## Introduction

The coupling of electronic excitations in molecular aggregates or in multi-chromophore molecules and metal complexes has long been an important research topic in chemistry and physics.1, 2 Of special interest are situations where the electric transition dipoles of the uncoupled excitations are geometrically arranged along a ‘helical’ path such that the coupled system exhibits electronic exciton circular dichroism (CD).3–7 Exciton CD reveals a great amount of information about the relative orientation of individual chromophores with respect to each other and about the distance between them. Exciton CD can be very strong, even if the individual chromophores have no intrinsic chirality. In leading order, and at large separation of the chromophores, the rotatory strengths of the coupled transitions (the integrated intensities of the CD of individual excitations) are determined simply by the lengths, relative orientations, and relative distances of the electric transition dipole vectors as well as by the energies of the uncoupled transitions.8–10 Equations for the ‘matrix method’ (MM) dipole coupling model8 are provided in the Discussion section to illustrate the case.

Berova and collaborators have experimentally detected very long-range exciton CD with chromophore separations up to 50 Å for exciton coupling between two porphyrin based chromophores, for instance for tetraphenylporphyrin (TPP).11 Tsubaki et al. have reported observations of TPP exciton CD of substituted chiral oligonaphthalenes at even larger distances of approximately 66 Å.12, 13 The strong coupling between TPP substituents in suitably derivatized biomolecules has allowed researchers to derive important information about their three-dimensional structure. Examples where porphyrin-based exciton CD has been investigated or used in this context include the use of magnesium porphyrin to determine the absolute stereochemistry of chiral alcohols,14 a theoretical study of Soret band coupling of bis-porphyrin derivatives using various exciton coupling models,15 a determination of absolute stereochemistry of cyclic α-hydroxyketones with zinc–TPP tweezers,16 and investigations of the conformational space of DNA.17 A study of bis-porphyrin dimers of various derivatized biomolecules, including a brevetoxin bis-TPP derivative shown in Figure 1, has established the very long-range nature of the exciton CD.11

In first-principles calculations of electronic absorption and CD spectra, the size of the system—dictating the number of basis functions, *B*—is one of the major limiting factors because of the scaling of the computational effort with *B*. Presently, the most frequently utilized method for electronic spectra calculations is (linear response) time-dependent density functional theory (TDDFT).18–20 Reasonably efficient wave function-based methods for the treatment of exciton CD are available as well.21 The attractiveness of TDDFT derives from the fact that it incorporates electron correlation at a computational cost that is comparable to Hartree–Fock (HF) theory, if hybrid functionals are used for the exchange. The scaling is formally of order *B*^{4} but in practice often lower. Non-hybrid functionals afford better scaling yet. Still, the computational demand can be prohibitive if one attempts a reliable first-principles theoretical modeling of large systems that are of interest in bio-chemistry, not only because of the scaling as a function of *B* but also because of very large conformational spaces that may be encountered. The application of a dipole coupling model for well separated chromophores, or a more refined model including higher orders of the multipole expansion, offers a way out of this dilemma as far as exciton coupling is concerned.15 For instance, exciton coupling models have been applied successfully in a recent study of the CD spectrum of bacteriorhodopsin.22 Moreover, input data for such a model can be calculated from first principles for each chromophore separately, and the coupled excitations are then obtained from the lower-level coupling model at essentially negligible computational overhead. Herein, we explore this computational route for the long-range exciton CD of TPP pairs as encountered, for instance, in the aforementioned study of derivatized brevetoxin,11 in conjunction with the ‘matrix method’ (MM). One aim of this work is to investigate the performance of such a two-level computational model by comparing the results of a dipole-coupling scheme using TPP monomer TDDFT data as input with full TDDFT dimer calculations.

TDDFT with popular standard hybrid and non-hybrid functionals may afford large errors in calculated excitation energies, if the excitation has an explicit or hidden charge-transfer (CT) character.23–27 For multi-chromophore systems, the CT problem may also create large numbers of spurious low-energy CT excitations.28, 29 The CT problem of TDDFT can be corrected effectively by employing hybrid functionals with range-separated exchange,30–35, 26 in particular if the functional goes to pure HF exchange asymptotically (full long-range correction [LC]). However, the range-separation parameter in the exchange functional is strongly system dependent36 and should therefore be determined system-specifically. Recently, there have been ways proposed of how to achieve a system-specific optimal ‘tuning’ of the range-separation parameter as well as other parameters in the exchange functional in an ab initio sense, based on criteria rooted in density functional theory (DFT).27, 37–43 Another aim of this work is to investigate whether such a functional tuning is beneficial in the description of the TPP excitation spectrum and in the calculation of exciton coupling CD of TPP dimers. It is shown that the use of a system-specific optimally tuned range-separation parameter significantly improves the calculated TPP absorption spectrum compared with a universal parametrization. A LC functional is also shown to be the best choice for the study of long-range exciton coupling in full TDDFT calculations of the coupled system, as it suppresses spurious CT excitations. At large separations of the chromophores, it is shown that a simple dipole coupling model based on TDDFT monomer input data gives excellent agreement with long-range corrected TDDFT dimer spectra.