ALBERT

Notes: Inverses and inverse square roots of overlap matrices are approximated by polynomials constructed on the basis of an integral transform of the metric resolvent and Kublanovskaya's conformal mapping technique. Given reasonable upper and lower bounds for the support of the overlap matrix spectral distribution, the described expansions have significantly better convergence properties than the well known power series methods, yet retain the simplicity of these schemes.

Notes: The algebra of the representation of the special unitary group SU(2), the universal covering of the proper rotation group SO(3), is studied in a nonstandard basis. We are using a basis adapted to a chain of type SU(2) ⊃ … ⊃ G″ ⊃ G′ ⊃ G. The introduction of such a chain enables us to label, at least partially, the elements of the irreducible tensorial sets under SU(2) with irreducible representations of G, G″ G″, …. We are thus led to introduce the restriction SU(2) → … → G″ → G′ → G in the Wigner-Racah algebra of the group SU(2). The physical interest of this machinery lies in the fact that the double group of any point symmetry group belongs, up to an isomorphism, to the considered chain. The formalism described in this paper thus appears to be useful in molecular and solid-state calculations. It is particularly efficient in the fields of vibrational-rotational and electronic spectroscopy of molecules. In Appendix A the master formulae, principally the Wigner-Eckart-Racah theorem, for the Wigner-Racah algebra of a chain of compact topological groups (discrete or continuous) are briefly discussed. Lastly, a programme for computing Clebsch-Gordan coefficients for a chain SU(2) ⊃ … ⊃ G″ ⊃ G′ ⊃ G and numerical results for chains isomorphic to SU(2) ⊃ O′ ⊃ D′4 ⊃ D′2 are described in Appendix B.

Notes: The space correlation factor is studied in LiH and Li2 molecules, using MCSCF wave-functions. The shape of the Fermi hole is related to the localization of molecular space orbitals, while the Coulomb hole study indicates the importance of symmetry properties of the molecular orbitals involved in excited configurations for the representation of the electronic correlation inside the chemical bond.

Notes: The influence of various correlation functions, multiplying the monoelectronic space function, is studied and applied to the Helium atom and its isoelectronic series. We used Slater-type basis orbitals and the ground and first excited states have been studied, taking into account the virial and cusp conditions. In the ground state, a very good value for the correlation energy is obtained, using a function of the type \documentclass{article}\pagestyle{empty}\begin{document}$$ 1 - \frac{1}{{1 + 2\alpha }}e^{ - \alpha r_{12} } $$\end{document}For the excited states, this type of function overestimates the assumed value of the correlation energy.

Notes: A method is given for generating the Yamanouchi-Kotani genealogical spin eigenfunctions which requires neither storage of eigenfunctions for smaller numbers of electrons, nor summations of large order, nor explicit use of results from the theory of representations of the symmetric group. An explicit formula is given for the coefficients of expansion in terms of spin products.

Notes: Accurate variational energies have been calculated for three lowest sigma states of the HeH+ ion. This includes the ground state (5 ≤ R ≤ 9 a.u.) which dissociates into He + H+, as well as the A 1Σ+ state (4 ≤ R ≤ 10) and the a 3Σ+ state (3 ≤ R ≤ 10) which both dissociate into He+ + H. The variational results are compared with those obtained using a perturbation theory expansion.

Notes: Semiorthogonalized orbitals are introduced as a basis of approximate ab initio molecular calculations. The semiorthogonalized orbitals are obtained from the usual overlapping atomic orbitals by means of a generalized Löwdin transformation. The new orbitals are characterized by two features: many overlaps between them are zero, and they are close to their corresponding initial atomic orbitals. On the basis of the semiorthogonalized orbitals, a scheme of integral approximations is proposed which allows one to carry out approximate ab initio calculations with less computational efforts and thus offers a possibility to perform calculations on much bigger molecules with limited computational facilities. Results are presented for model calculations on the anthracene π-electron system. Orbital energies and electron distribution are quite close to the rigorously calculated values.

Notes: The representation of an N-electron Schrödinger Hamiltonian on an orthonormal, spin-free (freon) orbital product space is exactly modeled by a second-degree polynomial in the infinitesimal generators of the unitary group. Symmetry adaptation of the orbital product space with respect to the symmetric group yields Gel'fand states which provide base for irreducible represntations of the unitary group. These exist in closed form as, in consequence, does the representation of the model Hamiltonian in this same basis. We retain as physical only those states characterized by tableaux with no more than two columns for which the spin labelling is one-half the difference in the lengths of the two columns. The unitary group formulation is equivalent to standard, number-conserving, second-quantized, many-body theory.

Notes: Quantum chemical CNDO/2 calculations for the conformational preference of the side chain of thioridazine as a function of angle indicated the crystallographically determined structure gave the lowest energy. There is also a small region of conformational flexibility within the first 90° of rotation of the side chain. This is commensurate with the results which we had obtained previously for our similar calculations for promazine and its Cl and CF3 derivatives, perazine and its Cl and CF3 derivatives, and for the hypothetical hitherto unknown N-piperidinopromazine and its Cl and CF3 derivatives. The conformational profile of thioridazine resembles that of the perazines. The calculated gross atomic populations on the alkyl nitrogen in thioridazine was within the range we had previously found necessary for neuroleptic activity.

Notes: The methodology developed in earlier papers is used to compute the 6j symbols and 3jm factors that arise in the group chain SO3 ⊃ T ⊃ C1. The relevant character theory is given and the 2j and 3j symbols calculated. Selection rules are used to predict which j symbols or jm factors are necessarily zero, and then a set of 6j fundamentals computed for T. The complete set of primitive 6j symbols are then computed by application of the orthogonality and Racah backcoupling relations. Primitive 3jm factors are calculated for SO3 ⊃ T and T ⊃ C3 and, from these, all the 3jm factors for T ⊃ C3 and some of those for SO3 ⊃ T computed. A complete table of non-equivalent 6j symbols for T and 3jm factors for T ⊃ C3 is given, together with a table for SO3 ⊃ T of all 3jm factors with j ≤ 2.

Notes: In this paper an iterative generalization of the minimum principle proposed for electronic systems by Hall, Hyslop, and Rees is investigated. It is shown that this generalization still retains the advantage of using members of a larger class of trial wave-functions, for example those with discontinuities, as initial approximations to the wave-functions. This scheme has the advantage that, at each stage of iteration, an upper bound is obtained which is at least as good as that obtained previously.The theory is first applied to the hydrogen atom. It is then adapted to estimate the Hartree-Fock energy of the helium atom, the Hartree-Fock limit being obtained after a relatively small number of iterations.

Notes: A method is proposed for characterizing pairs of finite dimensional subspaces of the Hilbert space. The method might be helpful for the proper selection of coordinate and projective spaces leading to reliable realizations of the Galerkin-Petrov method for use in quantum chemistry. In order to illustrate the method some numerical results are presented.

Notes: A new effective semiempirical method of optimization of one-electron wave-functions has been proposed without a necessity to determine any exact many-electron wave-function. The method designed for LCAO MO calculations is based on a concept of interacting quasi-particles and on nonorthogonal spin-orbitals. “Mixing” of the pure single-particle wave-functions is optimized by means of the procedure which requires only one empirical parameter for each considered state of a whole system. The procedure results in optimal distribution of total electron charge between the individual spin-orbitals. The general method has been illustrated with calculations for the ground state of lithium atom.

Notes: Coupled and uncoupled Hartree-Fock theories are used to compute the electric dipole polarizability of water, ammonia, and methane with three different GTO basis sets. Bounds for the geometric approximation to uncoupled polarizabilities are also computed to examine the accuracy of calculated values. The results are compared with those obtained by a variational-perturbation method proposed by Rebane. The numerical tests provide some information on the correlation terms affecting total computed polarizabilities. The computed values are shown to be in fair or good agreement with experimental data for the largest basis sets. The reliability of Rebane's method with respect to coupled HF procedure is also discussed.

Notes: The orthogonalized plane wave (OPW) method of calculating electronic continuum wave-functions is tested by the computation of photodetachment cross-sections and angular distributions for gaseous halide anions. The results are compared to those obtained by a related augmented plane wave (APW) method involving the exact solution of a single-particle Schrödinger equation containing a piecewise Coulombic potential energy. These comparisons, as well as other involving experimental and theoretical cross-sections from the literature, indicate that OPW cross-sections are, at best, only semi-quantitatively reliable for describing photodetachment even at low photon energies, and that OPW cross-sections should be calculated using the dipole length operator rather than the dipole velocity operator.

Notes: A computation scheme is proposed to determine the wave-functions of molecular systems within the framework of the CMC SCF theory and the APSG SCF approach. The orbital optimization is carried out by the refined first-order one-electron Hamiltonian method. Explicit expressions of the first and second energy derivatives are obtained. In the suggested scheme all the calculations are based on using the matrices of the “partial” Coulomb and the exchange operators constructed over the orbitals at the current iteration cycle.

Notes: The spatial angular correlation of electrons in the ground state of the helium atom has been examined using configuration interaction and Hylleraas wave-functions. It was found that, in general, the average angle between the electrons is not a maximum when the two electrons are at the same distance from the nucleus. For configuration interaction wave-functions there is a position of the electrons for which the average value of the angle between the electrons is a maximum. Hylleraas wave-functions do not show this behavior.

Notes: Interrelations between the local and the global aspects of the stability, continuity, and symmetry properties of variational wave-functions are discussed. The spherical limit of one-electron diatomic molecules and the Hartree-Fock approximation of the ground state of the two-electron atom are shown to exhibit the various concepts involved in an ab initio, yet sufficiently simple, manner.

Notes: The energy of the π-electron system in benzene has been obtained by means of valence bond calculations using orthogonal and nonorthogonal basis orbitals at different levels of configuration interaction, which shows the influence of the orthogonalization procedures.

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: Another possible application of a previously reported approximation theory for electron repulsion integrals using rigorous error bounds is considered by incorporating the electron density matrix in the approximation scheme. Error bounds are set on the contribution of a given integral to the total energy for a given molecular wave-function; the wave-function is then refined cyclically and additional integrals are computed exactly if necessary until convergence is achieved.

Notes: Matrix elements of the Runge-Lenz vector A are presented for those linear combinations of degenerate hydrogenic functions often referred to as hybrid orbitals. The uncertainties in the components of A for each type of wave-function are related to the distribution of classical Kepler orbits corresponding to each function. Matrix elements of A with respect to radially nodeless Slater functions are presented, as these functions are often used as a basis set in atomic and molecular calculations. The properties of A for a piecewise Coulombic central field are discussed in relation to the description of penetrating orbits in the old quantum theory. Simultaneous eigenfunctions of A and the Hamiltonian cannot be chosen for the piecewise Coulombic field because of a discontinuity in the radial derivative of the potential energy.

Notes: The PCILO method is used for the study of molecular systems with translational symmetry in a two-dimensional lattice. The calculation for the whole crystal is reduced to the calculation of unit cell pairs. By using the translational symmetry and the first-neighbour approximation, one shows that only four unit cell pairs have to be considered. The procedure described yields the ground-state energy and the charge distribution of the unit cell.

Notes: The instabilities, especially the singlet instabilities, of the conventional Hartree-Fock (HF) solutions for a variety of alternant and nonalternant hydrocarbons, some of which have been known to show lattice instabilities (bond-length alterations), are examined. The HF solutions for nonalternant hydrocarbons in the pentalene series larger than heptalene and [4n + 2]-annulenes larger than C22H22 are found to be singlet unstable and there appear new solutions lower in energy than the conventional HF solutions and characterized by charge-density waves exhibiting bond-order alterations. It is found that such symmetry-breaking solutions are energetically further stabilized by distorting the nuclear framework so that it may match up with the distribution of bond-order matrix elements of the charge-density wave, which means that in conjugated systems the singlet instability of the HF solution is always accompanied with the lattice instability. Further, it is shown that in conjugated systems, even when the HF solution is singlet stable, if it is not sufficiently stable as, for example, in pentalene and heptalene, there is every possibility for the occurrence of lattice instability. It is also shown that the singlet instability as well as the lattice instability arises from the existence of a sufficiently low-lying singlet excited state.

Notes: We have performed RHF, UHF, UHF with subsequent spin projection, and EHF calculations for the π-electrons of some polyene chains of different lengths in order to investigate the relations among these methods when the number of electrons increases. Special attention is paid to the importance of spin projection for different energetic quantities.

Notes: The densities of states and Fermi energies were accurately calculated from E(k) data obtained by the self-consistent-band Augmented Plane Wave (APW) method, using a quadratic energy expansion. The reliability of the Monte Carlo integration was investigated. It was sufficiently good to determine the anisotropy of the Fermi surface of lithium, sodium, and potassium.

Notes: A new method is proposed for the analysis of components of molecular interaction energy within the Hartree-Fock approximation. The Hartree-Fock molecular orbitals of the isolated molecules are used as the basis for the construction of Fock matrix of the supermolecule. Then certain blocks of this matrix are set to zero subject to specify boundary conditions of the supermolecule molecular orbitals, and the resultant matrix is diagonalized iteratively to obtain the desired energy components. This method can be considered as an extension of our previous method, but has an advantage in the explicit definition of the charge transfer energy, placing it on an equal footing with the exchange and polarization terms. The new method is compared with existing perturbation methods, and is also applied to the energy and electron density decomposition of (H2O)2.

Notes: The canonical orthornormalisation procedure is derived through an integral equation with a finite rank kernel. Its optimal properties in connection with the problem of approximate linear dependence are established.

Notes: Single-exponential Slater type orbitals of the form ψ1 = (1 + L1(r, θ) + L2(r, θ) +…+ Ln(r, θ)) exp (- αr) are examined for their potential use as one-center molecular orbitals. These are then to be used as molecular fragments in a LCMO study. The system examined is HeH+ + with calculated energies and dipole moments being compared to the exact values. These functions behave best in the region of chemical interest (the bonding region) and thus demonstrate a possible usefulness in LCMO calculations and in the field of one-electron diatomics.

Notes: The ground-state energy of the beryllium atom is calculated using a variational procedure in which the elements of the two-body reduced density matrix (particle-particle matrix) are the variational parameters. It is shown that, for this problem and with the limited number of spin-orbitals used, the trace condition and the simultaneous nonnegativity conditions on the particle-particle, the particle-hole, and the hole-hole matrices form a complete solution to the N-representability problem.The energy obtained is - 14.61425 a.u., practically identical to the value given by a configuration interaction calculation which uses the same states. The effects of weakening the nonnegativity conditions on each of the matrices in turn were also explored.

Notes: Inelastic collisions leading to rotationally excited molecular targets and involving protons as projectiles are theoretically examined from the point of view of both the scattering equations in the close coupling formalism and the interaction potentials between the partners.A phenomenological approach is suggested for constructing such interactions and computational results are reported for simple diatomic targets. The inadequacy of more traditional ‘static’ approaches, when called for explaining dynamic couplings with open channels and centrifugal effects, is also discussed.