Hamiltonian dressed molecular

To describe the shifts and intensities of the m-photon assisted collisional resonances with the microwave field Pillet et al. developed a picture based on dressed molecular states,3 and we follow that development here. As in the previous chapter, we break the Hamiltonian into an unperturbed Hamiltonian H(h and a perturbation V. The difference from our previous treatment of resonant collisions is that now H0 describes the isolated, noninteracting, atoms in both static and microwave fields. Each of the two atoms is described by a dressed atomic state, and we construct the dressed molecular state as a direct product of the two atomic states. The dipole-dipole interaction Vis still given by Eq. (14.12), and using it we can calculate the transition probabilities and cross sections for the radiatively assisted collisions. 

Hence, the determination of the eigenelements of K in Jf is reduced to the determination of those of B in //. When such a transformation C(0) can be found, the operator B is called the dressed Hamiltonian. Although it acts only on the molecular Hilbert space M, it contains the information on the photons, which dress the molecule. The transformation C(0) can be interpreted as a change of representation. We remark that the transformation C(x, 0), and thus the dressed Hamiltonian B, is clearly not unique since C(x, 0) can be composed with any unitary transformation that acts inside Jf. 

The polaron transformation, executed on the Hamiltonian (12.8)-( 12.10) was seen to yield a new Hamiltonian, Eq. (12.15), in which the interstate coupling is renormalized or dressed by an operator that shifts the position coordinates associated with the boson field. This transformation is well known in the solid-state physics literature, however in much of the chemical literature a similar end is achieved via a different route based on the Bom-Oppenheimer (BO) theory of molecular vibronic stmcture (Section 2.5). In the BO approximation, molecular vibronic states are of the form (/) (r,R)x ,v(R) where r and R denote electronic and nuclear coordinates, respectively, R) are eigenfunctions of the electronic Hamiltonian (with corresponding eigenvalues E r ) ) obtained at fixed nuclear coordinates R and... 

A direct consequence of the observation that Eqs. (12.55) provide also golden-rule expressions for transition rates between molecular electronic states in the shifted parallel harmonic potential surfaces model, is that the same theory can be applied to the calculation of optical absorption spectra. The electronic absorption lineshape expresses the photon-frequency dependent transition rate from the molecular ground state dressed by a photon, g) = g, hco ), to an electronically excited state without a photon, x). This absorption is broadened by electronic-vibrational coupling, and the resulting spectrum is sometimes referred to as the Franck-Condon envelope of the absorption lineshape. To see how this spectrum is obtained from the present formalism we start from the Hamiltonian (12.7) in which states L and R are replaced by g) and x) and Vlr becomes Pgx—the coupling between molecule and radiation field. The modes a represent intramolecular as well as intermolecular vibrational motions that couple to the electronic transition... 

Hamiltonian in an extended space, the direct product of the usual molecular Hilbert space, and the space of periodic functions of f e [0,T]. This extension of the Hilbert space can be made somewhat more transparent by introducing a new time-like variable, to be distinguished from the actual time variable t. This new time variable can be defined through the arbitrary phase of the continuous (periodic) field, as done in Ref. [28, 29]. A variant of the idea is found in the (f, t ) method developed by Peskin and Moiseyev [30] and applied to the photodissociation of HJ [31, 32]. We will continue with the more traditional and simpler formulation of Floquet theory here, as this is sufficient to bring out ideas of laser-induced resonances in the dressed molecule picture. 

The connection between the full molecular N-particle Hamiltonian of Equation 12.3 including rotational excitations and dressing fields, and the effective Hamiltonian of (Equation 12.1) can be made using a Born-Oppenheimer approximation. The diagonalization of the Hamiltonian [92] Hbo = - d,Ej - -... 

We derive a set of Born-Oppenheimer potentials by first separating Equation 12.19 into center-of-mass and relative coordinates, and diagonalizing the Hamiltonian //rei for the relative motion for fixed molecular positions. Within an adiabatic approximation, the corresponding eigenvalues play the role of an effective three-dimensional interaction potential in a given state manifold dressed by the external field. 

Presumably the most straightforward approach to chemical dynamics in intense laser fields is to use the time-independent or time-dependent adiabatic states [352], which are the eigenstates of field-free or field-dependent Hamiltonian at given time points respectively, and solve the Schrodinger equation in a stepwise manner. However, when the laser field is approximately periodic, one can also use a set of field-dressed periodic states as an expansion basis. The set of quasi-static states in a periodic Hamiltonian is derived by a Floquet type analysis and is often referred to as the Floquet states [370]. Provided that the laser field is approximately periodic, advantages of using the latter basis set include (1) analysis and interpretation of the electron dynamics is clearer since the Floquet state population often vary slowly with the timescale of the pulse envelope and each Floquet state is characterized as a field-dressed quasi-stationary state, (2) under some moderate conditions, the nuclear dynamics can be approximated by mixed quantum-classical (MQC) nonadiabatic dynamics on the field-dressed PES. The latter point not only provides a powerful clue for interpretation of nuclear dynamics but also implies possible MQC formulation of intense field molecular dynamics. 

Kato and Kono [4] were the first to develop an approximate method for computing the TDPES in an intense laser field. In their approach, the TDPES was defined in terms of instantaneous eigenvalues of the field-dressed electronic Hamiltonian. In other words, in the Kato-Kono approach, the TDPES is basically the adiabatic potential energy that is computed by including the instantaneous value of electron-field interactions in the Hamiltonian. This method has proved to be extremely helpful in understanding many intense field phenomena in realistic molecular systems. However, this method lacks the exact dynamical features of the intense laser-molecule interactions, and therefore, a need still exists for a method that is not computationally as expensive as the exact methods but captures the dynamical features of electrons/molecules in strong laser fields. 

The partitioning technique for solving secular equations is discussed. It is shown that the original secular equation may be transformed to a contracted secular equation referring to the specific subspace under consideration. The technique may be applied to hetero-atoms in a molecule, to central atoms in a crystal field, to chemical bonds in a molecular enviromnent, and to discuss the question of dressed and undressed particles. If the system under consideration is not complete, the addition of a complementary subspace leads to a new term in the Hamiltonian corresponding to a dressing of the system involved. 

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