Eikonal representation

INTRODUCTION DENSITY MATRIX TREATMENT Equation of motion for the density operator Variational method for the density amplitudes THE EIKONAL REPRESENTATION The eikonal representation for nuclear motions... 

The following treatment starts with the complete quantal equations and introduces an eikonal representation which allows for a formally exact treatment. It shows how a time-dependent eikonal treatment can be combined with TDHF... 

The density operator in the coordinate representation is given by the functions T q, Q, q Q, <), and can be expressed for Hamiltonian systems in terms of its amplitudes, which become the wavefunctions Q, <) I convenient to introduce the formally exact eikonal representation. 

The treatment presented so far is quite general and formally exact. It combines the eikonal representation for nuclear motions and the time-dependent density matrix in an approach which could be named as the Eik/TDDM approach. The following section reviews how the formalism can be implemented in the eikonal approximation of short wavelengths for the nuclear motions, and for specific choices of electronic states leading to the TDHF equations for the one-electron density matrix, and to extensions of TDHF. 

A formulation of electronic rearrangement in quantum molecular dynamics has been based on the Liouville-von Neumann equation for the density matrix. Introducing an eikonal representation, it naturally leads to a general treatment where Hamiltonian equations for nuclear motions are coupled to the electronic density matrix equations, in a formally exact theory. Expectation values of molecular operators can be obtained from integrations over initial conditions. 

Abstract The original approach called the Eikonal representation is used to study the... 

The new fundamental equations of the dynamical theory were written within the Eikonal representation. Such equations are valid for any kind and strength of a regular deformation and, in opposite to the THW equations, they describe only interbranch transitions of electrons. 

Our approach to the dynamics of complex electronic rearrangements, has been based on an eikonal representation of the molecular wavefunction. [40, 41, 42] In this representation, wavefunctions are written in the form x(q. Q, t)exp[iS(Q, t)/h], with a factorized exponential function of classical-like variables Q, where S is a classical-like mechanical action. It can be applied without detailed preliminary knowledge of electronic rearrangements,... 

Taking the nuclear coordinates Q to be classical-like, the eikonal representation gives the wavefunction k(g, Q, t) for an initial electronic state / as a superposition of functions, of the form... 

A formalism presented in this part unifies both approaches. Path integrals over projectile variables in momentum representation are evaluated in the quasiclassical limit separately before and after the turning point in the spirit of generalized eikonal method. The Faddeev-Popov method (Popov 1983) is used to fix classical trajectories with respect to the symmetry of the problem. The influence functional is treated as a pre-exponential factor. 

In most practical applications, discussed in this book, one dynamical approximation is used - so called generalized eikonal approach. It is very refined and effective combination of the uniform quasiclassical representation of the scattering amplitude and the eikonal approximation for propagator, that seems to be very natural compromise between the complexity of the problem and our ability to understand its main features aird tendencies. 

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