Nonadiabatic quantum molecular dynamics with hopping. I. General formalism and case study

An extension of the nonadiabatic quantum molecular dynamics approach is presented to account for electron-nuclear correlations in the dynamics of atomic many-body systems. The method combines electron dynamics described within time-dependent density-functional or Hartree-Fock theory with trajectory-surface-hopping dynamics for the nuclei, allowing us to take into account explicitly a possible external laser field. As a case study, a model system of ${\mathrm{H}}^{+}+\mathrm{H}$ collisions is considered where full quantum-mechanical calculations are available for comparison. For this benchmark system the extended surface-hopping scheme exactly reproduces the full quantum results. Future applications are briefly outlined.

Figure 1

Overview of notations, transformations, and representations of single-particle functions belonging (left) to the atomic basis $\left\{{\varphi }_{\alpha }(\mathbf{r}-{\mathbf{R}}_{A_{\alpha }})\right\}$ and (right) to the molecular basis $\left\{{\chi }_a^{\sigma }(\mathbf{r};\mathbf{R})\right\}$. Throughout the paper, the Greek indices $\alpha ,\beta ,\gamma ,\delta$ correspond to the atomic basis, whereas the Latin indices $a,b$ belong to the molecular basis. First row: Transformations between both basis functions. Second row: Expansions of the time-dependent KS functions in both basis sets. Third row: Transformations between the time-dependent expansion coefficients.

Figure 2

Schematic representation of the surface-hopping procedure in the adiabatic many-electron picture. Initially, at time $t_1$, the system is in its ground state. At some later time $t_2$, a hop occurs, leading to a singly excited state, and even later, another hop occurs, and the system ends up in a doubly excited state. In this way, many-particle transitions are realized as successive one-particle transitions. Defining the single-particle occupation configuration for each spin as $p_a^{\sigma }={\sum }_{j=1}^{N_{\mathrm{e}}}{c}_a^{j\sigma }$ [see (40)], we get a simple representation of the adiabatic many-electron states: $\left\{p^{\uparrow }\right\}=\left(111000\right)$ and $\left\{p^{\downarrow }\right\}=\left(111000\right)$ at $t_1$, $\left\{p^{\uparrow }\right\}=\left(101010\right)$ and $\left\{p^{\downarrow }\right\}=\left(111000\right)$ at $t_2$, $\left\{p^{\uparrow }\right\}=\left(101010\right)$ and $\left\{p^{\downarrow }\right\}=\left(011001\right)$ at $t_3$, etc.

Figure 3

Hierarchy of ab initio MD methods: QMD, NA-QMD, and NA-QMD-H. The left and right linkages indicate that both nonadiabatic approaches naturally merge into the adiabatic QMD limit under certain conditions (see text).

Figure 4

(a) Energy levels of ${\mathrm{H}}_2$${}^{+}$ in the adiabatic and diabatic representation as well as (b) the respective couplings as a function of the internuclear distance $R$. They are obtained from QMD ground-state calculations using the Gaussian basis set d-aug-cc-pV6Z (see [55, 56, 57]) for numerical construction of the $1s$ and $2s$ atomic orbitals of hydrogen. Only the $1s{\sigma }_{\mathrm{u}}$ and $2s{\sigma }_{\mathrm{u}}$ states are considered, as discussed in the text.

Figure 5

(a) Kinetic-energy loss $\Delta E$ as a function of the impact energy $E_{\mathrm{cm}}$ and (b) kinetic-energy spectra $P\left(E\right)$ taken at the three impact energies indicated in (a): $50\mathrm{eV}$ (I), $80\mathrm{eV}$ (II), and $129\mathrm{eV}$ (III).