13.4.2 NEO-DFT

NEO density functional theory (NEO-DFT)^{724, 152, 153} 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

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

(13.43)

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 ${\rho }^{\text{e}}({𝐫}_1^{\text{e}})=2{\sum }_{i=1}^{N_{\text{e}}/2}{|{\psi }_i^{\text{e}}({𝐫}_1^{\text{e}})|}^2$ and ${\rho }^{\text{p}}({𝐫}_1^{\text{p}})={\sum }_{I=1}^{N_{\text{p}}}{|{\psi }_I^{\text{p}}({𝐫}_1^{\text{p}})|}^2$ are the electron and proton densities, respectively, and ${\psi }_i^{\text{e}}({𝐫}_1^{\text{e}})$ and ${\psi }_I^{\text{p}}({𝐫}_1^{\text{p}})$ 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:

	$\left(-{\displaystyle \frac{1}{2}}{\nabla }^2+v_{\text{eff}}^{\text{e}}({𝐫}_1^{\text{e}})\right){\psi }_i^{\text{e}}={ϵ}_i^{\text{e}}{\psi }_i^{\text{e}}$		(13.44)
	$\left(-{\displaystyle \frac{1}{2m_{\text{p}}}}{\nabla }^2+v_{\text{eff}}^{\text{p}}({𝐫}_1^{\text{p}})\right){\psi }_I^{\text{p}}={ϵ}_I^{\text{p}}{\psi }_I^{\text{p}}.$		(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 (${\varphi }_{\mu }^{\text{e}}({𝐫}_{\text{e}})$ and ${\varphi }_{{\mu }^{\prime }}^{\text{p}}({𝐫}_{\text{p}})$). 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.¹⁰⁶ 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^{1084, 107} and epc19⁹⁵¹ 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:

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

(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:

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

In addition to the parameters $a$, $b$, and $c$ in the epc17-2 functional,¹⁰⁷ the epc19 functional⁹⁵¹ 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.