13.5 Nuclear-Electronic Orbital Method 13.5.2 NEO-Hartree-Fock 13.5.4 NEO-TDDFT

13.5.3 NEO-DFT

(February 4, 2022)

NEO density functional theory (NEO-DFT)^{854, 189, 190} is an extension of DFT to multicomponent systems within the NEO framework. The Hohenberg-Kohn theorems have been extended to multicomponent systems, where the reference is expressed as the product of electronic and nuclear Slater determinants composed of Kohn-Sham orbitals. The NEO-DFT total energy is

\displaystyle E[\rho^{\text{e}},\rho^{\text{p}}]=E_{\text{ext}}[\rho^{\text{e}% },\rho^{\text{p}}]+E_{\text{ref}}[\rho^{\text{e}},\rho^{\text{p}}]+E_{\text{% exc}}[\rho^{\text{e}}]+E_{\text{pxc}}[\rho^{\text{p}}]+E_{\text{epc}}[\rho^{% \text{e}},\rho^{\text{p}}]

(13.43)

In this equation, $E_{\text{ext}}[\rho^{\text{e}},\rho^{\text{p}}]$ is the interaction of the electronic and protonic densities with the external potential created by the classical nuclei, and $E_{\text{ref}}[\rho^{\text{e}},\rho^{\text{p}}]$ contains the electron-electron, proton-proton, and electron-proton classical Coulomb energies, as well as the noninteracting kinetic energies of the quantum particles. The terms $E_{\text{exc}}[\rho^{\text{e}}]$ , $E_{\text{pxc}}[\rho^{\text{p}}]$ , and $E_{\text{epc}}[\rho^{\text{e}},\rho^{\text{p}}]$ are the electron-electron exchange-correlation functional, the proton-proton exchange-correlation functional, and the electron-proton correlation functional, respectively. The quantities $\rho^{\text{e}}(\mathbf{r}_{1}^{\text{e}})=2\sum_{i=1}^{N_{\text{e}}/2}|\psi_{% i}^{\text{e}}(\mathbf{r}_{1}^{\text{e}})|^{2}$ and $\rho^{\text{p}}(\mathbf{r}_{1}^{\text{p}})=\sum_{I=1}^{N_{\text{p}}}|\psi_{I}^% {\text{p}}(\mathbf{r}_{1}^{\text{p}})|^{2}$ are the electron and proton densities, respectively, and $\psi_{i}^{\text{e}}(\mathbf{r}_{1}^{\text{e}})$ and $\psi_{I}^{\text{p}}(\mathbf{r}_{1}^{\text{p}})$ are the electronic and protonic Kohn-Sham spatial orbitals, respectively. These orbitals are obtained by solving two sets of coupled Kohn-Sham equations for the electrons and quantum protons:

	$\displaystyle\Big{(}-\frac{1}{2}\nabla^{2}+v_{\text{eff}}^{\text{e}}(\mathbf{r% }_{1}^{\text{e}})\Big{)}\psi_{i}^{\text{e}}=\epsilon_{i}^{\text{e}}\psi_{i}^{% \text{e}}$		(13.44)
	$\displaystyle\Big{(}-\frac{1}{2m_{\text{p}}}\nabla^{2}+v_{\text{eff}}^{\text{p% }}(\mathbf{r}_{1}^{\text{p}})\Big{)}\psi_{I}^{\text{p}}=\epsilon_{I}^{\text{p}% }\psi_{I}^{\text{p}}.$		(13.45)

The effective potentials $v_{\text{eff}}$ and $v_{\text{eff}}$ are obtained by taking the derivative of the total energy expression in Eq. (13.44) with respect to electron density and proton density, respectively. Analogous to NEO-HF, these electronic and protonic Kohn-Sham orbitals are expanded as linear combinations of electronic or protonic Gaussian basis functions ( $\phi^{\text{e}}_{\mu}(\mathbf{r}_{\text{e}})$ and $\phi^{\text{p}}_{\mu^{\prime}}(\mathbf{r}_{\text{p}})$ ). The extension to open-shell electron systems is analogous to the NEO-UHF method.

The practical implementation of the NEO-DFT method requires an electron-electron exchange-correlation functional, a proton-proton exchange-correlation functional, and an electron-proton correlation functional. Any conventional electron-electron exchange-correlation functional can be used within the NEO-DFT framework.¹³⁵ Because the proton-proton exchange and correlation are negligible in molecular systems, only the exchange at the NEO-Hartree-Fock level is included to eliminate self-interaction error in the NEO-DFT method. A suitable electron-proton correlation functional is essential for obtaining an accurate proton densities and energies, and the epc17-2^{1254, 136} and epc19¹¹¹³ functionals are designed to achieve this goal. These two functionals are based on the multicomponent extension of the Colle-Salvetti formalism. The epc17-2 functional is of the local density approximation (LDA) type with the functional form:

\displaystyle E_{\text{epc}}[\rho^{\text{e}},\rho^{\text{p}}]=-\int d\mathbf{r% }\frac{\rho^{\text{e}}(\mathbf{r})\rho^{\text{p}}(\mathbf{r})}{a-b[\rho^{\text% {e}}(\mathbf{r})\rho^{\text{p}}(\mathbf{r})]^{1/2}+c\rho^{\text{e}}(\mathbf{r}% )\rho^{\text{p}}(\mathbf{r})}.

(13.46)

The epc19 functional is its multicomponent generalized gradient approximation (GGA) extension that depends on the electron and proton density gradients and is of the form:

		$\displaystyle E_{\text{epc}}[\rho^{\text{e}},\rho^{\text{p}},\nabla\rho^{\text% {e}},\nabla\rho^{\text{p}}]=-\int d\mathbf{r}\frac{\rho^{\text{e}}(\mathbf{r})% \rho^{\text{p}}(\mathbf{r})}{a-b[\rho^{\text{e}}(\mathbf{r})\rho^{\text{p}}(% \mathbf{r})]^{1/2}+c\rho^{\text{e}}(\mathbf{r})\rho^{\text{p}}(\mathbf{r})}\times$		(13.47)
		$\displaystyle\bigg{\{}1-d\bigg{(}\frac{[\rho^{\text{e}}(\mathbf{r})\rho^{\text% {p}}(\mathbf{r})]^{-1/3}}{(1+m_{\text{p}})^{2}}\bigg{[}m_{\text{p}}^{2}\frac{% \nabla^{2}\rho^{\text{e}}(\mathbf{r})}{\rho^{\text{e}}(\mathbf{r})}-2m_{\text{% p}}\frac{\nabla\rho^{\text{e}}(\mathbf{r})\cdot\nabla\rho^{\text{p}}(\mathbf{r% })}{\rho^{\text{e}}(\mathbf{r})\rho^{\text{p}}(\mathbf{r})}+\frac{\nabla^{2}% \rho^{\text{p}}(\mathbf{r})}{\rho^{\text{p}}(\mathbf{r})}\bigg{]}\bigg{)}\text% {exp}\bigg{[}\frac{-k}{[\rho^{\text{e}}(\mathbf{r})\rho^{\text{p}}(\mathbf{r})% ]^{1/6}}\bigg{]}\bigg{\}}$

In addition to the parameters $a$ , $b$ , and $c$ in the epc17-2 functional,¹³⁶ the epc19 functional¹¹¹³ has the additional $d$ and $k$ parameters and also depends on the proton mass $m_{\text{p}}$ . Analogous to the NEO-HF analytic energy gradients, the NEO-DFT analytic gradients are also available for these two functionals.

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13.5.3 NEO-DFT