ALBERT

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

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

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

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

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

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

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

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

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

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

Notes: Double-helical polynucleotide conformations, poly(dA)·poly(dT), poly(d(A-T))·poly(d(T-A))·poly(dG)·poly(dC), and poly(d(G-C))·poly(d(C-G)) are analyzed by the atom-atom potential method. The energy optimization is carried out in the space of eight independent geometric parameters using analytical procedures for the constraints, taking into account the flexibility of the β-D-deoxyribose rings. At the first stage, the full screening of atomic partial charges was assumed. The structures of the calculated B and the A forms of DNA are characterized by low energy and absence of short contacts; the dihedral angles are near the average values in the monomers. With the typical energy difference of 3-5 kcal/mol nucleotide pairs in all cases, the B form is more preferable as compared to the A form. At the final step the effect of the Coulomb term is evaluated for poly(dA)·poly(dT) using various values of the effective dielectric constant (ε = 28, 24, 20, 18, 14, 12, 10, 8, 6, 4, and 1). If ε ≤24, the energy optimization leads A to B. We discuss the stereochemical details of the intermediate conformations on the A-B path and hypothesize the nature of stability of the A and the B forms and the mechanism of the A-B transition.

Notes: The energy of (H2O)3 has been calculated for 29 geometrical configurations of the trimer using the SCF LCAO MO method and extended as well as minimal basis sets of Gaussian functions. For two configurations two intermediate basis sets have also been tested. The results show the nonadditive component of the interaction energy to be small. They also indicate that fairly reliable results for the trimer can be obtained using minimal basis sets and the counterpoise method to eliminate the basis set superposition error. The nonadditive contribution to the interaction energy is shown to be mainly due to the long-range induction interaction.

Notes: We present two identities that permit a simplification of some lengthy derivations with spherical harmonics. Examples of their use are given.

Notes: The practical value of a wave function derives from its ability to estimate or predict the chemical and physical properties of the electronic structure which it describes. Reliability is defined as a property to be evaluated on the basis of (i) the magnitude of the difference between the measured and calculated values of the observable together with (ii) the sign of the difference, (ii) the number and nature of observables correctly estimated, and (iv) the number and type of electronic structures that are correctly estimated. Systematic statistical comparison implies the existence of both internally consistent sets of wave functions for homologous series of molecules and their corresponding experimental values together with reliable error estimates. The most complete data base currently available for comparison is the spectroscopic constants for the first and second row diatomic hydrides. Utilizing appropriate statistical comparison techniques, four approximations (CEPA, PNO-CI, GTO-SCF, and STO-SCF) are compared among themselves and against experimentally measured values. The CEPA approximation yields differences from experiment that approximate a normal error distribution, while the other approximations show systematic departures from experiment. Two values, ωe and ωeχe, for SH exhibit differences large enough to cast doubt upon the calculated value, the experimental value, or both.

Notes: The Fock generating function method is employed for the separation of spin variables into two-particle transition density matrices. Their spatial components introduced by McWeeny are expressed through Schrödinger's coordinate wave functions. Some nontrivial integral relations between these components and the charge and transition spin density matrices are obtained. The interrelation between the mentioned spatial components and the Matsen-Poshusta symmetrized density matrices of an arbitrary spatial function is found.

Notes: The SCF theory of multiplet monodeterminantal states is described from a comprehensive viewpoint. In this context, the formalism and derived theorems are developed, as well as the description of a SCF closed-shell-like structure, namely the hypermultiplet. Numerical applications are discussed taking as a model the H2O molecule and its first positive ion. Wave functions of double ζ-quality, with computed molecular properties of 16 states are given.

Notes: The Coulomb correlation hole distribution function has been computed with respect to various reference centers in the HCN molecule, using standard SCF+CI type wave functions. The extent to which statistical correlation between unlike-spin electrons is introduced into an SCF wave function through the inclusion of configuration interaction has been assessed by an examination of the range and depth of such holes, and compared with the behavior of analogous Fermi distribution functions. Our results show that the range of Fermi correlation is consistently longer than that of the corresponding Coulomb correlation.

Notes: The four-center nonplanar electron repulsion integrals over 1s Slater-type atomic orbitals are considered by a numerical Fourier-transform method. It is shown that the highly oscillating integrand appearing in the Fourier inversion formula could be successfully treated by using Tchebyscheff quadrature. The resulting formulas are thoroughly discussed with particular emphasis on their numerical features and convergence properties. It follows that the aforementioned integrals may be calculated with a good accuracy with a moderate amount of computing time.

Notes: This work describes the investigation of the space organization of DNA compact forms and DNA-protein complexes by the small-angle x-ray scattering method. A plane texture was disclosed in dehydrated DNP films. Compact DNA particles formed in the presence of poly(ethylene glycol) reveal ordered structures having a periodicity of 84 Å. Various morphological forms of crystals of CTA-DNA were used to obtain small-angle x-ray patterns of the single-crystal type.

Notes: Within the framework of the nonrelativistic Schrödinger equation, the Coulomb field case is used to assess the range of validity of the Thomas-Fermi statistical theory of atoms. In particular, attention is focused on (a) the x-ray scattering factor, which is the Fourier transform of the electron density; and (b) the momentum density. In each case the predictions of the statistical theory are compared with the exact results for the Coulomb potential. These can conveniently be calculated using earlier work of Fock. Some assessment is also made of the accuracy of the statistical approximation for the kinetic energy density; this is of interest in connection with the density functional approach. Finally, some brief comments are made on the relation between the self-consistent Thomas-Fermi method and the Hartree theory for atoms.

Notes: The theory of vibrations of a composite particle when vibrational amplitudes are not constrained to be small according to the Eckart conditions is developed using the methods of differential topology. A global classical Hamiltonian appropriate for this system is given, and for the case of the molecular vibration-rotation problem, it is transformed into a global quantum Hamiltonian operator. It is shown that the zeroth-order term in the global Hamiltonian operator is identical to the Wilson-Howard Hamiltonian; higher-order terms are shown to give successively better approximations to the large amplitude problem. Generalized Eckart conditions are derived for the global classical Hamiltonian; the quantum equivalent of these conditions along with the quantum equivalent of the Eckart conditions are given. The spectrum of the global Hamiltonian operator is discussed and it is shown that the calculation of the vibration-rotation energy states of the system reduces to the same straight-forward procedure, the solution of a secular determinant, as was carried out for the Wilson-Howard Hamiltonian at a later time by Nielsen.

Notes: Molecular polarizabilities are computed using ab initio SCF wave functions and second-order perturbation theory with special attention given to the use of a shifted denominator. Rather small basis sets are used, in order to obtain reasonable values at a reduced cost. Results are presented for H2, CO, H2O, C2H4, OCS, C6H6, Cl2, Br2, I2.

Notes: A mathematical model to relate order and rotational motion in lipid hydrocarbon chains in biological membranes is proposed. Schrödinger equations with suitable potential functions for C—C rotations in lipid hydrocarbons chains have been solved to obtain wave functions characterizing these motions. For a free chain a threefold potential function has been assumed. The effect of collisions between neighboring chains in a membrane is considered by including another term (δ function) in the potential function.

Notes: The individual absorption spectra of various ionic and tautomeric species of some derivatives of nucleic-acid components have been determined by computer handling of the summary spectra of these compounds measured at different pH values. These individual spectra have been decomposed into separate bands represented by log-normal curves and corresponding to different electronic transitions. Using the parameter system calculated earlier for the molecules of the vitamin B6 family, the calculation of the electronic spectra and electronic structure of various ionic and tautomeric species of the molecules studied have been carried out in the π-electron approximation. These properties have also been calculated by the all-valence electronic method CNDOS. The calculated results correlate well with both approximations and with our experimental spectral data. The localization of the molecular electronic excitation studied on the transition to the S1 and T1 states is considered in connection with the reactivity in these states.

Notes: A partitioning of the molecular Hamiltonian into the occupied and virtual orbital spaces and their orthogonal complement is introduced and used to develop a perturbation expansion of the exact ground-state energy relative to the Hartree-Fock energy computed using an incomplete basis set. The leading perturbation corrections to pair energies due to using the incomplete basis set are considered in detail. Summations of certain classes of pair contributions are discussed and a resummed correction is obtained.

Notes: The energy levels and perturbation expansions for the expectation values of arbitrary powers of position for a perturbed Morse oscillator are obtained by application of the hypervirial and Hellmann-Feynman theorems, solely in terms of the unperturbed energy. We obtain expressions for the first-order corrections for (1 - e-aq )m for 4 ≤ m ≤ 8 and the expressions to second and third order for the quartic perturbation. A numerical application to the CO molecule is made.

Notes: An ab initio Hartree-Fock-Slater procedure was applied to the calculation of the electronic structure of polysulfanes, H2Sn (n = 1-4) and the ions HSn- and Sn2-. Charge densities, overlap populations, and deprotonation energies are calculated; the latter appear well correlated with the first and second acidity constants.

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: In the present paper a new model for the transport through membranes, introduced previously by Hosur, is considered. In this model, the membrane is assimilated to a potential barrier, and the material flow through the membrane is determined from the transmission coefficient of the barrier, assuming the existence of an energy difference among the molecules placed at both sides of the membrane. An equation for the transport, in the case of small energy differences, is obtained, which is particularized to the cases that the energy difference is caused by a temperature gradient, a concentration gradient, and both gradients acting together. In all cases, under certain limitations, formally identical equations to those of the thermodynamics of irreversible processes are obtained.

Notes: The nonrigid molecule symmetry group for methylamine is determined as a set of permutations and permutation inversions for two situations. In the first case the internal rotations of the methyl group and NH2 group are included and the second case the transition of a hydrogen atom from the methyl group to NH2 group is also included. In both cases the energy level splitting of the rigid model of the methylamine is found, nuclear statistical weights of the split levels are calculated, and sections rules for the allowed dipole transitions are given.

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: A simple two-parameter analytic potential adjusted so as to reproduce the experimental energy levels is used to generate wave functions for the ground and excited states of the lithium atom. Using these wave functions in conjunction with the Born approximation and the Russell-Saunders LS-coupling scheme, we calculate the optical oscillator strengths for various excitations from the 1s22s(2S1/2) ground state. The results are compared to experiment and other calculations.

Notes: We present the results obtained within the CNDO/2 method for the diamagnetic susceptibilities of a series of conjugated molecules including benzene, fulvene, pyridine, pyrrole, and furan. The calculations have been carried out, according to Bley's method, in the framework of coupled Hartree-Fock perturbation theory, using gauge invariant atomic orbitals and the London approximation. The two usual CNDO parametrizations, due respectively to Pople, Santry, and Segal and to Del Bene and Jaffe, did not enable one to obtain the principal susceptibilities of the molecules under consideration. We investigated the effect of the CNDO parametrizations on the different contributions to the susceptibility, and gave a first improvement to Bley's method, which permits the calculation of the mean susceptibility of conjugated molecules with less than 9% error.

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: Effects of incident photon field coherence on resonant light scattering have been investigated. In order to obtain the scattering intensity and the photon counting rate, an expression for the reduced density matrix for the scattered field has been derived. The expression involves the first-order correlation function of the incident field. The relation between the line shape of the scattered light and the bandwidth of the incident field has been clarified. Model calculations of the photon counting rate have been performed in the case of an incident field without first-order coherence. In our treatment, the transverse and longitudinal relaxation constants have been taken into account by using the impact approximation.

Notes: The vibrational properties of a quantum system are determined by the density response matrix. In linear response theory this quantity is connected to the polarizability matrix, which can be expressed in terms of a double summation over one-particle energies and wave functions. In has been shown that this expression is not useful in the calculation of vibrational frequencies because of the very slow convergence of the summation in terms of the unoccupied states. In this paper, a different but equivalent expression is presented using a continued fraction. The resulting expression contains only one summation over the occupied states, solving in this way all the problems connected with the sum-over-states expression of the polarizability matrix. The elimination of all the unoccupied states via the use of the moment formula turns out to be a crucial step in the solution of the problem of the first-principles calculation of the vibrational spectra of molecules and solids.

Notes: Normal-mode vibrational frequencies were obtained for uracil and 4-hydroxyuracil with the MINDO/3 SCF method. MINDO/3 vibrational frequencies and the moments of inertia for STO-3G optimized geometries yielded a ΔST0 for U ⇄ U* of 1.06 e.u./mol by standard statistical thermodynamic equations. ΔGT0 is then 6.29 kcal/mol in the gas phase for U ⇄ U*. The reaction field continuum model of solvent effect lowers ΔGT0 to 5.3 kcal/mol in aqueous solution. The tautomeric equilibrium constant in solution is then 1.3 × 10-4, in good agreement with experimental values.

Notes: No. Abstract.

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 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: Neutron scattering spectra from methane molecules adsorbed on a surface of graphite were observed by Newbery et al. They found four inelastic peaks in the temperature range lower than 30°K and assigned them to transitions between rotational tunneling states of methane molecules. This system was investigated by introducing both the interaction between methane molecules and that between a methane molecule and carbon atoms of graphite in order to explain the observed spectrum. It is shown that the system can be analyzed in terms of an effective field with tetrahedral symmetry.

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: 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: The intermolecular energy of two conformations of stacked hydrated purine dimers is computed for a varying number of molecules of water around the stacks. The results obtained show that hydration can modify the relative stability of the dimers as calculated in vacuo. In addition, the optimized arrangements of the molecules of water around the dimers depict that purine molecules strongly interact with the solvent and that bridges made of water dimers or trimers are formed between the stacked bases.

Notes: In order to elucidate further details of RNA conformations we have studied base stacking in dinucleoside monophosphates (DNP's) using UV difference spectra and the hypochromic effect. SCF CI MO calculations according to PPP and MIM approximations were carried out for six pyrimidine-containing DNP's in each of several stacking geometries. The calculated and plotted difference spectra were fitted to the experimental spectra. Different DNP's showed distinct geometries in the range of the helix angle of 35°. From our results we conclude that there is a microstructure in the helix. This would imply an additional content of information in this macromolecule, a higher precision in nucleic acid interactions, and make possible the prediction of the conformation of polynucleotides.

Notes: The reaction mechanism of photoisomerization of isoxazoles to oxazoles through azirine intermediates is investigated theoretically by means of ab initio MO CI calculations. Azirine intermediates in S1 state [(n →*) state of the carbonyl chromophore] cause the C—N bond rupture of azirine ring and transform to isoxazoles via intersystem crossing to T1 state. On the other hand, azirine intermediates in S2 state [(n →*) state of the ketimine chromophore] lead to the C—C bond rupture of azirine ring and convert to oxazoles via intersystem crossing to T1 state.

Notes: The optical spectrum of a model oxyheme complex has been calculated using a new intermediate neglect of differential overlap (INDO-SCF-CI) method that allows for the inclusion of configuration interaction and transition metals. In addition to the porphyrin π→π* transitions common to all heme proteins, four weak x,y polarized transitions observed only in oxyheme complexes have been calculated and assigned to excitations involving the lowest-empty highly delocalized (Oπ, dπ) orbital. Two broad z-polarized bands observed in the single-crystal polarized absorption spectra of oxymyoglobin and hemoglobin have also been calculated. Controversy exists over the assignment of these transitions and, in particular, over the extent of involvement of the oxygen ligand. Our calculations assign the weaker near-IR visible band mainly to the d σ dπ→ dπ* excitations and the more intense UV band mainly to a2u → dσ* excitations. While significant participation (25%) of the highly delocalized (Oπ, dπ) virtual orbital is also found, these z-polarized transitions need not be totally unique to oxyheme complexes, in keeping with experimental observation.

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 correlation energies obtained by the fourth-order diagrammatic perturbation theory were analyzed for three diatomic molecules: N2, CO, and F2. The results were compared with correlation energies obtained previously for the ten-electron hydrides HF, H2O, and NH3. The relative importance of contributions which arise from the double excitations, from the quadruple excitations, as well as from the renormalization term was investigated. It is shown that for the diatomic molecules under study these contributions are considerably larger than for the ten-electron hydrides not only in absolute value but also in percentage: they represent about 3, 3, and 5%, respectively, of the valence shell correlation energy obtained by the perturbation theory up to the fourth order. A careful analysis of the fourth-order correlation effects is also presented for the reaction energy of the process ½H2 + ½F2 = HF.

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: The ground state and the first few excited states of an MnO69- cluster are calculated in the unrestricted Hartree-Fock model. The state ordering is 5B1 g, 5A1 g, 5B2 g, and 5Eg as can be expected from simpler models. Consistent with the results by the same method for copper complexes, we obtain d-d transition energies about one half or less of the experimental energies. The charge transfer spectrum is subject to a large spin polarization in the sense that the lowest charge transfer state (5Eu) has five unpaired spins on Mn.

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: 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 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: Variable electronegativity Pariser-Parr-Pople (VE-PPP) calculations have been carried out on the bases of nucleic acids and 5-fluorouracil. Directions of polarization for the various electronic transitions in these molecules have been calculated using the wave functions obtained by this and the CNDO/S-CI methods. The calculated excitation energies, oscillator strengths, and directions of polarization have been compared with the PPP results due to Pullman and co-workers and Bailey as well as the experimental ones. It has been found that a comparative study of this type is much more useful for understanding the transitions in the molecules than the individual calculations. The VE-PPP method is found to generally give the best results. The directions of polarization have been used to identify the transitions in the molecules and to compare the excitation energies obtained by different methods. It is shown that the transitions in these molecules do not have the characteristics of those of benzene derivatives. All the observed peaks in the molecules have been explained as π*→π.

Notes: The salient features of generalized second-quantization representations for nonrelativistic systems of composite particles are reviewed and their application to a reformulation of the quantum theory of reactive collisions is discussed. Such representations allow the properties of the bound composite states to be built explicitly into the algebra of states and observables. A single unperturbed Hamiltonian simultaneously describes the free propagation of the various species of bound composites as well as of their unbound constituents, while the interaction Hamiltonian describes only true scattering and reaction processes. The inclusion of the binding of all composites in the unperturbed Hamiltonian cures the divergences in the Born series arising from bound-state poles. Unstable composites can be included in a natural way, leading to an explicit representation for the kinematics and dynamics of their decay and their contribution to collision phenomena.

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: This paper proposes a direct, quantum-mechanical definition for Γ (effective action) using a two-step procedure for applying the variational principle to time-dependent quantum problems.

Notes: By using quantum-field theory techniques and the boson transformation method, we study systems that exhibit macroscopic ordered states. As an illustration, we apply the method to the study of superconducting systems by taking into account the quantization of the vector potential. The results of a dynamical calculation, performed in the pair approximation, are presented. The Higgs phenomenon is also discussed in the framework of a fully quantum theory.

Notes: Some aspects of fiber bundles, sheaf theory, and possible generalizations of the notion of differentiable manifolds are discussed.

Notes: Relativistic quantum-field theory provides the machinery for calculating wave functions or probability amplitudes depending upon space-time coordinates. The currently accepted theory, however, fails to provide position operators and a means of measuring particle coordinates that are consistent with Dirac's properties of physical observables. This is because it calls for a space position probability distribution at a specified time. This paper shows, however, that space-time event coordinate operators, together with a corresponding measurement procedure, can be found that are consistent with Dirac's requirements. This is done through a reinterpretation of the amplitudes computed by field theory and does not involve any change in that mathematical formalism. The measurement of the space-time coordinates of an event is accomplished by detecting the absorption of a photon by a particle from each of two light pulses designed to overlap at a given point at a given time. If a final emitted photon has an energy whose sum with the final particle energy approximately equals the sum of the mean energies of the pulses, then the absorption of the two pulse photons must certainly have taken place within a distance the order of a Compton wavelength of the small space-time region of overlapping pulses. This is clear from the fact that the high energy required to confine the pulses to very small volumes must throw a particle absorbing them far off the mass shell. Thus the absorption of the two photons throws the particle into a narrowly confined spatial wave function that must decay extremely rapidly - to within a Compton wavelength, a delta function in space-time. This delta function is the eigenfunction of space-time coordinate operators Xμ and is the scalar product of vectors in a Hilbert space spanned by spin-space-time kets large enough to contain the operators of the Poincaré group. These event operators transform properly under the action of Poincaré operators but do not commute with the mass. If the Compton wavelength is not negligible compared to the accuracy desired in the coordinate measurements, individual coordinate measurements are no longer possible. Nevertheless, a large number of repeated coordinate measurements can be carried out to produce a coordinate probability distribution. This distribution can be unfolded to find a true coordinate probability distribution if the charge form factor is known from basic theory. An analysis of laboratory particle detection techniques shows that they actually determine space coordinates and energy rather than spatial coordinates at a given time. When this fact is included, the Klein-Nishina formula can be derived using the electromagnetic four-vector potential as the photon probability amplitude wave. To clarify the meaning of the observables, a mass-momentum measurement is described.

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: 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: This paper contains the skeleton of one lecture, and of two raporteur's interventions, made at the Sanibel Symposium in February-March, 1979. The task asked of this lecturer was to present an outline of the program of nonequilibrium statistical mechanics, and to indicate some of the recent developments in our understanding of this program. Section I lists a few of the principal physical ingredients of the theory. The possible implementation of certain aspects of the program is demonstrated in Sec. II with the help of three exactly soluble models. Finally in Sec. III we extrapolate from these particular examples in two directions: the stochastic aspect of statistical mechanics, and some remarks on the theory of the quantum measurement process. The presence, at the Symposium, of several distinguished lecturers who presented their own original, and most interesting, contributions hopefully made appear less outrageous - at least at the time - the severe omissions due to the personal choices this particular speaker had to make to keep within the time limitations inherent to such a gathering. It is now hoped that, even when taken out of the full context of the Symposium, the rearranged text which follows might nevertheless help focusing attention on some of the important problems remaining to be solved. It is a pleasure to acknowledge here the stimulating hospitality of the Symposium Scientific Director, Professor P.-O. Löwdin.

Notes: Among the tensors used in the published literature to characterize a phenomenon of nonlinear optics, some are not invariant under a change of origin. In this paper, we show that this invariance can be obtained in a general manner, if the densities of electric and magnetic polarization are not separated but replaced by a density of magnetoelectric polarization whose quantum calculation is performed for a molecular crystal.

Notes: In the previous paper, a global tensor invariant under origin changes was introduced for all optical phenomena. By applying the results obtained, we determine invariant partial tensors which characterize refraction and the Faraday effect.

Notes: A systematic analysis of a diabatization method proposed in a recent publication is presented for the one-dimensional linear model of nonadiabatic transitions, and the method is extended to the exponential model. The asymptotic behavior of molecular properties is studied, and the usefulness of simple properties like the kinetic energy and multipole moments is examined in detail. The method is easy to use, and provides a reliable a posteriori procedure to deal with the existence of strong radial couplings between adiabatic states, when a program to calculate the radial couplings exactly is not available.

Notes: When calculating the correlation energy where the effect of higher than double excitations is included by means of many-body Rayleigh-Schrödinger perturbation theory (MB-RSPT) through fourth order the problem of cancellation of exclusion-principle-violating (EPV) diagrams arises. The cancellation of EPV diagrams in fourth order of MB-RSPT is demonstrated for diagrams which include the effect of tetraexcitations.

Notes: Scope and limitations of different approaches which enable π electronic systems of topologically equivalent structures to be compared have been examined. Particular attention has been paid to analyses in terms of molecular subsystems.

Notes: A complete development of the PCILO method is presented for the INDO approximation. The introduction of exchange terms in the theoretical treatment is discussed and a detailed analysis of the successive contributions to the total energy is given. The method is applied to various conformational problems (geometry optimization, rotation, and inversion barriers). Its ability to deal with biological systems is tested on the acetylcholine neurotransmitter. Inversion barriers are highly improved with respect to the corresponding CNDO algorithm. A better account of σ-π exchange phenomena can be expected with the presented PCILO-INDO development.