Method C

In the following approach to this problem, we try to eliminate the contribution to the matrices $\rho^{[2]}_l(r,r')$ from distant protons in a more elaborate way. We start from the matrices generated by averaging over all protons but also record distribution of the separation of the open ends $n(r)$ during the PIMC simulation. In the final analysis procedure, we used $n(r)$ to subtract the contribution from a uniform background of $N-1$ protons from the generate matrices $\rho_l({\bf r},{\bf r}')$. The idea behind it is that one imagines the electron to be in a certain orbital state at one proton. All other protons lead to additional contributions that need to be subtracted afterwards. Assuming that there is little correlation between the protons, one can model them by a uniform background. The results are shown in Tab. 5.2 and in Fig. 5.13. This method reproduces the high occupation of the 1s state and has also increased the level of the numerical noise. It represents one possible way to deal with problem of multiple pairs $({\bf r},{\bf r}')$ from different protons. However, this idea needs further investigation. One can also imagine other methods that would exclude contributions from distant protons, e.g one can introduce a cut-off or a localization function. The simplest system to test new proposals is composed of one electron and two protons at low temperature. The density matrix is then given by,

Figure 5.13: Natural orbitals calculated with method C for the system studied in Fig. 5.11.

$$\rho({\bf r},{\bf r}') = A \: \phi_{1s}^*({\bf r}-{\bf r}_{{... ... r}_{{\rm p}2})\:\phi_{1s}({\bf r}'-{\bf r}_{{\rm p}2}) \quad.$$ (229)

The next more advanced system would include free states, and one could add a free particle term $B \exp\{-({\bf r}-{\bf r}')^2/4\lambda\beta\}$. The coefficients A and B should be reproduced in the analysis procedure. The main question is to find an appropriate basis to decompose the many body density matrix, which can be diagonalized with a reasonable computational demand.