Antisymmetry in the Parameter Equations

The determinantal form for the VDM, Eq. 3.56, is correctly antisymmetric under exchange of identical particles. Since ion exchange effects are negligible at the temperatures considered here these are ignored.

The determinantal form leads to $N!$ terms in the equations of motion for the variational parameters presented in appendix A. It was originally hoped that exchange effects could be ignored in these equations while retaining the full determinantal form for the VDM but this leads to an instability in fermionic systems, e.g. it results in an unphysical strong attraction between two hydrogen molecules.

A practical means of treating all exchange terms, in particular terms involving the potential energy, in the variational parameter equations was not found. Instead it was necessary to use an approximation similar to that used in the real time computations (Knaup et al., 1999; Klakow et al., 1994b): only pair exchanges in the kinetic energy terms were retained. This will be illustrated for the hydrogen molecule after first giving the explicit form for this correction. It is stressed that, unlike the real time computations, once the variational parameters are determined the full determinantal form is then used in calculating the various averages.

For two particles with parallel spin, the correction term to the kinetic energy is given by,

$$\Delta K = \frac{N_I}{N_{AS}} \int \! {\bf d}{\bf R}\; \rho_{AS} \; \hat{K} \; \rho_{AS} \quad-\quad \int \! {\bf d}{\bf R}\; \rho_{I} \; \hat{K} \; \rho_{I}$$ (181)

$$\rho_{AS} = \rho_1({\bf r}_1)\rho_2({\bf r}_2) - \rho_2({\bf r}_1)\rho_1({\bf r}_2) \quad, \quad \rho_{I} = \rho_1({\bf r}_1)\rho_2({\bf r}_2)$$ (182)

$$N_{AS} = \int \! {\bf d}{\bf R}\; \rho_{AS}^2 \quad,\quad N_{I} = \int \! {\bf d}{\bf R}\; \rho_{I}^2$$ (183)

For the Gaussian ansatz in Eq. 3.56 it becomes,

$$\Delta K = - \frac{4 \lambda N_I }{w N_Q} \left[ \, 3 \left(1-\tilde{w}^2\right)-Q^2 \right] \quad,$$ (184)

$$w = w_1+w_2 \quad,\quad \tilde{w}= \frac{w}{2\sqrt{w_1 w_2}}\quad,\quad$$ (185)

$$Q^2 = \frac{2}{w} \left( {\bf m}_1-{\bf m}_2 \right)^2 \quad,\quad N_Q = \tilde{w}^3 e^{Q^2} - 1\quad.$$ (186)

The corrections to the norm matrix ${\mathcal{N}}$ are neglected in order to keep its analytically invertible form. The corrections to $H_{q_k}$ in Eq. 3.62 are given by

$$\Delta K_{q_k} = \frac{1}{2 N_I} \frac{\partial}{\partial {q_k}} \Delta K$$ (187)

The correction to dynamics of the parameters follow from Eq. 3.59 to 3.61,

$$\Delta \dot{w}_1 = -2 \, w_1 \left( \Delta K_{D} + \frac{4}{3} w_1 \: \Delta K_{w_1} \right)$$ (188)

$$\Delta \dot{{\bf m}}_1 = - w_1 \: \Delta K_{{\bf m}_1}$$ (189)

$$\Delta \dot{D} = -2 \left( \Delta K_{D} + w_1 \: \Delta K_{w_1} + w_2 \: \Delta K_{w_2} \right)\quad.$$ (190)

These equations lead to an effective repulsion between the Gaussians for two electrons with parallel spin if there is significant overlap. As a example of this effect the variational parameters for the singlet and triplet states of the hydrogen molecule are compared in Fig. 3.4. For the triplet state parameters, the solution including full exchange effects (long dashed line) are compared with those obtained in the kinetic pair exchange approximation (dot-dashed line). The approximation now prevents the Gaussian means for the same spin electrons from collapsing to the bond center at lower temperature and is numerically close to the solution for full exchange.

Figure 3.4: Effect of antisymmetry on the density matrix parameters, width and mean, for a hydrogen molecule. The protons (large black dots along x axis) are separated by $1.8$ and the initial electron positions $r_e(\beta =0)=\pm 1.5$ along the molecular axis. The solid line for the singlet state (electron spins anti-parallel) shows both electrons centered in the molecular bond at low temperatures (large $\beta$). In the triplet state (parallel electron spins), the electrons are centered close to the protons (long dashed line). The approximation that includes only kinetic exchanges (dot-dashed line) gives a similar result for the mean, with the electrons centered slightly inside the protons but overestimates the Gaussian width (left panel). At high temperature ($\beta \leq 4$), exchange is unimportant and the parameters are nearly the same for all cases.

Even at the lowest temperature considered here in the dense hydrogen simulations ($5000$ K) exchange effects between same spin electrons are negligible beyond a few angstroms, i.e. one or perhaps two nearest neighbors. Fig. 3.4 for the triplet state thus overestimates the effect likely in dense hydrogen. The main effect of including exchange in the parameter equations is probably to prevent the instability mentioned above.

Fig. 3.5 shows an energy comparison for the triplet ground state of the hydrogen molecule. First, we compare the Gaussian approximation using only the kinetic exchange term in the parameter equations. For the direct estimator, Eq. 3.49, one finds fairly good agreement with the accurate quantum chemistry result (Kolos and Roothan, 1969). The thermodynamic estimator gives a somewhat more repulsive triplet interaction for $R>2$. Considering also the Coulomb exchange terms in the Gaussian approximation leads to the dot-dashed line for the thermodynamic estimator. We conclude that leaving out the Coulomb exchange terms in the parameter equations for efficiency reasons is a reasonable approximation in many-particle simulations.

Figure 3.5: Energy of repulsion for the triplet ground state of the hydrogen molecule for bond length $R$. The thermodynamic (dashed line) and the direct estimator, Eq.  3.49, (circles with error bars) for the Gaussian approximation using the kinetic exchange term in the parameter equations are compared with the Kolos and Roothan results (solid line). The thermodynamic estimator for the Gaussian approximation with all exchange terms is shown by the dot-dashed line.