13.5 Nuclear–Electronic Orbital Method 13.5.2 NEO-Hartree-Fock 13.5.4 NEO-PCM

13.5.3 NEO-DFT

(April 13, 2024)

NEO density functional theory (NEO-DFT) ⁹²⁸ Pak M. V., Chakraborty A., Hammes-Schiffer S.
J. Phys. Chem. A
(2007), 111, pp. 4522. Link ^, ²⁰¹ Chakraborty A., Pak M. V., Hammes-Schiffer S.
Phys. Rev. Lett.
(2008), 101, pp. 153001. Link ^, ²⁰² Chakraborty A., Pak M. V., Hammes-Schiffer S.
J. Chem. Phys.
(2009), 131, pp. 124115. Link 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

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}}]\;.

(13.39)

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


$\displaystyle\rho^{\text{e}}(\mathbf{r}_{1}^{\text{e}})$	$\displaystyle=2\sum_{i=1}^{N_{\text{e}}/2}\|\psi_{i}^{\text{e}}(\mathbf{r}_{1}^% {\text{e}})\|^{2}$	(13.40a)
$\displaystyle\rho^{\text{p}}(\mathbf{r}_{1}^{\text{p}})$	$\displaystyle=\sum_{I=1}^{N_{\text{p}}}\|\psi_{I}^{\text{p}}(\mathbf{r}_{1}^{% \text{p}})\|^{2}$	(13.40b)

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:


$\displaystyle\Big{(}-\frac{1}{2}\nabla^{2}+v_{\text{eff}}^{\text{e}}(\mathbf{r% }_{1}^{\text{e}})\Big{)}\psi_{i}^{\text{e}}$	$\displaystyle=\epsilon_{i}^{\text{e}}\;\psi_{i}^{\text{e}}$	(13.41a)
$\displaystyle\Big{(}-\frac{1}{2m_{\text{p}}}\nabla^{2}+v_{\text{eff}}^{\text{p% }}(\mathbf{r}_{1}^{\text{p}})\Big{)}\psi_{I}^{\text{p}}$	$\displaystyle=\epsilon_{I}^{\text{p}}\;\psi_{I}^{\text{p}}\;.$	(13.41b)

The effective potentials $v_{\text{eff}}$ and $v_{\text{eff}}$ are obtained by taking the derivative of the total energy expression in Eq. (13.39) 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. ¹⁴³ Brorsen K. R., Schneider P. E., Hammes-Schiffer S.
J. Chem. Phys.
(2018), 149, pp. 044110. Link 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 accurate proton densities and energies, and the epc17-2 ¹³⁶⁰ Yang Y. et al.
J. Chem. Phys.
(2017), 147, pp. 114113. Link ^, ¹⁴⁴ Brorsen K. R., Yang Y., Hammes-Schiffer S.
J. Phys. Chem. Lett.
(2017), 8, pp. 3488. Link and epc19 ¹²¹³ Tao Z., Yang Y., Hammes-Schiffer S.
J. Chem. Phys.
(2019), 151, pp. 124102. Link 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})}\;.

(13.42)

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\begin{aligned} &\displaystyle E_{\text{epc}}[\rho^{\text{e}},% \rho^{\text{p}},\hat{\bm{\nabla}}\rho^{\text{e}},\hat{\bm{\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\\ &\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{\hat{\nabla}^{2}\rho^{\text{e}}(\mathbf{r})}{\rho^{\text{e}}(\mathbf{r})% }-2m_{\text{p}}\frac{\hat{\bm{\nabla}}\rho^{\text{e}}(\mathbf{r})\bm{\cdot}% \hat{\bm{\nabla}}\rho^{\text{p}}(\mathbf{r})}{\rho^{\text{e}}(\mathbf{r})\rho^% {\text{p}}(\mathbf{r})}+\frac{\hat{\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{\}}\;.% \end{aligned}

(13.43)

In addition to the parameters $a$ , $b$ , and $c$ in the epc17-2 functional, ¹⁴⁴ Brorsen K. R., Yang Y., Hammes-Schiffer S.
J. Phys. Chem. Lett.
(2017), 8, pp. 3488. Link the epc19 functional ¹²¹³ Tao Z., Yang Y., Hammes-Schiffer S.
J. Chem. Phys.
(2019), 151, pp. 124102. Link has the $d$ and $k$ parameters and also depends on the proton mass $m_{\text{p}}$ . Analogous to the NEO-HF analytical energy gradients, the NEO-DFT analytical gradients ¹²¹² Tao Z. et al.
J. Chem. Theory Comput.
(2021), 17, pp. 5110. Link are available for these two functionals, allowing geometry optimizations on the ground state vibronic potential energy surface. The NEO-DFT analytical Hessians ¹²¹² Tao Z. et al.
J. Chem. Theory Comput.
(2021), 17, pp. 5110. Link are available for the epc17-2 functional or when no electron-proton correlation functional is used and allow characterization of the stationary points.

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13.5.3 NEO-DFT