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.

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自由补体辅助理论：碳原子五重态的变分计算

免费补充 $s_{\mathrm{一世}Ĵ}$辅助 ${\mathrm{[R}}_{\mathrm{一世}Ĵ}$ （$\mathrm{足球俱乐部}{\mathrm{[R}}_{\mathrm{一世}Ĵ}{s}_{\mathrm{一世}Ĵ}^{ñ}$理论是作为变分方法开发的，用于解决原子和分子的薛定ding方程。这个理论只允许一个相关的${\mathrm{[R}}_{\mathrm{一世}Ĵ}$ 每个补函数（cf）和另一个 ${\mathrm{[R}}_{\mathrm{一世}Ĵ}$ 条款被替换为 $s_{\mathrm{一世}Ĵ}={\mathrm{[R}}_{\mathrm{一世}Ĵ}^2$项，以便在1至4电子积分内执行变分计算。我们开发了${\mathrm{[R}}_{\mathrm{一世}Ĵ}$扩展的Löwdin公式用于反对称化 ${\mathrm{[R}}_{\mathrm{一世}Ĵ}$包括非正交函数，并实现了本计算所需的一到四电子Slater原子积分代码。引入了cf选择技术以有效地减少自由度的数量，而不会损失太多精度。这些方法适用于最低的五重奏${}^5\mathrm{小号}^{Ø}（{sp}^3）$碳原子的状态，它是碳原子的激发态，但对化学键最重要。化学精度是通过绝对解决方案实现的$\mathrm{\Delta }Ë=0.215\mathrm{大卡}/\mathrm{摩尔}$ 根据估计的精确能量：使用的cf的数量为4577，但在利用cf选择技术获得化学精度后减少到仅129 $\mathrm{\Delta }Ë<\mathrm{1个}\mathrm{大卡}/\mathrm{摩尔}$。因此，如果所需的三电子和四电子积分实际上可用，则本理论可以实现具有紧凑波函数的多电子系统的精确变分计算。