We propose a new formulation of the correlation energy functional derived from the transcorrelated method in use in density functional theory (TC-DFT). An effective Hamiltonian, ${\mathcal{H}}_{TC}$, is introduced by a similarity transformation of a many-body Hamiltonian, $\mathcal{H}$, with respect to a complex function F: ${\mathcal{H}}_{TC}=\frac{1}{F}\mathcal{H}F$. It is proved that an expectation value of ${\mathcal{H}}_{TC}$ for a normalized single Slater determinant, Dⁿ, corresponds to the total energy: $E\left[n\right]=⟨{\mathrm{\Psi }}^n|\mathcal{H}|{\mathrm{\Psi }}^n⟩/⟨{\mathrm{\Psi }}^n|{\mathrm{\Psi }}^n⟩=⟨D^n|{\mathcal{H}}_{TC}|D^n⟩$ under the two assumptions: (1) The electron density $n\left(\mathbf{r}\right)$ associated with a trial wave function ${\mathrm{\Psi }}^n$ = DⁿF is $v$-representable and (2) ${\mathrm{\Psi }}^n$ and Dⁿ give rise to the same electron density $n\left(\mathbf{r}\right)$. This formulation, therefore, provides an alternative expression of the total energy that is useful for the development of novel correlation energy functionals. By substituting a specific function for F, we successfully derived a model correlation energy functional, which resembles the functional form of the screened exchange method. The proposed functional, named the extended screened exchange (ESX) functional, is described within two-body integrals and is parametrized for a numerically exact correlation energy of the homogeneous electron gas. The ESX functional does not contain any ingredients of (semi-)local functionals and thus is totally free from self-interactions. The computational cost for solving the self-consistent-field equation is comparable to that of the Hartree-Fock method. We apply the ESX functional to electronic structure calculations for a solid silicon, H⁻ ion, and small atoms. The results demonstrate that the TC-DFT formulation is promising for the systematic improvement of the correlation energy functional.

中文翻译：

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从互相关密度泛函理论推导的扩展筛选交换泛函

我们提出了一种从密度泛函理论（TC-DFT）中使用的互相关方法派生出的相关能量函数的新公式。一个有效的哈密顿量，${\mathcal{H}}_{ŤC}$是由多体哈密顿量的相似性变换引入的， $\mathcal{H}$关于复数函数F：${\mathcal{H}}_{ŤC}=\frac{\mathrm{1个}}{F}\mathcal{H}F$。证明期望值为${\mathcal{H}}_{ŤC}$对于归一化的单一Slater行列式D ⁿ，其对应于总能量：$E\left[ñ\right]=⟨{\mathrm{\Psi }}^{ñ}|\mathcal{H}|{\mathrm{\Psi }}^{ñ}⟩/⟨{\mathrm{\Psi }}^{ñ}|{\mathrm{\Psi }}^{ñ}⟩=⟨d^{ñ}|{\mathcal{H}}_{ŤC}|d^{ñ}⟩$ 在两个假设下：（1）电子密度 $ñ\left(\mathbf{[R}\right)$ 与试波功能相关 ${\mathrm{\Psi }}^{ñ}$= D ⁿ F为$v$-代表和（2） ${\mathrm{\Psi }}^{ñ}$和D ⁿ产生相同的电子密度$ñ\left(\mathbf{[R}\right)$。因此，该公式提供了总能量的替代表达式，可用于开发新型相关能量函数。通过将特定功能替换为F，我们成功地推导出了模型相关能量函数，该函数类似于筛选交换方法的函数形式。拟议的功能，称为扩展的屏蔽交换（ESX）功能，在两体积分中进行了描述，并针对均质电子气的精确数值相关能量进行了参数化。ESX功能块不包含任何（半）本地功能块的成分，因此完全没有自我交互。求解自洽场方程的计算成本与Hartree-Fock方法的计算成本相当。我们采用ESX的功能，以电子结构计算的固体硅，H ^-离子和小原子。结果表明，TC-DFT配方有望系统地改善相关能量功能。