Hartree-Fock theory plays a central role in Quantum Chemistry. It serves on the one hand as a basis for electron correlation approaches while on the other it is also essential for density functional theory. The evaluation of the HF exchange term is a rather time consuming operation for larger molecules and as a result for a wide range of chemically or biologically interesting systems. This motivates the development of various eﬃcient approaches to exchange evaluation. The Chain of Spheres exchange (COSX) algorithm, one of the most eﬃcient methods available for this purpose today, has been developed in the institute since 2009. The algorithm has favorable scaling and contraction properties which makes it especially suitable for larger systems. In combination with the resolution of identity for the Coulomb term (RI-J) it enables large scale HF and DFT calculations. To realize the full potential of any approximation to the SCF energy, it is inevitable to obtain the corresponding gradients and the Hessian. These can then be used in geometry optimizations and to calculate various properties. Gradients are determined in an approximate manner for the COSX algorithm, while for Hessian calculation its main advantage is in the CP-SCF step. The COSX method can also be used beneﬁcially in correlation methods. In these cases the SCF density is replaced by an eﬀective density in which the amplitudes of various excited determinants are contracted with molecular orbital coeﬃcients. Thus, the evaluation of the so called singles Fock term in the LPNO-CCSD method, and the external exchange term in the SCS-MP3, CCSD and EOM-CCSD methods can both be signiﬁcantly accelerated without signiﬁcant loss of accuracy. In the former case the contraction involves the singles, in the latter case the doubles amplitudes, but in both cases the approximated terms are form the bottleneck of the corresponding method.
In 2014, our group began to investigate various ways of implementing the equation of motion (EOM) coupled cluster (CC) theory for large molecules. Our ﬁrst eﬀort involved using COSX for the evaluation of the external exchange term, which lowered the costs of the calculation but did not change the scaling of the method which remained proportional to the sixth power of some measure of the system size. Similarly, domain-based local pair natural orbitals (DLPNO) can be employed to accelerate the ground state CC step, but since the amplitudes are back-transformed into the canonical basis before the EOM-CCSD code makes use of them, the overall scaling of the method remains unaﬀected. While the excited states would require state speciﬁc DLPNOs, it turns out that the ground state DLPNOs can be used to implement eﬃcient methods for the evaluation of ionization potentials and electron aﬃnities. These on the other hand are necessary ingredients for the so-called similarity transformed EOM or STEOM method, which reduces the the excitation manifold to the space of single excitations without neglecting terms based on a perturbative basis, as second order methods such as CC2 or ADC(2) do. Thus, DLPNOs for the ionization process and the STEOM method for the reduction of the excitation manifold in the canonical basis can be combined to yield the DLPNO-STEOM method, which can be used to obtain excitation energies and some of the spectroscopic properties of molecular systems using several thousand basis functions. In addition, the eﬀect of nuclear vibrations can be included into our description of molecular spectra. Our group has been involved in the eﬃcient implementation of a simple propagator approach to describe various rate constants that appear in the Jablonski diagram shown in Fig. 2. While the ﬂuorescence rate (kF) is determined by the transition dipole moment, which between singlets is only zero if spacial symmetry demands it, transitions between singlet and triplet states are spin forbidden in non-relativistic quantum mechanics. In this picture, the radiative process only competes with non-radiative internal conversion (kIC) in ﬂuorescent materials. If relativistic eﬀects are important, singlet-triplet transitions may become allowed depending on the strength of the SOC connecting the two states. Phosphoresce rates (kP) as well as intersystem crossing rates are proportional to SOC, and the challenge of designing strong emitters lies in ensuring that populating the triplet states and their radiative decay are fast (kISC and kP are large), while the non-radiative decay of the triples is slow (k0ISC is small). Thus, the combination of this propagator scheme with the DLPNO-STEOM method for the computation of vertical excitation energies and transition dipoles is an approach we plan to pursue in the future in what we hope to be a fruitful combination of method development and practical applications.