12.15.1 Introduction

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:

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., $t^{jb}=0$ 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 ($r_{\mathrm{cut}}$) so that transitions between neighboring fragments within a range of $r_{\mathrm{cut}}$ 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).