Theory of Cluster Dynamics

Time Dependent Density Functional Theory with Molecular Dynamics

TDLDA-MD:

Time-dependent local-density approximation plus ionic molecular dynamics

(This is a very short summary of our formal scheme. A most detailed description is found in [Rei03].)

The electron cloud is described by density functional theory at the level of TDLDA. The dynamical degrees of freedom are the set of occupied single-electron wavefunctions $\bgroup\color{red}$ \varphi_\alpha$\egroup$ . The ions are treated by classical MD and their degrees of freedom are the positions R_I and momenta P_I . The starting point is the total energy given by:

$\displaystyle E_{\rm total}$

$\displaystyle =$

$\displaystyle {\color{red} E_{\rm kin}(\{\varphi_\alpha\})+ E_{\rm C}(\rho)+ ......}_I,{P}_I\})} +E_{\rm ext}({\color{red} \rho},{\color{dgreen} {R}_I},t) \quad.$

The electronic kinetic energy $\bgroup\color{dgreen}$ {\color{red} E_{\rm kin}}$\egroup$ employs the single-electron wavefunctions $\bgroup\color{dgreen}$ {\color{red} \varphi_\alpha}$\egroup$ which maintains the quantum mechanical shell effects. All other electronic energies refer only to the local spin-densities or total density $\bgroup\color{dgreen}$ {\color{red} \rho=\rho_\uparrow+\rho_\downarrow}$\egroup$ ; the Coulomb energy $\bgroup\color{dgreen}$ {\color{red} E_{\rm C}}$\egroup$ naturally, and the exchange-correlation energy $\bgroup\color{dgreen}$ {\color{red} E_{\rm xc}}$\egroup$ by virtue of the LDA (often augmented by a self-interaction correction (SIC) [Leg02]). The electron-ion coupling $\bgroup\color{dgreen}$ E_{\rm el,ion}$\egroup$ is realized by pseudo-potentials, mostly soft local ones [Kue00]. The ionic part $\bgroup\color{dgreen}$ {\color{dgreen} E_{\rm ion}}$\egroup$ is composed of Coulomb interaction and kinetic energy. Excitation mechanisms (laser, ionic collisions) are described in $\bgroup\color{dgreen}$ E_{\rm ext}$\egroup$ as external time-dependent potentials.

The coupled equations of motion are obtained in standard manner by variation. They read

$\bgroup\color{dgreen}$\displaystyle {\color{red} \imath\partial_t\varphi_\alpha......M_I} \quad,\quad\partial_t{P}_I =-\nabla_{{R}_I}E_{\rm total}} \quad.$\egroup$

where $\bgroup\color{dgreen}$ {\color{red} \sigma_\alpha}$\egroup$ is the spin orientation of the state $\bgroup\color{dgreen}$ {\color{red} \alpha}$\egroup$ . The equations imply a non-adiabatic coupling which goes beyond usual Born-Oppenheimer approach. Non-adiabatic effects become crucial in cluster dynamics induced by strong fields. The numerical solution involves the representation of the wavefunctions on a spatial grid, time-splitting for the electronic propagation and the Verlet algorithm for MD, for details see [Cal00]. The obtained wavefunctions, densities, and ionic coordinates allow to compute a wide variety of observables, e.g. optical absorption spectra [Cal97], angular distributions [Poh04a], emission spectra [Poh03], or ionization [Ull98a] for electronic degrees of freedom. The ionic configurations can be measured indirectly through optical response and its dynamics with various pump and probe scenarios [And02].

Often, we use a semi-classical description for the electronic dynamics at the level of Vlasov-LDA, particularly for energetic processes and/or large clusters. Instead of the wavefunctions, the key ingredient becomes here the one-electron phase-space distribution $\bgroup\color{red}$ f({r},{p},t)$\egroup$ . The quantum-mechanical propagation for the electrons is replaced by the Vlasov equation

$\bgroup\color{red}$\displaystyle {\color{red} \partial_t f= \frac{{p}}{m}\nabl......{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}} \Big)\nabla_{p}f }$\egroup$

again non-adiabatically coupled to ionic motion as above. Note that formally the same Kohn-Sham potential $\bgroup\color{red}$ {\color{red} {\delta E_{\rm total}}\big/{\delta\rho_{\sigma_\alpha}}}$\egroup$ is employed. For a derivation and justification from TDLDA see [Dom97]. The Vlasov-LDA equation is solved with the test-particle method where the distribution function $\bgroup\color{red}$ {\color{red} f}$\egroup$ is represented as a sum of Gaussian test-particles which are propagated again by the Verlet algorithm [Gig02].

The semi-classical description makes it feasible to include dynamical correlations from electron-electron collisions. This is achieved by adding an Uehling-Uhlenbeck collision term leading to

$\bgroup\color{red}$\displaystyle {\color{red} \partial_t f= \frac{{p}}{m}\nabl......\delta\rho_{\sigma_\alpha}} \Big)\nabla_{p}f +I_{\rm UU}(f) }\quad. $\egroup$

The collision term $\bgroup\color{red}$ {\color{red} I_{\rm UU}}$\egroup$ is a non-linear functional of the distribution function $\bgroup\color{red}$ {\color{red} f}$\egroup$ . It contains terms up to third power in $\bgroup\color{red}$ {\color{red} f}$\egroup$ . It is constructed from local and instantaneous collisions which obey energy conservation, momentum conservation, and the Pauli principle [Gig02]. The resulting equation is called the Vlasov-Uehling-Uhlenbeck approach (VUU).