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

13.4.3 NEO-TDDFT

The NEO-TDDFT methodYang:2018 is a multicomponent extension of the TDDFT method within the NEO framework. It allows the simultaneous calculation of the electronic and proton vibrational excitation energies. In the NEO-TDDFT method, the linear response of the NEO Kohn-Sham system to perturbative external fields is computed. The NEO-TDDFT working equation is

[𝐀e𝐁e𝐂𝐂𝐁e𝐀e𝐂𝐂𝐂T𝐂T𝐀p𝐁p𝐂T𝐂T𝐁p𝐀p][𝐗e𝐘e𝐗p𝐘p]=ω[𝐈0000-𝐈0000𝐈0000-𝐈][𝐗e𝐘e𝐗p𝐘p] (13.48)

where

Aia,jbe =(ϵa-ϵi)δabδij+aj|ib+δ2EexcδPjbeδPaie+δ2EepcδPjbeδPaie (13.49)
Bia,jbe =ab|ij+δ2EexcδPjbeδPiae+δ2EepcδPjbeδPiae (13.50)
AIA,JBp =(ϵA-ϵI)δABδIJ+AJ|IB+δ2EpxcδPJBpδPAIp+δ2EepcδPJBpδPAIp (13.51)
BIA,JBp =AB|IJ+δ2EpxcδPJBpδPIAp+δ2EepcδPJBpδPIAp (13.52)
Cia,JB =-aB|iJ+δ2EepcδPJBpδPaie (13.53)

Here, the occupied electronic orbitals are denoted with indices i and j, whereas the unoccupied electronic orbitals are denoted with indices a and b. The upper case indices denote protonic orbitals. The solution of Eq. (13.48) provides the electronic and proton vibrational excitation energies ω, as well as the transition excitation and de-excitation amplitudes, 𝐗 and 𝐘, respectively. Analogous to the TDDFT method, the Tamm-Dancoff approximation (TDA) can be imposed within the NEO framework, defining the NEO-TDDFT-TDA method that is represented by

[𝐀e𝐂𝐂T𝐀p][𝐗e𝐗p]=ω[𝐗e𝐗p] (13.54)

The extension of the NEO-TDDFT and NEO-TDDFT-TDA approaches to open-shell electron systems is straightforward.Culpitt:2019 NEO-TDHF and NEO-CIS take the similar form as NEO-TDDFT and NEO-TDA without electron-proton correlation treatment and pure coulomb treatment for proton and electronic exchange correlation.