Energies and Expectation Values for Be by the Transcorrelated Method

Further calculations are reported using the transcorrelated method for energies and wavefunctions recently proposed by Boys and the author. In this method, approximate solutions to the Schrödinger equation are found by obtaining approximate solutions to the transcorrelated wave equation ${\mathrm{(C}}^{\text{−1}}\text{HC\hspace{0.17em}−\hspace{0.17em}W)Φ\hspace{0.17em}=\hspace{0.17em}0}$ , where ${\mathrm{C\hspace{0.17em}=\hspace{0.17em}\Pi }}_{\mathrm{i>j}}{\mathrm{f(r}}_{\mathrm{ij}},{{r}}_i,{{r}}_j)$ and ${\mathrm{\Phi \hspace{0.17em}=\hspace{0.17em}A(\phi }}_1^2{\phi }_2^2{\mathrm{···\phi }}_N^2)$ . The advantage of working with wavefunctions which explicitly include electron correlation seems to outweigh the difficulties of working with a nonvariational procedure. Two arbitrary parameters, similar to the Slater orbital exponents, occur in the expansion functions used here for ${\mathrm{f(r}}_{\mathrm{ij}},{{r}}_i,{{r}}_j$ . For nearly all values of these parameters, a nine‐term expansion for ${\mathrm{f(r}}_{\mathrm{ij}},{{r}}_i,{{r}}_j)$ gives at least 82% of the correlation energy, and a 19‐term expansion gives 88%. The transcorrelated orbitals φ are within 6% of the self‐consistent field orbitals and arguments are given which suggest that the true transcorrelated orbitals are much closer than this. It is also suggested that the method used here and in the previously reported LiH calculations is not the best, but that the variant used in the Ne calculations is preferable. In the second part, it is demonstrated how expectation values can be determined for transcorrelated wavefunctions without having to evaluate 3N‐dimensional integrals, and without using cluster expansion‐type formula and approximations for them. The method used the Hellmann–Feynman formula, and this is considered reasonable because of the great accuracy of the transcorrelated wavefunction. The general agreement between the expectation values reported here for Be by this method, and those obtained by configuration‐interaction‐type procedures, upholds this view. The transcorrelated method can therefore be used for the ab initio determination of highly accurate energies, wavefunctions, and expectation values.