Car-Parrinello Molecular Dynamics

The basic idea of the Car-Parrinello approach can be viewed to exploit the time-scale separation of fast electronic and slow nuclear motion by transforming that into classical-mechanical adiabatic energy-scale separation in the framework of dynamical systems theory. In order to achieve this goal the two-component quantum / classical problem is mapped onto a two-component purely classical problem with two separate energy scales at the expense of loosing the explicit time-dependence of the quantum subsystem dynamics. This is achieved by considering the extended Kohn-Sham energy functional ${\cal E}^{\rm KS}$ to be dependent on $\{ \Phi_i \}$ and ${\bf R}^N$ . In classical mechanics the force on the nuclei is obtained from the derivative of a Lagrangian with respect to the nuclear positions. This suggests that a functional derivative with respect to the orbitals, which are interpreted as classical fields, might yield the force on the orbitals, given a suitable Lagrangian.

Car and Parrinello postulated the following Lagrangian [1] using ${\cal E}^{\rm KS}$ .

$${\cal L}_{\rm CP}[{\bf R}^N, {\bf\dot R}^N, \{\Phi_i \}, \{ \dot \Phi_i \}] = \sum_I {1\over 2} M_I {\bf\dot R}_I^2 + \sum_i \mu \left< {\dot \... ...t. \right> - {\cal E}^{\rm KS}\left[ \{ \Phi_i \}, {\bf R}^N \right] \enspace .$$ (36)

The corresponding Newtonian equations of motion are obtained from the associated Euler-Lagrange equations

$${d\over {dt}} {{\partial {\cal L}_{\rm CP}} \over {\partial { {\bf\dot R}}_I}} = {{\partial {\cal L}_{\rm CP}} \over {\partial {\bf R}_I }}$$ (37)

$${d\over {dt}} {{\delta {\cal L}_{\rm CP}} \over {\delta \langle { \dot \Phi}_i \mid }} = {{\delta {\cal L}_{\rm CP}} \over {\delta \langle \Phi_i \mid }}$$ (38)

like in classical mechanics, but here for both the nuclear positions and the orbitals. Note that the constraints contained in ${\cal E}^{\rm KS}$ are holonomic [19]. Following this route of ideas, Car-Parrinello equations of motion are found to be of the form

$$M_I {\bf\ddot R}_I = - {\partial E^{\rm KS} \over {\partial {\bf R}_I}} + \sum_{ij} \L... ...a_{ij} {\partial \over {\partial {\bf R}_I}} \langle \Phi_i \mid \Phi_j \rangle$$ (39)

$$\mu \mid {\ddot \Phi}_i \rangle = - {\delta E^{\rm KS} \over {\delta \langle \Phi_i \mid }} + \sum_{j} \Lambda_{ij} \mid \Phi_j \rangle$$ (40)

where $\mu$ is the ``fictitious mass'' or inertia parameter assigned to the orbital degrees of freedom; the units of the mass parameter $\mu$ are energy times a squared time for reasons of dimensionality. Note that the constraints within ${\cal E}^{\rm KS}$ lead to ``constraint forces'' in the equations of motion. In general, these constraints will depend on both the Kohn-Sham orbitals and the nuclear positions through the overlap matrix of basis functions. These dependencies have to be taken into account properly in deriving the Car-Parrinello equations following from Eq. (36) using Eqs. (37)-(38).

The constant of motion is

$$E_{\rm cons} = \sum_I {1\over 2} M_I {\bf\dot R}_I^2 + \sum_i \m... ...right. \right> + {E}^{\rm KS}\left[ \{ \Phi_i \}, {\bf R}^N \right] \enspace .$$ (41)

According to the Car-Parrinello equations of motion, the nuclei evolve in time at a certain (instantaneous) physical temperature $\propto \sum_I M_I {\bf\dot R}_I^2$ , whereas a ``fictitious temperature'' $\propto \sum_i \mu \langle {\dot \Phi}_i \vert {\dot \Phi}_i \rangle$ is associated to the electronic degrees of freedom. In this terminology, ``low electronic temperature'' or ``cold electrons'' means that the electronic subsystem is close to its instantaneous minimum energy $\min_{\{\Phi_i\}} E^{\rm KS}$ i.e. close to the exact Born-Oppenheimer surface. Thus, a ground-state wavefunction optimized for the initial configuration of the nuclei will stay close to its ground state also during time evolution if it is kept at a sufficiently low temperature.

The remaining task is to separate in practice nuclear and electronic motion such that the fast electronic subsystem stays cold also for long times but still follows the slow nuclear motion adiabatically (or instantaneously). Simultaneously, the nuclei are nevertheless kept at a much higher temperature. This can be achieved in nonlinear classical dynamics via decoupling of the two subsystems and (quasi-) adiabatic time evolution. This is possible if the power spectra of both dynamics do not have substantial overlap in the frequency domain so that energy transfer from the ``hot nuclei'' to the ``cold electrons'' becomes practically impossible on the relevant time scales. This amounts in other words to imposing and maintaining a metastability condition in a complex dynamical system for sufficiently long times. How and to which extend this is possible in practice was investigated in detail in an important investigation based on well-controlled model systems [39] and with more mathematical rigor in Ref. [40].