Perturbation theory molecular crystals

Another important application of perturbation theory is to molecules with anisotropic interactions. Examples are dipolar hard spheres, in which the anisotropy is due to the polarity of tlie molecule, and liquid crystals in which the anisotropy is due also to the shape of the molecules. The use of an anisotropic reference system is more natural in accounting for molecular shape, but presents difficulties. Hence, we will consider only... 

In this paper a method [11], which allows for an a priori BSSE removal at the SCF level, is for the first time applied to interaction densities studies. This computational protocol which has been called SCF-MI (Self-Consistent Field for Molecular Interactions) to highlight its relationship to the standard Roothaan equations and its special usefulness in the evaluation of molecular interactions, has recently been successfully used [11-13] for evaluating Eint in a number of intermolecular complexes. Comparison of standard SCF interaction densities with those obtained from the SCF-MI approach should shed light on the effects of BSSE removal. Such effects may then be compared with those deriving from the introduction of Coulomb correlation corrections. To this aim, we adopt a variational perturbative valence bond (VB) approach that uses orbitals derived from the SCF-MI step and thus maintains a BSSE-free picture. Finally, no bias should be introduced in our study by the particular approach chosen to analyze the observed charge density rearrangements. Therefore, not a model but a theory which is firmly rooted in Quantum Mechanics, applied directly to the electron density p and giving quantitative answers, is to be adopted. Bader s Quantum Theory of Atoms in Molecules (QTAM) [14, 15] meets nicely all these requirements. Such a theory has also been recently applied to molecular crystals as a valid tool to rationalize and quantitatively detect crystal field effects on the molecular densities [16-18]. 

Micellar aggregates are considered in chapter 3 and a critical concentration is defined on the basis of a change in the shape of the size distribution of aggregates. This is followed by the examination, via a second order perturbation theory, of the phase behavior of a sterically stabilized non-aqueous colloidal dispersion containing free polymer molecules. This chapter is also concerned with the thermodynamic stability of microemulsions, which is treated via a new thermodynamic formalism. In addition, a molecular thermodynamics approach is suggested, which can predict the structural and compositional characteristics of microemulsions. Thermodynamic approaches similar to that used for microemulsions are applied to the phase transition in monolayers of insoluble surfactants and to lamellar liquid crystals. 

In order to correlate the solid state and solution phase structures, molecular modelling using the exciton matrix method was used to predict the CD spectrum of 1 from its crystal structure and was compared to the CD spectrum obtained in CHC13 solutions [23]. The matrix parameters for NDI were created using the Franck-Condon data derived from complete-active space self-consistent fields (CASSCF) calculations, combined with multi-configurational second-order perturbation theory (CASPT2). 

The problem with the theory of electronic transport in molecular crystals has been to deduce the transport, given a model Hamiltonian containing what one considers to be the essential physical interactions. Since several interactions may be comparable in size, simple perturbative methods fail. The method (12) adopted here yields a rather direct solution to the problem. 

Depending on the character of the molecular motions, one can distinguish several physical situations. In most cases, the molecules are trapped in relatively deep potential wells. Then, they perform small translational and orientational oscillations around well-defined equilibrium positions and orientations. Such motions are reasonably well described by the harmonic approximation. The collective vibrational excitations of the crystal, which are considered as a set of harmonic oscillators, are called phonons. Those phonons that represent pure angular oscillations, or libra-tions, are called librons. For some properties it turns out to be necessary to look at the effects of anharmonicities. Anharmonic corrections to the harmonic model can be made by perturbation theory or by the self-consistent phonon method. These methods, which are summarized in Section III under the name quasi-harmonic theories, can be considered to be the standard tools in lattice dynamics calculations, in addition to the harmonic model. They are only applicable in the case of fairly small amplitude motions. Only the simple harmonic approximation is widely used the calculation of anharmonic corrections is often hard in practice. For detailed descriptions of these methods, we refer the reader to the books and reviews by Maradudin et al. (1968, 1971, 1974), Cochran and Cowley (1967), Barron and Klein (1974), Birman (1974), Wallace (1972), and Cali-fano et al. (1981). 

Just as the perturbation theory described in the previous section, the self-consistent phonon (SCP) method applies only in the case of small oscillations around some equilibrium configuration. The SCP method was originally formulated (Werthamer, 1976) for atomic, rare gas, crystals. It can be directly applied to the translational vibrations in molecular crystals and, with some modification, to the librations. The essential idea is to look for an effective harmonic Hamiltonian H0, which approximates the exact crystal Hamiltonian as closely as possible, in the sense that it minimizes the free energy Avar. This minimization rests on the thermodynamic variation principle ... 

The Editor would like to thank the authors for their contributions, which give an interesting picture of the current status of selected parts of quantum chemistry. The topics in this volume range from studies of the Jahn-Teller effect, the quantum theory of tautomeric equilibria, over the dynamics of molecular crystals to coupled-cluster many-body perturbation theory. 

Crystal states, where one molecule is ionized, and the conduction band contains one electron, can be used as those intermediate states, as has been shown in (31) (the role of such intermediate states in theory of photoconductivity of molecular crystals has been discussed by Lyons (32)). The use of intermediate states becomes indispensable when the second-order perturbation theory is applied in the case of a degenerate term. According to (33), correct linear combinations of crystal states, containing one molecule in a triplet state and all remaining in the ground state, can be found by perturbation theory when in the corresponding secular equation the following effective Hamiltonian is used... 

Calculations of atomic and molecular hyperpolarizabilities usually proceed via time-dependent perturbation theory for the perturbed atomic states. Even for molecules of modest size, the calculation of the complete set of unperturbed wavefunctions, and exact calculation of the hyperpolarizabilities, is prohibitively difficult. Liquid crystals typically consist of organic molecules with aromatic cores, and there is considerable experimental [10] and theoretical [11, 12] evidence to indicate that the dominant contribution to the polarizabilities originates from the delocalized r-electrons in conjugated regions of these molecules. Even considering only r-electrons the calculations rapid-... 

McWeeny, R. (2001). Methods of molecular quantum mechanics (2nd reprinting). London Academic. McWeeny, R., 8c Diercksen, G. H. R (1968). Self-consistent perturbation theory. II. Extension to open shells./owrwaZ of Chemical Physics, 49,4852. McWeeny, R., Dodds, J. L., ScSadlej, A. J. (1977). Generalization for perturbation-dependent non-orthogonal basis set. Molecular Physics, 34,1779. Meier, W. M. (1961). The crystal structure of mordenite (ptilolite). Zeitschrift fur Kristallograhie, 115, 439. 

Another approximation scheme (strong electron-phonon coupling theory) starts from the opposite assumption that the electronic coupling Vei is a small perturbation, i.e., the molecular crystal can be seen as a collection of isolated molecules. Molecule j is deformed by the presence of a carrier and the set of vibronic states localized on j form the basis for a transformed Hamiltonian [69, 71,72]. In second quantization the transformed Hamiltonian can be written as... 

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