Fully self-consistent optimization of the Jastrow-Slater-type wave function using a similarity-transformed Hamiltonian

It has been well established that the Jastrow correlation factor can effectively capture the electron correlation effects, and thus, the efficient optimization of the many-body wave function including the Jastrow correlation factor is of great importance. For this purpose, the transcorrelated $+$ variational Monte Carlo ($\mathrm{TC}+\mathrm{VMC}$) method is one of the promising methods, where the one-electron orbitals in the Slater determinant and the Jastrow factor are self-consistently optimized in the TC and VMC methods, respectively. In particular, the TC method is based on similarity transformation of the Hamitonian by the Jastrow factor, which enables the efficient optimization of the one-electron orbitals under the effective interaction. Through test calculations of some closed-shell atoms, He, Be, and Ne, we find that the total energy is in many cases systematically improved by using better Jastrow functions. We find that even a one-shot $\mathrm{TC}+\mathrm{VMC}$ calculation, where the Jastrow parameters are optimized at the Hartree-$\mathrm{Fock}+\mathrm{VMC}$ level, can yield partial benefits from orbital optimization. It is also suggested that one-shot $\mathrm{TC}+\mathrm{VMC}$ can be a good alternative method for complex systems. Our study provides important insights for optimizing many-body wave function including the Jastrow correlation factor, which would be of great help for development of highly accurate electronic structure calculations.

Figure 1

Calculation procedure for (a) self-consistent $\mathrm{TC}+\mathrm{VMC}$ and (b) one-shot $\mathrm{TC}+\mathrm{VMC}$.

Figure 2

Total energy of a helium atom plotted against the inverse of the number of basis functions, $N_{\mathrm{basis}}^{-1}$, (a) for the HF method and (b) for the TC method ($E_{\mathrm{TC}}$) using the $S=S_{\mathrm{minimal}}=\left\{(1,0,0)\right\}$ Jastrow function in Eq. (26) with $a=1.5$ bohrs. Lines are guides for eyes.

Figure 3

Calculated total energies of a helium atom using the Jastrow functions with (a) $S=S_{\mathrm{minimal}}$, (b) $S=S_{\mathrm{ee}}$, and (c) $S=S_{\mathrm{een}}$, respectively. The broken lines show the exact total energy ($E_{\mathrm{exact}}=-2.90372$ Ht) estimated in Ref. [71].

Figure 4

Optimized Jastrow factor $F(x_1,x_2)=\exp [-u(x_1,x_2)]$, of a helium atom with constraints of $z_1=0, {\mathbf{r}}_2=(1.65,0,0)$ (bohrs), and ${\sigma }_1=-{\sigma }_2$ (see the main text). The $S_{\mathrm{ee}}$ and $S_{\mathrm{een}}$ Jastrow functions were used for panels (a) and (c) and (b) and (d), respectively. The Jastrow factors optimized by $\mathrm{HF}+\mathrm{VMC}$ and $\mathrm{TC}+\mathrm{VMC}$ are shown in panels (a) and (b) and (c) and (d), respectively. For all the panels, $a=1.5$ bohrs was used.

Figure 5

The same plot as Fig. 4 on the $y_1=z_1=0$ line. Each line represents the Jastrow factor $F$ shown in Figs. 4–4.

Figure 6

TC effective potential $V_{\mathrm{eff}}(x_1,x_2)$ as defined in Eq. (29), with constraints of $z_1=0, {\mathbf{r}}_2=(1.65,0,0)$ (bohrs), and ${\sigma }_1=-{\sigma }_2$ (see the main text), for a helium atom. (a) The bare potential, i.e., $V_{\mathrm{eff}}$ without ${\nabla }^{2}u$ terms, is shown, instead of the TC effective potential. (b), (c) The TC effective potential for several conditions, which are the same as those for Figs. 4 and 4. Contour lines are shown as guides for the eye.

Figure 7

The same plot as Fig. 6 on the $y_1=z_1=0$ line, corresponding to Figs. 6 and 6. The Coulomb potential from the nucleus, $-2/x_1$, is shown as a guide for the eye.

Figure 8

The radial wave function for the $1s$ state of a helium atom. The left orbital $\chi$ is shown for the BITC method. The right orbital $\varphi$ for the BITC method is not shown since a difference of $\varphi$ between TC and BITC is almost discernible.