Library

Notes: Various criteria that have been suggested for assessing the accuracy of complex-coordinate eigenvalues are critically tested in an application to the lowest helium 1S resonance, and found to be relatively ineffective. An alternative criterion is described, based on a generalized stabilization procedure, which seems better able to select the “best” resonance eigenvalue and to guide the construction of variational trial functions.

Notes: Adiabatic potential-energy surfaces in the lower excited states following the benzene⇌Dewar-benzene isomerization process were calculated by the INDO/S method postulating the concerted reaction mechanism which is proved to be valid in the ground state by the calculation of the intrinsic reaction coordinate. It was concluded that the benzene molecule in the 1B1u and 1E2u excited states readily isomerizes to Dewar-benzene in the condensed phase although the quantum yield may not be large. Bryce-Smith's proposal, that the isomerization to Dewar-benzene occurs only after the intersystem crossing to the 3B1u state in benzene molecule, will not be probable; for the 3B1u route is not favorable to the isomerization in comparison with these singlet routes. However, the rearomatization of Dewar-benzene in the ground state may produce the 3B1u benzene in small yield with higher yield of the ground-state benzene. The activation energy in the rearomatization is calculated to be 19.03 kcal/mol. These conclusions are consistent with the experimental facts. Molecular orbital correlations in the isomerization and the ionization potentials of Dewar-benzene were also discussed.

Notes: The transport of C6H5O- (or similarly charged moieties) through a lipoidal membrane may possibly be facilitated by forming complexes with the neutral compound. Thus, theoretical studies were performed on the model [C6H5OH⃛OC6H5]- molecular complex to obtain some information concerning the possible molecular and electronic structure of such complexes. Ab initio MODPOT/VRDDO SCF calculations were carried out on the neutral-anion dimer [C6H5OH⃛OC6H5] to optimize the equilibrium geometry. Electrostatic molecular potential contour maps have been generated from the ab initio MODPOT/VRDDO results in the molecular plane and in the plane perpendicular to the molecular plane and intersecting the hydrogen bond O⃛H—O. Difference maps have also been generated showing the change of potential on complex formation. There is a decrease of electrostatic interactions of the phenoxide anion upon complex formation with the neutral phenol. Counterpoise corrections for basis set size could not be made since calculation of the phenoxide anions in the basis set of the phenol plus the phenoxide anion led to an excited state for the phenoxide anion. This behavior is somewhat similar to that occurring in the stabilization method for excited states of negative ions as the size of the basis set is increased.

Notes: The s-p separation model CNDO-MO method has been introduced and developed for both the open- and closed-shell systems since 1975. This method has two chief advantages over most of the other CNDO methods. (1) The ns and np bases of the same atom may be considered independent to infer different bonding behavior. (2) Pariser-Parr and Nishimoto-Mataga approximations are applied to the Coulombic repulsion integrals, which not only simplifies the calculation and saves computer time but also gives reliable computational results. A series of nitrogen fluoride molecules such as NF, NF2, NF3, cis-N2F2, trans-N2F2, and N2F2 have been selected for this MO calculation. In each case, both ionization potential and dipole moment were calculated. The results are closer to the observed values than those reported in other works.

Notes: Calculations of molecular-orbital energies and x-ray photoelectron spectra have been carried out for the third-row oxyanions, transition-metal oxyanions, SiO2 and TiO2, by the discrete variational-Xα cluster method. The calculated orbital energies are consistent with those determined from the XPS experiments. Theoretical XPS line shapes with Gaussian are generally in good agreement with the observed spectra. However, underestimation has been found for the photopeak intensities in the low-binding-energy region of TiO2. The discrepancy is partially attributed to the use of inaccurate photoionization cross-section for the Ti3d orbital.

Notes: A general way for drawing the state correlation diagram and seeking the reaction path is presented. If a high-symmetry reacting system is given, its least-motion path that maintains the symmetry is primarily examined. For a given state, it is judged whether the least-motion path is symmetry allowed or forbidden. If allowed, it is called the direct process. If forbidden, the symmetry imposed on the system should be relaxed, resulting in the mixing of MO's. Then, the energy barrier of the avoided crossing for some excited states is removed and the possible reaction path is found. After this procedure, the symmetry-allowed paths may be sought by the geometry optimization with a suitable wave function. By the use of such a procedure, the dissociation of diazomethane and (3H-)diazirine is found to proceed via the Cs and C2 symmetries.

Notes: The differential cross-sections for high-energy electrons scattered from water have been measured over a wide range of momentum transfers. The effect of chemical binding was seen from the comparison between the experiment and the calculation for an independent-atom model. The ab initio calculation using SCF MO was carried out with respect to the elastic scattering. It was in a good agreement with the experiment and thus a reliable electron distribution in water was obtained.

Notes: Metal chelates [(C2H2X2)2M]n with ligating atoms X = NH, O, S and central atoms M = Be2+, Mg2+, Ni2+, Zn2+ (n = 0, ± 2), and M = Li+, Cu+ (n = ± 1, -3), have been studied in our Laboratory for some years by ab initio calculations. In this article it is shown that certain features of the complexes are the same for all the different metals in our series. These features include a drastic reduction of the energy gap between unoccupied and occupied orbitals, when electrons are added to the positively charged complexes. This change of the energy gap is shown to be an effect of the ligand dimer [(C2H2X2)2]n. But this dimer can only exist when a positive ion, e.g., a metal ion, forms a bridge between the two monomers. The reduced energy gap implies a strong bathochromic shift of the electronic spectrum and low electrical resistivity.

Notes: The photoisomerization of imine compounds is studied in terms of an ab initio MO CI calculation. The potential curves of the syn-anti isomerization via the rotation and the inversion are examined for benzaldimine. It is suggested that the photoisomerization is initiated through the rotation around the C—N bond in both singlet and triplet states. The ease of the photoisomerization is found to be determined by both the conformation of phenyl ring in the ground state and the energy difference of vertically excited states between two isomers.

Notes: To aid in designing new therapeutic iron chelating agents, the mechanism of iron binding to prototypic heterocyclic carboxaldehyde thiosemicarbazones has been studied. Based on molecular orbital and spectroscopic studies, iron (II) is found to bind in a covalent manner, while iron(III) seems to interact ionically. However, with both iron(II) and iron(III), chelate formation is dependent on charge interaction between the metal and the coordinating atoms of the ligand.

Notes: No. Abstract.

Notes: Molecules exposing considerable microsurface areas to the surrounding solvent such as amino acids, nucleotide bases in biopolymers, or various drug molecules, antigens, and substrates, tend to be pulled apart or pushed together as the case may be by the surrounding solvent medium. These solvophobic forces and their quantitative theory were introduced sometime earlier [O. Sinanoğlu, in Molecular Associations in Biology, B. Pullman, Ed. (Academic, New York, 1968), pp. 427-445; and references therein]. The forces involve both enthalpy and entropy effects. The case in water had been shown to involve particularly strong forces but not differing in kind from other solvents. There are still somewhat contradictory views on the aqueous case, referred to in the literature as “hydrophobic bonding” treated as a phenomena unique to water and previously thought to involve only entropic effects. The temperature dependence and volumetric effects on the earlier type of “hydrophobic bonding” are presently not always reconcilable with recent experimental evidence. By contrast, the solvophobic force theory allows the calculation of the full solvent effect on various association or isomerization equilibria using its quite rigorous relations and basic data derived with it from liquids and solutions. [cf., e.g., for an early application to cis-trans azobenzene isomerization, T. Halčǐoǧlu and O. Sinanoğlu, Ann. N.Y. Acad. Sci., 158, 308 (1969)]. The solvophobic force theory had also introduced as a measurable new quantity, “the thermodynamic microsurface area change of a reaction,” which is now finding considerable use in protein chemistry [cf., e.g., F.M. Richards and T. Richmond, in CIBA Symposium Proceedings on Molecular Interaction and Activity in Proteins (CIBA, New York, 1977)]. Solvophobic force theory has been tested and applied recently in detail in high pressure liquid chromatography (HPLC) by C. Horvàth, W. Melander, and I. Molnar [J. Chromatograph. 125, 123 (1976)] who verified the predicted temperature, molecular surface area, and salt concentration dependence. It worked well when applied in detail to the complicated methanol-water mixtures as well. The theory also gave the first a priori derivation of the experimentally well-known protein salting-in-salting-out curve [W. Melander and C. Horvath, Arch. Biochem. Biophys. 183, 200 (1977)]. We have now extended the theory further with new basic derivations which eliminate the need for the direct calculation of some cumbersome effects. The theory has also been recently applied to a number of new areas including drug-receptor interactions, aqueous amino acid interactions, and to multicomponent phase equilibria; the latter also of chemical engineering interest. Solvophobic interactions, although part statistical thermodynamical in nature, were shown to be usable as if they were an ordinary U(R) potential added on to quantum-chemical intrinsic in vacuo potentials for prediction of conformations in solution [O. Sinanoğlu, in The World of Quantum Chemistry, R. Daudel and B. Pullman, Eds. (Reidel, Dordrecht, 1974)]. There have also been some recent applications that illustrate the convenience of our solvophobic theory in correcting for solvent effects once quantum-mechanical molecular electronic structures and potential-energy surfaces are computed.

Notes: MINDO/3 calculations have been performed on the Clostridium MP flavodoxin active site (a complex of the redox active coenzyme flavin mononucleotide sandwiched between the side chains of methionine and tryptophan) at various redox levels using coordinates derived from x-ray diffraction studies of the holoenzyme. Frontier orbital indices were calculated and indicate that reduction of the flavin is accompanied by induced polar states in the amino acid side chains. This stabilization of charge by the amino acid side chains could account for the reaction rate enhancement of flavin reduction catalyzed by flavodoxin. Frontier orbitals for free flavin, for the flavodoxin bound flavin without the amino acid side chains, and for the oxidized Desulfovibrio vulgaris flavodoxin active site were computed for comparison.

Notes: A self-consistent modified extended Hückel (SC-MEH) MO method has been extended to study the electronic structure and bonding in the complex ions: PtC42-, Pt(NH3)42+, cis-Pt(NH3)2Cl2, trans-Pt(NH3)2Cl2, and Pt(CN)42-, in which the atomic basis functions were derived from the published relativistic Dirac-Fock functions. Both quasirelativistic (in the frozen spin approximation) and nonrelativistic calculations were made to include the 5s 5p 5d 6s and 6p levels of Pt and the nsnp valence levels of the ligands. Similar calculations have been completed in the nonrelativistic mode for Pt(NH3)2 (OH2)L and Pt(NH3)2L2, where L is the monodentated bond at the N-7 position of guanine in DNA and L2 are the bidentated bonds at the N-7 and O-6 guanine sites, respectively. Based on these initial results specific conclusions have been made regarding a proposed biochemical mechanism that has been advanced to explain the unique anticancer activity of cis-Pt(NH3)2 Cl2 (DDP).

Notes: The problem of the existence of a prostaglandin (PG) receptor is still controversial, but it seems to be clear that the main action of the PGs concerns the interface of the cellular membrane (which has a predominantly lipid character) and the extracellular liquid (which mainly consists of water). Also especially in the case of the exogenous PGs, the transport of these substances in the organism in influenced by the hydrophilic-hydrophobic properties of the reactive molecules. These properties can be estimated by MO studies using a solvation model and a semiempirical method of calculation. This paper is concerned with the molecules of PGA1 and PGE1. MO studies were performed for these molecules with the CNDO/SW method in order to calculate ionization potentials, dipole moments, and molecular polarizabilities. With these computed molecular properties, the electrostatic and van der Waals interactions with water and octanol molecules are evaluated and compared within the framework of the original model. This study allows the differentiation of the hydrophilic-hydrophobic character of the PGs investigated and the comparison of the results with experimental data.

Notes: The electronic structures of catechol derivatives, 3,4-dihydroxy-1-substituted benzenes, including epinephrine and norepinephrine, were calculated by means of the CNDO/2 and the MINDO/3 methods. Correlation between the maximum velocities divided by the molecular weight, as between the inverse of Michaelis-Menten constants in the O-methylation reaction catalyzed by catechol-O-methyltransferase and the electronic indices calculated on the basis of the frontier electron theory was found. The correlation coefficient between the observed maximum velocities divided by the molecular weight and the calculated ones was 0.873. The regression analysis showed that the Coulombic forces at the C3 and C4 positions and the electron acceptable indices at C6 and meta-O positions represented the best correlation with the biological activities (Vmax and Km). The electronic indices of the catechol moieties represent the rate-limiting factor of the O-methylation reaction.

Notes: Ligand-field theory is treated in a systematic and unified way by means of Lie algebra.

Notes: Binding energies of ionized donor-exciton complexes in crystals of CdSe, CdTe, and ZnSe are calculated using the most realistic effective potentials available for the electron-hole interaction.

Notes: The analysis of nonadditive effects in small palladium clusters via LCAO-MO-SCF calculations using the nonempirical pseudopotential model is presented. The results are tentatively correlated with previous studies on various metal clusters (Li, Be, Mg, etc.) for which a good knowledge of multibody terms has provided a valuable guide for understanding the cluster stabilities and their chemisorption capacity.

Notes: In Phillips' spectroscopic theory of semiconductors, the covalency and ionicity are derived empirically from the average band gap between the highest valence and lowest conduction bands. In this paper an explicit expression for the average band gap is derived based on a continued fraction representation of the polarizability matrix. Results of a calculation for six covalent and polar semiconductors, using the pseudopotential model, are presented and compare favorably with experimental values.

Notes: Exchange interaction parameters in several trinuclear chromium(III) complexes [Cr3O(R-COO)6(H2O)3]X·nH2O (R = CH3, C2H5) are determined in the study of luminescence, time-resolved luminescence (TRS), and decay time measurements over the temperature range 4-100°K. The presence of four inequivalent trinuclear clusters and a luminescence anomaly at 25 and 55°K are observed in the basic chromium(III) acetate prisms. Magnetic anomalies reported earlier should be attributed to the distortion and disordering of the hydrogen-bonded channels in crystals at low temperatures.

Notes: A molecular LCAO Hartree-Fock procedure was used to calculate total energies of axially stretched normal paraffins containing up to nine carbon atoms. The results are used to model the mechanical properties of polyethylene.

Notes: The wave function and total energy of ammonia crystal are determined from the results of a totally optimized STO-3G ab initio calculation of the tetramer, pattern of the crystal. The crystal electronic energy is expressed in terms of the interatomic distances and charge density matrix elements. The geometry obtained from optimization of the tetramer energy on geometric parameters is in very good agreement with the experimental data of the crystal.

Notes: It is shown that trial functions involving Padé approximants yield satisfactory results for the ground state of the helium atom. In particular, the five-parameter form reproduces the best variational function of the type Ψ = e-zsφ(u) obtained numerically, to a remarkable extent.

Notes: This investigation is a continuation of a study on the optimality of MO basis sets of Gaussian functions, when constructed from AO basis sets optimized for the neutral atom or for ions. A formal charge parameter Q is used to adjust AO basis sets to the molecular environment, by virtue of a simple quadratic equation. Calculations are performed on a series of seven C2 hydrocarbons (C2H2, C2H4, C2H6, C2H3+ (open), C2H3+ (bridged), C2H5+ (bridged), and C2H4- radical anion). A simple rule is formulated to give approximate values of the charge parameter Q.

Notes: Variational iterative solutions for multipole polarizabilities of the H atom have been carried out. The resulting numerical values of the Cauchy coefficients are in harmony with our recent work. The large frequency expansion of the multipole polarizabilities of the H atom is also calculated. The first term in such expansions, the multipole oscillator strength sum rule, is in harmony with earlier work.

Notes: In an attempt to apply band-structure calculations to polymers with “reduced” symmetries, and simultaneously to examine some of the causes of band gaps in semiconducting polymers, two types of symmetry reduction are applied on the all-trans polyacetylene chain. Using a neighboring-segment approximation the consequent perturbations were used to calculate changes to the band structure of the undistorted chain and the energy gaps compared with those obtained from a direct calculation on the distorted polymer.

Notes: The Hohenberg-Kohn theorem on the definition of the ground-state energy of a system of fermions is discussed.

Notes: An earlier version of the MCSCF procedure has been simplified for CI expansions involving orbital excitations only without decoupling. The computational time has been thereby cut down to a quarter of that taken by the earlier version.

Notes: Solutions to some practical problems that arise in multiple scattering calculations on large molecules are discussed. (1) Numerical instabilities near the zero of energy can be removed by rescaling the secular matrix. (2) The calculation of structure factors can be made much more efficient by the application of symmetry projection operators. (3) An energy search procedure is described that ensures that no states are inadvertently neglected. Test calculations incorporating these changes illustrate the improved numerical stability, and show decreases in computation time of 30-60%, when compared to previous codes. The procedures suggested here are applicable to both relativistic and nonrelatitistic calculations.

Notes: A solution of the wave equation for the nuclear motion of a diatomic molecule with an exponential potential function and the rotational term included has been performed by the Schrödinger-Infeld-Hull factorization method. Two different procedures have been followed for the factorization of the wave equation, and in each of them, as type-B factorization is incapable of producing wave functions, it has been transformed to type-C factorization, and a complete set of wave equations have been obtained. Of the two procedures, although the former appears to be more direct, the latter is the more accurate and generalized one.

Notes: Hydrogen 2p orbitals have been introduced into the basis set to calculate the Fermi contact term of spin-spin coupling constants using the FPT INDO method. Different coupling constants show different sensitivity to these hydrogen polarization functions. Some improvements are found for molecules containing N or F. Calculations of proton-proton geminal coupling constants give more negative results than those of FPT INDO, yielding a better agreement with experimental values. The π-transmission mechanism is notably exaggerated.

Notes: The electrostatic interaction for arbitrary symmetry (including non-simply reducible point groups), different coupling schemes, and several open shells is discussed from a common viewpoint using the particle-number representation (second quantization). The different coupling schemes simply arise from a different interpretation of the relevant symmetry group. Using the adjective CFPS of the preceding paper, the interaction within several open shells can be calculated. In the case of one-center expansions like ligand field theory the many-particle matrix elements are directly expressed by the radial Slater integrals using isoscalar factors. An example of the formalism is worked out. By the way the unitary transformations between strong and weak field coupling schemes are expressed recursively in the number of particles.

Notes: The one-center expansion (OCE) method is extended to evaluate molecular wave functions for molecules with heavy off-center nuclei. This extension is achieved through the use of model potentials (MP) to approximate the highly bound core orbitals. The remaining diffuse valence charge distribution is then rather easy to simulate using OCE. The formulation of the method is described. New molecular integrals are solved to a high degree of accuracy. Successful results are reported for H2O, H2S, and N2. The valence electron distributions and orbital energies are in good agreement with those obtained from more complete calculations. The method combines the computational economy of both OCE and MP procedures, resulting in a potentially useful package for further chemical applications.

Notes: A theoretical order-disorder model for the main phase transition of lipid monolayers is presented here. The model contains approximate treatments of hard-core repulsions and attractive forces between molecules. The lipid molecules are assumed as the square plates and then the entropy of the configurations is obtained using the placement method in a two-dimensional lattice. The approximation used to compute the thermodynamic behavior of the model turns out to be a kind of molecular-field approximation. The resultant pressure-density isotherms for the three-state model show the characteristic van der Waals loops.

Notes: A theoretical model is given that describes the nature of a membrane that is made up of several species of lipids. The model will partially simulate the biomembrane through numerical analysis.

Notes: The salt Pt6(NH3)10Cl10(HSO4)4(MGSPOS) with the average valence of Pt 2.33 was prepared in powder form by means of partial oxidization of [Pt(NH3)4][PtCl4](MGS) substituting HSO4- for 1/6 of both ligands NH3 and Cl-. The electrical conductivity σ has a maximum at about 230 K and a minimum at about 280 K. Therefore, the conductivity σ is semiconductive with the gap energy 0.04 eV below 230 K, and metallic in the sense of dσ/dT 〈 0 between 230 and 280 K, and again semiconductive above 280 K. The thermal analysis by DSC shows only on heating, three anomalies, G at ∼150 K, A at ∼ 200 K, and B at ∼250 K. The exothermic anomaly A succeeding to the weak anomaly G indicates that a glass transition takes place, and endothermic anomaly B corresponds to a structural transition. These two transitions may correspond to the two metal-semiconductor transitions found in the σ measurement. The ac calorimetry shows a specific heat anomaly also only on heating.

Notes: The fundamentals of the theory of density matrices are reviewed. The density matrix is introduced first in terms of state vectors for pure states over which a probability distribution is given. It is then shown that the applications of the density matrix do not require knowledge of the particular probability distribution over pure states that may have been used for its original determination. It is stressed that density matrices, like state vectors, describe ensembles rather than single systems, as determinism in quantum theory (except for some conservation laws) is of a statistical character only. The entropy of a quantum-mechanical system is defined as the average per system of the entropy of an ensemble of systems described by the quantum-mechanical (pure or mixed) state. For pure states this entropy is zero, in agreement with the fact that state vectors constitute the maximal information that can be given about a quantum-mechanical system. As the entropy should be independent of the particular probability distribution over pure states that may have been used to form the mixed state, it is calculated from statistical considerations about an imaginary ensemble in which the eigenvalues of the density matrix are probabilities over pure states given by the eigenfunctions of the density matrix. The entropy is calculated for a mixture of two nonorthogonal states. The time evolution of the density matrix, in general, is not a unitary transformation, as observable systems are not closed and therefore in the Hamiltonian contain unknown external parameters that are not the same for the entire mixture. Therefore transitions between pure states and mixed states take place. Those from pure to mixed states take place spontaneously and irreversibly under increase of entropy. Transitions from mixed states to pure states can be fabricated; an example is given. As Wigner's thought experiment of mixing coherently the two beams from a Stern-Gerlach apparatus according to estimates of Bohm should be impossible by loss of coherence in the gradient magnetic field, it is indicated how Wigner's ideas may be realized by using, instead, polarized light beams from birefringent crystals. After a brief review of the quantum theory of measurements and the various arguments usually given for the reducibility of state after the measurement, Wigner's argument against blind use of these arguments is interpreted as a limit of validity of the familiar probability rule of quantum theory, which should not be applied after an unsuccessful measurement. By splitting light beams coherently by glass plates, one can by the same apparatus have measurements both of the relative intensities of two coherent beams, and of interference effects between the two beams. This shows that complementarity applies to the individual systems in the beam, as they reach the glass plate where a choice must be made between two complementary possibilities of future measurement. Finally, by describing Schrödinger's cat by a density matrix and asking about its time dependence, it is shown that, if it were possible to measure the interference effects between living and dead cats used by some people as a justification for writing down a state vector for the cat, then this measurement could be used to revive some dead cats.

Notes: Let two strongly continuous one-parameter semigroups of contraction operators, {Z(t); t≧ 0} and {Z0(t); t ≧ 0}, determine physical evolutions realized in the Hilbert spaces ℋ and ℋ0, respectively. We consider the mappings under Z(t) and Z0(t), when the infinitesimal generators G and G0 belong to a product class, properly defined with respect to an H0-norm. The inversed transform of each side in the identity R(λ, G) = R(λ, -iH0)-iR(λ, -iH0)VR(λ, G), which represents a simple algebraic decomposition of the resolvent of G, converges on Ψ if and only if Ψ ∊ D(G). By iteration an asymptotic series emerges, when t → 0. Numerical considerations of this approximation in its first order may support the form of a deviation from the pure exponential decay when the semigroup is compared with a integral transform corresponding to a certain self-adjoint H0 in ℋ0. The deviation may hardly ever be observed and it is therefore most fruitful to discuss the results inside the framework of the enveloping algebra of the semigroups.

Notes: The nodes where the wave function changes sign in conventional quantum theory can alternatively be regarded as places where the wave function descends to zero but rises again without changing its sign. This behavior is accomplished by adding delta function barriers to the potential energy wherever a node occurs in the wave function. This model is supported by the recognition that the wave functions of a subsystem, which is not truly isolated, are really marginal amplitudes which implicitly average out interactions of the system with its environment. The finite effect of these interactions is to replace the delta function barriers by barriers of finite width and height.

Notes: The quantum-mechanical virial theorem (in diagonal and nondiagonal forms) for molecules in an external, weak, uniform electric field is used for obtaining the analytical expression of the potentialenergy surface in the Hartree-Fock-Roothaan one-determinantal approximation. The polarizability tensor components of the H2O molecule are computed on this basis. The dependence of basing the atomic orbitals upon the intensity of the applied field lead to good coincidence of results with the data of near Hartree-Fock calculations.

Notes: Intensities of a1Δg←X 3∑g- and b 1∑g+ ← X 3∑g- transitions in molecular oxygen have been calculated on the basis of the INDO method taking into account spin-orbit coupling by perturbation theory. The transitions are magnetic dipole in nature. The first of them (a - X) steals its intensity from 3Πg-3∑g- and 1Πg-1Δg transitions, which are determined by the orbital angular-momentum operator. This source is not the principal one for the intensity of the second (b-X) transition. Its intensity is stolen principally from microwave transitions between spin sublevels of the ground 3∑g- state. The last source explains the large difference in intensities of the a-X and b-X transitions. Calculated oscillator strengths are in a good agreement with experiment. The same integrals that determine the intensity also determine the parameters of the spin Hamiltonian for the ground 3∑g- state. These parameters are in a good agreement with experiment also, showing the validity of the whole calculation. In a condensed phase the investigated transitions are enhanced by intermolecular exchange interaction. It is known that an external heavy atom (EHA) enhances the b-X transition of oxygen in solution, but does not influence the a-X transition. In the collision complex O2-EHA, which has a geometry without inversion symmetry, the microwave transitions between spin sublevels of the “3∑g-” state obtain electric-dipole moments, which are stolen from the charge-transfer transition. This mechanism explains the selective effect of EHA.

Notes: Potential-energy curves for CH4·CH4 have been calculated using the SCF LCAO MO method. Six different mutual orientations of the molecules have been considered and several basis sets have been employed. An analytical potential has been fitted to the computed energy values and used in a Monte Carlo calculation to yield a spherically averaged CH4·CH4 potential. The potential, if corrected for dispersion interaction, agrees fairly well with that derived from thermophysical data. At short distances it disagrees with the potentials derived from scattering experiments which, however, are also incompatible with the experimental potential for large distances.

Notes: The ab-initio LCAO-HF method that was presented in a previous work is applied here to the study of graphite and hexagonal boron nitride monolayers. The dependence of total energy, band structure and density matrix on the computational parameters that control truncations in infinite sums over the translation vectors g of the direct lattice is first considered. For these systems it comes out that very good results can be obtained by neglecting all but the first few terms in the sums. For instance, for exchange contributions it is sufficient to consider integrals associated with the first 13 g vectors, suitably grouped, to obtain an error in total energy below 0.0001 a.u. The calculations were performed using an STO-3G basis set; results concerning conformational minimum, bond energy, symmetric force constant, band structure, density of states, and population analysis are presented and discussed.

Notes: A modified form of Frenkel's time-dependent variation principle, suggested by McLachlan for state vectors, is employed to discuss the optimal time evolution of a density operator ρ(t). An ansatz is made for this operator such that i(dρ/dt) = [S, ρ], where S(t) is a linear combination of operators belonging to a particular manifold of “basis operators.” The expansion coefficients are required to minimize the error Tr{[S - H, ρ]† [S - H, ρ]}. Linear response functions corresponding to the variationally determined density operator are compared to those derived by means of linear decoupling procedures for propagators based on the same operator manifold. The two approximation schemes are not equivalent, in general, and several consistency requirements must be fulfilled before it can be ascertained that a given linear decoupling procedure corresponds to an optimal time development of the density operator in the sense of McLachlan. Finally, the general applicability of the suggested variation principle is discussed.

Notes: The resonance energy of conjugated benzenoid systems is expressed as contributions arising from independent conjugated circuits. The scheme has been applied to numerous very large conjugated systems. In many cases, it was possible to find regularities in the increments for the resonance energy within a family of benzenoid systems as the number of benzene rings is increased.

Notes: A previously given iterative procedure to improve wave functions is analyzed. Its relationship with other well-known approximation methods is investigated. Hypervirial operators depending on a real parameter are proposed and their connection with the employment of an infinite number of hypervirial relations is analyzed. A way to use formulas that are valid for exact eigenfunctions in the case of approximate functions is presented. Formal results are applied to the harmonic oscillator and hydrogen atom models in order to show their practical utilization.

Notes: Ab initio MO calculations, using both minimal (STO-3G) and extended (Roos-Siegbahn) basis sets are reported for the systems methanethiol-imidazole, methanethiol-imidazole-formaldehyde, and methanethiol-imidazole-formamide, which, together with a point-change representation of a long α-helix, form models for the active site of papain. It is shown that the large electric field exerted by the helix in the active-site region is responsible for the presence of the essential residues Cys 25 and His 159 in the form of an ion pair RS- ··· ImH+, which is crucial for a recently proposed mechanism for the catalytic action of the enzyme. Also, an explanation is given for the anomalies in measured pK values for these residues. Detailed studies on the (sub)systems show that minimal basis sets lack the flexibility necessary for describing the type of proton transfer involved. We conclude that α-helices are essential parts of enzymes and that they play a significant role in the catalytic process.

Notes: The floating spherical Gaussian orbital (FSGO) method has been used to study the equilibrium geometries and electronic structures of some hydrogen-bridged systems containing lithium, beryllium, boron, and carbon. The predicted geometries are in good agreement with other theoretical estimates. The binding energies for these hydrogen-bridged systems are estimated and discussed. The FSGO total energies (EFSGO), for all systems studied here, are found to be well correlated to nuclear-nuclear repulsion (Vnn) and nuclear-electron attraction (Vne) terms by a relation of the type EFSGO =\documentclass{article}\pagestyle{empty}\begin{document}$ {\textstyle{3 \over 7}} $\end{document} (2 Vnn + Vne).

Notes: A solution of the wave equation for the rotational-vibrational motion in diatomic molecules with a new exponential potential function is carried out in detail. The solution gives simple expressions for the wave functions, eigenenergies, and other related spectroscopic constants. With these expressions Franck-Condon factors for the R branches of the A 1∑u--X 1∑g+ band system of Ca2, have been calculated which are in excellent agreement with experiment. Various strengths and weaknesses of the present method are also discussed.

Notes: Fully optimized lengthy Gaussian expansions of hydrogenic 1s and 2pz functions have been practically reproduced by generating the variational parameters by means of suitably chosen functions involving only few parameters. Hydrogenic 3dxy and 4fxyz functions have been approximated in a similar way. The usefulness of such generating functions is briefly discussed.

Notes: The qualitative theory of Shockley surface states is used to discuss the existence of surface states on three faces of niobium carbide: (001), (011), and (111). The results obtained for NbC are generalized to include other simple transition metal compounds MX, X = C, N, O. Surface states of these and related materials are of interest in chemisorption and catalysis. The qualitative theory of Shockley states appears to be useful tool in detecting strong covalency effects in solids.

Notes: We prove by elementary geometric methods and within the Born-Oppenheimer approximation that as the nuclei of a molecule are dissociated into spatially separated clusters, the discrete molecular energies approach sums of the energies of isolated subsystems. Our methods also show that the spectral projections associated with the discrete molecular spectrum asymptotically approach direct sums of suitable spectral projections for the isolated subsystems. These results apply to any system of particles interacting by asymptotically vanishing pair potentials. We prove that the 1/R expansion for discrete molecular potential curves is asymptotic as R → ∞, and we discuss the behavior of the coefficients of the 1/R expansion for the ground state of H2+.

Notes: This paper is devoted to the interpretation of DNA UDS caused by complexing DNA with Me ions using the correlation between the values of the spectral changes of the DNA absorption bands and the constants of reaction velocities of hydrated electrons with Me2+ and H3O+ ions.

Notes: In this paper, a fresh attempt has been made to discuss dual-symmetry groups by which the supra-antarafacial cycloaddition and the chelatropic reactions can be correctly described, as a direct consequence of which the corresponding energy correlation diagrams are successfully constructed. Furthermore, the energy correlation diagrams of sigmatropic reactions are also established by the use of another kind of single-symmetry group.

Notes: A comparison between the construction of symmetry-correlation diagrams and the perturbation method for studying chemical reactions is carried out. The perturbation method consists of decomposing the system Hamiltonian H into a sum, H = H0 + H′. Various symmetry correlation schemes appearing in the literature may be explained by the nonuniqueness of the decomposition scheme. All symmetry selection rules may be viewed as the varieties. By examining the symmetry-correlation diagrams, processes under investigation may be called “forbidden” or “allowed,” depending on the topological feature. Of particular importance is the topology associated with the “avoided crossing.” By making the comparison, we can establish the correspondence of the two methods and conclude that the perturbation order furnishes the origin of the “forbiddenness” of a process.

Notes: Ab initio molecular-orbital theory has been used to study the 1,3-sigmatropic hydrogen rearrangements: propene → propene, formic acic → formic acid, and vinyl alcohol → acetaldehyde. Fully optimized structures of stable molecules and transition states have been determined using gradient procedures and the 4-31G basis set. Improved energies have been obtained using a variety of techniques with basis sets up to the size of double-ζ plus polarization (DZP) and electron correlation up to the CEPA/DZP level. Although both polarization functions and electron correlation lead to a lowering of the calculated barriers, the values remain substantial for all three rearrangements.

Notes: As is known, the Schrödinger equation for a particle in the ring-shaped potential V(r,v) = ησ2(2a0/r-a02/r2 sin2v)ε0, defined in the whole space, has been solved exactly. Here the eigenfunctions are represented in a form which is advantageous for concrete evaluations. The spin-orbit interaction energy ELS in quasirelativistic approximation is determined analytically, for the first time with a nonspherically symmetric potential. The influence of spin-orbit interaction on the eigenvalues of the spin-free problem and on the selection rules for electrical dipole transitions are investigated as well as the dependence of ELS on the position and depth of the potential minimum. The model can be useful for investigations of axial symmetric subjects like the benzene molecule or related problems and may be easily extended to a many-electron theory.

Notes: A hierarchy of necessary conditions that an exact density matrix of a pure state or an ensemble has to satisfy is derived, namely the hermiticity of certain operators F(k). For k = 1 this reduces to the well-known Hartree-Fock condition. It is then shown that the kth set of conditions is equivalent to stationarity of the energy with respect to unitary k-particle transformations. k-Particle generalizations of Hartree-Fock theory are then discussed both in the spirit of k-particle pseudoeigenvalue equations and in the framework of a Newton-Raphson-type constructive scheme.

Notes: An electron pair in a molecule is a complex entity and is incompletely described by the independent particle picture. Electron pairs are treated as Bose particles to include correlation of intrapair interaction effectively by the use of the Dyson transformation. The use of a Bose-type Hamiltonian permits a straight forward interpretation for geminals and facilitates the calculation of electronic energies. A Bose-type representation method is applied to the calculation of electronic energies of a lithium hydride molecule.

Notes: A simple transition probability formula is applied to the core hole ionization in CO, satisfactorily predicting the ratio between the intensity of the two inner-shell peaks. The formula is also applied to the full ESCA spectra of water and methane; the theoretical results agree reasonably well with the experiment.

Notes: Global features of potential-energy curves are drawn using minimal basis SCF-CI method. Potential-energy surfaces are drawn at points of interest. The mechanism of the chemiluminescence is discussed along with the measurement of a rate constant exploiting the phenomena.

Notes: A theoretical study has been made on the translational or “rattling” motion of homonuclear diatomic molecules encaged in β-quinol clathrates. By the use of an adjustable parameter for the van der Waals radius, experimental results have been explained fairly well.

Notes: The steric and electronic structure of the most potent uncoupler of oxidative phosphorylation, SF 6847, was studied. From 1H NMR measurements, it was found that the benzylic protons meta to the hydroxyl group are equivalent in neutral form, and that these protons tend to become non-equivalent on acid dissociation of the phenolic proton. The CNDO/2 calculation of the dimethyl derivative of SF 6847 showed that in the most stable conformation the angle of the malononitrile group with respect to the benzene ring is about 40° and that the planar form is quite unstable. Results on electronic absorption of SF 6847 in various media showed that the acid dissociation of its phenolic group is affected very much by small changes in the nature of the surrounding environment. It was also found that the calculated total energy differences between the anionic and neutral forms are correlated well with the uncoupling activities of 4-substituted 2,6-di-tert-butylphenol derivatives.

Notes: Fluorescence spectra and their temperature dependence of amorphous anthracene films doped with naphthacene were observed. An amorphous film was prepared by vacuum evaporation technique on a quartz plate cooled down to 30-40°K. The obtained results were discussed in relation to the structural change in the amorphous film.

Notes: The range in ordinary space of a defect in a solid, polymer, or a large molecule is an important parameter in many physical and chemical situations like, e.g., substitutional and interstitial impurities in ionic crystals, deep and shallow energy levels in semiconductors, core holes in systems studied by photoelectron spectroscopy, and hydrogen in metals. Both the electronic structure and the lattice of the host molecule or solid are normally influenced by the defect, and it is desirable to treat, as far as possible, these two aspects together. An explicit procedure for investigating the degree of localization is proposed that is based on the direct calculation of Wannier functions and partitioning technique.

Notes: Exciton transfers in the two-dimensional and spin-canted antiferromagnet (C2H5NH3)2MnCl4 were studied by investigating the absorption spectra 6A1g → 4T2g(4D) of Mn2+. A magnon side band (exciton-magnon simultaneous excitation) which has an anomalous shape with the cutoffs at the low-energy as well as the high-energy sides was observed. This anomalous shape could be reproduced in calculation as magnon side band, considering both the intersublattice and intrasublattice exciton transfers. From the analysis of this band shape, the magnitudes of the intersublattice and intrasublattice exciton transfers are estimated to be 10.2 and 6.8 cm-1, respectively. In the case of simple antiferromagnets without spin canting, the intersublattice exciton transfer process is forbidden because of the spin angular momentum. However, in the case of (C2H5NH3)2MnCl4, the prohibition of the intersublattice exciton transfer is removed by the canted-spin arrangement.

Notes: Relativistically parametrized extended-Hückel (REX) calculations are reported for MX4 (M = Ti, Zr, Hf, C, Si, Ge, Sn, Pb; X = Cl, Br, I) and for M(CH3)4 (M = Ti, C, Si, Ge, Sn, Pb). Satisfactory agreement is obtained with experiment and with earlier assignments both for the energy levels and for their relativistic splittings. This encourages the use of REX for assigning PES spectra. Double-ζ radial functions (REX and EHT) are given for Cl, Br, and I.

Notes: The spin densities of the radical anions from 4-methylcatechol, 4-carboxylic catechol, and 3-methylcatechol were calculated by means of the INDO method with the geometrical parameter adjusting method. The observed protion hyperfine coupling constants were exactly reproduced.

Notes: Monoalkyl and dialkyl quarternary ammonium salt-type cationic surfactants containing a phenoxy group as aromatic chromophores at different positions of alkyl chains were synthesized. Monoalkyl-type surfactants formed micelles in aqueous solutions. The bilayer structure similar to that found in phospholipid liposomes was obtained in the aqueous dispersions of dialkyl-type surfactants. The phenoxy groups were partially oriented and aligned in these molecular assemblies. The interactions of aligned phenoxy groups in micelles or bilayers resulted in luminescence with lower energy than that of the monomer fluorescence or in a radiationless deactivation of excited states. Such interactions could be effectively prevented in bilayer structures composed of surfactant molecules containing a “spacer chain” between aligned chromophores. Evidence was obtained supporting energy migrations between aligned chromophores that were separated by a dodecyl chain in the bilayer structure.

Notes: Solvent effects and pH effects on CD and MCD spectra of phenylalanine and its derivatives were observed. CD spectra of N-acetyl-L-phenylalanine ethyl ester in various solvents were analyzed using the rotamer population determined by NMR. The magnitude of optical activity due to each rotamer was determined for two CD bands. The signs of Cotton effect of rotamers I and II are positive and that of rotamer II is negative. The optical activity is greater in polar than in nonpolar solvents. MCD spectra of phenylalanine reveal a remarkable pH effect. In the alkaline pH range it is of the toluene type, from which it turns to a weak spectrum in the acid pH range.

Notes: The properties of V-coupling coefficients from SO(3) to the octahedral group are discussed and, for the purpose of practical use in ligand field theory, the numerical values from j = 1/2 to j = 12 1/2 have been worked out through an Elliott-503 computer. The values for j ≤ 4 1/2 are listed.

Notes: This paper outlines a model for calculating the localized states of a 〈 100 〈 edge dislocation in α-Fe. The model used for the calculations is based on the multiple-scattering model (SCF-X α-SW). The purpose of this research is twofold: (1) To determine changes in electronic structure of the lattice near the core region of defects in α-Fe. (2) The variations of hydrostatic pressure about an edge dislocation produce a rearrangement of the conduction electrons. The question is what electrical interaction might be expected between a dislocation and a charged solute atom. The calculations show that the electrons tend to flow away from the compression side toward the dilated regions. The electrical contribution to the binding energy of a solute atom and a dislocation in α-Fe is of the order of 0.01 Ry/electronic unit charge of the atom.

Notes: The ion implantation of oxygen in either n or p-type gallium arsenide results in the formation of a semi-insulating layer that is thermally stable. There still is no reasonable explanation for this phenomenon (except one of us has made a preliminary investigationSee Ref. 1.). In this paper, we simulate the crystal of GaAs by a molecular cluster. As is well known the molecular-cluster model has a shortcoming in introducing “surface states” owing to the dangling of atomic bonds on the surfaces.See Ref. 2. In order to suppress this effect, the dangling bonds were saturated with hydrogen atoms as proposed by Larkins.See Ref. 3. Therefore, in this paper the perfect crystal of GaAs was simulated by a cluster of atoms with a composition Ga4As13H36 (which corresponds to a pseudocluster Ga13As13). In the authors' opinion, this kind of treatment may cause minor changes in the positions of a few energy levels, but the general energy diagram pattern will reflect the main features of the real crystal. The electronic states of the entire system are then computed by an LCAO-MO technique (extended-Hückel theory). The substitutional oxygen OAs, the arsenic vacancy VAs, and the OAs—VAs complex are treated similarly. The results of the calculation show that the substitutional oxygen (OAs) forms deep donor levels in the forbidden gap. The arsenic vacancy also forms donor levels in the forbidden gap. On the other hand, the OAs—VAs complex forms deep acceptor levels in the forbidden gap. Thus we have given a reasonable model for the mechanism of the formation of an insulating layer produced by oxygen ion implantation in GaAs.

Notes: The hypothesis is introduced, that nonrigid drug molecules interact with biological receptors in a well-defined but unknown conformation that need not belong to an energy minimum. A verification of this assumption has to include a match of conformation and chirality. Even the different action of pairs of nonrigid enantiomers allow one to recognize uniquely the important role of conformations because flexible molecules are partially able to avoid wrong attachment to the receptor. According to a simple model, a monotonic relation between biological activity and rotational barriers (and/or torque constants) can be expected and, by means of rank correlation analysis of adrenergic activities, has been found, too. In order to apply the latter method, particularities concerning quantitative structure-activity analyses of nonrigid and/or chiral molecules are worked out. We optimized conformations by means of analytically calculated first partials of the energy and a quasi-Newton method. Using a convenient numerical method, it was shown that coplanar conformations of catecholamines are generally unstable. Moreover, CNDO/2 proved to be a reliable method for conformational studies, in which rotations about conjugated bonds [1] are not considered.

Notes: Open-shell single-determinantal calculations are reported here for the molecular species H2, Li2+, N2, O2 (triplet), O22-;, O2-, O22+, O2+, and F2; corresponding closed-shell calculations are reported for the species H2, N2, O2 (singlet), O22-, O22+, and F2. The floating spherical Gaussian orbital (FSGO) method was employed. The calculated trend in bond lengths of isonucleic diatomic molecules is in agreement with experiment. For heteronucleic diatomic molecules, however, the experimental trend in bond lengths is not obtained; in this connection, the effect of lone pairs on bond length is discussed. The dissociation energies of H2 and Li2+ are evaluated. The energy gap between the triplet and singlet states of the oxygen molecule is calculated to be 8.96 eV compared to the experimental value of 4.54 eV.

Notes: Using a modified form of Sharma's method for the expansion of a Slater-type orbital in spherical harmonics about a displaced center, a general expression for the overlap integral between two orbitals is derived that is equivalent to that given by Sharma. By use of a simple kind of “computer algebra” this expression is developed here into a formula that is equivalent to earlier known ones and may have computational advantages when the generated formula coefficients are stored. This method is capable of extension to more complicated integrals.

Notes: The optimality of MO basis sets of Gaussian functions, when constructed from AO basis sets optimized for the neutral atom or for atom ions, is investigated. A formal charge parameter Q is defined and used to adjust the AO basis sets to the molecular environment, by virtue of a simple quadratic expression. Calculations on a series of C1 hydrocarbons (CH2, CH3, CH3+, CH3-, CH4) using 3G basis sets indicate considerable variations in the optimum Q value with the molecular species. The proposed method offers a simple alternative technique to a full molecular basis set optimization.

Notes: It is shown that the treatment of the superexchange in insulators, based on the spin projected extended Hartree-Fock (EHF) method may be considered as a direct generalization of Anderson's ligand field (or kinetic exchange) theory: the EHF methods permits to construct explicitly the effective orbitals for the magnetic electrons and to determine all the parameters occurring in Anderson's theory from only one variational problem.

Notes: The configuration interaction method where a single reference state is composed of a linear combination of reference configurations is analyzed in detail. In this method single and double replacements are constructed by applying annihilation and creation operators on the reference state. The analysis is based on the recently derived factorization of the direct CI coupling coefficients into internal and external parts. Using the internal coupling coefficients the integrals are transformed to new entities which are used in the diagonalization step. This two-step procedure differs significantly from the usual straightforward one-step direct CI procedure. A number of operations analysis shows that calculations with the present method should be feasible even with a large number of reference configurations in the reference state. Based on first-order perturbation theory the accuracy of the method is predicted to be close to the accuracy obtained with the usual CI method with many reference configurations.

Notes: By introducing a finite basis, the inhomogeneous equations of perturbation theory are approximated by systems of linear equations. We present and discuss a Padé-type algorithm suitable for large-order systems and having better convergence properties than the classical Jacobi procedure.

Notes: This paper reports some theoretical studies of PGA1 using conformational and molecular orbital techniques. The conformation energy (CE) calculations, using empirical potential-energy functions, for intrinsic torsional rotations around C12—C13 (θ), C7—C8 (Ψ), and C14—C15 (φ) show a number of energy minima. The relative value of the CE for these minima ranges from 4.09 to 8.01 kcal/mol. An additional rotation around C4—C5 (χ) giving “twist” to the carboxyl chains lowers the CE value by 2-3 kcal/mol and a conformation with CE value 2.39 kcal/mol less than crystallographic one is obtained. The interchain interaction energy showed changes with conformations. No significant change in the interchain interaction energy was observed due to “twist” in the carboxyl chain. The isopotential mapping study demonstrated the probable ionic binding site near the carboxyl and the ring (O9) oxygens. Conformational and molecular orbital results are discussed in the light of the reduced abortifacient potency of PGA1 with respect to PGF2α and the possible role of Ca2+ ions in this action.

Notes: It is assumed that the chemical reactivity may be described in terms of the appropriate energy density calculated for each atom. Indices of energy density for deprotonization of uracil, 5-nitro- and 5-aminouracil are presented. The results of a quantum chemical calculation of the deprotonization energy are in qualitative agreement with experimental pK and ΔH values.

Notes: The orbital approximation that polyelectronic atoms can be qualitatively described by using hydrogenlike orbitals was tested quantitatively. Hydrogenic atomic orbitals (HTO's) were investigated as best-atom minimum-basis set functions and were compared to best-atom Slater orbitals (STO's). The two sets of wave functions were compared in terms of the ground-state energies, the electron densities at the nuclei, charge distribution in space, electron populations, and atomic “radii.” In the atomic calculations the HTO's give somewhat poorer atom descriptions that STO's. SCF calculations on diatomic molecules (N2, CO, and HF) indicate that while HTO's give higher molecular energies, they closely mimic STO's in terms of electronic reorganization. The principal difference between HTO and STO results was attributed to the compact nature of the HTO 2s orbital.

Notes: A formulation of quantum mechanics is presented based on the theory of semigroups and the associated enveloping algebras of functions defined on countable subsemigroups. The existence of a unique *-involution is not assumed. The fundamental elements of a semigroup are identified with experimental precedures for the separation of subensembles from a given ensemble of experimental systems. Observables are represented as elements of enveloping algebras, and ensembles as density matrices within an enveloping algebra. The statistical properties of ensembles are expressed in terms of traces defined on the semigroup and its enveloping algebras. The elements and generators of the Poincaré group can be defined and interpreted in the usual way. A variety of applications is described, in which the theory of the density matrix plays an essential or effective role. Advantages associated with the resulting freedom from the limitations of Hilbert space are illustrated.

Notes: A scheme is presented for performing linear-combination-of-atomic-orbitals (LCAO) self-consistent-field (SCF) ab initio Hartree-Fock calculations of the electronic structure of periodic systems. The main aspects which characterize the present method are (i) a thorough discussion of both translational and local symmetry properties and the derivation of general formulas for the transformation of all the relevant monoelectronic and bielectronic terms under symmetry operators. (ii) The use of general yet powerful criteria for the truncation of infinite sums; in particular, the Coulomb electron-electron interactions are subdivided into terms corresponding to intersecting or nonintersecting charge distributions; the latter are grouped into shell contributions and the interaction is evaluated by multipolar expansions; the exchange interaction may be evaluated with great precision by retaining a relatively small number of two-electron integrals according to a truncation criterion which fully preserves its nonlocal character. (iii) The use of a procedure for performing integrals over k, as needed in the evaluation of the Fermi energy and in the reconstruction of the Fock matrix, which is particularly simple because it employs partially intersecting small spheres as integration subdomains where linear extrapolation is admitted. A comparison is finally made of our fundamental equations in the critical SCF stage with those obtainable by a recent proposal which uses Fourier transforms to express Coulomb and exchange integrals.

Notes: Pair correlation in the ground state of the Li isoelectronic sequence is studied through four approximate wave functions which incorporate inter- and intrashell pair correlation. Of these functions, two possess symmetry appropriate to a three-electron system, while two do not. The functions are not variational functions in the usual sense. They are instead fixed linear combinations of products of orbitals and pair functions for the appropriate states of two-electron atoms. They are considered here as zero-order approximations to the exact wave functions, and the corresponding zero-order Hamiltonians are obtained. The simplest of these functions is improved by the introduction of a screening parameter for the “outer” electron. This latter function is found to be a satisfactory compromise between accuracy and simplicity and is proposed for study via higher-order perturbation theory.