ab initio Ehrenfest Dynamics

From Li et.al.,2005, JCP

"The Born Oppenheimer (BO) and extended Lagrangian (EL) trajectories are founded on the assumption that a single electronic potential surface governs the dynamics. .. A major limitation of adiabatic trajectories is that they are not applicable to reactions involving nonadiabatic electronic processes, i.e., multiple potential-energy surfaces." (Ex: Conical Intersections)

To account for electronic adiabaticity, we solve the full time-dependent Schrödinger equation for both nuclear and electronic degrees of freedom. In Ehrenfest methodology, the adiabatic potential energy surface

$$ E_{eff} = langlePhi|hat{H_{el}}|Phirangle = sum_i |a_i|^2 E_i^2$$ Thus, the atoms evolve on an effective potential representing an average over the adiabatic states weighted by their state populations $|a_i|^2$. The method is also therefore referred to as mean-field approach. As a comparison, for BOMD/Ehrenfest dynamics:

• BOMD

$$hat{H}_{el} (mathbf{r}; mathbf{R}) Phi_k(mathbf{r}; mathbf{R}) = E^{el}_k(mathbf{R})Phi_k(mathbf{r}; mathbf{R})$$

$$M_Iddot{mathbf{R}}_I =-nabla_IE^{el}_k(mathbf{R})=-mathop{nabla_I}_{text{min }Phi_k}langlePhi_k|hat{H}_{el}|Phi_krangle$$

The electronic wavefunction $Phi_k(mathbf{r}; mathbf{R})$ is static (only implicit time-dependence) and the nuclear degrees of freedom are handled classically. The nuclear degrees of freedom are decoupled from electronic degrees of freedom, while for each MD step the electronic wavefunction has to be optimized for the ground state.

• Ehrenfest dynamics

$$ihbarfrac{partial Phi(mathbf{r};mathbf{R},t)}{partial t}= hat{H}_{el} (mathbf{r}; mathbf{R}) Phi(mathbf{r};mathbf{R},t) $$

$$M_Iddot{mathbf{R}}_I =-nabla_Ilanglehat{H}_{el}(mathbf{r};mathbf{R})rangle$$

Here we have an explicit time dependence of the electronic wavefunction. Electronic and nuclear time evolutions are propagated with a three-time-step integrator. The electronic wavefunction is evolved via TD-SCF approach.

2nd Generation CPMD

Car-Parrinello MD avoids repeatedly solving the electronic problem by propagating the orbitals as if they were particles governed by Newton's equations. This is much more efficient than having to solve at each time step as is done in Born-Oppenheimer MD, though at the cost of decreasing the maximum timestep for the dynamics (too large a step will lose the ground state), slightly reduced accuracy (not exactly at the ground state for each time step) and introducing a spurious "mass parameter" to describe the electronic motion.

To address these problems, Thomas Kuhne et al. developed the "Car-Parrinello like approach to BOMD", also referred to as second-generation CPMD. The key differences of this approach are:

• Rather than propagating the orbitals (or rather the MO coefficients $mathbf{C}$), 2nd-CPMD propagates the density $mathbf{P}$ (or $mathbf{PS}$ for nonorthogonal orbitals). The density seems to evolve more smoothly than the coefficients, making it easier to work with.
• A predictor-corrector method (in their paper, the Always Stable Predictor Corrector (ASPC) method, but in principle any such method) is used to propagate the density. This generates a prediction of the next coefficients $mathbf{C}^p(t_n)$ based on the previous $K$ density matrices. A corrected set of coefficients $mathbf{C}(t_n)$ is then formed as $$mathbf{C}(t_n)=omega text{MIN}[mathbf{C}^p(t_n)]+(1-omega)mathbf{C}^p(t_n)$$ $$omega=frac{K}{2K-1}$$ where $text{MIN}$ is a minimization and $K$ is a parameter which determines the accuracy $O(Delta t^{2K-2})$. This update procedure eliminates the need for the mass parameter.
• The nuclear dynamics become dissipative, possibly due to the nonsymplectic electron dynamics. This is corrected by performing a short validation run of dynamics to compute a damping coefficient $gamma$ for the system.

There are some finer details to make this scheme work (specific parameterization of $mathbf{C}$, how $gamma$ is obtained) which are available in the original papers (both of which have arXiv preprint versions available).

Implementations:

• CP2K

References:

1. Thomas D. Kühne, Matthias Krack, Fawzi R. Mohamed, and Michele Parrinello Phys. Rev. Lett. 98, 066401 DOI: 10.1103/PhysRevLett.98.066401
2. Kühne, T.D. (2014), Second generation Car–Parrinello molecular dynamics. WIREs Comput Mol Sci, 4: 391-406. DOI: 10.1002/wcms.1176

CPMD: Car-Parrinello Molecular Dynamics

An approximation of BOMD (Born-Oppenheimer MD) where fictitious dynamics is used on the electrons to keep them close to their ground state, so that we do not have to keep solving for their ground state at every single step. We start with Newton's 2nd law (as does classical MD), but instead of the force being calculated by a full-fledged ab initio calculation at every step, the force itself has an EOM (equation of motion) which below is given by Eq. eqref{eq:fictitious}. For one nucleus with position $vec{r}$ and several electrons with orbitals ${psi_i}$ we get:

begin{align} tag{1} vec{F} &= mvec{ddot{r}} \ - nabla , Eleft[{ psi_i } , vec{r} right] &= mvec{ddot{r}}tag{2}\ mu ddot{psi}_i(vec{r},t) &= - frac{delta E}{delta psi_i^*(vec{r},t)} + sum_j Lambda_{ij} psi_j(vec{r},t),tag{3}label{eq:fictitious} end{align}

where $Lambda_{ij}$ is a matrix of Lagrange multipliers to allow satisfaction of the constraint that the wavefunctions $psi_i$ must be orthogonal; and $E[{psi_i},vec{r}]$ is an energy functional (usually a Kohn–Sham energy one). For several nucleii, just make a new subscript for $vec{r}$ and change the functional to $E[{psi_i},{vec{r_I}}]$, then the equations are exactly the same.

Implemented in:

• CPMD: Literally named after the method! (open-source)
• CP2K: Might also be named after the method! (open-source)
• NWChem (open-source)