Variational Many-Particle Density Matrix

We represent the many-particle density matrix by a determinant of one-particle density matrices (Eq. 3.79). It can written as,

$$\rho({\bf R},{\bf R'},\beta) = \sum_{\cal{P}}\epsilon_{\cal{P}} \prod_{k} \rho_1({\bf r}_k,{\bf r}'_{{\cal{P}}_k},\beta)$$ (174)

$$= \sum_{\cal{P}}\epsilon_{\cal{P}} e^D \prod_{k} \:(\pi w_{{\cal{P}... ...ft\{ -\frac{1}{w_{{\cal{P}}_k}} ({\bf r}_k-{\bf m}_{{\cal{P}}_k})^2 \right\}\;,$$ (175)

where a factor $1/N!$ was dropped. The permutation sum is over all permutations of identical particles (e.g. same spin ($S_z$) electrons) and the permutation signature $\epsilon_{\cal{P}}=\pm 1$. The initial conditions for Eq. 3.19 are $w_k=0$, ${\bf m}_k = {\bf r}_k'$, and $D=0$. For this ansatz the generator of the norm matrix, Eq. 3.22 is,

$$N= \exp( D+D') \, \sum_{\cal{P}}\epsilon_{\cal{P}}\prod_{k... ...\bf m}_{{\cal P}_k} ')^{2}/(w_{k}+w_{{\cal P}_k}')\right\} \;.$$ (176)

For a periodic system the above equation is also summed over all periodic simulation cell vectors, ${\bf L}$, with ${\bf m}_{k}-{\bf m}_{{\cal P}_k}\to {\bf m}_{k}-{\bf m}_{{\cal P}_k}+{\bf L}$. If only the identity permutation is considered the norm matrix is easily inverted so that Eq. 3.19 gives

$$\dot{w}_k = - 2 w_k H_D -\frac{8}{3} w_k^2 H_{w_k}$$ (177)

$$\dot{{\bf m}}_k = - w_k H_{{\bf m}_k}$$ (178)

$$\dot{D} = - \left( \frac{3}{2} n + 1 \right) H_D - 2 \sum_{i=1}^n w_i H_{w_i} \quad,$$ (179)

$$\mbox{where} \quad\quad H_{q_k} = \frac{1}{2}\frac{\partial H}{\partial q_k}\;.$$ (180)

For systems of electrons and ions the full expression for $H_{q_k}$ and the norm matrix are derived in App. A.

Figure 3.3: Gaussian approximation for the ground state of a hydrogen molecule for bond length $R$. The top left panel shows the Gaussian mean parameter ${\bf m}$ for the two electrons. These stay in the center of the bond (${\bf m}=0$) until about $R=2$ and then attach themselves to the separating protons ($\pm\; R/2$). The width parameter, displayed in the lower left panel, makes the transition from the optimal value for a helium atom, $R=0$, to the hydrogen atom result $w=9\pi /8$ at large $R$. The right panel shows the dissociation energy for the singlet state computed from Eq. 3.49 (open circles with error bars) and the thermodynamic estimator ($-dD/d\beta$) (dashed line) compared to the exact results of Kolos and Roothan (solid line).

Application to an isolated hydrogen molecule at low temperature is shown in Figure 3.3. This is for the singlet state (anti-parallel electron spins). The triplet state is considered later after a discussion of how to treat permutation terms in the parameter equations. The bond length at minimum energy is 1.47, compared with the experimental value of 1.40. The direct energy estimator Eq. 3.49 gives a dissociation energy of 4.50 eV at the minimum compared to the experimental value of 4.75 eV. Beyond $R=2$, the energy rises quickly toward the value given by the Rayleigh-Ritz estimator $-dD/d\beta$.