Franz J. Vesely > CompPhys Tutorial > Selected Applications > Density Functional Dynamics  

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Part III: Ch. 7

7.4 Density Functional Molecular Dynamics (DFMD)

Car and Parrinello: "ab initio molecular dynamics" for the dynamical simulation of quantum mechanical systems [CAR 85].

Born-Oppenheimer approximation:
- the heavy atomic cores ("ions") consisting of nucleus and inner electronic shells are treated as classical particles
- the valence and conduction electrons are represented by wave functions which assume the configuration of least energy in the momentary field created by the ions and all other valence and conduction electrons.

Let $\Psi_{i}(r)$ denote the wave functions of the $N$ electrons; the electron density at position $r$ is then

$ n(r) \equiv \sum_{i=1}^{N} \left| \Psi_{i}(r)\right|^{2} $     (7.60)

The classical ions residing at positions $\{ R_{l} \}$ produce a potential field $U(r;\{ R_{l}\})$ in which the electronic wave functions take on their minimal energy form.

Total energy of the electrons:

$ E(\{\Psi_{i}\};\{R_{l}\})=E_{1}+E_{2}+E_{3}+E_{4} $     (7.61)


$ \begin{eqnarray} E_{1}&=& \sum_{i=1}^{N} \int \limits_{V}dr\, \Psi_{i}^{*}(r) \left[ -\frac{\hbar^{2}}{2m} \nabla^{2} \right] \Psi_{i}(r) \;\;\;\;\;\;\;\; {\rm kinetic \;\;\;energy} \\ && \\ E_{2}&=& \int\limits_{V} dr\, U(r;\{R_{l}\}) \, n(r) \;\;\;\;\;\;\;\; {\rm potential\;\;energy\;\;in\;\; ion\;\;field}\\ && \\ E_{3}&=& \frac{1}{2} \int\limits_{V} \int\limits_{V} dr \, dr'\, \frac{n(r)\,n(r')}{|r-r'|} \;\;\;\;\;\;\;\; {\rm electrostatic\;\;interaction\;\;with \;\;other\;\;electrons}\\ && \\ E_{4}&=&E_{xc}[n(r)] \;\;\;\;\;\;\;\; {\rm exchange \;\;and \;\;correlation \;\;\;energies} \end{eqnarray} $     (7.62-7.65)

Approximation for $E_{4}$: usually local density expression (see [CAR 85]).

  • Expand $\Psi_{i}(r)$ in plane waves,

    $ \Psi_{i}(r)=\sum_{k}c_{i}(k) e^{\textstyle i k\cdot r} $     (7.66)

    with a few hundred terms per electron.
  • Find that set of expansion coefficients $\{c_{i}(k) \}$ which minimizes the energy functional 7.61; as an additional requirement, maintain the orthonormality condition

    $ \int\limits_{V}\Psi_{i}^{*}(r,t)\,\Psi_{j}(r,t)\,dr =\delta_{ij} $     (7.67)

Apply variational calculus: $\Longrightarrow$ Kohn-Sham equations [KOHN 65]; then solve those by an iterative method.

Better: To find the variational minimum, use the Simulated Annealing method of Kirkpatrick et al. [KIRKPATRICK 83]. (See the chapter on Stochastics).

Still better: Dynamical Simulated Annealing by Car and Parrinello:

Let $\mu$ denote an abstract "mass" assigned to each electronic wave function $\Psi_{i}$. The "kinetic energy" due to the change of $\Psi_{i}$ in time is then

$ E_{kin}\equiv \frac{\mu}{2}\sum_{i=1}^{N}\int dr\, \left|\dot{\Psi}_{i} \right|^{2} $     (7.68)

Now introduce a Lagrangian function

$ \begin{eqnarray} L=\sum_{i}\frac{\mu}{2}\int\limits_{V}dr\, \left|\dot{\Psi}_{i}(r) \right|^{2}&+&\frac{M}{2}\sum_{l} \left| \dot{R}_{l}\right|^{2}-E(\{\Psi_{i}\};\{R_{l}\}) \\ &+& \sum_{i}\sum_{j}\lambda_{ij}\left[ \int \Psi_{i}^{*}\Psi_{j}-\delta_{ij} \right] \end{eqnarray} $     (7.69)

describing the "motion" of $\Psi_{i}(t)$. Here $M$ is the ionic mass, and the Lagrange multipliers $\lambda_{ij}$ allow for the conditions 7.67. From the Lagrangian the "equations of motion" are derived:

$ \begin{eqnarray} \mu\ddot{\Psi}_{i}(r,t)&=& -\frac{\delta E}{\delta \Psi_{i}^{*}(r,t)} +\sum_{j}\lambda_{ij}\Psi_{j}(r,t)\\ M\ddot{R}_{l}&=&-\nabla_{l}E \end{eqnarray} $     (7.70-7.71)

The second equation describes the classical dynamics of the ions. The first equation represents the abstract "motion" in the space of the electronic degrees of freedom.

By keeping the "kinetic energy" of this motion $(\mu/2) \sum \left\vert\dot{\Psi}_{i} \right\vert^{2}$ small we can constrain the $\Psi_{i}(t)$-subsystem always close to the momentary minimum of the energy surface defined by the slowly varying ionic configuration.

To control the temperature of the $\Psi_{i}$
- rescale all $\dot{\Psi_{i}}$ from time to time, or
- use one of the thermostats available from statistical-mechanical simulation; see Chapter 6.

The electronic degrees of freedom should adjust quickly to the varying energy landscape $\Longrightarrow$ mass $\mu$ small compared to the ionic masses; e. g. $\mu=1.0$ atomic mass unit.

- amorphous silicon [CAR 85]
- lithium [WENTZCOVICH 91]
microclusters of alkali metals [VITEK 89]
- molten carbon [GALLI 90B]
- ionic melts [GALLI 90A]

For current applications, see the web or reviews such as [VITEK 89,MAKRI 99,MAZZONE 99,OHNO 99].

vesely 2006

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