13.4 The Nuclear-electronic Orbital (NEO) Approach: Integrating Electronic and Nuclear Quantum Effects in Quantum Chemistry 13.4 The Nuclear-electronic Orbital (NEO) Approach: Integrating Electronic and Nuclear Quantum Effects in Quantum Chemistry 13.4.2 NEO-DFT

13.4.1 NEO-Hartree-Fock

The simplest method within the NEO framework is the Hartree-Fock (NEO-HF) method, where the total nuclear-electronic wavefunction is approximated as a product of electronic ( $\mathbf{\Phi}_{0}^{\text{e}}(\mathbf{x}_{\text{e}})$ ) and nuclear ( $\mathbf{\Phi}_{0}^{\text{p}}(\mathbf{x}_{\text{p}})$ ) Slater determinants composed of electronic and protonic spin orbitals, respectively:

$\displaystyle\mathbf{\Psi}_{\text{NEO-HF}}(\mathbf{x}_{\text{e}},\mathbf{x}_{% \text{p}})=\mathbf{\Phi}_{0}^{\text{e}}(\mathbf{x}_{\text{e}})\mathbf{\Phi}_{0% }^{\text{p}}(\mathbf{x}_{\text{p}}).$

(13.33)

In Eq. (13.33), $\mathbf{x}_{\text{e}}$ and $\mathbf{x}_{\text{p}}$ are collective spatial and spin coordinates of the quantum electrons and protons. The NEO-HF energy for a restricted Hartree-Fock (RHF) treatment of the electrons and a high-spin open-shell treatment of the quantum protons is

	$\displaystyle E_{\text{NEO-HF}}=$	$\displaystyle 2\sum_{i}^{N_{\text{e}}/2}h^{\text{e}}_{ii}+\sum_{i}^{N_{\text{e% }}/2}\sum_{j}^{N_{\text{e}}/2}\Big{(}2(ii\|jj)-(ij\|ij)\Big{)}$
		$\displaystyle+\sum_{I}^{N_{\text{p}}}h^{\text{p}}_{II}+\frac{1}{2}\sum_{I}^{N_% {\text{p}}}\sum_{J}^{N_{\text{p}}}\Big{(}(II\|JJ)-(IJ\|IJ)\Big{)}-2\sum_{i}^{N_{% \text{e}}/2}\sum_{I}^{N_{\text{p}}}(ii\|II).$		(13.34)

The $i, j \dots$ , indices denote spatial occupied electronic orbital, and the $I, J \dots$ , indices correspond to spatial occupied protonic orbitals. In Eq. (13.4.1), $h^{\text{e}}_{ij}$ and $(ij|kl)$ are conventional electronic core Hamiltonian and two-electron integrals, respectively, and the corresponding terms for quantum protons are defined analogously. The last term in Eq. (13.4.1) is the Coulomb interaction between the electrons and the quantum protons. The spatial electronic and protonic orbitals ( $\psi^{\text{e}}_{i}(\mathbf{r}_{\text{e}})$ and $\psi^{\text{p}}_{I}(\mathbf{r}_{\text{p}})$ ) are expanded as linear combinations of electronic or protonic Gaussian basis functions ( $\phi^{\text{e}}_{\mu}(\mathbf{r}_{\text{e}})$ and $\phi^{\text{p}}_{\mu^{\prime}}(\mathbf{r}_{\text{p}})$ ):

	$\displaystyle\psi^{\text{e}}_{i}(\mathbf{r}_{\text{e}})=$	$\displaystyle\sum_{\mu}^{N^{bf}_{\text{e}}}C_{\mu i}^{\text{e}}\phi^{\text{e}}% _{\mu}(\mathbf{r}_{\text{e}}),$		(13.35)
	$\displaystyle\psi^{\text{p}}_{I}(\mathbf{r}_{\text{p}})=$	$\displaystyle\sum_{\mu^{\prime}}^{N^{bf}_{\text{p}}}C_{\mu^{\prime}I}^{\text{p% }}\phi^{\text{p}}_{\mu^{\prime}}(\mathbf{r}_{\text{p}}).$		(13.36)

The lower-case Greek letters without and with primes denote basis functions for electrons and protons, respectively, and $C_{\mu i}^{\text{e}}$ and $C_{\mu^{\prime}I}^{\text{p}}$ are electronic and protonic MO expansion coefficients, respectively.

Analogous to the conventional electronic Hartree-Fock method, the electronic and protonic coefficients are determined by variationally minimizing the energy in Eq. (13.4.1) via the self-consistent field (SCF) procedure. This procedure leads to a set of coupled electronic and protonic HF-Roothaan equations:

	$\displaystyle\mathbf{F}^{\text{e}}\mathbf{C}^{\text{e}}=$	$\displaystyle\mathbf{S}^{\text{e}}\mathbf{C}^{\text{e}}\mathbf{E}^{\text{e}},$		(13.37)
	$\displaystyle\mathbf{F}^{\text{p}}\mathbf{C}^{\text{p}}=$	$\displaystyle\mathbf{S}^{\text{p}}\mathbf{C}^{\text{p}}\mathbf{E}^{\text{p}},$		(13.38)

where $\mathbf{S}^{\text{e}}$ and $\mathbf{S}^{\text{p}}$ are electronic and protonic overlap matrices, respectively. The electronic and protonic Fock elements in Eqs. (13.37) and (13.38) are given by

	$\displaystyle F^{\text{e}}_{\mu\nu}=$	$\displaystyle h^{\text{e}}_{\mu\nu}+\sum_{\rho\lambda}P^{\text{e}}_{\lambda% \rho}\Big{(}(\mu\nu\|\rho\lambda)-\frac{1}{2}(\mu\lambda\|\rho\nu)\Big{)}-\sum_{% \mu^{\prime}\nu^{\prime}}P^{\text{p}}_{\nu^{\prime}\mu^{\prime}}(\mu\nu\|\mu^{% \prime}\nu^{\prime}),$		(13.39)
	$\displaystyle F^{\text{p}}_{\mu^{\prime}\nu^{\prime}}=$	$\displaystyle h^{\text{p}}_{\mu^{\prime}\nu^{\prime}}+\sum_{\rho^{\prime}% \lambda^{\prime}}P^{\text{p}}_{\lambda^{\prime}\rho^{\prime}}\Big{(}(\mu^{% \prime}\nu^{\prime}\|\rho^{\prime}\lambda^{\prime})-(\mu^{\prime}\lambda^{% \prime}\|\rho^{\prime}\nu^{\prime})\Big{)}-\sum_{\mu\nu}P^{\text{e}}_{\nu\mu}(% \mu^{\prime}\nu^{\prime}\|\mu\nu).$		(13.40)

The electronic and protonic density matrix elements in Eqs. (13.39) and (13.40) are defined as

	$\displaystyle P^{\text{e}}_{\nu\mu}=$	$\displaystyle 2\sum_{i}^{N_{\text{e}}/2}C_{\nu i}^{\text{e}}C_{\mu i}^{\text{e% }*},$		(13.41)
	$\displaystyle P^{\text{p}}_{\nu^{\prime}\mu^{\prime}}=$	$\displaystyle\sum_{I}^{N_{\text{p}}}C_{\nu^{\prime}I}^{\text{p}}C_{\mu^{\prime% }I}^{\text{p}*}.$		(13.42)

The generalization to the unrestricted Hartree-Fock (NEO-UHF) treatment of electrons is accomplished by introducing separate spatial orbitals for $\alpha$ and $\beta$ electron spins.

The analytic gradients of the NEO-HF energy¹⁰¹⁷ with respect to the classical nuclear coordinates (or coordinates of the centers of the quantum proton basis functions) can be derived and implemented. These gradients allow geometry optimizations within the NEO framework.

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13.4.1 NEO-Hartree-Fock