Hamiltonian equation dipole approximation

Here Hq is the molecular Hamiltonian, and fi e(t) is the interaction between the molecule and the laser field in the dipole approximation, where (i is the transition dipole moment of the molecule. Time evolution of the system is determined by the time-dependent Schrodinger equation,... 

From a quantum mechanical perspective, the treatment of the laser-adsorbate interaction can be straightforwardly included in the equations of motion for the reduced density matrix. Since the molecules are only of modest size compared to the wavelength of the incoming electric field, F t), the interaction can be treated within the semi-classical dipole approximation, i.e. the system Hamiltonian is replaced by... 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

The Born-Oppenheimer approximation allows us to decouple the electronic and nuclear motions of the free molecule of the Hamiltonian Hq. Solving the Schrodinger equation //o l = with respect to the electron coordinates r = r[, O, gives rise to the electronic states (r, R) = (r n(R)), n = 0,..., Ne, of respective energies En (R) as functions of the nuclear coordinates R, with the electronic scalar product defined as (n(R) n (R))r = j dr rj( r. R) T,-(r, R). We assume Ne bound electronic states. The Floquet Hamiltonian of the molecule perturbed by a field (of frequency co, of amplitude 8, and of linear polarization e), in the dipole coupling approximation, and in a coordinate system of origin at the center of mass of the molecule can be written as... 

The first term results from the Fermi contact interaction, while the second represents the long-range dipole-dipole interaction. In the equations above, ge is the free-electron g factor, /Xe the Bohr magneton, gi the nuclear gyromagnetic ratio, and /xi the nuclear moment. Moreover, the nucleus is located at position R, and the vector r has the nuclear position as its origin. Finally, p (r) = p (r) — p (r) is the electron spin density. The only nontrivial input into these equations is precisely this last quantity, i.e. Ps(r), which can be computed in the LSDA or another DFT approximation. The resulting Hamiltonian can be used to interpret the hyperfine structure measured in experiments. A recent application to metal clusters is reported in Ref. [118]. 

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