Periodic Hamiltonian

The coherent potential approximation (CPA) was originally developed by Soven for the electronic problem and by Taylor for the phonon problem. The basic assumption of CPA is that a random medium with diagonal disorder can be described by an efiiective periodic Hamiltonian of the type... 

Floquet theory provides a generalised form for the propagators of systems with periodically time dependent Hamiltonians [90,91]. The propagator for a doubly periodic Hamiltonian in the BMFT representation maybe written as [92, 93]... 

Define the periodic Hamiltonian in the Hilbert space and transform it to the time independent Fioquet Hamiltonian. 

A discussion about the effects of the higher-order van Vleck correction terms will be given at a later stage, where we deal with the BMFT approach. However, a comment about Average Hamiltonian Theory (AHT) must be made at this point. This powerful theoretical approach is valid for single frequency periodic Hamiltonians. 

The Average Hamiltonian Theory (AHT) [23,26] is applied to express the propagator of a periodic Hamiltonian T-Lmtit + krt) = Hintit) in terms of a time independent effective Hamiltonian %). When the Hamiltonian is also cyclic and Uint kTt) = Uint(0) = 1, the propagator gets the generalised form... 

In the semiclassical model the molecule is treated quantum mechanically whereas the held is represented classically. The held is an externally given function of time F that is not affected by any feedback from the interaction with the molecule. We consider the simplest case of a dipole coupling. The formalism is easily extended to other types of couplings. The time dependence of the periodic Hamiltonian is introduced through the time evolution of the initial phase F = F(0 + oat) = electric held and oa is its frequency. The semiclassical Hamiltonian can be, for example, written as... 

Hamiltonians (for finite- or infinite-dimensional M ). However, in the case of several incommensurate frequencies (quasi-periodic Hamiltonian), reducibility is not always satisfied, even for finite-dimensional Jtf [19,33-36], We remark that for finite-dimensional ffl, reducibility is equivalent to the property of K having no continuous spectrum [19],... 

The equation is written in velocity gauge. Atomic units are used. The particle has charge unity and mass m in units of the free electron mass. V is the constant potential energy appropriate for the interval under consideration. The vector potential is supposed to be spatially constant at the length scale of the structure. With such a vector potential, the A2 term contributes an irrelevant phase factor which can be omitted. For a one-mode field A(t) is written as Ao cos(ut). The associated electric field is 0 sin(ut), with 0 = uAq. px is the linear momentum i ld/dx. For such a time-periodic Hamiltonian, a scattering approach can be developped, with a well-defined initial energy, and time-independent transition probabilities for reflection and transmission. 

Figure 2. This figure shows how alkane would be converted to a Hamiltonian used in this model (a — b), and then how this periodic Hamiltonian could be further reduced (b c) to result in the "string of pearls" H below it. The new energies and couplings in this chain have the effects of the side groups renorrml-...

The eigenvectors of the lattice periodic Hamiltonian of the Kohn-Sham-Dirac equation (28) are Bloch states fen) with crystal momentum k and band index n. They are expressed by the ansatz... 

The power of ultrafast MAS can easily be understood with AHT, as is explained in the seminal paper by Maricq and Waugh [52]. The solution to the periodic Hamiltonian problem H t) is obtained with a Magnus expansion that provides an effective Hamiltonian Hgg- acting on the spin system during a rotor period. This is relevant in the case of stroboscopic observation, that is, when a spectral window of or a sampling dwell time equal to Tr= 1/Vr is used HefF governs the shapes of the sidebands in the MAS spectrum and, indirectly, the resolution that can be achieved. On the other hand, the decay of the rotational echo is responsible for the shape of the spinning sideband pattern [52]. 

While combining the relation (3.23) with the periodic Hamiltonian of the crystalline system further result the identities ... 

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. 

The Floquet theorem, when apphed to the quantum mechanics [370], implies the stationarity of Floquet states imder a perfectly periodic Hamiltonian. We define the electronic Floquet operator as 7ft = Hf — ihdt and the Floquet states as its periodic eigenstates which satisfy Tlt x t)) = A] A(f))- The above mentioned stationarity states that the solution of time-dependent Schrodinger equation ] t) can be expanded as... 

For time-periodic Hamiltonians Hit) = Hit + T), the Floqnet theorem allows a solution of the time-dependent Schrodinger equation (TDSE) in the following form ... 

Figure 1. Density of states for the random and ideal periodic Hamiltonians defined in text with N 1000.

The most elementary assumption which one can make is that adding A and B does not change n E) at all. This is the rigid band approximation, often used in the interpretation of experimental results. It is, in the present case, equivalent to the slightly more sophisticated model defined by the virtual crystal approximation, in which the Hamiltonian is averaged over the ensemble defined by the above probability distribution, to define a periodic Hamiltonian of the form (46), with... 

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