Computational treatment of quantum dynamics in molecules is often based, in one form or another, on the solution of the time-dependent Schrodinger equation. Due to extreme complexity of this problem, a lot of efforts were directed on the development of efficient approaches which allow to tackle dynamics in realistic systems. One of the most promising recent ideas is to use a specific ansatz for the full molecular wavefunction which is known as the exact factorization approach [1,2]. Within this formalism the correlated motion of nuclei and electrons in a system is represented by the two coupled equations containing terms responsible for the electron-nuclear couplings. Despite the bright promise of the exact factorization, the appearing equations of motion are extremely complicated to solve [3]. Namely, one of the key complexities is the necessity to calculate the logarithmic derivative of the time-dependent nuclear wave packet. Although at first glance the logarithmic derivative seems to be a straightforward quantity to calculate, it becomes clear upon closer inspection that its direct numerical evaluation is ill-defined. Here we present a new approach for the explicit propagation of the logarithmic derivative in time. The developed methodology combines Schrodinger picture with de Broglie-Bohm formulation of the quantum theory. In particular, we show how the polar representation of the wavefunction can be used to propagate the logarithmic derivative utilizing the topological phase of the wavefunction. Although the developed approach were initially designed to calculate the logarithmic derivative, it turned out to be an interesting tool to explore connections between Schrodinger and Bohmian mechanics. We hope that our study will stimulate further theoretical research aiming at developing computational approaches based on the factorized form of the molecular wavefunction.

[1] L. S. Cederbaum, J. Chem. Phys., vol. 128, no. 12, p. 124101, Mar. 2008.
[2] A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett., vol. 105, no. 12, Sep. 2010.
[3] A. Abedi, N. T. Maitra, and E. K. U. Gross, J. Chem. Phys., vol. 137, no. 22, 22A530, Dec. 2012.