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

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[ρe,ρp]=Eext[ρe,ρp]+Eref[ρe,ρp]+Eexc[ρe]+Epxc[ρp]+Eepc[ρe,ρp] (13.43)

In this equation, Eext[ρe,ρp] is the interaction of the electronic and protonic densities with the external potential created by the classical nuclei, and Eref[ρe,ρp] 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 Eexc[ρe], Epxc[ρp], and Eepc[ρe,ρp] are the electron-electron exchange-correlation functional, the proton-proton exchange-correlation functional, and the electron-proton correlation functional, respectively. The quantities ρe(𝐫1e)=2i=1Ne/2|ψie(𝐫1e)|2 and ρp(𝐫1p)=I=1Np|ψIp(𝐫1p)|2 are the electron and proton densities, respectively, and ψie(𝐫1e) and ψIp(𝐫1p) 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:

(-122+veffe(𝐫1e))ψie=ϵieψie (13.44)
(-12mp2+veffp(𝐫1p))ψIp=ϵIpψIp. (13.45)

The effective potentials veff and veff 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 (ϕμe(𝐫e) and ϕμp(𝐫p)). 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 epc17-21084, 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:

Eepc[ρe,ρp]=-𝑑𝐫ρe(𝐫)ρp(𝐫)a-b[ρe(𝐫)ρp(𝐫)]1/2+cρe(𝐫)ρp(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:

Eepc[ρe,ρp,ρe,ρp]=-d𝐫ρe(𝐫)ρp(𝐫)a-b[ρe(𝐫)ρp(𝐫)]1/2+cρe(𝐫)ρp(𝐫)× (13.47)

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 mp. Analogous to the NEO-HF analytic energy gradients, the NEO-DFT analytic gradients are also available for these two functionals.