Time-dependent many-electron problem

In the Time Dependent Density Functional Theory (TDDFT) [16], the correlated many-electron problem is mapped into a set of coupled Schrodinger equations for each single electronic wavefunctions (o7 (r, t),j= 1, ), which yields the so-called Kohn-Sham equations (in atomic units)... 

On the other hand, there are many dynamic phenomena whose quantitative description cannot be achieved via a stationary-state formalism, whose hallmark, as already indicated, is the form of Eq. (1) or (2) for the eigenfunction. In other words, now, the complete solution of the TDSE for all t cannot be written as a product of two terms, one of which is the phase that contains time and the other is a time-independent eigenfunction in coordinate space. In these cases, in most real situations one faces a genuine time-dependent many-electron problem (TDMEP), whose solution must be based on the quantitative knowledge of time-dependent, nonstationary (unstable) states, l> q, t). 

The first volume contained nine state-of-the-art chapters on fundamental aspects, on formalism, and on a variety of applications. The various discussions employ both stationary and time-dependent frameworks, with Hermitian and non-Hermitian Hamiltonian constructions. A variety of formal and computational results address themes from quantum and statistical mechanics to the detailed analysis of time evolution of material or photon wave packets, from the difficult problem of combining advanced many-electron methods with properties of field-free and field-induced resonances to the dynamics of molecular processes and coherence effects in strong electromagnetic fields and strong laser pulses, from portrayals of novel phase space approaches of quantum reactive scattering to aspects of recent developments related to quantum information processing. 

Resonances are common and unique features of elastic and inelastic collisions, photodissociation, unimolecular decay, autoionization problems, and related topics. Their general behavior and formal description are rather universal and identical for nuclear, electronic, atomic, or molecular scattering. Truhlar (1984) contains many examples of resonances in various fields of atomic and molecular physics. Resonances are particularly interesting if more than one degree of freedom is involved they reflect the quasi-bound states of the Hamiltonian and reveal a great deal of information about the multi-dimensional PES, the internal energy transfer, and the decay mechanism. A quantitative analysis based on time-dependent perturbation theory follows in the next section. 

For such nonequilibrium processes, the direct mapping of the electron propagator methods to calculations of electric current becomes inapplicable because of the time-dependent nature of electric current in both phenomena. A time-dependent problem requires the further development of the theory of Green s functions to electron dynamics in which e-e correlation effects are taken into account. Such methodology already exists in physics in which many-body ideas have been developed for time-dependent problems. This theory is based on nonequilibrium Green s or Keldysh functions [2,5, 6, 40-46]. 

Because of the apparent chaos in Fig. 6.5, simple analytical solutions of the driven SSE system probably do not exist, neither for the classical nor for the quantum mechanical problem. Therefore, if we want to investigate the quantum dynamics of the SSE system, powerful numerical schemes have to be devised to solve the time dependent Schrddinger equation of the microwave-driven SSE system. While the integration of classical trajectories is nearly trivial (a simple fourth order Runge-Kutta scheme, e.g., is sufficient), the quantum mechanical treatment of microwave-driven surface state electrons is far from trivial. In the chaotic regime many SSE bound states are strongly coupled, and the existence of the continuum and associated ionization channels poses additional problems. Numerical and approximate analytical solutions of the quantum SSE problem are proposed in the following section. 

Fig. 28.1. In a genetic switch, the total state of the system depends on two variables whether the DNA site is occupied or not and the number of copies of the transcription factor. In the left panel, the logarithm of the steady state probability for the occupancy state and protein number is plotted. This probability acts like an effective potential. In the right panel, the effective potential for a charge transfer or two site polaron is plotted as a function of the enviromnent polarization for the two electronic states. The governing time-dependent eqnations for the two problems share many similarities...

All these extensions of DFT to time-dependent, magnetic, relativistic and a multitude of other situations involve more complicated Hamiltonians than the basic ab initio many-electron Hamiltonian defined by Eqs. (2) to (6). Instead of attempting to achieve a more complete description of the many-body system under study by adding additional terms to the Hamiltonian, it is often advantageous to employ the opposite strategy, and reduce the complexity of the ab initio Hamiltonian by replacing it by simpler models, which focalize on specific aspects of the full many-body problem. Density-... 

As is clear from this overview the present discussion focuses completely on the RKS scheme for stationary ground state problems of many-electron systems. Nevertheless it seems worthwhile to mention some further work on RDFT which is beyond this scope. A time-dependent generalization of the... 

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