Two-center Repulsion Integrals Research Articles

Quantum chemical “Aufbau” principles: how to estimate the shape of highly flexible (bio-)polymers? A recursively extendable “chemion picture” of Euler-Hückel-type

An outline is given of how to split the n-dimensional space of torsion angles occurring in flexible (bio-)polymers (like alkanes, nucleic acids, or proteins, for instance) into n one-dimensional potential curves. Forthcoming applications will focus on the “protein folding problem,” beginning with polyglycine.ContextIn accordance with Euler’s topology rules, molecules are considered to be composed of “vertices” (atoms, ligands, bonding sites, functional groups, and bigger fragments). Following Hückel, each vertex is represented by only one basis function. Starting from the “monofocal” hydrids CH4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_4$$\\end{document}, NH3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_3$$\\end{document}, OH2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document}, FH, and SiH4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_4$$\\end{document}, PH3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_3$$\\end{document}, SH2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document}, ClH as anchor units, “chemionic” Hamiltonians (of individual “chemion ensembles” and proportional nuclear charges) are constructed recursively, together with an appropriate basis set for the first five (normal) alkanes and some related oligomers like primary alcohols, alkyl amines, and alkyl chlorides.MethodsStandard methods (“Restricted Hartree-Fock RHF” and “Full Configuration Interaction FCI”) are used to solve the various stationary Schrödinger equations. Two software packages are indispensable: “SMILES” for integral evaluations over Slater-type orbitals (STO), and “Numerical Recipes” for matrix diagonalizations and inversions. While managing with only two-center repulsion integrals, “implicit multi-center integrations” lead us to the non-empirical fundament of Hoffmann’s “Extended-Hückel Theory.”

Efficient Multipole Model and Linear Scaling of NDDO-Based Methods

Fast growth of computational costs with that of the system's size is a bottleneck for the applications of traditional methods of quantum chemistry to polyatomic molecular systems. This problem is addressed by the development of linear (or almost linear) scaling methods. In the semiempirical domain, it is typically achieved by a series of approximations to the self-consistent field (SCF) solution. By contrast, we propose a route to linear scalability by modifying the trial wave function itself. Our approach is based on variationally determined strictly local one-electron states and a geminal representation of chemical bonds and lone pairs. A serious obstacle previously faced on this route were the numerous transformations of the two-center repulsion integrals characteristic for the neglect of diatomic differential overlap (NDDO) methods. We pass it by replacing the fictitious charge configurations usual for the NDDO scheme by atomic multipoles interacting through semiempirical potentials. It ensures invariance of these integrals and improves the computational efficiency of the whole method. We discuss possible schemes for evaluating the integrals as well as their numerical values. The method proposed is implemented for the most popular modified neglect of diatomic overlap (MNDO), Austin model 1 (AM1), and PM3 parametrization schemes of the NDDO family. Our calculations involving well-justified cutoff procedures for molecular interactions unequivocally show that the proposed scheme provides almost linear scaling of computational costs with the system's size. The numerical results on molecular properties certify that our method is superior with respect to its SCF-based ancestors.

Transition moment calculations on small peptides

Electric transition moments of small peptides with known structure are calculated with a special version of the CNDO method. In this version called CNDO/OPTIC, a special procedure of configuration interaction (CI) is used which avoids strong oscillations in the results arising with changing the CI size. It is recommended that in the framework of the suggested method the Pariser approximation be used for two-center repulsion integrals when calculating optical properties of peptides near transitions. A description of electronic transition symmetry and transition moment directions is given. It is shown that in the case of myristamide the substitution of the hydrogen atom connected to the nitrogen atom by a methyl group changes the direction of the electric transition moment of the ππ* transition qualitatively in the same way as by the dimerization of this compound. The results are in good agreement with available experimental data.

A semiempirical model for the two-center repulsion integrals in the NDDO approximation

A self-consistent formalism is proposed for the two-center electron repulsion integrals in the NDDO approximation, based on their expansion in terms of multipole-multipole interactions and free from adjustable parameters.

SCF MO calculations of polyene spectra

A computationally simple "variable fl" procedure within the Pariser-ParrPople formalism has been applied with success to the calculation of aromatic hydrocarbon spectra [1]. The method has now been extended to encompass linear polyenes and cr co-diphenylpolyenes. In the variable/7 approach the precise molecular geometry need not be specified and it is not necessary to assume bond alternation. Bond lengths were all set at 1.40 tix and bond angles at 120 ~ The core resonance integrals,/~ ---Alp + Ao, were adjusted at each iteration but the twocenter repulsion integrals, ypq = 14.394/(1.294 + rpq) eV were kept fixed. A 1 --0.51 eV is appropriate for all C-C bonds [1] and a good fit for the trans-butadiene spectrum is obtained with A 0 = -2 . t0 eV. Since this latter quanti ty is close to the value used for benzene, it was employed for all bonds in the polyenes and diphcnylpolyenes. PLATT has classified the polyene transitions by means of configuration interaction t rea tment [2]. The transitions to 1B, ~D, and 1F (PLATT'S 1D2) are allowed. Labeling the occupied orbitals l, 2, 3 . . . (in order of decreasing energy) and the vacant orbitals t ' , 2', 3' . . . (in order of increasing energy), 1B +1A is essentially I -~ t ' , but I -+ 2' and 2 -+ 1', degenerate in the zeroth order, are mixed by configuration interaction to form the IC and 1L c states. Transitions to both of these states arc forbidden in the all-trans isomers, but 1C +1A becomes allowed when the center of symmetry is destroyed (cis band) I f the SCF MO method with extensive configuration interaction is to be useful for routine computations, the results cannot be greatly dependent on the amount of configuration interaction. To obtain the higher transition energies it is useful to include the nine configurations arising from the set 3, 2, t --, t ' , 2', 3'. For convenience in machine calculations additional configurations were included. This only results in an increase in the number of low-lying states of 1L character, to which transitions are forbidden in all configurations, but the energies and intensities of the allowed transitions are essentially unaltered. The linear polyene results are compared with experiment in Tab. 1. This comparison is complicated by solvent effects [3], but the results are comparable to those obtained by assuming bond alternation [3, 4]. The oscillator strengths calculated by a "molecules in molecules" method [3] were smaller than those observed, but in the present work the calculated f nos. exceed experiment, a result

Semi-empirical Self-consistent Field Molecular Orbital Calculation of the Electronic Structure of the Base Components of Nucleic Acids

Abstract SCF MO calculations have been carried out for adenine, guanine, thymine, cytosine, uracil, 9-methyladenine and 1-methylthymine. In evaluating the two-center repulsion integral, the Pariser-Parr and Mataga-Nishimoto approximations have both been used, and the results compared with each other and with the experimental data on the optical properties. In the calculation of thymine, 9-methyladenine, and 1-methylthymine, the hyperconjugation of the methyl group has been taken into account. The calculated wavelength and the oscillator strength of the singlet-singlet transition for the above-mentioned molecules explain the experimental values well except for thymine and 1-methylthymine. The theoretically determined directions of the polarization of the transition moments have been compared with the experimental results for adenine, 9-methyladenine and 1-methylthymine. With regard to the direction of the polarization of the transition moments of the first and the second absorptions of adenine and 9-methyladenine, the results are questionable. From the theoretical point of view, it is difficult to determine which assignment is correct, however, in view of the wavelength and the oscillator strength, the first absorption may reasonably be assigned to the long-axis polarized band. The calculated and the experimentally-determined direction of polarization for 1-methylthymine agree well with each other. The energies of the highest and lowest vacant orbitals and the π-electron densities for the molecules treated in the present paper have been calculated and compared with the result obtained by the Hückel MO method.

Theory of Electronic Excitation and Reorganization in the Formaldehyde Molecule

The use of a previously described iterative method of separated electron pairs is demonstrated by calculations on the formaldehyde molecule. Investigation is made, for example, of detailed changes in the polarity of sigma bonding electron wave functions in this molecule when a pi or lone-pair electron is excited or ionized. The molecule is treated as a six-electron problem (four electrons in the C=O bond plus two lone-pair electrons on O); matrix elements involving the ``core'' are obtained by fitting experimental data. The use of Pariser-type one-center repulsion integrals is tested in conjunction with various sets of two-center repulsion integrals. Although one combination gives reasonable results, none of the combinations is found to be entirely satisfactory, and a procedure is outlined from which could be obtained one-center integrals approximately incorporating Arai-Hurley type corrections. It is demonstrated (with reservations) that the constant sigma structure assumption, usually made in calculations of pi electron excitations and ionizations, is not always good with regard to wave functions, but that it may lead to only small errors in calculated energies.