Integrals dispersion energy

Kumar A, Fairley G R G and Meath W J 1985 Dipole properties, dispersion energy coefficients and integrated oscillator strengths for SFg J. Chem. Phys. 83 70... 

Kumar A, Fairley GRG, Meath WJ (1985) Dipole properties, dispersion energy coefficients, and integrated oscillator strengths for SF6. J Chem Phys 83 70-77... 

Here a and b are occupied MO s of systems A and B. Equation (6,32) is easily expressible in terms of integrals over atomic basis functions and elements of the density matrix. In eqn. (5.31) two terms may be distinguished. The first one is due to single electron excitations of the type a r") and (b —->s"), where a and r", respectively, are occupied and virtual MO s in the system A, and b and s" are occupied and virtual MO s in the system B, Contribution of these terms corresponds to the classical polarization interaction energy, Ep, Two-electron excitations (a r", b — s"), i.e. simultaneous single excitations of either subsystem, may be taken as contributions to the second term - the classical London dispersion energy, Ep, If the Mjiller-Ples-set partitioning of the Hamiltonian is used, Ep may be expressed in... 

London s eqn. (15) for the dipole-dipole dispersion energy is not a simple product of properties of the separate atoms. A partial separation was achieved in 1948 by Casimir and Polder who expressed the /r dispersion energy as the product of the polarizability of each molecule at the imaginary frequency iu integrated over u from zero to infinity. The polarizability at imaginary frequencies may be a bizarre property but it is a mathematically well behaved function that decreases monotonically from the static polarizability at m = 0 to zero as u—> oo. 

This coupling is not convenient. As will become clear below it is better to first couple the transition moments on each center. The dispersion energy becomes a sum of which the summand can be expressed with the use of Eq. (2) as a Casimir-Polder integral... 

The problem of computing dispersion energies is reduced to the computation of polarizabilities for a sufficient number of frequencies, so that the Casimir-Polder integral can be obtained by numerical quadrature [93]. An alternative to this quadrature is the substitution of the product of the polarizabilities by a sum over Hartree-Fock orbitals... 

In the general vdW-DF framework, such an NL correlation (dispersion) energy takes the form of a double-space integral... 

The dispersion integrals need only be evaluated once for each pair of molecules, since all the information about the relative positions of the molecules is contained in the interaction functions not in the dispersion integrals. This is therefore an efficient way to calculate the dispersion energy. 

The susceptibility formula Eq. (2.20) can be used to describe the dispersions energy between macroscopic particles 1 and 2 along two alternative routes. We may replace each atom of particles 1 and 2 by a set of dipole oscillators and integrate the dispersion energy between any pair of atoms, or we may treat the particles as a whole as harmonic oscillators. 

The dispersion energy between two parallel cylinders of length L obeys a d relationship for small separations d. By inserting the one-dimensional weight function (2.42) into the interaction integral (2.40) for halfspaces we find straight away... 

The dispersion energy between two half-spaces of surface area 1 is proportional to the inverse square of their separation. The above parallel treatment of interaction integrals between spheres, cylinders and halfspaces shows that the power law to be expected is clearly related to the number of directions in which the particles under consideration exhibit curvatures. If their curvature vanishes in n directions, we may readily carry out the integration over these directions and are then left with the... 

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