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13.5 Nuclear–Electronic Orbital Method

13.5.6 NEO-CC

(July 14, 2022)

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

|ΨNEO-CC=eT^|0e0p, (13.53)

where T^ is the cluster operator that incorporates the correlation effects between quantum particles, and |0e0p 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

T^=T^1+T^2=taiaia+tAIaIA+14tabijaijab+14tABIJaIJAB+taAiIaiIaA=αtαaα, (13.54)

where aα=aα={aia,aIA,aijab,aIJAB,aiIaA} are the excitation operators expressed in terms of creation/annihilation (ap/ap) fermionic operators, and α is the excitation rank. Here, the i,j, indices denote occupied electronic orbitals, the a,b, indices denote unoccupied electronic orbitals, and the p,q, indices denote general electronic orbitals. The protonic orbitals are denoted analogously using the capitalized indices. The unknown tα wave function parameters (amplitudes) are determined by solving the set of nonlinear equations for each α: 913 Pavošević F., Culpitt T., Hammes-Schiffer S.
J. Chem. Theory Comput.
(2018), 15, pp. 338.
Link

0e0p|aαe-T^1-T^2H^NEOeT^1+T^2|0e0p=0. (13.55)

In this equation, HNEO=hqpapq+12grspqapqrs+hQPapq+12gRSPQaPQRS-gqQpPapPqQ is the second-quantized NEO Hamiltonian, where hqp=q|h^e|p and grspq=rs|pq are conventional electronic core Hamiltonian and two-electron integrals, respectively. The remaining protonic (hQP and gRSPQ) and electron-proton (gqQpP) integrals are defined analogously. Lastly, the NEO-CCSD energy is calculated from

ENEO-CCSD=0e0p|e-T^1-T^2H^NEOeT^1+T^2|0e0p. (13.56)

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

(μν|ρσ)=μρ|νσXY(μν|X)(X|Y)-1(Y|ρσ), (13.57)

where (μν|X) and (X|Y) are three-center and two-center two-electron integrals, respectively. In this equation, μ,ν, and X,Y, indices denote electronic and auxiliary electronic basis functions, respectively. The four-center two-proton integrals are approximated analogously by

(μν|ρσ)=μρ|νσXY(μν|X)(X|Y)-1(Y|ρσ), (13.58)

where primed indices denote protonic basis functions and (μν|X) and (X|Y) are three-center and two-center two-proton integrals, respectively. Finally, the four-center electron-proton integrals are approximated as

(μν|μν)=μμ|ννXY(μν|X)(X|Y)-1(Y|μν). (13.59)

By employing the DF approximation, the memory requirements for storing four-center two-particle integrals are reduced from Nbf4 to Nbf2×Naux, where Nbf and Naux 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