By W. Andrzej Sokalski
Layout of recent molecular fabrics is rising as a brand new interdisciplinary learn box. Corresponding reviews are scattered in literature, and this publication constitutes one of many first makes an attempt to review ongoing learn efforts. It offers uncomplicated info, in addition to the main points of conception and examples of its software, to experimentalists and theoreticians attracted to modeling molecular homes and placing into perform rational layout of latest fabrics.
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Additional info for Molecular Materials with Specific Interactions - Modeling and Design (Challenges and Advances in Computational Chemistry and Physics)
Stone derived 155 a general scheme of reducing a Cartesian tensor of rank n into several spherical components and investigated in detail properties of Cartesian-spherical transformation coefficients 156 . The truncated multipole expansion, N VN = Vn n R n=1 (1-116) can be used to define the Van der Waals constants. By applying the RayleighSchrödinger perturbation theory to the Schrödinger equation with the Hamiltonian HN H N = H0 + V N (1-117) and using 1/R as the expansion parameter Ahlrichs 145 has shown that the Van der Waals constants entering the asymptotic expansion (1-104) can be computed from the following recursive formulas, n Cn = 0 Vk n−k (1-118) k=1 and n n =− ˆ 0 C k − Vk R n−k (1-119) k=1 where the superscript n at n denotes the order in 1/R.
The operator VlA lB can be written as VlA lB = XlA lB lA +lB m=−lA −lB lA +lB −1 m C−m R MlA ⊗ MlB lA +lB m (1-109) where XlA lB = −1 lB 2lA + 2lB 2lA 1/2 (1-110) and the spherical multipole moment operator is given by, m m MlXX = Zp rplX ClXX rp p∈X (1-111) 38 Robert Moszynski Here, the summation index p runs over all particles, both nuclei and electrons, of the molecule X, Zp are the charges of those particles, and Clm r is a spherical harmonic in the Racah normalization. 148 The irreducible tensor product m of two multipole moment tensors MlA = MlA A mA = −lA +lA and MlB = mB MlB mB = −lB +lB is defined as, MlA ⊗ MlB l m lA = lB m mA =−lA mB =−lB m MlA A MlB B lA mA lB mB lm (1-112) Here l1 m1 l2 m2 lm is the Clebsch-Gordan coefficient.
132–137). The authors of Refs. (132–137) have shown that the complete dispersion energy, including the charge-overlap effects, can be expressed, via the Casimir-Polder type integral, in terms of the polarization propagators of the isolated monomers 2 Edisp = − 1 k1 m1 k2 m2 v v 4 l1 n1 l2 n2 + l1 l2 k1 k2 − n1 n2 m1 m2 i −i d (1-92) In the above expression we assumed that k1 k2 l1 l2 and m1 m2 n1 n2 label the orbitals of monomers A and B, respectively. We also introduced the following notation for the Coulomb integrals: k m vl11n11 = l1 1 n1 −1 2 r12 k1 1 m1 2 (1-93) Equation (1-92) is very important since in the region of the Van der Waals minimum the charge-overlap contribution to the dispersion energy is always substantial.