Extensions of the Gaussian Ansatz

There are several ways the VDM based on the Gaussian ansatz Eq. 3.27 can be improved. First of all, one can write it as a two step path integral,

$$\rho({\bf R},{\bf R}';\beta) \ = \int {\bf d}{\bf R}'' \; \rho({\bf R},{\bf R}'';\beta/2) \; \rho({\bf R}'',{\bf R}';\beta/2)$$ (192)

$$= \int {\bf d}{\bf R}'' \; \rho({\bf R},\vec{q}({\bf R}'';\beta/2)) \; \rho({\bf R}'',\vec{q}({\bf R}';\beta/2)) \quad.$$ (193)

This has the advantage of using a density matrix at higher temperatures corresponding to $\frac{\beta}{2}$, which makes the method more accurate. In a MC simulation sampling $\mbox{Tr}\rho$, one would keep two sets of coordinates ${\bf R}={\bf R}'$ and ${\bf R}''$ as done for path integrals, integrate the parameter equations with the initial conditions at ${\bf R}$ and ${\bf R}''$ up to $\frac{\beta}{2}$ and evaluate the integrand in Eq. 3.75 rather than calculating $\rho({\bf R},{\bf R};\beta)$ as done in all simulations discussed previously. The method of matrix squaring cannot be applied further without introducing minus signs, which lead to the fermion sign problem and would also require nodes.

Alternatively, one can try an approximation to the convolution in Eq. 3.74,

$$\rho({\bf R},{\bf R}';\beta) \ \approx \int {\bf d}{\bf R}'' \; \rho({\bf R}'',{\bf R};\beta/2) \; \rho({\bf R}'',{\bf R}';\beta/2)$$ (194)

$$\approx \int {\bf d}{\bf R}'' \; \rho({\bf R}'',\vec{q}({\bf R};\beta/2)) \; \rho({\bf R}'',\vec{q}({\bf R}';\beta/2)) \quad,$$ (195)

which has the advantage of being symmetric in ${\bf R}$ and ${\bf R}'$ unlike the ansatz used throughout this work, which was discussed in section 3.5.2. It would also simplify an MC simulation, because one would need only one set of coordinates ${\bf R}$, derive one set of variational parameters and could perform convolution analytically,

$$\rho({\bf R},{\bf R};\beta) \ \approx \int {\bf d}{\bf R}'' ... ..., \rho({\bf R}'',\vec{q}({\bf R};\beta/2)) \, \right]^2 \quad.$$ (196)

Furthermore, the Gaussian ansatz can be improved by including additional variational parameters, e.g. to use a sum of Gaussians. In the zero temperature limit, this would lead to a solution closer to the Hartree Fock result. To go beyond one needs to include additional correlations in the ansatz and derive modified equations for the parameter from Eq. 3.19. Correlations are usually introduced with a Jastrow factor, which can be generalized to finite temperature. The new ansatz then reads,

$$\rho({\bf R},{\bf R'};\beta)=\left\vert \begin{array}{ccc} \... ...eft \{ -\frac{1}{2}{\sum_{i<j}u(r_{ij})+u(r'_{ij})} \right\} ,$$ (197)

where $u$ depends on the type of pair (electron-electron, electron-ion). It contains extra parameters, that depend in temperature but we suggest to consider them as fixed in the variational principle. The consequence for the parameter equation is that there are additional terms to the norm matrix and, further, most integrals cannot be expressed analytically.

Alternatively, one can use an unitary correlation operator as suggested by Schnack (1996). The idea is to applied a short range correlation operator to an uncorrelated state in order to generate a correlated state. The fundamental difference to the Jastrow type ansatz above that correlations are introduced by the operator.

Another improvement for many-particle simulations would be to consider all $N!$ terms from the permutations, which would make contributions at very high levels of degeneracy. The scaling of a full exchange method can be reduced to $N^4$ as suggested by Schnack (1996) for real time wave packet molecular dynamics but the application of this method to the VDM imaginary time remains to be done.

To summarize, one can say that the VDM approach provides a way to systematically improve the many-particle density matrix. Already the simplest ansatz using one Gaussian to describe the single particle density matrices gives a good description of hydrogen in the discussed range of temperature and density. The method includes the correct high temperature behavior and shows the expected formation of atoms and molecules. The thermodynamic variables are in reasonable agreement with PIMC simulations and lead to a good approximation of the Hugoniot function. Further one can use this essentially analytic density matrix to furnish the nodal surface in PIMC simulations, replacing the free particle nodes by a density matrix that already includes the principle physical effects. Results will be discussed in the next chapter.