# How can the fictitious mass in the Car-Parrinello method reproduce the “real” dynamics?

In the Car-Parrinello method, to solve simultaneously the classical equations of motion for the atoms and the Kohn-Sham equations for the electrons, the following effective Lagrangian is used: $$\tag{1} \mathcal L_{\text{CP}} = \frac{1}{2} \sum_I^{\text{nuclei}} M_I \dot{\textbf{R}}_I^2 + \frac{1}{2} \mu \sum_i^{\text{electrons}} \int d \textbf r | \dot\phi_i(\textbf r,t)|^2 - E_{\text{KS}}[\{ \phi_i\},\{R_I\}],$$ where $E_{\text{KS}}[\{ \phi_i\},\{R_I\}]$ is the Kohn-Sham energy, usually given in the form: $$\tag{2} E_{\text{KS}}[\{ \phi_i\},\{R_I\}] = \\ \sum_i \epsilon_i - \frac{1}{2} \int d\textbf r d\textbf r' n(\textbf r)W(\textbf r,\textbf r')n(\textbf r') + E^{XC}[n] - \int d\textbf r \,n(\textbf r)\frac{\delta E^{XC}}{\delta n(\textbf r)},$$ with $W$ the electron-electron electrostatic interaction energy and $E^{XC}$ the exchange-correlation energy.

The fictitious mass $\mu$ in (1) is used instead of the electron mass $m_e$, and the wikipedia article says that in the limit $\mu \to 0$ the equations of motion approach Born-Oppenheimer molecular dynamics.

1. Why is the fictitious mass $\mu$ used instead of the real electronic mass?
2. How can we simulate the real motion while using fictitious masses?

Looking for other information, I came across this unanswered question. Maybe the OP found elsewhere the answer or she/he is not interested anymore. But, since Car Parrinello method is one of the most used algorithm to perform ab-initio Molecular Dynamics, I feel worth to provide an answer.

1. Role of $$\mu$$

First of all, even if people often say and write that the Car-Parrinello fictitious mass $$\mu$$ is used in place of the real electronic mass $$m_e$$, this is simply wrong. The reason is that $$\mu$$ and $$m_e$$ appear in different formulae and have completely different physical dimensions (thus, are measured with different units). Indeed the physical dimension of $$m_e$$ is mass, while from the Lagrangian (eqn (1) in the question) it is evident that, since the term $$\frac{1}{2} \mu \int d \textbf r | \dot\phi_i(\textbf r,t)|^2$$ must have dimension of energy, and being $$\int d \textbf r | \phi_i(\textbf r,t)|^2$$ dimensionless (it is a probability), $$\left[ \mu \right]= {energy}\times time^2=mass\times lenght^2$$.

What is true is that $$\mu$$ is a parameter playing the same role of a mass: it directly controls the acceleration of the corresponding degrees of freedom (the Kohn-Sham wavefunctions) and consequently the range of frequencies where there is a non negligible density of states.

1. How can we simulate the real motion while using fictitious masses?

This is the core of Car-Parrinello algorithm. It is based on the adiabatic decoupling between fast variables (the Kohn-Sham wavefunctions) and slow variables (ionic positions). When their frequencies become very different and the lowest orbital frequency is significantly higher than the higest ionic frequency, if the system starts its dynamics with the electronic degrees of freedom very close to their minimum, clssical mechanics ensures that they'll keep moving around the instantaneous minimum, without need of an explicit minimization at each time step. Of course, wavefunctions will oscillate around the instantaneous minimum, but, provided $$\mu$$ is low enough: - time everage of the forces on the ions will remain close to the Bohr-Oppenheimer forces (i.e. those obtained at the minimum of the electronic functional); - transfer of energy from the ionic to the wavefunction degrees of freedom will be weak and the system remains very close to the adiabatic evolution for very long times.

• well, this would have been great information to know four years ago when I had the exam on this stuff =). Jokes aside, you absolutely don't need to justify answering old questions on SE! That's the beauty of it: questions stay around until someone comes about to answer them, and the answers are more about the wandering googlers than about the askers themselves (or at least that's how I see it). – glS Jan 18 at 19:42