The free complement $s_{ij}$-assisted $r_{ij}$ ($\mathrm{FC}{r}_{ij}{s}_{ij}^n$) theory was developed as a variational method for solving the Schrödinger equations of atoms and molecules. This theory permits only a single correlated $r_{ij}$ term in each complement function (cf) and the other $r_{ij}$ terms are replaced with $s_{ij}=r_{ij}^2$ terms so that the variational calculations are performed within one- to four-electron integrals. We developed the $r_{ij}$-extended Löwdin formula for the antisymmetrization of $r_{ij}$-included nonorthogonal functions and implemented one- to four-electron Slater atomic integral codes necessary for the present calculations. The cf selection technique was introduced to reduce the number of degrees of freedom efficiently without much loss of accuracy. These methods were applied to the lowest quintet ${}^{5}S^o\left({sp}^3\right)$ state of a carbon atom, which is an excited state of a carbon atom but most important for chemical bonds. The chemical accuracy was achieved with the absolute solution of $\mathrm{\Delta }E=0.215\mathrm{kcal}/\mathrm{mol}$ from the estimated exact energy: The number of the cf's used was 4577 but reduced to only 129 after utilizing the cf selection technique for obtaining the chemical accuracy $\mathrm{\Delta }E<1\mathrm{kcal}/\mathrm{mol}$. Thus, the present theory can realize accurate variational calculations of many-electron systems with compact wave functions if the required three- and four-electron integrals are practically available.

• Received 29 September 2020
• Accepted 29 October 2020

DOI:https://doi.org/10.1103/PhysRevA.102.052835