Energy conserving approximations to the quantum potential: Dynamics with linearized quantum force

Abstract

Solution of the Schrödinger equation within the de Broglie-Bohm formulation is based on propagation of trajectories in the presence of a nonlocal quantum potential. We present a new strategy for defining approximate quantum potentials within a restricted trial function by performing the optimal fit to the log-derivatives of the wave function density. This procedure results in the energy-conserving dynamics for a closed system. For one particular form of the trial function leading to the linear quantum force, the optimization problem is solved analytically in terms of the first and second moments of the weighted trajectory distribution. This approach gives exact time-evolution of a correlated Gaussian wave function in a locally quadratic potential. The method is computationally cheap in many dimensions, conserves total energy and satisfies the criterion on the average quantum force. Expectation values are readily found by summing over trajectory weights. Efficient extraction of the phase-dependent quantities is discussed. We illustrate the efficiency and accuracy of the linear quantum force approximation by examining a one-dimensional scattering problem and by computing the wavepacket reaction probability for the hydrogen exchange reaction and the photodissociation spectrum of ICN in two dimensions.

Semiclassical nonadiabatic dynamics based on quantum trajectories for the O(P3,D1)+H2 system

“…The resulting potential is exact for Gaussian wave packets and can describe tunneling and zero-points energy effects in chemical systems as was shown for twodimensional H 3 and ICN model systems. 22 This approximation can be improved by linearizing r on subspaces 23 or by including higher-order basis functions representing r. For example, and exponential function gives the exact description of the Morse oscillator eigenstates. 18 The mixed wavefunction representation, ͑x,t͒ = ͑x,t͒ ͑x,t͒, where the polar part ͑x , t͒ is represented in terms of quantum trajectories and the coordinate space prefactor ͑x , t͒ is computed approximately for each trajectory, gives a cheap description of the nodes in the wave-function density 24 or of the nonadiabatic effects.…”

Bohmian dynamics on subspaces using linearized quantum force

“…As we have shown earlier, 30 Ũ evaluated at the optimal values of s ជ satisfies Eq. ͑6͒ and, therefore, conserves the total energy of a closed system regardless of the functional form of g (n) .…”

“…Recently, we have computed the wave packet transmission probabilities and studied the isotope effect for this system using the LQF. 30 The system is described in the Jacobi coordinates of reactants where the kinetic energy is diagonal. The Hamiltonian, coordinates, and potential surface are the same as in Ref.…”

“…Therefore, the wave packet bifurcations are describable within this approach, as was shown by computing the wave packet transmission probabilities for onedimensional scattering and for the collinear hydrogen exchange reaction. 30 To go beyond the LQF within the framework of approximating the nonclassical momentum, one may approach a full quantum-mechanical description by increasing the flexibility of the fitting functions. If the fitting function can be represented in terms of a complete basis set, the description will be formally exact, though optimization with complicated fitting functions will be a nonlinear problem which significantly increases computational costs.…”

Incorporation of quantum effects for selected degrees of freedom into the trajectory-based dynamics using spatial domains

“…Thus a cheap approximation which is insensitive (even if inaccurate) to the singularities, yielding linear quantum force (LQF) has been developed. 23 The LQF is obtained from a global fitting of the nonclassical momentum components,…”