Local field method

This chapter presents a consistent description of the local field method and some of its results. First, we should make a brief note. 

We can use the local field method to find the dielectric constant tensor for systems of this type. This method goes back to Lorentz who used it to derive the well-known formula for the optical refractive index in isotropic media (the Lorenz-Lorentz formula). Let us recall the derivation of this formula. 

We show below how these inaccuracies in the Lorenz-Lorentz formula can be eliminated. We employ the local field method to modify this formula and apply it to anisotropic organic crystals of complex structure. We discuss below also a variety of their optical properties which were earlier analyzed in a less general form only in the framework of exciton theory. We explain how it has... 

In the preceding sections of this chapter the local field method was applied mainly to the theory of optical properties of crystals and crystalline solutions. In this section we will investigate the influence of the local field corrections on the resonance interaction between the impurity molecules. Just this interaction determines the transfer of intramolecular (electronic or vibrational) excitation energy from one impurity molecule to another. 

To illustrate how we can take into account the higher multipoles in the framework of the local field method let us consider, along with the dipole polarization, also the quadrupole qij and the octupole polarizations of the molecule. In this approximation, the operator for the energy of interaction of the molecule with the external monochromatic electric field E(r,t) is... 

Equations (5.72), which are a generalization of eqn (5.2), show that when we take into account higher multipoles within the framework of the local field method we must find not only the local field amplitude but also the amplitudes of its derivatives. Bearing in mind the above discussion and the results reported by Born and Huang (4) and Khokhlov (10), we can write the local field and its derivatives in the following form ... 

The same question arises in the theory of the dielectric tensor in the framework of semiphe-nomenological local field method described in Ch. 5. 

Both spin-lattice (motional) and spin-spin processes contribute to TjpCC). Experimental cross-polarization transfer rates from protons in the local dipolar field to carbons in an applied rf field can be used to determine the relative contributions quantitatively. This measurement also requires a determination of the proton local field. Methods for making both measurements have been developed in the last few years [1,2]. For polystyrenes, the spin-lattice contribution to TjpCCO s is by far the larger. This means that the TipCCVs can be interpreted in terms of rotational motions in the low-to-mid-kHz frequency range. 

To circumvent problems associated with the link atoms different approaches have been developed in which localized orbitals are added to model the bond between the QM and MM regions. Warshel and Levitt [17] were the first to suggest the use of localized orbitals in QM/MM studies. In the local self-consistent field (LSCF) method the QM/MM frontier bond is described with a strictly localized orbital, also called a frozen orbital [43]. These frozen orbitals are parameterized by use of small model molecules and are kept constant in the SCF calculation. The frozen orbitals, and the localized orbital methods in general, must be parameterized for each quantum mechanical model (i.e. energy-calculation method and basis set) to achieve reliable treatment of the boundary [34]. This restriction is partly circumvented in the generalized hybrid orbital (GHO) method [44], In this method, which is an extension of the LSCF method, the boundary MM atom is described by four hybrid orbitals. The three hybrid orbitals that would be attached to other MM atoms are fixed. The remaining hybrid orbital, which represents the bond to a QM atom, participates in the SCF calculation of the QM part. In contrast with LSCF approach the added flexibility of the optimized hybrid orbital means that no specific parameterization of this orbital is needed for each new system. 

For excited state calculations, significant progress has been made based on the GW method first introduced by Hybertsen and Louie. [29] By considering quasi-partide and local field effects, this scheme has allowed accurate calculations of band gaps, which are usually underestimated when using the LDA. This GW approach has been applied to a variety of crystals, and it yields optical spectra in good agreement with experiment. 

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