Chapter 4 Static ion structure in warm dense matter
4.1.5 Numerical simulation techniques
Numerical simulations, such as molecular dynamics (MD), can precisely predict the ionic structure in strongly coupled systems but require large numerical efforts [Hansen and McDonald,1978, 1981]. With a given inter-particle potential, the MD method simulates the time dependent behaviour on a microscopic scale by solving the classical equation of motion in the force field considered [Fehske et al., 2008]. The simulations were used here to verify the accuracy of the HNC results since they intrinsically include all correlations, especially the bridge diagrams neglected in the HNC approach. All simulations were performed by Dr Zoltán Donkó from the Hun- garian Academy of Sciences in Budapest. For each set-up, several thousand ions contained in a cubic volume with periodic boundary conditions were considered. When extracting the structural information, the average over many independent configurations was taken to reduce the numerical noise due to finite system size.
The accuracy of MD simulations strongly depends on the chosen interaction potential, which limits its capability in the WDM regime as the effective potential it not exactly known for such complex many-particle systems. An alternative way is the combination of electronic structure methods and MD simulations. Based on the Born-Oppenheimer approximation, the electron motion can be decoupled from the ion motion. This allows the calculation of the forces acting on the nuclei from a given electronic structure which can be directly used in the MD simulations. The electron configuration for a given ion arrangement is calculated, e.g., by density- functional theory. This method is know asab initio MD simulations as the effective interaction potential is directly obtained from the electron configuration. Thus, no further approximations are required which made this method a powerful tool in many scientific areas [Marx and Hutter,2005].
To calculate the electronic structure in the WDM regime, density functional theory (DFT) has proven to yield accurate results as it incorporates quantum effects as well as correlation effects between the electrons. The basic principle of DFT is the Hohenberg-Kohn theorem which proves that the physical system is completely de- termined by the ground state density [Hohenberg and Kohn,1964]. With this “basic variable” all properties of the many-particle system can be written as a functional of the density, for example, the many-body wave functionψ =ψ[n]. The practical application of this fundamental theorem was dramatically improved by the Kohn- Sham ansatz, which connects the original interacting many-particle system with an auxiliary, non-interacting system by demanding that the ground state density is kept [Kohn and Sham, 1965]. The Hamiltonian of the auxiliary, non-interacting system is then given by (in atomic units) [Martin,2004]
ˆ HKSσ =−1 2∇ 2+Vσ KS(r), (4.38) where Vσ
KS(r) characterises the effective potential of the Kohn-Sham system, which
reproduces the exact ground state density of the original interaction system. This can be decomposed as
VKSσ (r) =Vext(r) +VHartree[n] +Vxcσ[n]. (4.39)
The first term describes the external potential due to the nuclei or any other external fields, the second contribution is the Hartree potential and the last term characterises the exchange-correlation potential. This approach is an exact description as long as the exchange-correlation potential, which contains all the many-body effects, is known. However, in most correlated systems this quantity is not known and must be approximated. Many studies have been made to find reasonable approximations for the exchange-correlation potential which leads to sufficient results for various systems. Some widely used approximations are the local density approximation (LDA) and the generalised gradient approximation (GGA) [Martin,2004]. The first approach assumes that the exchange-correlation energy for each particle is equivalent to the energy of a homogeneous electron gas with the same density. This can be expanded to the more general local spin density approximation (LSDA). The GGA still approximates the exchange-correlation energy locally, but it takes the gradient of the density into account. This leads to improvements for many cases. With an approximate exchange-correlation potential, the Kohn-Sham equations can be solved in a self-consistent way: with an initial ground state density, the effective potential
(4.39) can be estimated. Then, the generalised Schrödinger equation [Martin, 2004] −1 2∇ 2+Vσ KS(r) ψiσ(r) =ǫσiψiσ(r) (4.40) can be solved which leads to a new expression for the electron density. These newly obtained functions can be used as an improved guess until convergence is achieved.
The original DFT approach was derived for a pure ground state in theT = 0 limit, which is not applicable in the WDM regime. Here, temperature effects have to be included, which can be done based on the work of Mermin [Mermin, 1965], who expanded the Hohenberg-Kohn theorem to non-zero temperature systems.
R. Car and M. Parrinello combined later the electron structure calculation DFT with MD simulations [Car and Parrinello,1985]. For an initial ionic structure, the electron configuration is calculated via DFT, which allows the determination of the interaction potentials required for the MD simulations. According to the forces, the ions are moved to a new configuration and the loop can start again. As for classical MD simulations, the system needs to reach thermodynamic equilibrium in the beginning of each run, before physical properties can be extracted.
All the DFT-MD simulations presented in this thesis were performed by J. Vorberger applying several packages, such as VASP [Blöchl, 1994; Kresse and Hafner, 1993,1994a,b; Kresse and Furthmüller,1996a,b; Kresse and Joubert, 1999; Perdew and Zunger, 1981;Perdew et al., 1992, 1993, 1996, 1997; Vanderbilt, 1990] and abinit [Gonze et al.,2002,2005,2009] which implement density functional the- ory and combine it with a MD solver for the ions. DFT-MD exactly meets the requirements to describe fully interacting quantum systems such as WDM, but it is a very computationally intensive method. Run times of DFT-MD simulations on high performance computers can easily exceed a couple of days, which limits their applicability as an analysis tool for experimental support. Here, DFT-MD simu- lations are mainly used to benchmark the HNC results and, thus, investigate the effective inter-particle potential in WDM.