Time-dependent Variational Principle TDVP

Obviously, the BO or the adiabatic states only serve as a basis, albeit a useful basis if they are determined accurately, for such evolving states, and one may ask whether another, less costly, basis could be Just as useful. The electron nuclear dynamics (END) theory [1-4] treats the simultaneous dynamics of electrons and nuclei and may be characterized as a time-dependent, fully nonadiabatic approach to direct dynamics. The END equations that approximate the time-dependent Schrddinger equation are derived by employing the time-dependent variational principle (TDVP). 

Conventional presentaticsis of DFT start with pure states but sooner w later encounter mixed states and d sities (ensemble densities is the usual formulation in the DFT literature) as well. These arise, for example in formation or breaking of chemical bonds and in treatments of so-called static correlation (situations in which several different one-electron configurations are nearly degenerate). Much of the DFT literature treats these problems by extension and generalization from pure state, closed shell system results. A more inclusively systematic treatment is preferable. Therefore, the first task is to obtain the Time-Dependent Variational Principle (TDVP) in a form which includes mixed states. 

Computational strategies can be based on variational procedures using the Dirac-Frenkel time-dependent variational principle (TDVP). Introducing a shorthand notation so that... 

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,... 

Euler-Lagrange equations, electron nuclear dynamics (END), time-dependent variational principle (TDVP) basic ansatz, 330-333 free electrons, 333-334 Evans-Dewar-Zimmerman approach, phase-change rule, 435... 

As is well-known, the time-dependent variational principle (TDVP) applied to the quantum mechanics action, when fully general variations in state vector space are possible, yields the time-dependent Schrodinger equation. However, when the variations take place in a limited space determined by the choice of an approximate form of wavefunction the result is a set of coupled first-order differentiS equations that govern the time-evolution of the wavefunction parameters (27). 

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