13.5.6 NEO-CC

An alternative route for inclusion of correlation effects between quantum particles (i.e., electrons and protons) is with wave functions methods that are systematically improvable and parameter-free. ¹²⁵⁴ Webb S. P., Iordanov T., Hammes-Schiffer S.
J. Chem. Phys.
(2002), 117, pp. 4106. Link ^{, 914} Pavošević F., Culpitt T., Hammes-Schiffer S.
Chem. Rev.
(2020), 120, pp. 4222. Link Among the various developed multicomponent wave function methods, the NEO coupled cluster (NEO-CC) methods have been particularly successful. ⁹¹³ Pavošević F., Culpitt T., Hammes-Schiffer S.
J. Chem. Theory Comput.
(2018), 15, pp. 338. Link ^{, 915} Pavošević F., Hammes-Schiffer S.
J. Chem. Phys.
(2019), 151, pp. 074104. Link ^{, 916} Pavošević F., Rousseau B. J. G., Hammes-Schiffer S.
J. Phys. Chem. Lett.
(2020), 11, pp. 1578. Link ^{, 917} Pavošević F., Tao Z., Hammes-Schiffer S.
J. Phys. Chem. Lett.
(2021), 12, pp. 1631. Link The NEO-CC wave function is given by

where $\widehat{T}$ is the cluster operator that incorporates the correlation effects between quantum particles, and $|0^{\text{e}}{0}^{\text{p}}⟩$ is the NEO-HF reference wave function. In the NEO-CCSD method, ⁹¹³ Pavošević F., Culpitt T., Hammes-Schiffer S.
J. Chem. Theory Comput.
(2018), 15, pp. 338. Link the cluster operator is given by

where $a^{\alpha }=a_{\alpha }^{†}=\{a_i^a,a_I^A,a_{ij}^{ab},a_{IJ}^{AB},a_{iI}^{aA}\}$ are the excitation operators expressed in terms of creation/annihilation ($a_p^{†}/a_p$) fermionic operators, and $\alpha$ is the excitation rank. Here, the $i,j,\mathrm{\dots }$ indices denote occupied electronic orbitals, the $a,b,\mathrm{\dots }$ indices denote unoccupied electronic orbitals, and the $p,q,\mathrm{\dots }$ indices denote general electronic orbitals. The protonic orbitals are denoted analogously using the capitalized indices. The unknown $t_{\alpha }$ wave function parameters (amplitudes) are determined by solving the set of nonlinear equations for each $\alpha$: ⁹¹³ Pavošević F., Culpitt T., Hammes-Schiffer S.
J. Chem. Theory Comput.
(2018), 15, pp. 338. Link

In this equation, $H_{\text{NEO}}=h_q^p{a}_p^q+\frac{1}{2}{g}_{rs}^{pq}{a}_{pq}^{rs}+h_Q^P{a}_p^q+\frac{1}{2}{g}_{RS}^{PQ}{a}_{PQ}^{RS}-g_{qQ}^{pP}{a}_{pP}^{qQ}$ is the second-quantized NEO Hamiltonian, where $h_q^p=⟨q|{\widehat{h}}^{\text{e}}|p⟩$ and $g_{rs}^{pq}=⟨rs|pq⟩$ are conventional electronic core Hamiltonian and two-electron integrals, respectively. The remaining protonic ($h_Q^P$ and $g_{RS}^{PQ}$) and electron-proton ($g_{qQ}^{pP}$) integrals are defined analogously. Lastly, the NEO-CCSD energy is calculated from

To increase the computational efficiency and reduce the memory requirements for the NEO-CCSD method, the two-particle integrals can be approximated with the density fitting (DF) approximation, ⁹¹⁷ Pavošević F., Tao Z., Hammes-Schiffer S.
J. Phys. Chem. Lett.
(2021), 12, pp. 1631. Link in which the two-particle four-center integrals are factorized into a sum of products of three-center and two-center two-particle integrals. In particular, the four-center two-electron integrals are approximated by

where $(\mu \nu |X)$ and $(X|Y)$ are three-center and two-center two-electron integrals, respectively. In this equation, $\mu ,\nu ,\mathrm{\dots }$ and $X,Y,\mathrm{\dots }$ indices denote electronic and auxiliary electronic basis functions, respectively. The four-center two-proton integrals are approximated analogously by

where primed indices denote protonic basis functions and $({\mu }^{\prime }{\nu }^{\prime }|X)$ and $(X^{\prime }|Y^{\prime })$ are three-center and two-center two-proton integrals, respectively. Finally, the four-center electron-proton integrals are approximated as

By employing the DF approximation, the memory requirements for storing four-center two-particle integrals are reduced from $N_{\text{bf}}^4$ to $N_{\text{bf}}^2\times N_{\text{aux}}$, where $N_{\text{bf}}$ and $N_{\text{aux}}$ are the number of electronic or protonic basis functions and auxiliary basis functions, respectively. ⁹¹⁷ Pavošević F., Tao Z., Hammes-Schiffer S.
J. Phys. Chem. Lett.
(2021), 12, pp. 1631. Link