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12.15 ALMO-CIS/TDA and Its Charge-Transfer Correction

12.15.1 Introduction

(May 7, 2024)

The ALMO-CIS 238 Closser K. D. et al.
J. Chem. Theory Comput.
(2015), 11, pp. 5791.
Link
and ALMO-CIS+CT 405 Ge Q. et al.
J. Chem. Phys.
(2017), 146, pp. 044111.
Link
methods are local variants of Configuration Interaction Singles (CIS) for excited states, which are formulated based on the locality of Absolutely Localized Molecular Orbitals (ALMOs). The ALMO-CIS method shares same spirit with the TDDFT(MI) method 794 Liu J., Herbert J. M.
J. Chem. Phys.
(2015), 143, pp. 034106.
Link
(Section 12.14), but it was originally designed to calculate a large number of excited states in atomic/molecular clusters, e.g., the entire n=2 band in helium clusters that contain hundreds of atoms. 238 Closser K. D. et al.
J. Chem. Theory Comput.
(2015), 11, pp. 5791.
Link
, 405 Ge Q. et al.
J. Chem. Phys.
(2017), 146, pp. 044111.
Link

In ALMO-CIS and ALMO-CIS+CT, one solves a truncated non-orthogonal CIS eigenvalue problem:

Aia,jbtjb=ωSia,jbtjb (12.76)

The use of ALMOs allows associating each MO index (i, a, j, or b) to a fragment. In ALMO-CIS, only the CIS amplitudes corresponding to intrafragment transitions are retained, i.e., tjb=0 if the occupied orbital j and the virtual orbital b reside on two different fragments. The Hamiltonian and overlap matrix are also truncated, with i (j) and a (b) belonging to the same fragment. This approximation excludes interfragment charge transfer (CT) excitations entirely, which sometimes turns out to be insufficiently accurate. In ALMO-CIS+CT, the CT effect is reintroduced by providing a distance-based cutoff (rcut) so that transitions between neighboring fragments within a range of rcut are allowed, i.e., i (j) and a (b) that are on a pair of neighboring fragments are also included in Eq. (12.76). In both ALMO-CIS and ALMO-CIS+CT, the dimension of the eigenvalue problem scales linearly with the system size rather than having a quadratic scaling as in standard CIS. Because of the reduction of matrix size, it is computationally feasible to explicitly build the Hamiltonian and directly diagonalize it to obtain a full band of excited states for relatively large systems. The overall scaling of the diagonalization step in ALMO-CIS/ALMO-CIS+CT is cubic, in contrast to the sixth-order scaling of standard CIS for full-spectrum calculation. To accelerate the construction of the CIS Hamiltonian (the 𝐀 matrix in Eq. (12.76)), the resolution-of-the-identity (RI) technique is employed to evaluate some of the 2-electron terms (see Ref.  238 Closser K. D. et al.
J. Chem. Theory Comput.
(2015), 11, pp. 5791.
Link
for details).

Besides the full-spectrum calculations described above, use of the Davidson algorithm is also available for ALMO-CIS and ALMO-CIS+CT, which targets a few lowest excited states as in standard CIS/TDDFT calculations. This iterative method, unlike the original full-spectrum version, also supports the ALMO variant of linear-response TDDFT within the Tame-Dancoff approximation (TDA), 542 Hirata S., Head-Gordon M.
Chem. Phys. Lett.
(1999), 314, pp. 291.
Link
which is referred to as ALMO-TDA and shares the same working equation (Eq. 12.76).