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
In this equation, is the interaction of the electronic and protonic densities with the external potential created by the classical nuclei, and 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 , , and are the electron-electron exchange-correlation functional, the proton-proton exchange-correlation functional, and the electron-proton correlation functional, respectively. The quantities and are the electron and proton densities, respectively, and and 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:
The effective potentials and 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 ( and ). 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:
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:
In addition to the parameters , , and in the epc17-2 functional,107 the epc19 functional951 has the additional and parameters and also depends on the proton mass . Analogous to the NEO-HF analytic energy gradients, the NEO-DFT analytic gradients are also available for these two functionals.