Method A

Figure 5.11: Natural orbitals calculated with method A for a system of 32 protons and 32 spin-polarized electrons at $T=10\,000\,\rm K$ and $r_s=6$.

Table: Occupation numbers for the electron eigenstates in a system of 32 spin-polarized hydrogen atoms at $T=10\,000\,\rm K$ and $r_s=6.0$ estimated by methods A, B, and C.

Method	1s	2s	2p	3s	3p	3d	4s	4p	4d	4f
A	0.535	0.056	0.049	0.042	0.036	0.032	0.027	0.022	0.019	0.017
B	0.973	0.009	0.003	0.002	0.002	0.002	0.000	0.000	0.000	0.007
C	0.919	0.012	0.010	0.006	0.008	0.008	0.003	0.002	0.003	0.006

In an off-diagonal PIMC simulation of spin-polarized hydrogen, there are $N$ protons, $N-1$ closed electron paths and one open electron path that can be permuted with the others electrons. The simplest approach to generalize the natural orbital method to systems with several protons would be to loop over all protons $r_{{\rm p}i}$ and to add all pairs of $({\bf r}_e-{\bf r}_{{\rm p}i},{\bf r}'_e-{\bf r}_{{\rm p}i})$ to the $l$ matrices. Then the analysis proceeds like in the case of the single hydrogen atom. The result for a system of 32 protons and spin-polarized electrons at $T=10\,000\,\rm K$ and $r_s=6$ is shown in Fig. 5.11. We chose to study a system of electrons in the same spin state, which prevents the formation of molecules and simplifies the following analysis. Under this condition, one expects the electrons mainly to be in the 1s ground state at one of the protons because the ideal Saha equation predicts an occupation probability of 0.9998 for the ground state. However, the described analysis procedure (eigenvectors in Fig. 5.11 and occupation numbers in Tab. 5.2) does not reproduce this result. Instead, it leads to a far too low occupation of 0.535 and one finds a significant contribution from higher $l$ components. These contributions can be interpreted as an artifact of this analysis procedure, in which we averaged over all protons, as can be understood from the following argument. If one imagines the electron fixed in the 1s state at proton 1 the pairs of $({\bf r}_e-{\bf r}_{{\rm p}i},{\bf r}'_e-{\bf r}_{{\rm p}i})$ from another distant proton always give a small angle $\theta$. The distribution of the angles $\theta$ is very non-uniform since it is localized around $\theta=0$. Therefore those contributions cannot be expressed as an s state.