizsäcker and Hodges gradient expansion corrections to the kinetic energy term. In this paper we make use of the respective electron densities and calculate the first gradient expansion corrections to the Dirac exchange term. The results show that, for the Na atom, the magnitude of this correction is about 1% of the magnitude of the total binding energy.

Notes: We discuss how the local convergence of Newton-Raphson and fixed Hessian MCSCF iterative models may be rationalized in terms of a total order of convergence in an error vector and a corresponding error term. We demonstrate that a sequence of N Newton-Raphson iterations has a total order of convergence of 2N and that a sequence of N fixed Hessian iterations has a total order of convergence of N + 1. We derive the error terms of a Newton-Raphson and a fixed Hessian sequence of iterations. We discuss the implementation of the fixed Hessian and the Newton-Raphson approaches both when linear and nonlinear transformations of the variables are carried out. Sample calculations show that insight into the structure of the local convergence of Newton-Raphson and fixed Hessian models can be based on an order of convergence and an error term analysis.

Notes: The description of electron density in solids by quantum theory and by x-ray diffraction experiments should closely coincide. A formalism for describing the electronic structure of single crystals in terms of a density matrix is presented. Iterative equations for calculating the density matrix in a basis of Bloch orbitals using measured crystallographic intensity data are derived. Numerical results are presented. The Bloch orbital formalism explicitly accounts for the mutual interaction of atoms in different unit cells, and is useful for insulators, semiconductors, and metals.

Notes: The Löwdin basis functions are introduced and their permutation symmetry is studied in detail. A simple relation between the Löwdin basis and the orthogonal basis constructed by means of Young-Yamanouchi units is established. Both analytical and recurrence formulas for the overlap integrals of the Löwdin and some relevant auxiliary functions are derived. The symmetry of the overlap matrix of the Löwdin basis functions is briefly discussed.

Notes: The definitive breakdown of the CBO potential picture in consequence of the vibronic coupling of two electronic states is discussed on the basis of our model. The present treatment differs from that given in the literature.

Notes: The electrostatic potentials and fields created by the thiol protease, papain, are computed on the surface envelope of the protein and in the region of its active site using a technique based on accurate multipole expansions of the electron density of appropriately chosen subunits. The effect of binding counterions to the protein is considered specifically. The possible role of the “electrostatic environment” in the functioning of the enzyme is discussed and it is found that this environment can favor the proton transfer between Cys 25 and His 159 that is thought to initiate its mode of action, but that the reasons for this appear more complicated than earlier models would suggest.

Notes: The bond length alternation problem in cyclic polyene models as described by the Pariser-Parr-Pople Hamiltonian and an empirical quasiharmonic π-core potential is investigated using the one-parameter alternant molecular orbital (AMO) method. It is shown that in contrast to the unrestricted Hartree-Fock (UHF) results, which lead to symmetric equidistant structures, the one-parameter AMO results yield bond length alternating structures similar to those obtained with the restricted HF approach. The correlation energy recovered by the AMO method is examined for the symmetric polyenes in the whole range of coupling constants for both the Pariser-Parr-Pople and Hubbard Hamiltonians and compared with exact full configuration interaction (FCI) results. For the first member of the cyclic polyene series we also compared the FCI and AMO correlation energies for different nuclear framework distortions. This comparison indicates that in contrast to the UHF results the fraction of the correlation energy recovered by the AMO approach is very uniform over the range of nuclear distortions considered. The AMO results thus strongly indicate the dimerization in the polyenic chains.

Notes: For most structures (molecules, graphs, lattices) a count of random walks for nonequivalent sites will give different numbers, particularly for walks of many steps. Occasionally one finds the same count of walks for nonequivalent sites. These have been termed “unusual walks” and have been closely examined in the case of trivalent graphs. While it remains to be understood what structural factors are critical, some regularities have been observed and are discussed. Unusual walks within a single structure signal “isospectural” points in a graph. A number of structures possessing unusual walks have been displayed, and a few constructive steps which do not alter the “unusual” characteristics of selected vertices have been indicated.

Notes: Boundaries of electronic energy level sets in the nuclear charge space wZ can be parametrized as single valued functionals for any nuclear geometry of molecules, which property, combined with the general convexity of such level sets, leads to various electronic energy inequalities.

Notes: In connection with the reinterpretation of Hund's multiplicity rules for molecules, a detailed study has been made of the energy differences in the total energy and its components for the triplet and singlet Πu states of the hydrogen molecule and the analogous states of the four- and six-membered hydrogen atom rings. For the hydrogen molecule, both SCF and CI studies indicated that the outer electron is considerably more contracted in the triplet than in the singlet state. In both approximations, the energy difference is dominated for all bond distances of chemical and physical significance by the electron-nuclear attraction component and not by the electron repulsion component as predicted by simple first-order perturbation theory. Although the correlation energy for each of the states is of the same magnitude as the energy differences considered here, the difference of the correlation energies is much smaller. It had little effect on the qualitative differences between these states of the hydrogen molecule. For the four- and six-membered rings, SCF studies were made on the lowest singlet and triplet states where one electron was promoted from the σg to a Πu orbital. Even though the coupled electrons were more delocalized in these cases, the electron repulsion became relatively more important. However in all cases, the lower state had the highest electron repulsion energy and lower electron-nuclear attraction. The triplet state continued to have the more contracted outer open-shell orbital.

Notes: Basic integrals arising in the momentum space formulation of the van der Waals forces are derived for arbitrary two atoms. These integrals are essentially the matrix elements of the Fourier transformed Coulomb operator and the Fourier transformed square of the Coulomb operator between any two Slater orbitals. The derivation is completely analytic and the results are expressed as finite series expansions in terms of auxiliary integrals. Recursion relations among the auxiliary integrals are developed.

Notes: Every Slater determinant D may be uniquely analyzed in terms of spin components Dl = OlD which are pure spin eigenfunctions, so that S2Dl = l(l+1)D. Every component Dl = OlD may in turn be written as a sum of symmetric combinations of Slater determinants, Tk = [αμ-kβk‖αkβν-k], and the coefficients ck(l) in the expansion OlD = ∑k ck(l) Tk are known as the “Sanibel coefficients.” By using the relation S2Dl = l(l+1)D, a recursion formula for the coefficients ck(l) is derived, which is then explicitly solved in the special case when Sz has the pure quantum number m = 0.

Notes: The concept of atomic valency is looked at as a specific response of an atomic system to an arbitrary environment. It is shown, on the basis of a mathematical result in group actions theory, how valency might be related both to previously found Hartree-Fock deformed states and to the isotropy groups (up to a conjugation) of the extrema of the electronic energy functional.

Notes: A biological system usually operates under nonstable values of relevant parameters, such as pH, temperature, and ionic strength. The question therefore poses whether such fluctations do affect its relevant processes. Experimental studies on the role of random temperature fluctuations on functional encounters of biostructural polymer molecules, and consequent self-assembly of supramolecular structures, have evidenced an additional, noise-induced order of these structures. This type of effect is the result of nonlinearity in physical systems, and the case of a biosystem is especially interesting. As recent experiments have shown, spinodal decomposition resulting from thermodynamic instability may favor the onset of the supramolecular ordering process. If the random fluctuations of temperature are imposed in such conditions of thermodynamic instability, their ordering effect is further enhanced.

Notes: There are significant differences between the conditions for chemical and biochemical reactivity. There-fore, models for treating chemical reactions are mostly not suitable for investigating interactions and transformations of bio(macro)molecules. Common features of numerous processes occurring in vivo and in vitro (such as the role of water, ions, and colloids and the significance of Helmholtz energy surfaces) are outlined. Some characteristics of a model suitable for studying van der Waals interactions between biomacromolecules, based on Brownian dynamics and the Lifshitz theory, are described.

Notes: Quantum chemical studies (INDO-RHF-SCF) have been made for the resting state active sites of three closely related heme proteins, cytochrome c peroxidase (CCP), metmyoglobin (MMB), and catalase (CAT). The relative energies of the germane sextet, quartet, and doublet spin-states of each active site were calculated. Both CCP and MMB have similar heme units, consisting of an Fe(III)-protoporphyrin-IX with an imidazole and water as axial ligands. Our calculations show that the larger doming of the porphyrin, greater out-of-planarity of the iron, and the shorter iron-water distance in MMB leads to a sextet ground state with a low-lying quartet state. By contrast, the order of these two states is reversed in CCP, when a neutral imidazole is used as the endogenous axial ligand. An imidazolate ligand, on the other hand, which is an extreme representation of the H-bonding believed to occur in CCP with a nearby aspartate residue, leads to a sextet ground state with a low-lying quartet state. Assuming at least a partially anionic ligand in the intact protein, it follows that the quartet contribution to the ground state properties will be larger in CCP than in MMB. These predictions are consistent with the observed differences in the temperature-dependent magnetic susceptibility for these two proteins. The present results suggest that the experimentally observed Mössbauer resonance spectra of CCP should be reinterpreted in terms of sextet and quartet state contributions to the electric field gradient. Calculations for catalase, which has a single phenolate ligand, result in a sextet ground state with a low-lying quartet state consistent with available Mössbauer and magnetic susceptibility data. Our calculations of the Im- form of CCP show that it more closely resembles CAT. Thus, the effect of proton transfer in CCP can account at least in part for the similarities between CCP and CAT function. Minor differences in ground spin-state and electronic properties calculated for CCP and MMB, however, cannot explain why MMB does not have significant peroxidase activity. The different functions of MMB and CCP must then be due in part to other known differences in their protein environment such as polar residues around the distal ligand binding pocket of CCP, which are absent in MMB, and could help its transformation to an active oxidizing state.

Notes: Recent studies performed in our group on a classical problem of quantum chemistry, with strong implications for theoretical biochemistry and pharmacology, are here summarized. Ab initio descriptions of noncovalent interactions, and in particular H bonds and acid-base couples, have been reexamined using as novel tools the decomposition of ΔE with the inclusion of CP corrections and a further decomposition of the ΔE components into group contributions. Some results of systematic analyses performed over H-bonded dimers are reported, supplemented by a successful application of this approach to a problem of noticeable economic importance (the identification of catalysts for the industrial synthesis of tensioactives). A new feature, presented here for the first time, is the extension of the CP-corrected decomposition of ΔE to bimolecular interactions in solution.

Notes: Two new numerical methods for solving the Schrödinger equation for an N-well periodic potential are presented. One is a diagonalization method based on Floquet-Bloch formalism, and the other is a renormalized Numerov-Cooley method with periodic boundary conditions. The numerical superiority of these methods over the standard diagonalization technique is demonstrated. The methods are illustrated by applications to internal rotation.

Notes: Ab initio MO-LCAO-SCF calculations using an STO-3G basis set were performed to find the most stable conformations of L-serine phosphate and hydrated L-serine phosphate. The most favorable conformation of L-serine phosphate is found to be one where the bond sequence O—C—C—C is trans and P—O—C—C gauche, and a very short hydrogen bond is formed between an oxygen atom of the phosphate group and a hydrogen atom of the ammonium group.For hydrated L-serine phosphate, a bridge-type hydration in which a water molecule links a phosphate oxygen and an ammonium hydrogen displays particularly low energy. In the four-hydrated L-serine phosphate anion, the most favorable conformation is such a bridged one having a rather extended configuration with regard to the bond sequences O—C—C—C and P—O—C—C.

Notes: A many-body diagrammatic perturbation theory of rotation-vibration spectra is elaborated. The present approach is based on two many-body techniques, namely on the second quantization formalism (a rotating-vibrating molecule is formally treated here as a system of interacting vibrons, obeying the Bose-Einstein statistics) and the many-body diagrammatic theory of a model Hamiltonian, initially suggested in the microscopic theory of nuclei and in the last decade very frequently exploited in the accounting for the correlation effects in many electron systems. In the framework of this theory, the rotation-vibration energies are determined as the eigenvalues of a finite-dimensional model eigenproblem.

Notes: Previous discussions of the bases and projection operators for projective representations are analyzed and alternatives are proposed. Detailed proofs are provided for a result which is often assumed or proved within unacceptable limitations, namely that the form of the projection operators which is standard for vector representations is also valid for unitary projective representations. These proofs provide necessary conditions for this result, and they are constructed in terms of the definition given for the bases of projective representations. The calculation of Clebsch-Gordan coefficients for projective representations by means of the projection operators is discussed. Whereas in the method of Dirl the work is carried out entirely in terms of the matrix representations, and the symmetrization of the bases has to be considered in a second step, all the work of this paper is conducted starting from the symmetrization of the bases of the projective representations, so that those two steps are carried out simultaneously.

Notes: Two approaches in treating the weak representability problem for density operators are studied. In the first approach this problem is reduced to the minimization of a certain distance function. The second approach is based on inductive generation of linear inequalities for diagonal elements of the second-order density operators. Generalization of Kuhn-Yoseloff's results is also discussed.

Notes: The approximation of hyperpolarizabilities through the use of operator inequalities is sketched. Previously applied methods to ordinary polarizabilities are extended. They lead to expressions involving moments of oscillator strength distributions related through sum rules to ground state properties. Systematic applications of new formulae are suggested.

Notes: A generalization of the Gibbs-Bogoliubov inequality F ≤ F0 + 〈H - H0〉0 for the free energy F is studied which leads to a variation principle for this quantity that may be of importance in certain computational applications to quantum systems. This approach is coupled with a study of the perturbation expansion of the free energy for a canonical ensemble with H = H0 + λV in the general case when H0 and V do not commute. The second- and high-order derivatives of the free energy with respect to the perturbation parameter λ are calculated. From the second-order term is finally obtained a second-order correction to the previous variational minimum for the free energy.

Notes: The ab initio calculation has been performed with the addition pathways of HLi and its dimer to acetylene at the RHF/3-21G basis set. It shows that the reaction mechanisms of these two reactions are rather similar. In either of two reaction pathways, there is a meta-stable molecular complex near the isolated reactant state. This kind of addition can be treated approximately as the unimolecular reaction in which the molecular complex rearranges into the product. We have estimated the activation entropies and the statistical A factors of these two reactions by the use of RRKM theory. Frontier molecular orbital analysis of these two transition states reveals their HOMOS to be formed from both HOMO-LUMO and HOMO-HOMO interactions.

Notes: In order to apply the TDHF equations in the matrix form we established previously, we conceive a model for which the static Hartree-Fock equations are rigorously solved; so we study problems only linked to the TDHF method, independent of any further approximation (for example, of LCAO type). This model is made of a spherical box containing electrons subject to a particular potential. We solve, numerically, the TDHF equations at different orders, and we calculate the third order hyperpolarizability tensor. We then compare our results with the ones given by a variation-perturbation method.

Notes: A explicit expression for the unitary group Clebsch-Gordan coefficients, which couple two fully antisymmetric single-column states into the two-column Gel'fand-Tsetlin states, is given in terms of isoscalar factors for the canonical subgroup chain U(n) ⊃ U(n - 1) ⊃ … ⊃ U(1). The isoscalar factors are expressed through the step numbers labeling canonical basis states and enable a straightforward construction of Gel'fand-Tsetlin states in the Clifford algebra unitary group approach, without the use of the tables for the symmetric group outer-product reduction coefficients.

Notes: In this article, the Hamiltonian for the scattering of the He + H2 system is given by using the interaction potential V(X, Y, Θ) determined by experiments and the semiclassical method. From this Hamiltonian we find a dynamical algebra h6. The statistical expectation of the energy and the transition probability of H2, Pn→m, are derived; therefore, selection rules have been found easily.

Notes: Recently we extended our strategy for MRD-CI (multireference double excitation-configuration interaction) calculations based on localized/local orbitals and an “effective” CI Hamiltonian for molecular decompositions of large molecules to breaking a chemical bond in a molecule in a crystal or other solid environment. Our technique involves solving a quantum chemical ab-initio SCF explicitly for a system of a reference molecule surrounded by a number of other molecules in the multipole environment of more distant neighbors. The resulting canonical molecular orbitals are then localized and the localized occupied and virtual orbitals in the region of interest are included explicitly in the MRD-CI with the remainder of the occupied localized orbitals being folded into an “effective” CI Hamiltonian. The MRD-CI calculations are carried out for breaking a bond in the reference molecule. This method is completely general. The space treated explicitly quantum chemically and the surrounding space can have voids, defects, deformations, dislocations, impurities, dopants, edges and surfaces, boundaries, etc. We previously applied this procedure successfully to the H3C—NO2 bond dissociation of nitromethane in a nitromethane crystal with extensive testing of the number of molecules that have to be included explicitly in the SCF and how many molecules have to be represented by more distant multipoles. The results indicated that it took more energy to dissociate the H3C—NO2 bond when the nitromethane molecule was in the crystal than it did to dissociate that bond in the free nitromethane molecule. In this present study we have investigated the effect of voids (both in the nitromethane molecules treated explicitly in the SCF and those in the environment represented by multipoles) on the calculated H3C—NO2 bond dissociation energies.

Notes: The magnetooptical properties (B terms) vibronically induced have been calculated for a series of carbonyl compounds in the region of the first absorption band. The rules deduced experimentally for the signs and sizes of the B terms induced by vibrations of different symmetry are generally confirmed by these calculations.

Notes: A particularly compact form of the orthogonally spin-adapted coupled-cluster equations involving all singly and doubly excited clusters is derived for the general case of a non-Hartree-Fock closed-shell reference determinant. The diagrammatic approach based on the graphical methods of spin algebras is applied. The relationship of different diagrammatic procedures for spin-adaptation, employing both bare and spin-adapted two-electron interaction vertices, is discussed. A comparison with the results obtained with algebraic spin-adaption approaches is also given.

Notes: The electronic properties of a one-dimensional diatomic crystal have been analyzed by using the MO-LCAO method in the tight binding approximation, with mathematical techniques involved in setting up and solving difference equations. The approach gives the exact sets of analytic solutions for both localized and nonlocalized states. The theory of surface states is developed as a characteristic value problem. To illustrate the method the surface states for a semiinfinite crystal which contains a local imperfection at the surface were examined. It appears that this method has advantages over previous methods developed to solve surface problems in crystalline lattices.

Notes: Energy levels of I2 ⃛ X van der Waals (VDW) molecules, where X is a rare gas atom (X = He, Ne) are obtained in an approximate way by using cofocal elliptic coordinates and taking the I2 internuclear axis as a quantization axis. This is the starting point to study vibrational predissociation (VP) of larger I2 ⃛ Xn (n ≥ 2) clusters, where a good energetic description may provide accurate initial conditions from which trajectory calculations can be carried out.

Notes: Pariser-Parr-Pople Hartree-Fock crystal orbital calculations (both restricted and unrestricted versions) are performed for one-dimensional models of π-electron polymers. The π-electron band structures corresponding to symmetric and nonsymmetric solutions have been calculated. To investigate the influence of the form of the interelectronic interaction, the calculations were performed with the Mataga-Nishimoto (MN) formula and a modified Mataga-Nishimoto (MMN) formula for two-electron integrals. We have found that curves of the points of the minimum total energy per unit cell for the alternating models are very similar, but band structures and energy gaps are quite different when the MN formula was substituted with the MMN formula for two-electron integrals.

Notes: When viewed as a square two-indexed matrix, the array of atomic orbital-based, two-electron integrals (ij|kl) is a positive semidefinite array. Beebe and Linderberg showed, in 1977, that actual or near linear dependencies often exist within the types of atomic orbital basis sets employed in conventional quantum chemical calculations. In fact, large (i.e., higher quality) bases were shown to be substantially more redundant than smaller or more spatially separated bases. In situations where there exists significant basis near redundancy, the rank (r) of the (ij|kl) ≡ Vl,J matrix of integrals will be significantly smaller than the matrix dimension M. When this occurs, it proves computationally tractable to decompose the M-dimensional matrix V into components L (V = LLT) which contain all of the information needed to form the full V matrix. The Cholesky algorithm allow such a decomposition to be carried out and forms the basis of the work described here. The method is found to be highly successful in reducing the number of integrals and integral derivatives that must actually be calculated. In particular, results on the C2 molecule indicate that the algorithm can be superior to traditional methods of integral derivative generation if the orbital basis is large enough to contain appreciable near redundancy. In contrast, results on benzene with a more spatially delocalized basis show that conventional methods are preferred whenever substantial basis (near) redundancy is not present.

Notes: An analysis of the pathways and intermediates for the epoxidation of olefins using a Ru(IV)-oxo model complex is presented. Simple bonding concepts, coupled with INDO/1 semiempirical MO calculations, provide a description of the metal-oxo/olefin interaction that agrees well with experimental facts. Concerted [1 + 2] and [2 + 2] pathways are investigated and found to lead to unfavorable orbital interactions as with organic analogues. Nonconcerted process (those in which one bond is formed before the second), which connect the two concerted pathways, are preferred. A nonconcerted [1 + 2] pathway is preferred over a nonconcerted [2 + 2] pathway on the basis of less steric repulsions between the olefin and the metal ligands. Also, all open structures (those with one C—O bond formed) investigated minimized to bound epoxides. In the most favorable pathway, a nonconcerted [1 + 2] pathway, radical cation character on the terminal carbon increases as the nonconcerted process is induced. Preference for the bound epoxide intermediate over the oxometallocycle is simply explained by partial charge considerations.

Notes: Since the overlap integral between two functions in position space is the same as the overlap integral between their counterparts in momentum space, there is an intimate connection between orthonormalization procedures in the two spaces. It is pointed out that in certain cases this situation can be used to simplify the orthogonalization.

Notes: After a brief review of the historical development of the concept “antiparticle,” some recent theoretical results on CP violation are reviewed.

Notes: The ground (N) state and the 1B1u(V) excited state of planar ethylene have been studied at the CPF and MR-SDCI levels of theory, using an extended CGTO basis set of the ANO type. The investigation especially addresses the near-degeneracy problem in the ground state and the coupling between the diffuse character of the π* orbital and the amount of correlation included in the wave function of the V state. The MR-SDCI results yield a vertical excitation energy in the range 7.8-8.0 eV, whereas the CPF result is 7.9 eV. The best MR-SDCI result for 〈1π*‖z2‖1π*〉 is 7.8, whereas CPF calculations based on MR-SDCI INOS give the value 6.7. It is clear from the results that these numbers have not converged and that more extended calculations than was possible in the present work would yield an even more compact wave function.

Notes: Self-consistent-field (SCF) wave functions are used to calculate cross sections for the elastic and inelastic scattering of fast electrons and x-rays from CH4 and C2H2 molecules. The effects of basis set choice and free rotation on these cross sections are investigated. The utility of an approximate scheme to correct SCF inelastic cross sections for the effects of electron correlation is examined. The probability density for the interelectronic distance, or radial intracule density, is obtained and discussed.

Notes: The problem of the validity of the mass-velocity operator in computational quantum chemistry is discussed. The opinion that the mass-velocity operator is incorrect is shown not to be well founded.