ALBERT

Notes: In this paper a calculation of the static quadrupole polarizabilities and the shielding factors of the atoms and ions of the 2p open shell systems is presented. A fully coupled Hartree-Fock method using the Roothaan formalism is adopted. Calculations have been done for the ground as well as for some valence excited states. An antishielding effect is observed for all the atoms and ions under study.

Notes: A new integral transform function: \documentclass{article}\pagestyle{empty}\begin{document}$$ {\rm exp }\left({{\rm - }\zeta {\rm r}} \right)\int_0^\infty {s(n)r^n dn} $$\end{document} where s(n) is a positive definite shape function, is proposed and investigated. A test on H2+ shows that it does not suffer from the inadequacies of earlier integral transform functions, of the form: \documentclass{article}\pagestyle{empty}\begin{document}$$ r^n \int_0^\infty {c(\zeta ){\rm exp}({{-}\xi r})d\zeta } $$\end{document} where c(ζ) is a positive definite shape function.

Notes: Weyl's eigenvalue theory for ordinary second-order differential equations is discussed for the case of a continuous spectrum. It is demonstrated that the spectral density function obtained from a suitably averaged Green's function, equal to the Weinstein function, can be directly related to the Weyl-Titchmarsh m-function. The explicit connections with scattering theory are derived and it is found that the Weyl and Jost solutions are proportional; the proportionality factor being the reciprocal value of the latter at the origin. The physical interpretation of the complex poles of the spectral density is discussed in relation to Gamow's exponentially growing functions. The advantage of using a formulation that allows for a “perturbation” of boundary conditions is pointed out.

Notes: Combined CI-HY method techniques have been employed in obtaining a 57-term CI-HY wave function with an energy of -14.66632 a.u. A method due to Brown has been adopted for obtaining this wave function and various shorter expansions. A 44-term expansion with an energy of -14.66606 a.u. is analyzed in terms of various pair effects, and qualitative arguments are presented for understanding these effects.

Notes: Mø-Plesset theory, in which electron correlation energy is calculated by perturbation techniques, is used in second order to calculate energies of the ground states of atoms up to neon. The unrestricted Hartree-Fock (UHF) Hamiltonian is used as the unperturbed system and the technique is then described as unrestricted Mø-Plesset to second order (UMP2). Use of large Gaussian basis sets suggests that the limiting UMP2 energies with a complete basis of s, p, and d functions account for 75-84% of the correlation energy. Preliminary estimates of the contributions of basis functions with higher angular quantum numbers indicate that full UMP2 limits give even more accurate total energies.

Notes: A smooth interpolation method for the calculation of Fourier transforms of numerical atomic orbitals is presented and the method is used to calculate two-center overlap integrals. Examples applying various definitions of smoothness are given.

Notes: A general variational principle for transition and density matrices is proposed. The principle is closely related to Rowe's variational treatment of the equations-of-motion method. It permits the simultaneous construction of coupled approximations for two eigenstates, and it is a straightforward extension of the usual variational method.

Notes: In this paper we discuss an extension of the standard multiple scattered wave method for molecular electronic structure calculations to the case of arbitrarily shaped molecules. A general theory is presented where the difficulties arising from the matching problem across an arbitrary boundary are avoided. The secular equation to determine the energy spectrum is obtained and the wave functions are calculated in all regions.

Notes: Reformulation, generalization, and application of Löwdin's symmetric orthonormalization, of orthogonalization of two sets of vectors, and of McWeeny's variational method of a density matrix, etc., by means of an operator for forming ket vectors on which are imposed some conditions, are presented.

Notes: Analytical formulas are derived for integrals of the orbit-orbit operator in the Breit-Pauli Hamiltonian. The present method differs from an earlier one in the introduction of a new auxiliary function Gn(a) which is an integral of Shavitt's Fm(t) function. A method for its evaluation is discussed and some numerical examples are given.

Notes: A new method is presented for obtaining the spin eigenfunctions of 2n electron systems in the spin state S and MS = 0. Using a modified Young operator the function for the state S, MS ( = S) of the system is first projected out. The projected function is then symmetrized over the last n particles and the weight lowering operator Š-S is then applied to it, resulting in a projection of the state S, 0. To within a multiplicative factor, the resulting function is identical with the one resulting from the vector coupling methods.

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: In this communication we discuss a method of incorporating corrections to the coupled Hartree-Fock (CHF) formalism by introducing the so-called “rearrangement effect.” In this we take account of the relaxation of the core orbitals when excitations from a starting Hartree-Fock wave function occur. The magnitude of this correction numerically is found to be quite significant for the polarizabilities of two-electron atomic systems, results for which are reported.

Notes: A perturbation approach based on resolvent technique and Padé approximants is proposed. The eigenvalue of interest is part of a solution of two nonlinear algebraic equations. The nonlinear equations are arrived at by considering two different expression of the expectation value of the resolvent of an outer projection of the Hamiltonian. The first expression is based on the spectral resolution of the resolvent, and the second one is obtained by a power series expansion analogous to that applied in the derivation of the energy expression in the Brillouin-Wigner perturbation theory. The truncated power series is extrapolated by Padé approximants of type II. The method is tested on a CI calculation of the energy of the lowest 1Σ state of the B2 molecule.

Notes: Complementary variational functionals are derived which impose upper and lower bounds on transition probabilities. These functionals are used to yield bounds in terms of sets of sum rules, and illustrative calculations are presented for hydrogen, helium and krypton atoms.

Notes: Dynamical screening in π-electron systems is studied by the equations of motion method. By using a partitioning technique on the equations of motion we can obtain simple expressions for the effect of dynamical screening directly on the transition energies and transition moments in π-electron systems. These results are used to study the effect of such screening in the N → V transition in ethylene. This procedure can be used to extend the equations-of-motion method to larger π-electron systems.

Notes: We report an attempt of ab initio computation for the ground-state wave-function of a few organic semiconductors (namely TCNQ salts and the PDC charge transfer complex) for which we show that electrical conduction appears to look strongly like that of a band-gap semiconductor.

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: Some applications of an optimization technique due to Levenberg in theoretical submolecular biology are described. This technique has proved a very useful tool for the generation of three-dimensional structures of molecules having cyclic components or other constrained regions. The generated molecular structures provide the starting point for quantum chemical calculations of great value in solving problems of biological interest. The mechanism of transport of alkali metal cations by valinomycin is one such example and this paper describes briefly how various conformations of valinomycin were obtained using the optimization technique. A variant of the same method has been suggested as a new technique for the detection of topographical similarities between apparently dissimilar molecules which are substrates for the same enzyme or receptor site. The technique is illustrated with a comparison of a tricyclic antidepressant with norepinephrine both of which have been postulated to compete for a common receptor site in the “amine pump” of peripheral adrenergic nerve cell membranes. Another area in which the optimization technique has proved useful is in developing an optimum separable localized bond function method designed for rigorous calculations on large saturated molecules. This method is briefly described.

Notes: Ab initio SCF-LCAO-MO calculations have been performed for TCNQ and its positive and negative ions in various electronic states. A basis set consisting of 412 primitive Gaussian type orbitals contracted to 180 was used in the investigation. The electron density distribution in TCNQ and the negative ions, and the redistribution upon ionization has been illustrated by plotting difference density contour maps. The quinone structure of the neutral TCNQ system undergoes a transformation to a benzenoid structure when electrons are added. Electronic transitions, ionization potentials and electron affinities have also been calculated.

Notes: The extended Hartree-Fock equations of the spin-projected scheme are derived in a form suitable for the construction of a surely convergent method of solution using successive optimization of the individual orbitals. The derivation is based on a specific form of the generalized Brillouin theorem.

Notes: Valence-bond type wavefunctions based on contracted Gaussian atomic orbitals afford an adequate description of the potential of the ground 1Σg+ state of H2 for a wide range of internuclear separations. Semiempirical techniques for scaling the potential curve are evaluated. An efficient method for optimizing non-linear variables in the wavefunctions is explored.

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.