Linear-Scaling Time-Dependent Density Functional Theory Based on the Idea of “From Fragments to Molecule”

Abstract

To circumvent the cubic scaling and convergence difficulties encountered in the standard top-down localization of the global canonical molecular orbitals (CMOs), a bottom-up localization scheme is proposed based on the idea of "from fragments to molecule". That is, the global localized MOs (LMOs), both occupied and unoccupied, are to be synthesized from the primitive fragment LMOs (pFLMOs) obtained from subsystem calculations. They are orthonormal but are still well localized on the parent fragments of the pFLMOs and can hence be termed as "fragment LMOs" (FLMOs). This has been achieved by making use of two important factors. Physically, it is the transferability of the locality of the fragments that serves as the basis. Mathematically, it is the special block-diagonalization of the Kohn-Sham matrix that allows retention of the locality: The occupied-occupied and virtual-virtual diagonal blocks are only minimally modified when the occupied-virtual off-diagonal blocks are annihilated. Such a bottom-up localization scheme is applicable to systems composed of all kinds of chemical bonds. It is then shown that, by a simple prescreening of the particle-hole pairs, the FLMO-based time-dependent density functional theory (TDDFT) can achieve linear scaling with respect to the system size, with a very small prefactor. As a proof of principle, representative model systems are taken as examples to demonstrate the accuracy and efficiency of the algorithms. As both the orbital picture and integral number of electrons are retained, the FLMO-TDDFT offers a clear characterization of the nature of the excited states in line with chemical/physical intuition.

Side chain directly participates in the solar absorption of fullerene derivative PC61BM

“…So the side chain directly contributes 10.4% [= (0.025135 + 0.013867)/0.374075] to the solar absorption of the PC 61 BM molecule. We note in passing that, at variance with the present rough estimate of the side-chain contribution to excitations, a rigorous and pictorial analysis in terms of fragment localized molecular orbitals [11,15] should in principle be made. However, such a tool is not at our disposal.…”

An efficient localization procedure for large systems using a sequential transformation strategy

“…41,47 Very different strategies have also been proposed for getting localized orthonormal HF orbitals without using optimization techniques. 16,[34][35][36]38,39,42 The least-change algorithm proposed by Ziółkowski et al 35 is one such strategy. The algorithm minimizes the size of the transformation matrix from the atomic orbital basis to the HF optimized orbital basis, which removes the canonical condition when performing a global Hartree-Fock optimization.…”

Characterization and Generation of Local Occupied and Virtual Hartree–Fock Orbitals

“…Such a strategy for generating a set of local orbitals is denoted a topdown strategy [37][38][39] emphasizing that the Hartree-Fock optimization step (the top step) is followed by an orbital localization step (the down step). The generation of local orbitals using the top-down strategy will be discussed later.…”

“…39 The locality of orbitals generated from bottom-up approaches is seen to be similar to that obtained using the traditional Boys and Pipek-Mezey localization functions. 30,[37][38][39] In this review we restrict ourselves to consider only how local orbitals may be generated when orbital localization functions (top-down strategies) are used.…”

“…Liu and co-workers have also shown how their bottom-up orbitals may be used as starting orbitals for further localization. 37,38 For local correlation and frozen core calculations, the core and valence space localizations should be performed separately to avoid a mixing of the core and valence orbitals. Høyvik et al have generalized the approach of Ziolkowski et al to include a separation of the core and valence orbital spaces.…”