Ab initio Molecular Dynamics

In the remainder of this section the two most popular extension of classical molecular dynamics to include first-principles derived potential functions are discussed. The focus is on the Kohn-Sham method of density functional theory [34,35], as this is the method used in CPMD.

The total ground-state energy of the interacting system of electrons with classical nuclei fixed at positions $\{ {\bf R}_I \}$ can be obtained

$$\min_{\Psi_0} \left\{ \left< \Psi_0 \left\vert {\cal H}_{\rm e} \right\vert \Psi_0 \right> \right\} = \min_{\{ \phi_i \} } E^{\rm KS} [\{ \phi_i \}]$$

as the minimum of the Kohn-Sham energy [36,37]

$$E^{\rm KS} [\{ \phi_i \} ] = T_{\rm s} [ \{ \phi_i \}] + \int d {\bf r} \enspace V_{\rm ext} ({\bf r}) \; n ({\bf r})$$

$$+ {1\over 2} \int d {\bf r} \enspace V_{\rm H} ({\bf r}) \; n ({\bf r}) + E_{\rm xc} [n] + E_{\rm ions}({\bf R}^N) \enspace ,$$ (17)

which is an explicit functional of the set of auxiliary functions (Kohn-Sham orbitals) $\{ \phi_i ({\bf r})\}$ that satisfy the orthonormality relation

$$\langle \phi_i \mid \phi_j \rangle = \delta_{ij} \enspace .$$ (18)

This is a dramatic simplification since the minimization with respect to all possible many-body wavefunctions $\{ \Psi \}$ is replaced by a minimization with respect to a set of orthonormal one-particle functions. The associated charge density

$$n ({\bf r}) = \sum_i^{\rm occ} f_i \mid \phi_i ({\bf r}) \mid^2$$ (19)

is obtained from a single Slater determinant built from the occupied orbitals, where $\{ f_i \}$ are integer occupation numbers.

The first term in the Kohn-Sham functional Eq. (17) is the kinetic energy of a non-interacting reference system

$$T_{\rm s} [\{ \phi_i \}] = \sum_i^{\rm occ} f_i \left<\phi_i \left\vert -{1\over 2} \nabla^2 \right\vert \phi_i \right>$$ (20)

consisting of the same number of electrons exposed to the same external potential as in the fully interacting system. The second term comes from the fixed external potential $V_{\rm ext} ({\bf r})$ , in most cases the potential due to the classical nuclei, in which the electrons move. The third term is the classical electrostatic energy of the electronic density and is obtained from the Hartree potential

$$V_{\rm H} ({\bf r}) = \int d{\bf r}^\prime \enspace {{ n( {\bf r}^\prime )} \over { \mid {\bf r} - {\bf r}^\prime \mid }} \enspace ,$$ (21)

which in turn is related to the density through

$$\nabla^2 V_{\rm H} ({\bf r}) = - 4 \pi n ({\bf r})$$ (22)

Poisson's equation. The second last contribution in the Kohn-Sham functional, is the exchange-correlation functional $E_{\rm xc}[n]$ . The electronic exchange and correlation effects are lumped together and basically define this functional as the remainder between the exact energy and its Kohn-Sham decomposition in terms of the three previous contributions. Finally, the interaction energy of the bare nuclear charges is added in the last term.

The minimum of the Kohn-Sham functional is obtained by varying the energy functional Eq. (17) for a fixed number of electrons with respect to the orbitals subject to the orthonormality constraint. This leads to the Kohn-Sham equations

$$\left\{ - {1\over 2} \nabla^2 + V_{\rm ext} ({\bf r}) + V_{\rm H}... ...{{\delta E_{\rm xc} [n]} \over {\delta n ({\bf r}) }} \right\} \phi_i ({\bf r}) = \sum_j \Lambda_{ij} \phi_j ({\bf r})$$ (23)

$$\left\{ - {1\over 2} \nabla^2 +V^{\rm KS} ({\bf r}) \right\} \phi_i ({\bf r}) = \sum_j \Lambda_{ij} \phi_j ({\bf r})$$ (24)

$$H^{\rm KS} \phi_i ({\bf r}) = \sum_j \Lambda_{ij} \phi_j ({\bf r}) \enspace ,$$ (25)

which are one-electron equations involving an effective one-particle Hamiltonian $H^{\rm KS}$ with the local potential $V^{\rm KS}$ . Note that $H^{\rm KS}$ nevertheless embodies the electronic many-body effects by virtue of the exchange-correlation potential

$${{\delta E_{\rm xc}[n]} \over {\delta n ({\bf r})}} = V_{\rm xc}({\bf r}) \enspace .$$ (26)

A unitary transformation within the space of the occupied orbitals leads to the canonical form

$$H^{\rm KS} \phi_i({\bf r}) = \epsilon_i \phi_i({\bf r})$$ (27)

of the Kohn-Sham equations, with the eigenvalues $\{ \epsilon_i\}$ . This set of equations has to be solved self-consistently in order to yield the density, the orbitals and the Kohn-Sham potential for the electronic ground state. The functional derivative of the Kohn-Sham functional with respect to the orbitals, the Kohn-Sham force acting on the orbitals, can be expressed as

$${{\delta E^{\rm KS} } \over {\delta \phi_i^\star }({\bf r})} = f_i H^{\rm KS} \phi_i({\bf r}) \enspace .$$ (28)

Crucial to any application of density functional theory is the approximation of the unknown exchange-correlation functional. Investigations on the performance of different functionals for different type of properties and applications are abundant in the recent literature. A discussion focused on the framework of ab initio molecular dynamics is for instance given in Ref. [38]. Two important classes of functionals are the ``Generalized Gradient Approximation'' (GGA) functionals

$$E_{\rm xc}^{\rm GGA} [n] = \int d {\bf r} \enspace n ({\bf r}) \enspace \varepsilon_{\rm xc}^{\rm GGA} ( n({\bf r}) ; \nabla n({\bf r}) ) \enspace ,$$ (29)

where the functional depends only on the density and its gradient at a given point in space, and hybrid functionals, where the GGA type functionals are combined with a fraction of exact exchange energy from Hartree-Fock theory.