Temperature-dependent quantum pair potentials and their application to dense partially ionized hydrogen plasmas

Extending our previous work [A.V. Filinov et al., J. Phys. A 36, 5957 (2003)], we present a detailed discussion of accuracy and practical applications of finite-temperature pseudopotentials for two-component Coulomb systems. Different pseudopotentials are discussed: (i) the diagonal Kelbg potential, (ii) the off-diagonal Kelbg potential, (iii) the improved diagonal Kelbg potential, (iv) an effective potential obtained with the Feynman-Kleinert variational principle, and (v) the “exact” quantum pair potential derived from the two-particle density matrix. For the improved diagonal Kelbg potential, a simple temperature-dependent fit is derived which accurately reproduces the “exact” pair potential in the whole temperature range. The derived pseudopotentials are then used in path integral Monte Carlo and molecular-dynamics (MD) simulations to obtain thermodynamical properties of strongly coupled hydrogen. It is demonstrated that classical MD simulations with spin-dependent interaction potentials for the electrons allow for an accurate description of the internal energy of hydrogen in the difficult regime of partial ionization down to the temperatures of about 60 000 K. Finally, we point out an interesting relationship between the quantum potentials and the effective potentials used in density-functional theory.

Figure 1

Temperature dependence of the fit parameter for the binary interactions: electron-proton, ${\gamma }_{ep}\left(T\right)$, and electron-electron (no exchange), ${\gamma }_{ee}\left(T\right)$. Symbols show the $\gamma$ values obtained with the least-square fit of the IDKP to the “exact” pair potential without exchange. Solid curves correspond to the Padé approximation (22, 23).

Figure 2

The “exact” off-diagonal density matrix $\rho (\mathbf{r},{\mathbf{r}}^{\prime };\varphi )$ for an electron-proton pair vs the density matrix calculated with the diagonal (DKP) and off-diagonal (ODKP) Kelbg potentials. In all figures, results for three angular values are given as $\varphi =0$ (upper curves), $\varphi =\pi ∕2$ (middle), and $\varphi =\pi$ (lower curves). The proton is located at the origin, and the vector ${\mathbf{r}}^{\prime }$ (initial electron position) is fixed, $\mid {\mathbf{r}}^{\prime }\mid =0.25$; 1.0; 2.0. The vector $\mathbf{r}$ (final electron position) is varied; $\varphi$ is the angle between the vectors $\mathbf{r}$ and ${\mathbf{r}}^{\prime }$.

Figure 3

(a) Diagonal effective electron-proton potential (in units of Ha) for several cases: the DKP ${\Phi }^0(\mathbf{r};\beta )$ (4), the improved DKP $\Phi (\mathbf{r};\beta )$ (8), variational potential $W_1^{\Omega ,{\mathbf{x}}_m}$ (35), and pair potential $U_p$ (27) corresponding to the “exact” density matrix. Each potential is given at three temperature values: 5000, 40 000, 125 000, and 320 000 K. (b) The function $U\left(\beta \right)+\beta \partial U\left(\beta \right)∕\partial \beta$ for the same approximations and temperatures. The ovals include a set of curves corresponding to the denoted temperature.

Figure 4

Pair distribution function for a pair of electrons. (a) In a triplet state (parallel spins, $S=1$). Cases compared are Eq. (14), using for $U_{ee}$ the “exact” pair potential, $g\propto e^{-\beta U_{ee}^T}$, the Kelbg potential, $g\propto e^{-\beta {\Phi }_{ee}^T}$, and Eq. (17) with the Kelbg potential, $g\propto e^{-\beta {\Phi }_{ee,0}^T}$. (b) In a singlet state (antiparallel spins, $S=0$), Eq. (14) with the “exact” pair potential, $g\propto e^{-\beta U_{ee}^S}$, and Eq. (17) with the Kelbg potential, $g\propto e^{-\beta {\Phi }_{ee,0}^S}$. The pair distribution functions are shown for temperatures, $T=31250$, 125 000, 250 000, and 1 000 000 K.

Figure 5

Proton-electron pair correlation functions, $g_{ep}\left(\mathbf{r}\right)$, from PIMC simulations with the “exact” pair density matrix (diamonds), the DKP (dots), and the ODKP (full line). Temperature values are as indicated in the figure parts: $T=5208$, 10 416, 31 250, and 62 500 K. For comparison, $T=0$ “GST” denotes the pair correlation function (shown by crosses) corresponding to the ground state of a hydrogen atom.

Figure 6

PIMC results for the internal energy of the proton-electron pair using “exact” pair density matrix, DKP, ODKP, and Improved Kelbg potential (IDKP) vs different number of factorization factors $M$.

Figure 7

Relative error in the internal energy of the proton-electron pair from PIMC simulations with the diagonal, off-diagonal and improved Kelbg potential (IDKP) vs temperature argument in the two-particle density matrix $1∕\tau$.

Figure 8

Semiclassical MD results (full lines) for the internal energy per hydrogen atom at densities ${\mathbf{r}}_s=4$,6 vs temperature. The results of restricted PIMC simulations by Militzer [41] are shown for comparison (dashed lines). Symbols indicate the five temperatures for which MD simulations have been performed: $T=31250$, 50 000, 62 500, 125 000, and 166 670 K (solid lines). The pair approximation breaks down around 50 000 K at the molecule binding energy.

Figure 9

Electron-electron and proton-proton radial pair distribution functions for a correlated hydrogen plasma with ${\mathbf{r}}_s=4$ (left row) and ${\mathbf{r}}_s=6$ (right row) for $T=125000$, 61 250, and 31 250 K (from top to bottom).

Figure 10

Electron-proton radial pair distribution functions at ${\mathbf{r}}_s=4$ (left figure) and ${\mathbf{r}}_s=6$ (right figure) and five temperatures: $T=166667$, 125 000, 62 500, 50 000, and 31 250 K.

Figure 11

Electron-proton radial pair distribution functions multiplied by ${\mathbf{r}}^2$. Same data as in Fig. 10.