Rotating wave transformation

Resonant effects that prevent convergence of the perturbation theory and that appear as small denominators will be treated specifically by rotating wave transformations (RWT) in Section III.C. 

For very small field amplitudes, the multiphoton resonances can be treated by time-dependent perturbation theory combined with the rotating wave approximation (RWA) [10]. In a strong field, all types of resonances can be treated by the concept of the rotating wave transformation, combined with an additional stationary perturbation theory (such as the KAM techniques explained above). It will allow us to construct an effective Hamiltonian in a subspace spanned by the resonant dressed states, degenerate at zero field. 

As opposed to the KAM-type transformation eeW, the transformation Ri is not close to the identity. It is named rotating wave transformation (RWT) (or equivalently resonant transformation) in contrast with the usual RWA for which... 

In this subsection we will combine the general ideas of the iterative perturbation algorithms by unitary transformations and the rotating wave transformation, to construct effective models. We first show that the preceding KAM iterative perturbation algorithms allow us to partition at a desired order operators in orthogonal Hilbert subspaces. Its relation with the standard adiabatic elimination is proved for the second order. We next apply this partitioning technique combined with RWT to construct effective dressed Hamiltonians from the Floquet Hamiltonian. This is illustrated in the next two Sections III.E and III.F for two-photon resonant processes in atoms and molecules. 

We will apply specific rotating wave transformations R that will allow us to identify resonant terms and to eliminate the nonresonant ones. We obtain an effective one-mode Floquet Hamiltonian of the form... 

We use the rotating wave transformation dressing the state 2) with minus one coi photon ... 

With the technique combining the rotating wave transformations and contact transformations developed in Section III.C, one can treat accurately the dynamical resonances and construct approximately the quasi-energies. If we take into account the first two dynamical resonances by appropriate RWTs [associated with path (b)], one obtains the following explicit expression for the dressed energy surfaces ... 

To obtain the effective Floquet Hamiltonian, we apply the rotating wave transformation (RWT)... 

Eq. (6.26) is the TDSE in the Schrodinger picture. In general, it proves more convenient to discuss the time evolution of the driven system in a rotating frame, such as the frame rotating with the laser carrier frequency q- After transformation into the carrier frequency picture and application of the rotating wave approximation (RWA), the TDSE takes the form [92]... 

We have seen that the molecular electronic and vibrational wave functions el and vib each transform according to the irreducible representations of the molecular point group. We now consider the rotational wave function ptot. 

After transformation into the interaction picture and application of the rotating-wave approximation [46, SO, 54] the population dynamics can be calculated numerically by solving the time-dependent three-level Schrodinger equation or (if phenomenological relaxation rates are considered) by solving the density matrix equation (3) for the molecular system. The density matrix equation is given by... 

Cs2HgBr4 is representative of the /3-K2S04-type structure exhibiting several phase sequences upon cooling.580 The phase transformations are explained by a rotation wave through the HgBr4 tetrahedron. 

We are not going to review here the transformation properties of spatial wave functions under the symmetry operations of molecular point groups. To prepare the discussion of the transformation properties of spinors, we shall put some effort, however, in discussing the symmetry operations of 0(3)+, the group of proper rotations in 3D coordinate space (i.e., orthogonal transformations with determinant + 1). Reflections and improper rotations (orthogonal transformations with determinant -1) will be dealt with later. 

In the same way that we dealt with the transformation of the vibrational, case (a) spin and rotational wave functions in section 6.9.3, it is easy to show that... 

Note that Eqs (9.46) are completely identical to the set of equations (9.6) and (9.7). The problem of a single oscillator coupled linearly to a set of other oscillators that are otherwise independent is found to be isomorphic, in the rotating wave approximation, to the problem of a quantum level coupled to a manifold of other levels. There is one important difference between these problems though. Equations (9.6) and (9.7) were solved for the initial conditions Co(Z = 0) = 1, Cz(Z = 0) = 0, while here a t = 0) and h/(Z = 0) are the Schrodinger representation counterparts of fl(Z) and Still, Eqs (9.46) can be solved by Laplace transform following the route used to solve (9.6) and (9.7). 

We assume that the molecule has no permanent dipole moment, that is, VaaiR) = Rn>(R) — 0- Moreover, we assume the Condon approximation fiab(R) = nba(R) = constant. Non-Condon terms would introduce interesting new effects, particularly if n(R) has a node. However, that is beyond the scope of the current treatment. We transform the representation to a rotating frame and adopt the rotating-wave approximation. Then... 

Now, we are amazed to see that Eq. (6.23) is identical (ef., p. 199) to that which appeared as a result of the transformation of the Schrddinger equation for a rigid rotator. Y denoting the corresponding wave function. As we know from p. 200, this equation has a solution only if A = —J J + 1), where / — 0,1, 2,... Since Y stands for the rigid rotator wave function, we now concentrate exclusively on the function Xk which describes vibrations (changes in the length of R). 

Upon transforming to an interaction representation (i.e., a reference frame rotating at frequency w) in which the density matrix is defined by Equation 7, and invoking the rotating wave approximation which consists of dropping all high-frequency motions with respect to u . Equation 25 becomes... 

For computational purposes it is convenient to work with canonical MOs, i.e. those which make the matrix of Lagrange multipliers diagonal, and which are eigenfunctions of the Fock operator at convergence (eq, (3.41)). This corresponds to a specific choice of a unitary transformation of the occupied MOs. Once the SCF procedure has converged, however, we may chose other sets of orbitals by forming linear combinations of the canonical MOs. The total wave function, and thus all observable properties, are independent of such a rotation of the MOs. 

The probability that J has a wave vector K relative to I in HD + is given by the momentum transform of the wave function for the vibrational and rotational interactions in HD +. The probability that I is captured by X with a wave vector k is given by the momentum transform of the wave function for the rotational and vibrational interactions in XI+. 

If hd+ (i ) and pxi+ (k) denote the Fourier transforms of the indicated rotational and vibrational wave functions, the expression for the differential cross-section is... 

Wigner rotation/adiabatic-to-diabatic transformation matrices, 92 Electronic structure theory, electron nuclear dynamics (END) structure and properties, 326-327 theoretical background, 324-325 time-dependent variational principle (TDVP), general nuclear dynamics, 334-337 Electronic wave function, permutational symmetry, 680-682 Electron nuclear dynamics (END) degenerate states chemistry, xii-xiii direct molecular dynamics, structure and properties, 327 molecular systems, 337-351 final-state analysis, 342-349 intramolecular electron transfer,... 

Wigner rotation/adiabatic-to-diabatic transformation matrices, 92 Renner-Teller effect, triatomic molecules, 624 wave functions, molecular systems, 202—205... 

---