On the relation between Peierls' perturbation theory for projected states and exchange perturbation theory

The derivations of a rather general class of exchange perturbation theories (or symmetry-adapted perturbation theories) and Peierls' perturbation theory for projected states are discussed. While the approaches are different, the equivalence of the resulting formalisms is established.     

Self-consistent perturbation theory for conjugated molecules

Self-consistent bond polarizabilities are defined and computed for butadiene, benzene, naphthalene and anthracene. A self-consistent derivation of the bond order-bond length relationship is given and self-consistent formulae for force constants are obtained. Theoretical bond lengths for butadiene, naphthalene and anthracene are calculated in two ways; firstly by using the same values of _rs and _rs for all bonds and secondly by allowing these to vary with bond length. The agreement with experiment is very satisfactory although in some respects the first set of results is to be preferred. Force constants for ethylene and benzene are found which have the correct orders of magnitude although the detailed agreement is not always very good and the interaction force constant between meta bonds in benzene is predicted to have the wrong sign.

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Zusammenfassung Des polarisabilités de liaison self-consistantes sont définies et calculées pour le butadiène, le benzène, le naphtalène et l' anthracène. Une dérivation self-consistante de la relation entre l'indice de liaison et la longueur de liaison est obtenue, ainsi que des formules self-consistantes pour les constantes de force. Les longueurs théoriques des liaisons pour le butadiène, le naphtalène et l'anthracène sont calculées de deux manières: tout d'abord en utilisant les mêmes valeurs de _rs et _rs pour toutes les liaisons, puis en les faisant varier avec la longueur de la liaison. L'accord avec l'expérience est très satisfaisant, quoique, d'un certain point de vue, le premier groupe de résultats doit être préféré à l'autre. Les constantes de force calculées pour l'éthylène et le benzène ont un ordre de grandeur correct quoique dans le détail l'accord ne soit pas toujours très bon et que la constante de force d'interaction entre liaisons en méta du benzène soit obtenue avec le signe contraire.

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Résumé Es werden selbstkonsistente Bindungspolarisierbarkeiten definiert und für Butadiene, Benzol, Naphthalin und Anthrazen berechnet. Ebenso wird in diesem Rahmen eine Ableitung für eine Beziehung zwischen Bindungsordnung und Bindungslänge sowie ein Ausdruck für die Kraftkonstanten angegeben. Die Bindungslängen werden auf zwei Wegen berechnet: einerseits mit gleichen _rs und _rs für alle Bindungen und andererseits mit Werten, die von der Bindungslänge abhängen. Letztere ergeben sich in befriedigender Weise, während bei den Kraftkonstanten in bezug auf Einzelheiten Abweichungen zu verzeichnen sind.

     

Nonstationary perturbation theory for open-shell molecules

Nonstationary perturbation theory equations have been obtained for open-shell molecules. The equations were formulated in terms of a density matrix in the MO-LCAO method. The first variant is coupled perturbation theory in the framework of the restricted Hartree-Fock method for open shells, and the second variant is variational perturbation theory for ground and excited electronic states of molecules, in which the perturbed wave function of the system is constructed in the form of a superposition of the ground and singly excited configurations composed of the Hartree-Fock orbitals of the open shell. A calculation of the Cauchy moments of the dynamic dipole polarizability of several molecules of conjugated open-shell hydrocarbons, viz., doublet states of odd alternant hydrocarbons, as well as triplet excited states and doublet states of radical ions of even alternant hydrocarbons, has been carried out in the framework of both methods.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 21, No. 1, pp. 18–27, January–February, 1985.     

Self-consistent perturbation theory for conjugated molecules

The first order changes in the orbital energies of a number of hydrocarbons due to perturbations at particular atoms are given. These are useful for discussing ionization potentials, polarographic reduction and oxidation potentials, and electron affinities of heteromolecules.

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Zusammenfassung Die Änderungen erster Ordnung von Einelektronenenergien bei Störung an bestimmten Atomen werden für eine Reihe von Kohlenwasserstoffen angegeben. Sie erweisen sich als brauchbar für die Diskussion von Ionisationspotentialen, polarographischen Oxydations- und Reduktionspotentialen und Elektronenaffinitäten von Heteromolekülen.

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Résumé Les variations au premier ordre des énergies des orbitales, sous l'influence de perturbations sur des atomes déterminés, sont données pour un certain numbre d'hydrocarbures. Elles sont utiles pour la discussion des potentiels d'ionisation, des potentiels d'oxydation et de réduction polarographiques et des affinités électroniques des molécules hétéroatomiques.

     

Self-consistent perturbation theory for conjugated molecules

Since an external field produces a drift of the mobile -electrons in the plane of a conjugated molecule the electric polarizability of these molecules is anisotropic. The S.C.F. perturbation theory is used to calculate the -electron contribution to the polarizability tensor for a number of conjugated molecules. Using previous theoretical and semi-empirical values for the polarizabilities of C-C and C-H bonds the contribution is estimated and total polarizabilities for the molecules obtained as the sum of the and contributions. There is reasonable agreement between these theoretical values and the available experimental ones.

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Zusammenfassung Ein äußeres elektrisches Feld verschiebt die -Elektronen in der Molekülebene, so daß die elektrische Polarisierbarkeit von Molekülen mit konjugierten Bindungen anisotrop ist. Mittels SCF-Störungsrechnung wurde der Beitrag der -Elektronen zum Tensor der Polarisierbarkeit für eine Anzahl von Molekülen berechnet und der Beitrag des -Bindungsgerüstes aus semiempirischen Größen ermittelt. Die Übereinstimmung mit gemessenen Werten ist zufriedenstellend.

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Résumé La polarisabilité électrique des molécules conjuguées est anisotrope car un champ extérieur provoque un déplacement des électrons mobiles dans le plan moléculaire. La théorie des perturbations SCF est utilisée pour le calcul de la contribution des électrons au tenseur de polarisabilité d'un certain nombre de molécules conjuguées. En utilisant des valeurs théoriques et semi-empiriques antérieures concernant les polarisabilités des liaisons C-C et C-H on peut évaluer la contribution des électrons . Les polarisabilités totales sont obtenues comme la somme des contributions et . L'accord avec les données expérimentales disponibles est raisonnable.

     

Self-consistent perturbation theory for conjugated molecules

Using self-consistent perturbation theory the first order changes in the spin densities of positive and negative ions of substituted hydrocarbons are calculated. By comparison with ESR hyperfine splitting constants the values of the inductive parameters for aza and methyl substitution are obtained. For aza substitution the most reliable estimate is –1.73 eV in agreement with the best theoretical estimate of –1.85 eV. For methyl substitution in anthracene the data can best be explained by using different values for the positive and negative ions so as to allow for hyperconjugation. The values obtained are –0.3 for the positive ion and –0.15 for the negative ion. The theoretical values are –0.33 and –0.11 .

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Zusammenfassung Mittels selbstkonsistenter Störungsrechnung wurden die Spindichteänderungen erster Ordnung von substituierten Kohlenwasserstoffen berechnet. Durch Vergleich mit ESR-Hyperfeinstrukturkonstanten erhält man dann die induktiven Parameter für Aza- und Methylsubstitution. Für Azasubstitution ist der wahrscheinlichste Wert –1,73 eV, in guter Übereinstimmung mit dem als dem besten geltenden Wert von –1,85 eV. Bei Methylsubstitution im Anthrazen benötigt man für positive und negative Ionen verschiedene Werte, um die Hyperkonjugation zu berücksichtigen, und zwar –0,3 für das positive Ion und –0,15 für das negative. Theoretische Werte sind –0,33 bzw. –0,11 .

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Résumé La théorie des perturbations self-consistantes est utilisée pour le calcul de la variation du premier ordre des densités de spin des ions positifs et négatifs des hydrocarbures substitués. Les valeurs des paramètres inductifs pour les dérivés aza ou méthylés sont obtenues par comparaison avec les constantes de séparation hyperfine en R.P.E. Pour une substitution aza l'estimation la plus sûre donne –1,73 eV en accord avec la meilleure valeur théorique de –1,85 eV. Pour la substitution du méthyle dans l'anthracène les données expérimentales sont le mieux expliquées en utilisant des valeurs différentes pour les ions positifs et négatifs afin de tenir compte de l'hyperconjugaison. Les valeurs obtenues sont –0,3 pour l'ion positif et –0,15 pour l'ion négatif. Les valeurs théoriques sont –0,33 et –0,11 .

     

Realization of the transformation operator in the Jansen—Byers brown perturbation theory for exchange interactions

We obtain the general form and exhibit simple expressions for the transformation operator in the Jansen—Byers Brown exchange perturbation theory, without the use of an orthonormalization process.     

Third-order interactions in symmetry-adapted perturbation theory

We present an extension of many-body symmetry-adapted perturbation theory (SAPT) by including all third-order polarization and exchange contributions obtained with the neglect of intramonomer correlation effects. The third-order polarization energy, which naturally decomposes into the induction, dispersion, and mixed, induction-dispersion components, is significantly quenched at short range by electron exchange effects. We propose a decomposition of the total third-order exchange energy into the exchange-induction, exchange-dispersion, and exchange-induction-dispersion contributions which provide the quenching for the corresponding individual polarization contributions. All components of the third-order energy have been expressed in terms of molecular integrals and orbital energies. The obtained formulas, valid for both dimer- and monomer-centered basis sets, have been implemented within the general closed-shell many-electron SAPT program. Test calculations for several small dimers have been performed and their results are presented. For dispersion-bound dimers, the inclusion of the third-order effects eliminates the need for a hybrid SAPT approach, involving supermolecular Hartree-Fock calculations. For dimers consisting of strongly polar monomers, the hybrid approach remains more accurate. It is shown that, due to the extent of the quenching, the third-order polarization effects should be included only together with their exchange counterparts. Furthermore, the latter have to be calculated exactly, rather than estimated by scaling the second-order values.     

SCF perturbation theory and intermolecular interactions

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.     

Möller-plesset perturbation theory for excited states of molecules

We consider the criteria that should be met by the perturbation theory for excited states to preserve the advantages of the Möller-Plesset version for the ground state. A few zero-approximation Hamiltonians are proposed and discussed. Expressions for the first-order corrections to the wave function and for the second-order corrections to the energy of excited states are obtained. The expressions are consistent with the conditions of orthogonality of states with the same symmetry. It is shown that the perturbation theory for excited states proposed in this work is identical to the Möller-Plesset perturbation theory for the ground state.     

Generalized perturbation theory of effective valence shell hamiltonians

The second quantized effective valence shell hamiltonian of Iwata and Freed is generalized to incorporate valence orbital energies into the perturbative (matrix) energy denominators, eliminating convergence difficulties in calculations of atomic valence shell hamiltonians. When the matrix energy denominators are taken to be the simplest form our generalized effective hamiltonian reduces to Brandow quasi-degenerate theory.     

Supplement to spherical tensor theory of long-range interactions between two molecules

We simplify a recently obtained approximate formula for the third-order interaction energy between two molecules expressed in terms of irreducible spherical polarizabilities and multipole moments of interacting systems. Similarly, we can simplify analogous higher-order expressions. Then we briefly examine the magnitudes of the various third-order contributions in comparison with the second-order categories of long-range interactions for some specific systems.     

Nonadditivity of orientational interactions and perturbation theory for dipole hard spheres

The procedure of approximate summation is applied to a series of many-body perturbation theory for the internal energy of a dipolar hard sphere (DHS) fluid to produce an expression similar to the mean field approximation in the Widom-Rowlinson penetrable sphere model, which is well consistent with the experiment at moderate or high densities. Similar results are obtained from the hindered rotation model generalized for arbitrary density. The critical parameters ρ_c, P _c and T _c of both models are consistent with the data of machine experiments and are close to the parameters of the percolation transition in the DHS system.     

Monte carlo simulation and thermodynamic perturbation theory for mixtures of diatomic molecules

The thermodynamic properties and site—site distribution functions of mixtures of non-spherical molecules are obtained by Monte Carlo simulation. A non-spherical reference-system perturbation theory based on the RISM equation is developed to predict these results. The agreement between theory and simulation for the thermodynamic properties is encouraging. Important differences in the relative peak heights of the site—site distribution functions from theory and simulation are attributed to the role of attractive forces in determining local structure in the fluid mixtures, where the volumes of the components are similar but the well depths differ.     

Møller-Plesset (MP2) perturbation theory for large molecules

Summary A novel formulation of MP2 theory is presented which starts from the Laplace transform MP2 ansatz, and subsequently moves from a molecular orbital (MO) representation to an atomic orbital (AO) representation. Consequently, the new formulation is denoted AO-MP2. As in traditional MP2 approaches electron repulsion integrals still need to be transformed. Strict bounds on the individual MP2 energy contribution of each intermediate four-index quantity allow to screen off numerically insignificant integrals with a single threshold parameter. Implicit in our formulation is a bound to two-particle density matrix elements. For small molecules the computational cost for AO-MP2 calculations is about a factor of 100 higher than for traditional MO-based approaches, but due to screening the computational effort in larger systems will only grow with the fourth power of the size of the system (or less) as is demonstrated both in theory and in application. MP2 calculations on (non-metallic) crystalline systems seem to be a feasible extension of the Laplace transform approach. In large molecules the AO-MP2 ansatz allows massively parallel MP2 calculations without input/output of four-index quantities provided that each processor has in-core memory for a limited number of two-index quantities. Energy gradient formulas for the AO-MP2 approach are derived.Dedicated to Prof. W. Kutzelnigg whose books on theoretical chemistry aroused my interest in this field     

Generalized perturbation theory: Quality of the first-order wave function

A recently proposed version of generalized perturbation theory, in which the whole energy correction is taken care of in second order, is investigated with respect to the quality of its first-order wave function. It is demonstrated that the overlap of the wave function generated in this procedure with the exact solution is in most cases much closer to unity than those of the Rayleigh-Schroedinger or Brillouin-Wigner perturbation theories. Certain approximations, by means of which realistic systems become amenable to investigation within the presently discussed framework, are studied.Based on a section of a thesis to be submitted by N. M. to the senate of the Technion — Israel Institute of Technology, in partial fulfilment of the requirements for the D.Sc. degree, and presented in the Second International Congress of Quantum Chemistry, New Orleans, April 1976.