Interaction Hamiltonian multipole expansion

Consequently, we introduce the second approximation which is to use an approximate electrostatic potential in Eq.(4-21) to determine inter-fragment electronic interaction energies. Thus, the electronic integrals in Eq. (4-21) are expressed as a multipole expansion on molecule J, whose formalisms are not detailed here. If we only use the monopole term, i.e., partial atomic charges, the interaction Hamiltonian is simply given as follows ... 

In practical applications of the sapt approach to interactions of many-elect ron systems, one has to use the many-body version of sapt, which includes order-by-order the intramonomer correlation effects. The many-body SAPT is based on the partitioning of the total Hamiltonian as H = F+V+W, where the zeroth-order operator F = Fa + Fb is the sum of the Fock operators for the monomers A and B. The intermolecular interaction operator V = H — Ha — Hb is the difference between the Hamiltonians of interacting and noninteracting systems, and the intramonomer correlation operator W = Wa + Wb is the sum of the Moller-Plesset fluctuation potentials of the monomers Wx — Hx — Fx, X — A or B. The interaction operator V is taken in the non-expanded form, i.e., it is not approximated by the multipole expansion. The interaction energy components of Eq. (1) are now given in the form of a double perturbation series,... 

In quantum mechanical terms, the interaction Hamiltonian associated with a perturbing external field is approximated by the multipole expansion... 

Longuet-Higgins [43] first drew attention to the fact that the dispersion interaction between two molecules could be calculated directly in terms of charge density functions, without making the usual multipole expansion of the interaction terms in the Hamiltonian. The charge density operator for molecule A at point r may be defined as (using a for particle index, a for nuclei, i for electrons)... 

Investigations of the linear and nonlinear optical properties of molecules, polymers, and clusters generally adopt the semi-classical approach. In this approach, the particles are treated quantum mechanically while a classical treatment is applied to the radiation so that the Hamiltonian is written as the sum of two types of terms, one representing the isolated system (Hq) and one being the radiation-molecule interaction term (Hi). For sufficiently large wavelengths with respect to the system dimensions. Hi can be expressed under the form of a multipole expansion ... 

Here the four-component potential is expressed in terms of the vector potential A(r,w) and scalar potential (j) r,u), and e is the electron charge. It is assumed that the interaction Hamiltonian has incoming photon field time dependence Using a multipole expansion of the vector poten-... 

As a consequence, the multipole moments of a molecule are also changed (see details in the Chaps. 3-5).Similarly, the expansion of in Eq. (2.2.2) in terms of spherical harmonics allows us to define the interaction Hamiltonian H through the spherical 2 -pole electrical moments of interacting molecules A and B (see details, for instants, in [1, 2]) ... 

Similar to the distributed-multipole expansion of molecular electrostatic fields, one can derive a distributed-polarizability expansion of the molecular field response. We can start by including the multipole-expansion in the perturbing Hamiltonian term W = Qf(p, where we again use the Einstein sum convention for both superscripts a, referencing an expansion site, and subscripts t, which summarize the multipole components (/, k) in just one index. Using this approximation for the intermolecular electrostatic interaction, the second-order energy correction now reads ... 

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