Wave function time-dependent variational principle

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,... 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

The END theory was proposed in 1988 [11] as a general approach to deal with time-dependent non-adiabatic processes in quantum chemistry. We have applied the END method to the study of time-dependent processes in energy loss [12-16]. The END method takes advantage of a coherent state representation of the molecular wave function. A quantum mechanical Lagrangian formulation is employed to approximate the Schrodinger equation, via the time-dependent variational principle, by a set of coupled first-order differential equations in time to describe the END. 

It is possible that a slight improvement in the treatment of the nuclear motion, based on the time-dependent variational principle, will accurately predict the interference signal on the short timescale necessary to observe geometric phase development, without suffering the instabilities of the locally quadratic method [36, 37]. Such an improvement may come at the cost of describing the excited state wave function as a superposition of... 

Let us assume that at r = 0 the wave function If is given in MCTDH form, i.e., given by equation (16). (The question of how to define and generate an initial-state wave function is addressed below.) We want to propagate If while preserving its MCTDH form. As was done above, we derive first-order differential equations (equations of motion) for A and by employing the time-dependent variational principle equation (5). But before doing so, we partition the Hamiltonian H into a separable and residual part ... 

There are two ways to determine the parameters that enter into the specification of the wave function. The long (and more correct) way is via a variational principle (53, 90-94). For our purpose it is sufficient to require that the ansatz (Eq. (50)) (50) for the wave function satisfies the time-dependent Schrodinger equation. To do so consider the equation of motion for the moments ... 

The regular wave function (t) > satisfies a generalized time-dependent Frenkel variational principles [24] for an arbitrary variation W... 

The variational principle for the action integral is derived starting from the observation that, if the time-dependent wave function 4/(r, 0 is a solution of the time-dependent Schrodinger equation, then it corresponds to a stationary point of the quantum-mechanical action integral ... 

In principle, the ultrasonic techniques described for solid-liquid flow measurement can be applied to measure air flow rate and particle velocity. Direct measurement of air flow rate by measuring upstream and downstream transit times has been demonstrated. But, the Doppler and cross-correlation techniques have never been applied to solid/gas flow because the attenuation of ultrasound in the air is high. Recent developments have shown that high-frequency (0.5-MHz) air-coupled transducers can be built and 0.5-MI Iz ultrasound can be transmitted through air for a distance of at least 1 in. Thus, the cross-correlation technique should be applicable to monitoring of solid/gas flow. Here, we present a new cross-correlation technique that does not require transmission of ultrasonic waves through the solid/gas flow. The new technique detects chiefly the noise that interacts with the acoustic field established within the pipe wall. Because noise may be related to particle concentration, as we discussed earlier, the noise-modulated sound field in the pipe wall may contain flow information that is related to the variation in particle concentration. Therefore, crosscorrelation of the noise modulation may yield a velocity-dependent correlation function. 

---