13.4 The Nuclear-electronic Orbital (NEO) Approach: Integrating Electronic and Nuclear Quantum Effects in Quantum Chemistry 13.4.2 NEO-DFT 13.4.4 Job Control for the NEO-SCF methods

13.4.3 NEO-TDDFT

The NEO-TDDFT method¹⁰⁸⁵ 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

\displaystyle\begin{bmatrix}\mathbf{A}^{\text{e}}&\mathbf{B}^{\text{e}}&% \mathbf{C}&\mathbf{C}\\ \mathbf{B}^{\text{e}}&\mathbf{A}^{\text{e}}&\mathbf{C}&\mathbf{C}\\ \mathbf{C}^{\text{T}}&\mathbf{C}^{\text{T}}&\mathbf{A}^{\text{p}}&\mathbf{B}^{% \text{p}}\\ \mathbf{C}^{\text{T}}&\mathbf{C}^{\text{T}}&\mathbf{B}^{\text{p}}&\mathbf{A}^{% \text{p}}\end{bmatrix}\begin{bmatrix}\mathbf{X}^{\text{e}}\\ \mathbf{Y}^{\text{e}}\\ \mathbf{X}^{\text{p}}\\ \mathbf{Y}^{\text{p}}\end{bmatrix}=\omega\begin{bmatrix}\mathbf{I}&0&0&0\\ 0&-\mathbf{I}&0&0\\ 0&0&\mathbf{I}&0\\ 0&0&0&-\mathbf{I}\end{bmatrix}\begin{bmatrix}\mathbf{X}^{\text{e}}\\ \mathbf{Y}^{\text{e}}\\ \mathbf{X}^{\text{p}}\\ \mathbf{Y}^{\text{p}}\end{bmatrix}

(13.48)

where

$\displaystyle A_{ia,jb}^{\text{e}}$	$\displaystyle=(\epsilon_{a}-\epsilon_{i})\delta_{ab}\delta_{ij}+\langle aj\|ib% \rangle+\frac{\delta^{2}E_{\text{exc}}}{\delta P^{\text{e}}_{jb}\delta P^{% \text{e}}_{ai}}+\frac{\delta^{2}E_{\text{epc}}}{\delta P^{\text{e}}_{jb}\delta P% ^{\text{e}}_{ai}}$	(13.49)
$\displaystyle B_{ia,jb}^{\text{e}}$	$\displaystyle=\langle ab\|ij\rangle+\frac{\delta^{2}E_{\text{exc}}}{\delta P^{% \text{e}}_{jb}\delta P^{\text{e}}_{ia}}+\frac{\delta^{2}E_{\text{epc}}}{\delta P% ^{\text{e}}_{jb}\delta P^{\text{e}}_{ia}}$	(13.50)
$\displaystyle A_{IA,JB}^{\text{p}}$	$\displaystyle=(\epsilon_{A}-\epsilon_{I})\delta_{AB}\delta_{IJ}+\langle AJ\|IB% \rangle+\frac{\delta^{2}E_{\text{pxc}}}{\delta P^{\text{p}}_{JB}\delta P^{% \text{p}}_{AI}}+\frac{\delta^{2}E_{\text{epc}}}{\delta P^{\text{p}}_{JB}\delta P% ^{\text{p}}_{AI}}$	(13.51)
$\displaystyle B_{IA,JB}^{\text{p}}$	$\displaystyle=\langle AB\|IJ\rangle+\frac{\delta^{2}E_{\text{pxc}}}{\delta P^{% \text{p}}_{JB}\delta P^{\text{p}}_{IA}}+\frac{\delta^{2}E_{\text{epc}}}{\delta P% ^{\text{p}}_{JB}\delta P^{\text{p}}_{IA}}$	(13.52)
$\displaystyle C_{ia,JB}$	$\displaystyle=-\langle aB\|iJ\rangle+\frac{\delta^{2}E_{\text{epc}}}{\delta P^{% \text{p}}_{JB}\delta P^{\text{e}}_{ai}}$	(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 $\omega$ , as well as the transition excitation and de-excitation amplitudes, $\mathbf{X}$ and $\mathbf{Y}$ , 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

\displaystyle\begin{bmatrix}\mathbf{A}^{\text{e}}&\mathbf{C}\\ \mathbf{C}^{\text{T}}&\mathbf{A}^{\text{p}}\end{bmatrix}\begin{bmatrix}\mathbf% {X}^{\text{e}}\\ \mathbf{X}^{\text{p}}\end{bmatrix}=\omega\begin{bmatrix}\mathbf{X}^{\text{e}}% \\ \mathbf{X}^{\text{p}}\end{bmatrix}

(13.54)

The extension of the NEO-TDDFT and NEO-TDDFT-TDA approaches to open-shell electron systems is straightforward.¹⁹² 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.

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13.4.3 NEO-TDDFT