12.19.1 Introduction

The ALMO-CIS²¹² and ALMO-CIS+CT³⁵⁵ methods are local variants of configuration interaction singles (CIS), and are formulated through the use of absolutely localized molecular orbitals (ALMOs). They share the same spirit with the TDDFT(MI) method (Section 12.18), but were originally designed to target large numbers of excited states in atomic/molecular clusters, such as the entire $n=2$ band in helium clusters that contain hundreds of atoms.

In ALMO-CIS and ALMO-CIS+CT, we solve a truncated non-orthogonal CIS eigen-equation:

The use of ALMOs allows associating each MO index ($i$, $a$, $j$ or $b$) to a fragment. In ALMO-CIS, approximation is made such that only the amplitudes corresponding to intrafragment transitions are non-zero, 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 and sometimes may lead to insufficient accuracy. In ALMO-CIS+CT, the CT effect is reintroduced by setting a distance cutoff $r_{\mathrm{cut}}$, so that transitions between neighboring fragments within a distance smaller than $r_{\mathrm{cut}}$ are allowed ($i$ ($j$) and $a$ ($b$) belonging to such a pair of fragments are also included in the eigen-equation). In both ALMO-CIS and ALMO-CIS+CT, dimension of the eigenvalue problem scales linearly with respect to system size, instead of 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 diagonalize it to get a full band of eigenstates for relatively large systems, and the overall scaling of ALMO-CIS/ALMO-CIS+CT is cubic, in contrast with the sixth order scaling of standard CIS for a full-spectrum calculation.

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. This implementation, unlike the original full-spectrum version, also supports the local variants of TDDFT/TDA calculations that share the same working equation (eq. 12.73).