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: Ab initio calculations using an STO-3G Gaussian basis set are performed in order to obtain the binding energy to the 06 and N7 of guanine of such amino acid models as the guanidinium ion for arginine, the ammonium ion for lysine, the methanol for serine, and the formamide for glutamic acid. The binding of formamide to adenine is also investigated. The charged ions exhibit a much higher binding energy to the bases, as expected, than the uncharged molecules. The order of binding strength is NH+4 〉 guanidinium+ 〉 formamide 〉 methanol, and for formamide, guanine 〉 adenine.

Notes: This paper presents a survey of published and unpublished ab initio calculations of the vibrational structures of the ten lowest electronic singlet states of the hydrogen molecule up to the H(n = 1) + H(n = 2) dissociation limit. The data are based on adiabatic potential functions (clamped-nuclei electronic energies and nuclear-mass-dependent diagonal corrections). Nonadiabatic coupling has been treated ab initio within the five states. of 1Λg+ symmetry (X,EF, GK, HH̄) and 1Σ+g I.1Πg. The accuracies of the theoretical energies are determined by comparisons with experimental data for H2, HD, and D2. The level shifts and predissociation probabilities of the excited 1Σ+g states, generated by nonadiabatic coupling with the discrete and continuous vibrational structure of the ground state, and radiative properties have also been calculated.

Notes: We report in this paper the results of outer and inner valence IP calculations for the HF molecule using two different many-body methods for the direct evaluation of energy differences. The first is the nonperturbative coupled-cluster based linear response theory (LRT) and the second is the hermitian open-shell many-body perturbation theory (MBPT). A Huzinaga-Dunning (9s5p→ 5s3p/3s) basis has been used. LRT uses an “ionization operator” S as in the equation of motion method (EOM) to generate the ionized states from a coupled-cluster type of ground state. S is chosen to consist of single ionization and ionization-cum-shake-up operators, thus treating the Koopmans as well as the shake-up states on equal footing. LRT would thus be capable of computing both the outer and the inner valence regions with equal facility. This is borne out by the results. For the open-shell MBPT, the model space is chosen to be spanned by the singly ionized determinants. The convergence of the results for the inner valence region is slow, and the results obtained from the [2, 1] Pade' approximants are presented. Unlike the LRT, the inner valence region is not reproduced with full complexity in MBPT, indicating that it is essential to modify the theory by way of expanding the model space to contain the shake-up determinants also.

Notes: In this paper, the unitarily invariant decomposition of Hermitian operators is performed by means of irreducible tensor operators to give the explicit expression of the coupling coefficients for [1m] X [r-n] → [2s, 1t] with respect to the group structure \documentclass{article}\pagestyle{empty}\begin{document}$$ {\rm SU}^Q (2) \times U(r) $$ \end{document} with the Gel'fand chain of subgroups \documentclass{article}\pagestyle{empty}\begin{document}$$ U(r) \supset U(r - 1) \supset \cdots \supset U(1) $$\end{document}.

Notes: The general problem of relativistic corrections to the kinetic energy in quasirelativistic theories, is discussed and related formulas are developed. It is shown that the well-known mass-velocity operator, Hmv = (-α2/8)p4, is incorrect and does not provide any proper relativistic corrections.

Notes: Counterpoise corrections for the basis set superposition error to the components of the bimolecular interaction energy are defined for three methods of decomposition. The results for the case of the NH3 + BH3 interaction are presented and discussed.

Notes: The correlation hole of Coulson and Nielson and its extension to momentum space by Banyard and Reed is studied by using an exactly solvable model. For this model all relevant quantities pertaining to the correlation hole have been calculated exactly. We use this model to study the relationship between the fit to the correlation hole for an approximate wave function and the closeness of the approximate energies to the exact ones. We show that, although in general the better the fit the closer are the approximate physical quantities to the exact ones, there are exceptions where that is not the case. Also, we present a convenient method for the calculation of the two particle distribution in momentum space and generalize the concept of the correlation hole by defining it in the pseudophase space of position and momentum.

Notes: A hermitian, variational open-shell coupled-cluster method is described and applied to the calculation of H2O and N2 ionization potentials in the T ≈ T2 approximation. A nonvariational calculation is also carried out, with the inclusion of T1 and T3 in addition to T2. Both methods give fair agreement with experiment when only T2 is taken into account. T3, which is included at present in the nonvariational scheme only, has a considerable effect on the results and gives good agreement with experiment.

Notes: The definition and properties of the bond order and valence indices calculated from ab initio wave functions are summarized. Their physical interpretation relationships to the exchange effects in bonding and generalization to correlated wave functions are also discussed. Some examples with typical bond order and valence values are shown.

Notes: We explore a generalized weak-field model for the description of the electronic properties of a partly filled shell(S) ion in a crystalline field. Such a model corresponds to the one developed, in parts III and IV of this series, for dN and fN ions in cubical symmetry except that the constraint relations are relaxed. This leads to a fourteen-parameter weak-field model for dN ions in octahedral symmetry and to a 33-parameter weak-field model for fN ions in octahedral symmetry. The latter two models are completely equivalent to the corresponding strong-field models as developed by Griffith and by Tanabe, Sugano, and Kamimura. The constraint relations of parts III and IV are further discussed. In particular, the role they play in fitting procedures is examined. As a conclusion, the weak-field model of III and IV appears as a phenomenological version of the generalized weak-field model introduced in this paper.

Notes: The [2s + 2a] cycloaddition of ethylene and acetylene has been studied. A transition structure of C2 symmetry was located on the potential surface. Activation energies for the process are also reported.

Notes: The kinetic-energy anisotropies of fifteen diatomic hydrides AH with A = H, Li, Be, B, C, N, O, F, Na, Mg, Al, Si, P, S, Cl are calculated from self-consistent-field wave functions constructed from extended basis sets of Slater-type orbitals. It is found that there is no consistent ordering of the bond-parallel and bond-perpendicular components of the kinetic energy with respect to separated atom values. An analysis of the orbital contributions reveals that nonbonding π orbitals make large contributions to the total kinetic-energy anisotropy. This makes it difficult, if not impossible, to deduce anything about the nature of the chemical bond from the total anisotropy. However, certain dimensionless orbital kinetic-energy anisotropies are useful for interpretative studies because, in free atoms, these quantities have fixed values that depend only on the symmetry of the orbital.

Notes: The minimal energy conformations of o-benzosemiquinone anion radical were calculated for several cases of assignments by using the INDO method and the molecular geometry adjusting method. In order to know the effect of lithium ion in the solvent, the minimal energy conformations of the system of Li—O—H and o-benzosemiquinone anion radical were calculated. The calculations of the minimal energy conformations of this radical in t-butyl alcohol, alkaline aqueous ethanol, alkaline water, neutral methanol, and acetonitrile were carried out. The total energies of the minimal energy conformations in the assignment |A3| 〉 |A4| were lower than those in the assignment |A3| 〈 |A4|.

Notes: A state-specific approach to the calculation of transition moments between molecular electronic states requires that the wavefunction for each state is expanded in its optimum one-electron basis and that nonorthonormal basis techniques are used for the calculation of the transition moment integrals. A method has been developed for carrying out such nonorthonormal basis calculations, based on the corresponding orbitals transformation and appropriately defined density matrices, which may be used with configuration interaction (CI) wavefunctions. Further improvements of the method have resulted in a decrease in the time required for the calculations and thus allow its application with moderately large CI expansions for each state. Nonorthonormal basis calculations on transition moments in H2O have been carried out using the above method. The results are in agreement with those of large MRD-CI calculations.

Notes: The photoelectron spectra of ozone have been recorded and the first three electronic band systems reassigned on the basis of observed vibrational structure and calculations reported in the literature. The systems X̃,Ã, and B̃ at 12.75, 13.03, and 13.57 eV are assigned as 2A1, 2A2, and 2B2, respectively.

Notes: An important class of polymers is comprised, in part, of 1,4-disubstituted phenylene groups. It is widely believed that large amplitude phenylene torsional motions play a critical role in determining such physical properties as the toughness and degree of crystallinity of these polymers. We have studied what is perhaps the simplest polymer in this class, polyphenylene oxide (PPO), using ab initio quantum chemistry methods to determine the conformational properties, torsional potential energy functions, and vibrational frequencies. From our calculations on dimer (diphenyl ether) and trimer (para-diphenoxy benzene) fragments emerges a qualitative description of the mechanism of phenylene rotation in the polymeric material.

Notes: We present recent advances with the quantum Monte Carlo (QMC) method in its application to molecular systems. The QMC method is a procedure for solving the Schrödinger equation statistically, by the simulation of an appropriate random process. The formal similarity of the Schrödinger equation with a diffusion equation allows one to calculate quantum mechanical expectation values as Monte Carlo averages over an ensemble of random walks. We have previously obtained highly accurate correlation energies for a number of molecules, as well as the singlet-triplet splitting in methylene and the barrier height for the H + H2 exchange reaction. Recently we have begun a program of extending the QMC approach to the calculation of analytic derivatives of the energy. A brief description of the approach is presented here, together with some preliminary results. In addition, we are now computing expectation values of properties other than the energy. We summarize how standard QMC must be modified, and present some results for H2 and N2. Finally, we describe preliminary work toward the goal of obtaining accurate molecular excited states through QMC.

Notes: It is commonly assumed that the nonlinear absorption of two photons is a process involving the simultaneous capture of two radiation quanta. The purpose of this paper is to evaluate semiclassically the spectroscopic consequences of relaxing this assumption by permitting dephasing in the virtual level. Semiclassical wave-packet propagation theory is used to model the vibronic consequences of finite, virtual-state population times. We demonstrate that extremely short virtual-state lifetimes (1-10 femtoseconds) can have a profound effect on two-photon excitation line shapes and total vibronic envelopes. We provide experimental evidence suggesting that virtual state dephasing has an important influence on the two-photon excitation spectrum of the polyene chromophore of isotachysterol. Our analysis suggests that dephasing in the virtual state is of poetntial importance in defining the vibronic development of two photon spectra of many polyatomic molecules in solution.

Notes: Molecular orbitals for Si2 and Ge2 have been optimized in hyper-HF calculations and utilized in valenxe CI treatments to describe the low-lying states of the molecules. The calculational results reveal pronounced similarities between the electronic structures of Si2 and Ge2. Thus, for both molecules the two lowest-lying electronic states, 3Σ-g(σ2gΠu3) and 3πu(∑g1πu3), have crossing potential energy curves, and the two lowestlying states of 1∑g+ symmetry exhibit crossing of configurations. The Sequence of the low-lying electronic states can be rationalized on basis of a simple molecular-orbital picture in which the σg and the πu valence orbitals are almost degenerate. The spectroscopic constants derived from the present work compare favorably with the results of more elaborate calculations. It appears that transition energies derived in valence CI calculations between states of identical configurations are improved in large CI calculations, whereas this is not the case for transition energies between states of different configurations. The valence CI calculations based on the molecular orbitals optimized in hyper-HF calculations appear to effer reliable descriptions of the chemical bonds as well as of the electronic structures of the molecules Si2 and Ge2.

Notes: Observation of trends in computed spin-orbit splittings for relatively light molecules leads to the conclusion that relativistic corrections to the electronic charge distribution are important when treating molecules containing heavy atoms (Z 〉 18). In order to preserve the nature of the successful computational techniques currently applied to light molecules in so far as possible, particularly to allow for the treatment of correlation effects in an efficient CI procedure on an equal footing with relativistic effects, emphasis is placed on the development of a two-component formalism for this purpose. A first attempt in this direction consists of formulating a spin-free quantum mechanical operator that reflects relativistic kinematics. The mass-velocity term in the Breit-Pauli Hamiltonian is not appropriate for a variational treatment, however, since it drastically alters the spectrum and gives results that are not bounded from below. To avoid this problem the relativistic free particle energy has been used directly for the representation of the kinetic energy, and in addition the Darwin term has been included as a correction to the potential energy. This approach can be justified with reference to the Foldy-Wouthuysen reduction of the Dirac equation, but the class of basis functions used in a variational procedure with this Hamiltonian must be restricted to avoid the formation of a node in the wavefunction at the nucleus; the same problem is circumvented in the Cowan-Griffin method by imposing Dirac boundary conditions on the wavefunction. With this method, accurate spin-orbit splittings have been computed for Br, I, Xe+, CBr, and XeF, but the resulting total energies are found to be overly sensitive to the representation of the inner shells of these systems. Improved results for both valence and inner shells are then shown to follow from the use of the no-pair equation, which provides a variationally tractable two-component method employing a momentum dependent potential that gives a realistic description of relativistic effects for atoms and molecules over a suitably large range of Z.

Notes: A method is outlined for the calculation of the multiplet ligand-field states of transition metal complexes. The procedure involves the use of MS-Xα wave functions, in connection with irreducible tensor operators, and allows the calculation of the elements of the many-electron CI matrices. Comparison of the calculated and experimental multiplet state energies of CrF3-6, CrCl3-6, and MnF4-6 allows one to conclude that the method is useful for the prediction of ligand-field spectra of transition-metal complexes.

Notes: The concept of an atom in a molecule in the context of density functional theory is used to analyze different levels of approximation to the description of interatomic interactions. Such an approach strongly suggests the use of Kohn-Sham atomic densities as an alternative to Hartree-Fock atomic densities in the electron gas model of Gordon and Kim. The results for rare gases and ionic crystals show that both densities lead to similar results.

Notes: Molecular difference densities (DD) are conventionally constructed using spherically averaged atomic densities at the appropriate positions. For atoms in degenerate ground states, this is an unphysical choice, and artifacts dominate the DD. We suggest the extraction of both the position and the orientation of an atom with an open valence shell from x-ray scattering or molecular density data. Subtracting the oriented atoms yields a uniquely defined, as well as chemically meaningful, DD. Covalent bonds to electronegative atoms such as O are no longer exceptional but show bond charges of normal magnitude. Lone pairs are characterized by a dipolar density shift from the bond to the back side of the atomic core.

Notes: The bond critical points of the binary hydrides formed by the elements of the first two rows of the periodic table have been calculated. Particular attention has been paid to the basis-set dependence of the bond critical points at the experimental equilibrium geometries, or where necessary at model geometries. With the exception of H2S, stepwise extension of the basis set leads to a smooth convergence of the bond critical points to a set of values which appear to converge to the Hartree-Fock limit. For H2S it is shownb that the position of the bodn critical point is not only more sensitive to the presence of polarization functions in the basis set, but depends strongly on the orbital exponents of the polarization functions. Extensive optimizations of the exponents of the polarization functions have been carried out with the (12s9p/5s) basis set for second-row hydrides. The effects of contracting the Huzinaga basis sets have been examined.

Notes: Heats of reaction and barrier heights have been computed for H + CH2CH2 → C2H5, H + CH2O → CH3O, and H + CH2O → CH2OH using unrestricted Hartree-Fock and Møller-Plesset perturbation theory up to fourth order (with and without spin annihilation), using single-reference configuration interaction, and using multiconfiguration self-consistent field methods with 3-21G, 6-31G(d), 6-31G(d,p), and 6-311G(d,p) basis sets. The barrier height in all three reactions appears to be relatively insensitive to the basis sets, but the heats of reaction are affected by p-type polarization functions on hydrogen. Computation of the harmonic vibrational frequencies and infrared intensities with two sets of polarization functions on heavy atoms [6-31G(2d)] improves the agreement with experiment. The experimental barrier height for H + C2H4 (2.04 ± 0.08 kcal/mol) is overestimated by 7-9 kcal/mol at the MP2, MP3, and MP4 levels. MCSCF and CISD calculations lower the barrier height by approximately 4 kcal/mol relative to the MP4 calculations but are still almost 4 kcal/mol too high compared to experiment. Annihilation of the largest spin contaminant lowers the MP4SDTQ computed barrier height by 8-9 kcal/mol. For the hydrogen addition to formaldehyde, the same trends are observed. The overestimation of the barrier height with Møller-Plesset perdicted barrier heights for H + C2H4 → C2H5, H + CH2O → CH3O, and H + CH2O → CH2OH at the MP4SDTQ/6-31G(d) after spin annihilation are respectively 1.8, 4.6, and 10.5 kcal/mol.

Notes: LCGTO-LSD model potential calculations have been performed for CO interacting with two-, four-, and eight-atom clusters of Pd, chosen to model the bridge site of the (100) surface. The geometry and vibrational frequencies are not very sensitive to the cluster size. For Pd8 + CO we obtain dC - O = 1.18 Å (1.13 ± 0.1 exp.), dPd - C = 1.87 Å (1.93 ± 0.07 exp.), and (uncoupled) estimates for ωC - O = 1828 cm-1 (1895 exp.) and ωPd - CO = 454 cm-1 (339 exp.) Binding energies of 4.8, 3.8, and 2.6 eV are calculated, respectively, for Pd2 + CO, Pd4 + CO, and Pd8 + CO which may be compared with the experimental initial heat of adsorption of 1.6 eV. Ionization potentials for CO-derived levels are in excellent agreement with experiment (relative to ∊F: 4σ (-11.0 eV, -11.2 exp.); 5σ (-8.0, -8.2 exp.); 1π [-7.5 (b1), -7.3 (b2), -7.5 exp.]). The main negative ion states of 2π* character are calculated at 2.8 eV (b1) and 2.7 eV (b2) above EF. Other states with appreciable 2π* character are found near 5 eV. These may be compared with inverse photoemission results which show a broad peak centered at 4.8 eV. Interactions of the 4σ, 5σ, 1π, and 2π* orbitals of CO with the metal are discussed. The 4σ and 5σ levels are highly mixed, each receiving appreciable contributions from the 4σ and 5σ orbitals of isolated CO. This is discussed in relation to the dispersion of the 4σ and 5σ levels observed in UPS and to the photon-energy dependent intensities of the 4σ and 5σ resonances. The 2π* component of the backbonding comes through several levels in the upper part of the d band which contain small 2π* contributions in bonding combination with Pd d orbitals. The main 2π* orbitals (contaminated by small antibonding contributions from the metal) are empty (see above).

Notes: We have used the MINDO/SR molecular orbital method in order to model the migration of hydrogen atoms over a Ni(100) surface. The present calculations indicate the existence of two different states for adsorbed hydrogen, a result which is in agreement with experimental thermal desorption data and LEED. A detailed analysis of the electronic factors involved in this process is presented.

Notes: The preferred conformations of the thyrotropin-releasing hormone (TRH) have been calculated by the global optimization method proposed earlier by us. [G. Subba Rao, R. S. Tyagi, and R. K. Mishra, J. Theor. Biol. 90, 377 (1981)]. The potential function used comprises the electrostatic, nonbonded, torsional and hydrogen-bonding terms. The results are in good agreement with the crystal structures of TRH. No intramolecular hydrogen bonding is found to occur.

Notes: Several Fe(H2O)n2+ clusters, with n up to 20, have been studied, both by energy minimization in the pairwise approximation and by Monte Carlo simulation. In the last case the calculations have been carried out at three different temperatures in order to investigate the effect of thermal agitation. The most interesting result which can be deduced from this work is the existence of eight water molecules in the first hydration shell of the iron (II) ion. A microscopic analysis has shown that the minimum energy structure of the Fe(H2O)82+ cluster presents a D4d symmetry. This structure is slightly distorted as far as the temperature is increased. The validity of these theoretical predictions is discussed.

Notes: The appearance of the maximum linewidth by the vibrational predissociation of I2He van der Waals (vdW) molecule as a function of the initial vibrational quantum number of I2 is predicted by the dynamic perturbation theory. The kinetic energy perturbation is introduced in addition to perturbation potential. The linewidth agrees quite well with experiment.

Notes: The aim of our work was to find an unambiguous connection between irreversible macroscopic dynamics and reversible microdynamics that makes it possible to manifest irreversibility on a submacroscopic level without the use of coarse graining arguments. On this level the state of a physical system can be described by a field amplitude Ψ and the time evolution of this system is determined by a field equation for Ψ. For conservative systems, this field equation is formally identical with the linear Schrödinger equation, which can be constructed with the help of the classic Hamiltonian function for the corresponding problem. Regarding irreversible phenomena like damping due to a frictional force, for those dissipative systems no classic Hamiltonian function exists. Therefore the corresponding field equation cannot be obtained in the usual way. Nevertheless, also for dissipative systems it is possible to obtain a field equation in an unambiguous way using only Newton's form of classic mechanics. The result of our method is a nonlinear Schrödinger-type field equation with a logarithmic nonlinearity. We discuss in more detail the properties of our logarithmic nonlinearity that corresponds to a macroscopic frictional force in a unique way. A figurative interpretation in terms of environment and interaction can be given. From a more formal point of view, the compatibility of our nonlinear operator with principles known from the theory of linear operators is investigated. One of the surprising results is the fact that although our nonlinear Hamiltonian HNL is “Hermitean” in the usual sense, in contrast to the linear theory an operator exp(i · HNL) is not unitary. Furthermore, in our theory the time-derivative of the mean value of an operator is not only essentially determined by (the mean value of) its commutator with the Hamiltonian. There also occurs an additional term that causes irreversibility of the evolution and is connected with the feature of our theory that (in general) time derivative and construction of the mean value are no longer commuting operations. This fact shows some similarity with coarse graining theories, but in our theory the reason can be traced back unambiguously to an irreversible physical phenomenon.

Notes: A brief general survey of the current state of the art of quantum chemistry is given with some aspects also towards the future. It is emphasized that, if one wants to incorporate such concepts as temperature, entropy, and free energy into quantum chemistry, it is necessary to make a transition from pure quantum mechanics based on wavefunctions to the more general “quantum statistics” based on density matrices and system operators. In addition to the Schrödinger equations, one obtains the Liouville equations, and it is shown that both the time-dependent and the time-independent equations may be solved in both cases by using analogous Hilbert-space methods. Some of the methods for solving the time-independent eigenvalue problems are reviewed, and the need for giant “number crunchers” in this connection are discussed. It is shown that the resolvent methods combined with the “inner projection” technique for the calculations provide a powerful tool for handling the eigenvalue problems in the future in both the Hamiltonian and Liouvillian formalisms. It is stressed that, by going over to supercomputers, one may gain a factor of 100, and that one may gain another factor of 100 by going over to more powerful theoretical methods; however, for programming reasons, it will take a long time before one can reach the combined efficiency factor 100 ✗ 100.

Notes: Let f1, ⃛, fn be a basis of unit vectors and g1, ⃛, gn an orthonormal basis in a Hilbert space. We consider and solve the problem of finding an orthonormal basis e1, ⃛, en such that a weighted average of the distance of ej from fj and gj is minimized in the sense of least squares.

Notes: A theory of electron transfer at low temperatures has been developed. The mechanism of electron transfer consists in the fact that donor ionization and electron capture by the acceptor take place in a tunnel manner.

Notes: The reaction pathway of fulminic acid (HCNO) and acetonitrile oxide (CH3CNO) with methanol as a nucleophile (RCNO + CH3OH → RC(OCH3)=NOH) and the formation of H-bonded complex with methanol have been studied using the MNDO method. MNDO-SCF calculations were performed with complete geometry optimization using the Davidon-Fletcher-Powell method. The reaction pathways were studied by varying all the bond lengths, the bond angles and the twist angles, using the distance C3—O2(R) between the carbon of the 1,3-dipoles and the oxygen of the methanol molecule as the reaction coordinate. The reaction is exothermic and proceeds in two steps. The first step is the formation of a five-centered hydrogen-bonded complex (INT) and is the rate-determining step of the reaction. The second step involves the rearrangement of the H-bonded complex to the product, and this step requires a very small amount of activation energy. Thus, there is an intermediate on the reaction pathway, and therefore, the reaction is stepwise. Acetonitrile oxide is less reactive (activation energy 34.59 kcal/mol) relative to fulminic acid (activation energy 28.91 kcal/mol).