Understanding the properties of electronically excited states is a challenging task that becomes increasingly important for numerous applications in chemistry, molecular physics, molecular biology, and materials science. A substantial impact is exerted by the fascinating progress in time-resolved spectroscopy, which leads to a strongly growing demand for theoretical methods to describe the characteristic features of excited states accurately. Whereas for electronic ground state problems of stable molecules the quantum chemical methodology is now so well developed that informed nonexperts can use it efficiently, the situation is entirely different concerning the investigation of excited states. This review is devoted to a specific class of approaches, usually denoted as multireference (MR) methods, the generality of which is needed for solving many spectroscopic or photodynamical problems. However, the understanding and proper application of these MR methods is often found to be difficult due to their complexity and their computational cost. The purpose of this review is to provide an overview of the most important facts about the different theoretical approaches available and to present by means of a collection of characteristic examples useful information, which can guide the reader in performing their own applications.

The core part of the program system COLUMBUS allows highly efficient calculations using variational multireference (MR) methods in the framework of configuration interaction with single and double excitations (MR-CISD) and averaged quadratic coupled-cluster calculations (MR-AQCC), based on uncontracted sets of configurations and the graphical unitary group approach (GUGA). The availability of analytic MR-CISD and MR-AQCC energy gradients and analytic nonadiabatic couplings for MR-CISD enables exciting applications including, e.g., investigations of π-conjugated biradicaloid compounds, calculations of multitudes of excited states, development of diabatization procedures, and furnishing the electronic structure information for on-the-fly surface nonadiabatic dynamics. With fully variational uncontracted spin-orbit MRCI, COLUMBUS provides a unique possibility of performing high-level calculations on compounds containing heavy atoms up to lanthanides and actinides. Crucial for carrying out all of these calculations effectively is the availability of an efficient parallel code for the CI step. Configuration spaces of several billion in size now can be treated quite routinely on standard parallel computer clusters. Emerging developments in COLUMBUS, including the all configuration mean energy multiconfiguration self-consistent field method and the graphically contracted function method, promise to allow practically unlimited configuration space dimensions. Spin density based on the GUGA approach, analytic spin-orbit energy gradients, possibilities for local electron correlation MR calculations, development of general interfaces for nonadiabatic dynamics, and MRCI linear vibronic coupling models conclude this overview.

Five paradigmatic polycyclic aromatic hydrocarbons (PAHs) (pyrene, circum-1-pyrene, coronene, circum-1-coronene, and circum-2-coronene) are used for studying the performance of three single-reference methods {scaled opposite-spin-algebraic diagrammatic construction to second-order [SOS-ADC(2)], time-dependent (TD)-B3LYP, and TD-Coulomb-attenuating method (CAM)-B3LYP} and three multireference (MR) methods [density functional theory/multireference configuration interaction (DFT/MRCI), strongly contracted-n-electron valence state perturbation theory to second order (NEVPT2), and spectroscopy oriented configuration interaction (SORCI)]. The performance of these methods was evaluated by comparison of the calculated vertical excitation energies with experiments, where available. DFT/MRCI performs best and thus was used as a benchmark for other approaches where experimental values were not available. Both TD-B3LYP and NEVPT2 agree well with the benchmark data. SORCI performs better for coronene than for pyrene. SOS-ADC(2) does reasonably well in terms of excitation energies for smaller systems, but the error increases somewhat as the size of the system gets bigger. The natural transition orbital analysis for SOS-ADC(2) results indicated that at least two configurations were essential to characterize most of the lower-case electronic states. TD-CAM-B3LYP gives the largest errors for excitation energies and also gives an incorrect order of the lowest two states in circum-1-pyrene. A strong density increase of dark states was observed in the UV spectra with increasing size except for the lowest few states which remained well separated. An extrapolation of the UV spectra to infinite PAH size for S₁, S₂, and the first bright state based on the coronene series was made. The extrapolated excitation energies closest to experimental measurements on graphene quantum dots were obtained by TD-CAM-B3LYP.

A theoretical study at the multireference singles and doubles configuration‐interaction level is reported for the six lowest electronic states of BeH⁺. Of the four new states described, the ¹Π and ³Π are bound states supporting at least 8 and 12 vibrational levels, respectively. Two ³Σ⁺ states are repulsive. For the two known ¹Σ⁺ states improved results are also presented. For all these states, dipole moments and transition moments have been computed. For the bound states, besides the vibrational levels, transition probabilities, radiative lifetimes, vibrationally averaged dipole moments, and a set of spectroscopic constants are also reported.

The study of electronically excited states of stacked polycyclic aromatic hydrocarbons (PAHs) is of great interest due to promising applications of these compounds as luminescent carbon nanomaterials such as graphene quantum dots (GQDs) and carbon dots (CDs). In this study, the excited states and excitonic interactions are described in detail based on four CD model dimer systems of pyrene, coronene, circum-1-pyrene and circum-1-coronene. Two multi-reference methods, DFT/MRCI and SC-NEVPT2, and two single-reference methods, ADC(2) and CAM-B3LYP, have been used for excited state calculations. The DFT/MRCI method has been used as a benchmark method to evaluate the performance of the other ones. All methods produce useful lists of excited states. However, an overestimation of excitation energies and an inverted ordering of states, especially concerning the bright HOMO–LUMO excitation, are observed. In the pyrene-based systems, the first bright state appears among the first four states, whereas the number of dark states is significantly larger for the coronene-based systems. Fluorescence emission properties are addressed by means of geometry optimization in the S₁ state. The inter sheet distances for the S₁ state decrease in comparison to the corresponding ground-state values. These reductions are largest for the pyrene dimer and decrease significantly for the larger dimers. Several minima have been determined on the S₁ energy surface for most of the dimers. The largest variability in emission energies is found for the pyrene dimer, whereas in the other cases a more regular behavior of the emission spectra is observed.

The development of efficient organic-based photovoltaic devices is a vibrant area of research with the potential of providing a cheap source of sustainable energy to society. The attainable power conversion efficiencies could be strongly enhanced via the singlet fission (SF) mechanism, a quantum mechanical phenomenon that potentially doubles the number of electron–hole pairs in a photoexcitation process by splitting a high energy singlet into two triplets. Biradicaloid molecules are particularly appealing for SF applications due to the possibility of controlling the balance between open-shell and closed-shell resonance structures via chemical modifications, which open new opportunities to fine tune the singlet and triplet excitation energies, and thus maximize the SF efficiency. Recently, we have shown that doping acenes with boron (B) or nitrogen (N) atoms leads to a large modulation in its biradicaloid nature at the ground-state. Herein, this previous study is extended to the case of asymmetric substitutions by introducing a BN-pair in a tetracene molecule to form azaborine analogues of acenes. The consequences of the chemical doping on the excitonic properties of tetracene are investigated through high-level multireference calculations. From a pool of 60 proposed BN-tetracene chromophores, we identify 15 new promising candidates for SF as they satisfy the energy level matching conditions involving the low-lying singlet and triplet states of a monomer. Still, some of these compounds show good chemical stability as evidenced by their modest biradical character. These results are interpreted in terms of aromaticity changes, charge transfer effects and exciton properties. More generally, this study shows how the energetics of singlet fission materials can be dramatically altered by using fairly simple chemical substitutions and provides detailed insight into the underlying relationships between the molecular structure, the electronic structure, and the excited state energies.