Time-dependent electron localization functions for coupled nuclear-electronic motion

M. Erdmann, E. K. U. Gross, V. Engel
2004 Journal of Chemical Physics  
We study the quantum dynamics in a model system consisting of two electrons and a nucleus which move between two fixed ions in one dimension. The numerically determined wave functions allow for the calculation of time-dependent electron localization functions in the case of parallel spin and of the time-dependent antiparallel spin electron localization functions for antiparallel spin. With the help of these functions, it becomes possible to illustrate how electronic localization is modified
more » ... ion is modified through the vibrational wave-packet motion of the nucleus. In 1990, Becke and Edgecombe introduced a simple measure of electron localization in atomic and molecular systems in their such entitled article. 1 The authors defined the so-called electron localization function ͑ELF͒ within the Hartree-Fock theory and for systems with parallel spin. Up to now, many applications of the ELF to analyze atomic shell structure and bonding situations in molecules were presented, for a review see Ref. 2. Most recently, the ELF was employed in the time-domain to study the electronic dynamics of acetylene in a strong laser pulse and the collision of a proton with ethylene with classically moved nuclei. 3 In this letter we elaborate on the idea of the ELF, extending the concept to a more general situation. In doing so, we exploit a model of two electrons and a nucleus all of which are allowed to move quantum mechanically in a single dimension between two ions, the latter being fixed in space. This particle configuration is sketched in Fig. 1 . We solve the time-dependent Schrödinger equation numerically exact to obtain wave functions (x,y,R,t), where x, y, R are the electronic and nuclear coordinates, , are the spin coordinates of the electrons, and t denotes time. The wave functions serve as the starting point to calculate quantities which characterize electron localization. To proceed, we follow the strategy of Becke and Edgecombe, and start from the diagonal of the time-dependent density matrix of the two-electron/one-nucleus system, given by This quantity is the probability density for finding, at time t, the two electrons with spins and at x and y, respectively, and the nucleus at position R. Although we might, in principle, investigate the localization of all three particles, we concentrate here on the electrons. An average of the diagonal part of the density matrix over the nuclear degree of freedom yields ͑2͒ The time-dependent probability density ͑the electronic spin density͒ to find one electron with spin at point x, independently of where the other electron and the nucleus are located, is obtained by integration ͑3͒ Furthermore we define the conditional probability density to find an electron with spin at y, if we know with certainty that another electron with spin is located at x, by We now have to distinguish the cases of parallel ͑␣␣͒ and antiparallel ͑␣␤͒ spin of the two electrons. Since in the latter case, the coordinate-space wave function is symmetric with respect to exchange of the electrons, we may use the function as a measure of localization. Accordingly, P ␣␤ (x,t)dx is the conditional probability in the volume element dx to find one electron at time t at point x, if we know with certainty that the other electron with opposite spin is at the same place. Unfortunately, this relation is an indirect one. P ␣␤ (x,t) is small, if the electron at x is strongly localized. Following the example given by Becke and Edgecombe for the ELF, we define an inverse quantity denoted as time-dependent antiparallel spin electron localization function ͓TDALF(x,t)͔ as
doi:10.1063/1.1806812 pmid:15538889 fatcat:nqp6yqhcg5ezjlsfsybm3ud7gy