Scaling transformations Hamiltonian

The classical dynamics of the FPC is governed by the Hamiltonian (1) for F = 0 and is regular as evident from the Poincare surface of section in Fig. 1(a) (D. Wintgen et.al., 1992 P. Schlagheck, 1992), where position and momentum of the outer electron are represented by a point each time when the inner electron collides with the nucleus. Due to the homogeneity of the Hamiltonian (1), the dynamics remain invariant under scaling transformations (P. Schlagheck et.al., 2003 J. Madronero, 2004)... 

One must note that q and p are coordinates and momenta up to a scale transformation. A more general form of the Hamiltonian // is... 

However, although the locally scaled transformed wavefunctions preserve the orthonormality condition, they fail to comply with Hamiltonian orthogonality. Of course, one can recombine the transformed wavefunctions so as to satisfy the latter requirement, by solving once more the eigenvalue problem... 

Now we should specify the response Hamiltonian hres r, t). For this aim, we use the scaling transformation and define the perturbed many-body wave function of the system as... 

Therefore the scaling transformation of the quantum-mechanical force field is an empirical way to account for the electronic correlation effects. As far as the conditions listed above are not always satisfied (e.g. in the presence of delocalized 7r-electron wavefunctions) the real transformation is not exactly homogeneous but rather of Puley s type, involving n different scale constants. The need of inhomogeneous Puley s scaling also arises due to the fact that the quantum-mechanical calculations are never performed in the perfect Hartree-Fock level. The realistic calculations employ incomplete basis sets and often are based on different calculation schemes, e.g. semiempirical hamiltonians or methods which account for the electronic correlations like Cl and density-functional techniques. In this context we want to stress that the set of scale factors for the molecule under consideration is specific for a given set of internal coordinates and a given quantum-mechanical method. 

There is a close connection between the set H, L, V and the set (T3, L, A obtained from it via the scaling transformation. The so(4) Lie algebra generated by L, V is the dynamical invariance (symmetry) algebra for the hydrogenic Hamiltonian, whereas L, A plays the same role for the T3 operator. In fact... 

In this section we shall discuss in some detail the formalism needed to apply the so(4, 2) algebraic methods to problems whose unperturbed Hamiltonian is hydrogenic. First a scaling transformation is applied to obtain a new Hamiltonian whose unperturbed part is just the so(2, 1) generator T3, which has a purely discrete spectrum. Next we use the scaled hydrogenic eigenfunctions of T3 as a basis for the expansion of the exact wave function. This discrete basis is complete with respect to the expansion of bound-state wave functions whereas the usual bound-state eigenfunctions do not form a complete set continuum functions must also be included to ensure completeness (cf. Section VI,A)-... 

After some scaling transformation, we have the following Hamiltonian H for this system ... 

Because local-scaling transformations preserve the orthonormality of basis functions, condition (54) is immediately fulfilled. Hamiltonian orthogonality (Eq. (55)), however, is not satisfied. For this reason, one must solve the eigenvalue equation (51]... 

The form of the Hamiltonian for the atom in a strong magnetic field suggests a scaling transformation r = (7B)2/3r and p = (7B)-1/3p [564]. With this transformation, the EBK quantisation condition becomes... 

In order to appreciate the fine points in this analysis, we therefore return to the domain issues, i.e. how to define the operator and the basis functions so that the scaling operation above becomes meaningful. Following Balslev and Combes [3], we introduce the N-body (molecular) Hamiltonian as H = T + V, where T is the kinetic energy operator and V is the (dilatation analytic) interaction potential (expressed as sum of two-body potentials Vy bounded relative Ty = Ay, where the indices i and j refers to particles i and j respectively). As a first crucial point we realize that the complex scaling transformation is unbounded, which necessitates a restriction of the domain of H note that H is normally bounded from below. Hence we need to specify the domain of H as... 

Eq. (5) shows that the classical dynamics depends on the scaled energy e = E Y-1/2. As it is clear from Eq. (5) the Hamiltonian has the singularity at f = 0. This singularity can be removed by performing the following transformations... 

The LVC model further allows one to introduce coordinate transformations by which a set of relevant effective, or collective modes are extracted that act as generalized reaction coordinates for the dynamics. As shown in Refs. [54, 55,72], neg = nei(nei + l)/2 such coordinates can be defined for an electronic nei-state system, in such a way that the short time dynamics is completely described in terms of these effective coordinates. Thus, three effective modes are introduced for an electronic two-level system, six effective modes for a three-level system etc., for an arbitrary number of phonon modes that couple to the electronic subsystem according to the LVC Hamiltonian Eq. (7). In order to capture the dynamics on longer time scales, chains of such effective modes can be introduced [50,51,73]. These transformations, which are briefly summarized below, will be shown to yield a unique perspective on the excited-state dynamics of the extended systems under study. 

---