Showing papers in "International Journal of Quantum Chemistry in 1972"

Théorie quantique des phénomènes de biréfringence electrique et magnétique des molécules

Abstract: A general formalism is set up to apply the variation-perturbation method to the quantum theory of non-linear optical effects, in particular to the electric and magnetic birefringence (Kerr and Cotton-Mouton effects). The quantities describing these effects are obtained from the wave function Ψ of the ground state of the unperturbed molecule and the different vectors and tensors which are determined by a variation method. With this method an approximate computation of the Kerr and Cotton-Mouton constants of some simple molecules has been made.

The Renner effect in four-atomic molecules

Abstract: The theory of the Renner effect in X2Y2 and X2YZ linear molecules is considered. The Renner equations are obtained with the help of the Born-Oppenheimer method. It is shown that these equations contain two Renner parameters for an X2Y2 molecule and three Renner parameters for an X2YZ molecules. Assuming these parameters to be small enough, an approximate solution of the Renner equations is obtained with the help of perturbation theory. Some general qualitative results are obtained in the first order of the perturbation theory; the second order calculations are made for some specific cases. The theoretical results are used to consider experimental data concerning the vibrational structure of the 1IIu - 1Σ electronic transition in C2H2 and C2D2 molecules. The frequencies of the bending vibrations and the Renner parameters are calculated for the electronic 1IIu-state mentioned.

The non‐crossing rule in molecular quantum mechanics

Abstract: Previous mathematical proofs of the non-crossing rule concerning the potential energy curves of a diatomic molecule are examined and criticised. A new treatment is presented which is simple and mathematically rigorous.

Electronic structure of long cumulene chains

Abstract: The analysis of the equations of the unrestricted Hartree–Fock (UHF) method for polyenes CNHN+2 with even and odd N » 1 is carried out. The equations of the UHF method are shown to be the same in both cases. The comparison of the UHF method with the extended Hartree–Fock (EHF) method applied to large systems is performed. The ground state and π-electron spectra of long cumulene chains CNH4 are treated by the EHF Method. The end effects are taken into consideration. It is shown that the EHF method gives a finite value of the first optical transition frequency and, at the same time, zero value of torsion barrier of end CH2–groups in long cumulene chains (N ) in contrast to previous calculations of cumulenes by the Huckel method and the restricted Hartree–Fock method.

Branch point singularities in the energy of the delta-function model of one-electron diatoms

Abstract: Two different perturbation series (the polarization expansion and a united-atom expansion) of the ground state energy of the delta-function model for one-electron diatoms are studied and the radii of convergence are determined For both expansions the singularity in the energy which limits the radius of convergence is a branch point with exponent one-half The physical significance of the branch point is that for particular values of the perturbation parameter, two different energy eigenvalues coalesce The positions of the branch points are computed as a function of the internuclear separation R For all values of R, both series converge for all physical values of the perturbation parameters A lower bound to the radius of convergence of the polarization expansion has been computed previously by Claverie It is proved in the present paper that the lower bound calculation is in fact an exact determination of the radius of convergence The results of the model study are applied to real one-electron diatoms to suggest the possible location of a branch point singularity in the energy of the ground state

Calculation of molecular quadrupole moments and a demonstration of the importance of overlap densities in the theory of polyatomic molecules

Abstract: A computational method for calculating quadrupole moments from molecular wave functions in a Slater orbital basis set is described. Using both IEHT and CNDO wave functions quadrupole moments for a series of polyatomic molecules are calculated. They are compared with experimental results and the IEHT wave functions are found to give agreement with experiment while CNDO wave functions do not. The importance of bicentric densities (overlap densities) in the calculation of multipole moments is shown. This is followed by a discussion of the usefulness of these wave functions for a quantitative characterization of the electronic structure of large molecules.

Study of the electronic structure of molecules. XV. Comments on the molecular orbital valency state and on the molecular orbital energies

Abstract: The valency state (vs) concept is analyzed in the Hartree–Fock approximation. A valency state “standard” is defined for atoms at infinite separation. A molecular orbital valency state (Movs) is defined from a partitioning technique (bond energy analysis) previously introduced for the Hartree–Fock molecular wave functions. The Movs for a given atom in a molecule is much higher in energy than the vs and its energy varies from molecule to molecule depending on the exact field of the surrounding atoms. The examples selected in the discussion are the CH4 CH3F, CH2F2, CHF3 and CF4 molecules. An analysis of the orbital energies is then given in terms of the bond energy. The importance of the rearrangement effects following ionization of inner shell electrons (simulation of ESCA type experiments) is illustrated with computations of the positive ion for methane and its fluoroderivatives. It is concluded that rearrangement following ionization from inner shells is as important as rearrangements following ionization from valency electrons. A direct consequence is that the orbital energies should not be equated to the inner shell ionization potentials. The computation of such ionization potentials agrees to about 99.5% with ESCA data, when the energy of both the neutral and ionic species are computed; the use of the orbital energies limits this agreement to about 95%.

Intermolecular interactions in the region of small overlap

Abstract: In order to treat the interaction energy of two molecules a standard Rayleigh-Schrodinger perturbation theory is developed. The Hartree–Fock functions of the separated molecules are used as one particle basis functions, the initial set of states being truncated and non-orthogonal. The non-orthogonality is included into the Hamiltonian by orthogonalization of the basis set. The unperturbed Hamiltonian is chosen so that it possesses the correct symmetry properties with respect to the electron permutations between different molecules. The procedure of this kind automatically results in the appearance of charge transfer states. A graphical technique is elaborated which is a modified version of the Feynman–Goldstone technique and provides a convenient representation of the interaction energy contributions of any order. As an example the first- and the second-order diagrams are considered. A correct expression for the dispersion energy is obtained which differs by a factor from that of the theory using a nonsymmetrical zero approximation.

SCF perturbation theory and intermolecular interactions

Abstract: A self-consistent perturbation theory is derived in the framework of Roothaan's MOLCAO procedure for closed shell systems. Contrary to previous investigations which have considered only one particle perturbations, two particle perturbation operators are considered. Expressions for the first-order density matrix and first- and second-order energy corrections are obtained. A diagram formulation of the complete perturbation expansion is presented. The results are applied to the treatment of the intermolecular interaction problem. The interaction energy is represented as a sum of several contributions: Coulomb, exchange, resonance, polarization and exchange repulsion. A semi-empirical version of the theory is suggested which explicitly involves all the physically significant energy terms and may be useful for the investigation of complex systems.

Internal symmetry groups of non-rigid molecules

Abstract: Hougen has established, for quasi-rigid molecules, the relationship between permutationinversions acting on the molecular Hamiltonians written in Cartesian co-ordinates and permutation-rotations (perrotations) of symmetry acting on nuclear equilibrium configurations. We extend these relations to the case of non-rigid molecules. For this, we introduce kinetic perrotations which act on nuclear equilibrium configurations in the same way as do Altmann's isodynamic operators. We show that isodynamic operators do not always form a group. Moreover, their action cannot be extended to the electrons. They cannot be used for the classification of molecular wave functions. This classification is achieved by using the group of Longuet-Higgins and the group of the corresponding feasible perrotations.

Self consistent calculation of dynamic multipole polarizabilities and excited state wave functions of open shell ions: Li sequence

Abstract: A variational formulation of the Hartree–Fock–Roothaan (HFR) theory for open shell ions in presence of time dependent perturbations is presented. The theory has been used to calculate the dynamic polarizabilities of the first three ions of the Li sequence. The polarizability values, extrapolated to zero frequency, show good shell by shell agreement with the corresponding static results. The polarizability graphs display resonance behaviour at the transition frequencies of the ions, and a study of these points leads to analytic HF wave functions for their low lying excited states. The calculated transition frequencies are in excellent accord with the experimental values. The calculated oscillator strengths for the 2s → np transitions are in reasonable agreement with the extensive multiconfiguration calculations of Weiss and the available experimental results.

Accurate calculation of Fourier transform of two-center Slater orbital products†

Abstract: A method is presented which leads to accurate Fourier transform values of any 1s−1s Slater-type orbital overlap distribution. The numerical merits are discussed and illustrated by some examples.

Vibrational matrix elements of the quadrupole moment functions of H2, N2 and CO†

Abstract: The quadrupole moment functions (molecular quadrupole moment versus internuclear distance) have been determined by quantum mechanical calculations for H2 (by Kolos and Wolniewicz), N2 (by Wahl and Nesbet), and CO (by Nesbet). These functions are used with numerical vibrational wave functions to compute matrix elements which are useful for calculations of scattering cross sections, energy transfer rates and excitation probabilities, and infrared intensities of forbidden bands.

Molecular-orbital calculations on transition metal complexes, charge-transfer spectra and the sequence of metal and ligand orbitals

Abstract: An outline is given of a semi-empirical scheme for molecular-orbital calculations on transition-metal complexes according to a revised INDO procedure. To judge the reliability of the results of the calculations, the charge-transfer transitions of a number of complexes have been calculated and compared with experimental data. Both for the excitation energies and for the oscillator strengths the agreement is very satisfactory. With respect to the sequence of occupied metal and ligand orbitals it was found that for closed-shell d6 and d8 systems the molecular orbitals which are mainly composed of metal d orbitals have a lower energy than the orbitals built up from ligand p orbitals. Calculations by the extended Huckel method and other similar schemes give the d orbitals as the highest occupied ones as a result of a bad approximation of the diagonal elements of the Fock matrix. The consequences for the interpretation of photo-electron spectra of transition-metal complexes are discussed.

Analysis of reduced density matrices in the co-ordinate representation. III. Electron density and correlation in the ground states of H2 and the H6 ring system within some approximations of the simple LCAO type

Abstract: The electron density and spatial correlation as given by the MO, VB and AMO methods for H2 and H6 are studied by means of diagrams. For comparison, diagrams representing accurate wave functions for H2 are also given. The study of model functions representing localized bonds leads to results concerning the nature of localized electron pairs in agreement with those of Lennard-Jones.

Valence-Bond calculations and matrix elements between two tableau functions of non-orthogonal orbitals

Abstract: An integral formula is derived involving a new type of determinantal function which reproduces the effect of the Young operator θNPN. This result is used to calculate matrix elements between tableau functions of non-orthogonal orbitals. Matrices which transform this representation into the traditional valence-bond scheme are also given.

Structure electronique des complexes acide‐base de Lewis. I. Structure electronique et conformation moléculaire des molécules F3P·BH3 et F2HP·BH3

Abstract: The electronic structure and preferred conformations of F3P·BH3 and F2HP·BH3 are investigated in the framework of the CNDO/2 approximation. In complete agreement with microwave data, the staggered conformations are predicted to be the most stable ones. The barriers to internal rotation are in good agreement with experimental values (F3P·BH3: calc. = 3.03 kcal/mole, exp. = 3.24 ± 0.15 kcal/mole; F2HP·BH3: calc. = 3.63 kcal/mole, exp. = 4.05 ± 0.45 kcal/mole) and a bicentric energy partitioning shows that the variations of the total energy are completely reflected by the only variation of the interaction energy between phosphorus and H atoms bonded to boron. The analysis of the electron densities reveals the importance of the 3s(P) → 2px(B) transfer in the formation of the co-ordination. Finally, the computed dipole moment value and direction agree with corresponding experimental data.

All electron calculations of open‐shell polyatomic molecules. III. Perturbation treatment of the spin polarization in vinyl and methyl radicals

Abstract: A first order perturbation treatment starting with SCF-MO'S in canonical or equivalent quasi-localized form is presented for the hyperfine coupling constants of vinyl and methyl radicals. The spin-polarisation contribution to hyperfine splittings is found to be large, negative for the proton of the radical center in both radicals and positive for the β protons of vinyl.

The MC SCF theory: Method of one-electron Hamiltonian

Abstract: The one-electron Hamiltonian method is developed to solve the variational equations of the MC SCF theory. The many-parameter family of the one-electron Hamiltonian is derived and conditions for parameters to provide convergence of the SCF procedure to the energy minimum are obtained. A computation scheme based on the use of the one-electron Hamiltonian is described.

The general theory of the extended method of calulation of atomic structures

Abstract: A general theory of the extended method of calculation of atomic structures having complex configurations is presented. It is based on the correspondence between the radial integrals of the ordinary method and the linear combinations of those of the extended method. The rules for going over from the ordinary method to the extended one are given.

Molecular electronic structure of metal phthalocyanines

Abstract: The molecular–electronic structure of the metal phthalocyanines (Fe, Co, Ni and Cu) has been determined by the molecular orbital treatment. Coulomb integrals of the metal atom occurring in the secular determinants have been approximated equivalent to the valence state ionization energy (VSIE) of a metal orbital for a particular charge configuration. The calculated π-electron charge densities have been found to be higher on the nitrogen atoms as compared to the other atoms in the molecule. This is in agreement with the e.s.r. studies of the metal phthalocyanines. To test the correctness of the molecular orbital calculations, the π-π* transitions (14,000 cm−1 − 30000 cm−1), d-d* transitions (20000 cm−1 − 60000 cm−1) and charge transfer transitions (15000 cm −1 − 30000 cm−1) have been calculated in the metal phthalocyanine molecules. The calculated frequencies have been compared with the observed ones and found in fair agreement.

Long range interaction energies using Gaussian basis sets and a one center method

Abstract: A one center method, based on the work of Karplus and Kolker, is discussed and used to calculate the induction energy, through O(R−8), for the H(ls) – H+ interaction employing two types of Gaussian basis sets constructed from functions of the form {rje−αr2}. The effective hydrogen atom excitation energies and transition multipole moment matrix elements generated in these calculations are used to calculate the dispersion energy for the H(ls) – H(ls) interaction, through O(R−10), and the R−9 triple dipole energy corresponding to the interaction of three H(ls) atoms. The results indicate that Gaussian functions can form good basis sets for obtaining long range forces for a variety of multipole interaction energies.

Quasi-relativistic approximation in the SCF-MO-LCAO method

Abstract: A quasi-relativistic approach to the MO-LCAO method is formulated taking into account the relativistic effects with an accuracy up to (v/c)2 terms, the relativistic part of the electronic interaction in the Hamiltonian being neglected. In the framework of this approximation a set of SCF equations of the Roothaan form is derived; here only the relativistic analogue to the closed shell systems with one-determinant wave functions is considered. In so doing three types of relativistic corrections arise which are quite similar to those of the Pauli equation for one-electron atoms. The new matrix elements appearing due to these corrections can be reduced to some common integrals, which have to be calculated with relativistic radial atomic functions. The method allows a semi-empirical approach to the problem and does not require the Dirac four-component atomic functions (unknown in the most cases), thus making possible approximate quasi-relativistic electronic structure calculations of heavy-atom compounds.

On the nature of paramagnetism in macromolecules with conjugated cc bonds

Abstract: The paramagnetic properties of one-dimensional macromolecules with conjugated CC bonds are discussed on the basis of low-lying quasi-homeopolar triplet excitations A good agreement with the experimental results is shown

Molecular one‐electron integrals over slater‐type atomic orbitals and irregular solid spherical harmonics

Abstract: The one-electron integral over Slater-type atomic orbitals centered at A and B and irregular solid spherical harmonics as operator centered at C is evaluated analytically by using elliptical coordinates and translation of the solid spherical harmonics from center C to either focus A or B. A special case is the three-center nuclear attraction integral which is also evaluated by means of the Neumann expansion and expressed without associate Legendre functions of the second kind. The strict observation of the charge distribution concept leads to compact expressions for the integral. It has the further advantage that the charge density distributions which have been developed for the two-center cases and used in calculations for diatomic molecules can be utilized.

The cusp condition: Constraint on the electron density matrix

Abstract: Attention is called to the attractive cusp conditions as useful constraints in fixing the elements of the electron density matrix. An equation for determination of pure state densities satisfying arbitrary constraints is reviewed, and a detailed formalism for using cusp conditions with this equation is displayed. A calculation is done using the nitrogen molecule as an example.

Variable electronegativity SCF-MO calculations on the electronic structure and spectra of some substituted benzenes I. Monosubstitutions

Abstract: π-electron SCF-MO theory in its variable electronegativity formalism has been applied to some monosubstituted benzenes. Calculated charge densities and bond orders for the ground and the first excited electronic states are correlated with chemical reactivity and the changes in molecular geometry on electronic excitation. The calculated results for spectra are compared with those obtained using the PPP method and also with the available experimental data.

Comparison of several expansions in the calculation of static electric dipole π polarizability of conjugated molecules by pertubation theory. The ground and the first excited singlet states

Abstract: Ground and excited singlet state dipole electric π polarizabilities of a set of conjugated molecules are calculated. Second order perturbation theory is used in the Epstein–Nesbet and Moller–Plesset versions. Huckel and SCF-LCAO-MO are used alternatively as a basis. The Moller–Plesset–SCF–LCAO–MO calculation appears well related to experimental values.

The symmetrized powers of group representations

Abstract: Tables of symmetrized powers of the irreducible representations of point groups are presented together with a derivation of the formula used to obtain them. It is shown how these tables may be applied to various quantum-chemical problems.