Comparison with quantum simulations

Density functional molecular dynamics (DFT-MD) (see section 4.1.5) describe the strong correlations of the ions as well as the degeneracy of the electrons in a consistent way. As it is an ab initio simulation, it uses Coulomb forces modified for technical applicability only. It self-consistently calculates the effective inter-particle forces from electronic structure methods. This treatment meets the requirements of warm dense matter exactly, but such simulations demand high computing power.

By benchmarking the results of the HNC approach against DFT-MD data, the effective inter-particle potential within the system can be investigated [Schwarz et al., 2010]. In particular, the applicability of the effective quantum potentials, introduced in the previous section, can be studied. Once the effective interaction is understood, the ionic structure in WDM can be determined very efficiently by the HNC approach.

Fig. 4.8 presents ionic radial distribution functions obtained by HNC and DFT-MD simulations for warm dense beryllium under several compression levels changing its density. The temperature and the ion charge state for the HNC runs are set toT = 13 eV and Z = 2 which gives plasma parameters similar to recently performed scattering experiments on beryllium [Glenzer et al.,2003a,2007;Lee et al., 2009].

0.2 0.4 0.6 0.8 1.0 1.2 gii (r) 0.0 0.5 1.0 1.5 2.0 r [A] = 2.5 0 (c) 0.0 0.5 1.0 1.5 2.0 2.5 r [A] DFT-MD HNC-Y+SRR HNC-Y HNC-KK HNC-Deutsch = 3 0 (d) 0.2 0.4 0.6 0.8 1.0 1.2 gii (r) = 0 (a) = 2 0 (b)

Figure 4.8: Ion-ion pair distribution functions for warm dense beryllium with differ- ent densities obtained by HNC calculations applying different effective interaction potentials and DFT-MD simulations. The normal solid density is̺0=1.848 g/cm3,

temperature isT = 13 eV and the ion charge isZ = 2.

The HNC calculation applying the Klimontovich-Kraeft potential uses the Kelbg potential between the electrons and the Coulomb potential between the ions as the Klimontovich-Kraeft potential can describe only the electron-ion interactions. It can be seen, that the results from the two-component HNC method using effective quantum potentials lead to pair distributions that rise less sharply than the data obtained by the DFT-MD simulations. Furthermore, the pair distribution function,

gii(r), are shifted to the right. This is the typical behaviour for too strongly cou- pled ions or, equivalently, less effectively screened ions. As already discussed in the previous section, the quantum potentials, in particular the Klimontovich-Kraeft potential, generate weak electron-ion interactions, yielding almost OCP-like result where no screening is incorporated.

In contrast, the model which considers only ions via linearly screened Coulomb forces, that is, the Yukawa model (labelled HNC-Y), works rather well when com-

0.2 0.4 0.6 0.8 1.0 1.2 Sii (k) 0 2 4 6 8 10 k [A-1] DFT-MD HNC-Y+SRR HNC-Y HNC-KK HNC-Deutsch = 3 0

Figure 4.9: Ion-ion static structure factor for threefold compressed warm dense beryl- lium obtained by HNC calculations applying different effective interaction potentials and DFT-MD simulations. The plasma conditions are the same as in Fig.4.8(d).

paring its outcome to the DFT-MD data for lower densities. As it can be seen in Fig. 4.8 (a), the large distance behaviour as well as the shoulder of the pair dis- tribution function is well described. With the increase of the density, however, the Yukawa model seems to underestimated the coupling for smaller distances, as it cannot reproduce the rising peak. Nevertheless, the results from the comparison indicates that screening can be considered to be linear for larger distances.

The underestimation of the smallr behaviour can be understood by consid-

ering the electronic configuration of warm beryllium: with the degree of ionisation

Z = 2, the beryllium ions still have an intact 1s2 shell. In simple terms, the wave

functions of the core electrons are not allowed to overlap due to the Pauli exclusion principle. Therefore, an additional repulsion force occurs for distances smaller than the binding radius of the 1s states. This effect can be modelled by an additional

short-range repulsion (SRR) term added to the Debye potential. We suggested a Lennard-Jones-like structure of the form [Wünsch et al.,2009a]

V_ijY+SRR(r) =a r 4 + ZiZje 2 r exp (−κr). (4.44)

A fit to the potential directly extracted from the DFT-MD simulations yields the power of the SRR contribution. The parameter, a, is a fit parameter to match

the HNC results to the DFT-MD data. It defines the strength of the short-range repulsion and stays constant for the same material under conditions where the charge state does not change.

0.2 0.4 0.6 0.8 1.0 gii (r) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 r [A]

(a) pair distribution functions

0.0 0.2 0.4 0.6 0.8 1.0 1.2 Sii (k) 0 1 2 3 4 5 6 k [A-1] DFT-MD HNC-Y HNC-KK HNC-Deutsch

(b) static structure factors

Figure 4.10: Pair distribution functions and static structure factors for the ions in a lithium plasma obtained by HNC calculations applying different effective interaction potentials and DFT-MD simulations. The plasma parameters are given byT = 5 eV,

̺= 0.85 g/cm3 andZ = 1.6.

Applying the modified Debye potential (4.44) in the HNC approach (labelled HNC-Y+SRR) yieldsg(r)’s with a larger correlation hole and a steeper rise at small

distances. In particular, for the highly compressed beryllium in Fig. 4.8 (d), the additional repulsion is sufficient to reproduce the correct inter-particle spacing and, thus, a pair distribution function which is now in very good agreement with results from DFT-MD simulations.

In Fig.4.9, the static ion-ion structure factor for beryllium with three-times the solid density, that is, the case from Fig. 4.8(d), is displayed. The data from the DFT-MD simulations cannot describe the very smallk-behaviour as the simulations are restricted to kvalues larger then 2π/L, where Lcharacterises the length of the

simulation box. Therefore, HNC calculations are a valuable method if the effective ionic interaction is understood as they can predict the small wavelength behaviour. The Klimontovich-Kraeft potential used in the HNC method leads a OCP-like behaviour as already described, that is, Sii(k = 0) ≈ 0. Similarly, the application of the Deutsch potential in the HNC calculations yields a structure factor with a small value at the origin. As for the pair distribution function, the best agreement with the DFT-MD data can be achieved by the use of the Debye potential with an additional short-range repulsion (labelled HNC-Y+SRR).

Fig. 4.10 shows the predicted microscopic structure of warm dense lithium obtained by DFT-MD simulations and HNC calculations applying various inter- particle potentials. The plasma parameters are ̺ = 0.85 g/cm3_{, T = 5 eV and}

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 gii (r) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 r [A]

(a) pair distribution functions

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Sii (k) 1 2 3 4 5 6 7 8 9 k [A-1] DFT-MD HNC-Y+SRR HNC-Y HNC-OCP

(b) static structure factors

Figure 4.11: Pair distribution functions and static structure factors for the ions in an aluminium plasma obtained by HNC calculations applying different effective interaction potentials and DFT-MD simulations. The plasma parameters are given byT = 1.1 eV,̺= 3.4 g/cm3 and Z = 3.

matter [García Saiz et al.,2008]. Once more, the HNC results applying the quantum potentials, namely, the Klimontovich-Kraeft potential and the Deutsch potential, yield distribution functions and structure factors which are too strongly coupled. Thus, these potentials fail to describe the effective screening in warm dense lithium correctly. In contrast, the HNC approach with the Debye interaction leads to results that are in very good agreement to the simulation data. Here, a modification of the Debye potential with an additional short-range repulsion is not required as the lithium ions do not have an intact inner 1s2 shell for the charge state considered.

As a last example, warm dense aluminium is considered for plasma parameters

̺= 3.4 g/cm3,T = 1.1 eV andZ = 3. The corresponding pair distribution functions

and static structure factors for the ions are presented in Fig. 4.11. The DFT-MD simulation leads to a pair distribution function with well-pronounced maxima and a large correlation hole which is a typical behaviour for strongly coupled systems expected under the plasma parameter considered. A similar shape, however slightly shifted to the right, is obtained by HNC calculations when unscreened Coulomb interactions are applied between the ions (labelled HNC-OCP).

The negligence of screening effects in the OCP model systems seems, however, physically questionable. Thus, HNC calculations were performed for the Yukawa model to account for linear screening. The resulting pair distribution strongly un- derestimates the correlations in warm dense aluminium when being compared to DFT-MD data. However, so far the effects of the inner shells of the aluminium ions were neglected. These bound electrons lead, in the same way as discussed for

the beryllium plasma, to an additional repulsion for small distances. After taking the SRR contribution (4.44) into account, the ion-ion pair distribution functions, obtained by the HNC approach, also show the typical characteristics of strongly coupled systems and agree well with the simulation data. Thus, core electrons in warm dense aluminium influence the ionic structure by raising correlations for small distances if the inter-particle distance is comparable to the bound state radius.

In summary, the comparisons presented here lead to two main conclusions: Firstly, the quantum potentials, which mimic quantum effects in classical methods like HNC, exceed their applicability in the warm dense matter region. They strongly underestimate the effective screening in the system due to a weak electron-ion in- teraction. This effect then leads to too strongly coupled ions. Secondly, the ionic structure in WDM can be described by a simple linearly screened Coulomb potential. If partially ionised ions are considered, an additional short range repulsion due to the forbidden overlap of the wave functions of bound electrons has to be incorporated in the description. An easy algebraic expression is capable to mimicking this effect and, thus, allows the application of the HNC approach to efficiently calculate the microscopic structure in warm dense matter.