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Luminescence lifetimes of Ruthenium (II) polypyridine compounds is thought to be controlled by the barrier to conversion of triplet metal-ligand charge transfer ($^{3}$MLCT) state to a non-luminescent triplet metal-centered ($^{3}$MC) state. This work builds on earlier work [J. Photochem. Photobiol. A 276, 8 (2014)] and [J. Photochem. Photobiol. A 348, 305 (2017)] that derived several orbital-based luminescence indices of which the third (LI3) was based upon frontier-molecular-orbital-like ideas and correlated linearly with values of $E_{ave}$. $E_{ave}$ is a large underestimate of the true $^3$MLCT$\rightarrow$ $^3$MC TS barrier height in the case of the tris bipyridine ruthenium(II) cation {[Ru(bpy)$_3]^{2+}$}, but accurate TS barrier heights are difficult to obtain experimentally, it was judged useful to verify the ideas used to derive the LI3 index by calculating the energetics of the gas-phase $^{3}$MLCT $\rightarrow$ $^{3}$MC reaction for four complexes $\{$[Ru(N$^\wedge$N)$_3$]$^{2+}$ with N$^\wedge$N = bpy ({6}), 4,4'-dm-bpy ({70}), 4,4'-dph-bpy ({73}), and 4,4'-DTB-bpy ({74}) $\}$ using the same density functional and basis sets used in calculating LI3. We examine the trans dissociation mechanism in detail at the B3LYP/6-31G+LANLDZ(Ru) level and uncover a two-part mechanism. In the first part, the electron is transferred to a single ligand rather than symmetrically to all three ligands. It is the two Ru-N bonds to this ligand which are equally elongated in the transition state. The intrinsic reaction coordinate then continues down a ridge in hyperspace and bifurcates into one of two symmetry-equivalent $^3$MC structures with elongated trans bonds. Interestingly, no significant difference is found for the TS barriers for the four complexes treated here. Instead, LI3 is linearly correlated with the energy difference $\Delta$ E = E($^{3}$MLCT) - E($^{3}$MC).

Static correlation is a difficult problem for density-functional theory (DFT) as it arises in cases of degenerate or quasi-degenerate states where a multideterminantal wave function provides the simplest reasonable first approximation to the true interacting wave function. This is also where Kohn-Sham DFT may also fail to be noninteracting v-representible (NVR). In contrast, Kohn-Sham DFT typically works well for describing the missing dynamic correlation when a single-determinantal reference wave function provides a good first approximation to the true interacting wave function. Multiplet sum method (MSM) DFT [Theor. Chim. Acta 4, 877 (1977)] provides one of the earliest and simplest ways to include static correlation in DFT. MSM-DFT assumes that DFT provides a good description of single-determant energies and uses symmetry and simple ansatzes to include the effects of static correlation. This is equivalent to determining the off-diagonal matrix elements in a small configuration interaction (CI) eigenvalue problem. We have developed a diagrammatic approach to MS-DFT facilitates comparison with wave function CI and so allows educated guesses of off-diagonal CI matrix elements even in the absence of symmetry. In every case, an additional exchange-only ansatz (EXAN) allows the MSM-DFT formulae to be transformed into wave function formulae. This EXAN also works for transforming time-dependent DFT into time-dependent Hartree-Fock. Although not enough to uniquely guess DFT formulae from wave function formulae, the diagrammatic approach and the EXAN provide important constraints on any guesses that might be used. Some alternative guesses are tried out for problems concerning the ground and excited states of H2 , LiH, and O2 in order to assess how much difference might be involved for different DFT guesses for off-diagonal matrix elements.

One of us (MEC) developed a hands-on workbook for density-functional theory (DFT) during the summer of 2020. The idea was to have something that could be used to provide practical teaching for students at the Masters or advanced undergraduate level that would be free, could be used on a student's own personal computer, and would complement formal course work. The workbook is also very much intended to encourage students to explore program options, discover theory limitations, puzzle out what to do when the program does not work as expected, and to help students transition to thinking and using quantum chemistry programs as a researcher might do. After describing the structure of the workbook, we describe how the workbook has been used thus far as a teaching tool and as a useful step towards research-level problems., Comment: 15 pages, 3 Figures, Conference: International Congress on Molecular Electronic Structure (MES 2022), Monastir, Tunisia, 2-6 September 2022. Pure Appl. Chem.

Singlet oxygen (O2) comes in two flavors -- namely the dominant lower-energy a 1 Delta g state and the higher-energy shorter-lived b 1 Sigma + g state -- and plays a key role in many photochemical and photobiological reactions. For this reason, and because of the large size of the systems treated, many papers have appeared with density-functional theory (DFT) treatments of the reactions of 1 O 2 with different chemical species. The present work serves as a reminder that the common assumption that it is enough to fix the spin multipicity as unity is not enough to insure a correct treatment of singlet oxygen. We review the correct group theoretical treatment of the three lowest energy electronic states of O 2 which, in the case of 1 O 2 is often so badly explained in the relevant photochemical literature that the explanation borders on being incorrect and prevents, rather than encourages, a correct treatment of this interesting and important photochemical species. We then show how many electronic structure programs, such as a freely downloadable and personal-computer compatible Linux version of deMon2k, may be used, together with the multiplet sum method (MSM), to obtain a more accurate estimation of the potential energy curves (PECs) of the two 1 O 2 states. Various strengths and weaknesses of different DFAs emerge through our application of the MSM method. In particular, the quality of the a 1 Delta g excitation energy reflects how well functionals are able to describe the spin-flip energy in DFT while the quality of the b 1 Sigma + g excitation energy reflects how well functionals are able to describe the spin-pairing energy in DFT. Finally we note that improvements in DFT-based excited-state methods will be needed to describe the full PECs of 1 O 2 including both the equilibrium bond lengths and dissociation behavior.

Density-functional tight binding (DFTB) has become a popular form of approximate density-functional theory (DFT) based upon a minimal valence basis set and neglect of all but two center integrals. We report the results of our tests of a recent long-range correction (lc) for time-dependent (TD) lc-DFTB by carrying out TD-lc-DFTB fewest switches surface hopping (FSSH) calculations of energy and charge transfer times using the relatively new DFTBaby program. An advantage of this method is the ability to run enough trajectories to get meaningful ensemble averages. Our interest in the present work is less in determining exact energy and charge transfer rates than in understanding how the results of these calculations vary with the value of the range-separation parameter (Rlc = 1/{\mu}) for a model organic solar cell heterojunction consisting of a van der Waals complex P/F made up of single pentacene (P) molecule together with a single buckminsterfullerene (F) molecule. The default value of Rlc = 3.03 a0 is found to be much too small as neither energy nor charge transfer is observed until Rlc ~ 10 a0. Tests at a single geometry show that best agreement with high-quality ab-initio spectra is obtained in the limit of no lc (i.e., very large Rlc.) A plot of energy and charge transfer rates as a function of Rlc is provided which suggests that a value of Rlc ~ 15 a0 yields the typical literature charge transfer time of about 100 fs. However, energy and charge transfer times become as high as ~ 300 fs for Rlc ~ 25 a0. A closer examination of the charge transfer process P*/F to P+/F- shows that the initial electron transfer is accompanied by a partial delocalization of the P hole onto F which then relocalizes back onto P, consistent with a polaron-like picture in which the nuclei relax to stabilize the resultant redistribution of charges.

Exciton formation leads to J-bands in solid pentacene. Describing these exciton bands represents a challenge for both time-dependent (TD) density-functional theory (DFT) and for its semiempirical analogue, namely for TD density-functional tight binding (DFTB) for three reasons (i) solid pentacene and pentacene aggregates are bound only by van der Waals forces which are notoriously difficult to describe with DFT and DFTB, (ii) the proper description of the long-range coupling between molecules, needed to describe Davydov splitting, is not easy to include in TD-DFT with traditional functionals and in TD-DFTB, and (iii) mixing may occur between local and charge transfer excitons, which may, in turn, require special functionals. We assess how far TD-DFT and TD-DFTB have progressed towards a correct description of this type of exciton by including both a dispersion correction for the ground state and a range-separated hybrid functional for the excited state. Analytic results for parallel-stacked ethylene are derived which go beyond Kasha's exciton model in that we are able to make a clear distinction between charge transfer and energy transfer excitons. This is further confirmed when it is shown that range-separated hybrids have a markedly greater effect on charge-transfer excitons than on energy-transfer excitons in the case of parallel-stacked pentacenes. TD-DFT calculations with the CAM-B3LYP functional and TD-lc-DFT calculations lead to negligeable excitonic corrections for the herringbone crystal structure, possibly because of an overcorrection of charge-transfer effects. In this case, TD-DFT calculations with the B3LYP functional or TD-DFTB calculations parameterized to B3LYP give the best results for excitonic corrections for the herringbone crystal structure as judged from comparison with experimental spectra and with Bethe-Salpeter equation calculations from the literature.

Novel low-band-gap copolymer oligomers are proposed on the basis of density functional theory (DFT) quantum chemical calculations of photophysical properties. These molecules have an electron donor-accepter (D-A) architecture involving poly(3-hexylthiophene-2,5-diyl) (P3HT) as D units and furan, aniline, or hydroquinone as A units. Structural parameters, electronic properties, highest occupied molecular orbital (HOMO)-lowest unoccupied molecular orbital (LUMO) gaps and molecular orbital densities are predicted. The charge transfer process between the D unit and the A unit one is supported by analyzing the optical absorption spectra of the compounds and the localization of the HOMO and LUMO.

Gas phase density-functional theory (DFT) and time-dependent DFT (TD-DFT) calculations are reported for a data base of 98 ruthenium(II) polypyridine complexes. Comparison with X-ray crystal geometries and with experimental absorption spectra measured in solution show an excellent linear correlation with the results of the gas phase calculations. Comparing this with the usual chemical understanding based upon ligand field theory (LFT) is complicated by the large number of molecular orbitals present and especially by the heavy mixing of the antibonding metal e*$_{g}$ orbitals with ligand orbitals. Nevertheless, we show that a deeper understanding can be obtained by a partial density-of-states (PDOS) analysis which allows us to extract approximate metal t$_{2g}$ and e*$_{g}$ and ligand \pi* orbital energies in a well-defined way, thus providing a PDOS analogue of LFT (PDOS-LFT). Not only do PDOS-LFT energies generate a spectrochemical series for the ligands, but orbital energy differences provide good estimates of TD-DFT absorption energies. Encouraged by this success, we use frontier-molecular-orbital-theory-like reasoning to construct a model which allows us in most, but not all, of the cases studied to use PDOS-LFT energies to provide a semiquantitative relationship between luminescence lifetimes at room temperature and liquid nitrogen temperature.

We give a fairly comprehensive review of wavelets and of their application to density-functional theory (DFT) and to our recent application of a wavelet-based version of linear-response time-dependent DFT (LR-TD-DFT). Our intended audience is quantum chemists and theoretical solid-state and chemical physicists. Wavelets are a Fourier-transform-like approach which developed primarily in the latter half of the last century and which was rapidly adapted by engineers in the 1990s because of its advantages compared to standard Fourier transform techniques for multiresolution problems with complicated boundary conditions. High performance computing wavelet codes now also exist for DFT applications in quantum chemistry and solid-state physics, notably the BigDFT code described in this chapter. After briefly describing the basic equations of DFT and LR-TD-DFT, we discuss how they are solved in BigDFT and present new results on the small test molecule carbon monoxide to show how BigDFT results compare against those obtained with the quantum chemistry gaussian-type orbital (GTO) based code deMon2k. In general, the two programs give essentially the same orbital energies, but the wavelet basis of BigDFT converges to the basis set limit much more rapidly than does the GTO basis set of deMon2k. Wavelet-based LR-TD-DFT is still in its infancy, but our calculations confirm the feasibility of implementing LR-TD-DFT in a wavelet-based code., Comment: 45 pages, 10 figures, book chapter in Theoretical and Computational Methods in Modern Density Functional Theory, Editor: A.K. Roy

Linear-response time-dependent (TD) density-functional theory (DFT) has been implemented in the pseudopotential wavelet-based electronic structure program BigDFT and results are compared against those obtained with the all-electron Gaussian-type orbital program deMon2k for the calculation of electronic absorption spectra of N2 using the TD local density approximation (LDA). The two programs give comparable excitation energies and absorption spectra once suitably extensive basis sets are used. Convergence of LDA density orbitals and orbital energies to the basis-set limit is significantly faster for BigDFT than for deMon2k. However the number of virtual orbitals used in TD-DFT calculations is a parameter in BigDFT, while all virtual orbitals are included in TD-DFT calculations in deMon2k. As a reality check, we report the x-ray crystal structure and the measured and calculated absorption spectrum (excitation energies and oscillator strengths) of the small organic molecule N-cyclohexyl-2-(4-methoxyphenyl)imidazo[1,2-a]pyridin-3-amine.

Almost all time-dependent density-functional theory (TDDFT) calculations of excited states make use of the adiabatic approximation, which implies a frequency-independent exchange-correlation kernel that limits applications to one-hole/one-particle states. To remedy this problem, Maitra et al.[J.Chem.Phys. 120, 5932 (2004)] proposed dressed TDDFT (D-TDDFT), which includes explicit two-hole/two-particle states by adding a frequency-dependent term to adiabatic TDDFT. This paper offers the first extensive test of D-TDDFT, and its ability to represent excitation energies in a general fashion. We present D-TDDFT excited states for 28 chromophores and compare them with the benchmark results of Schreiber et al.[J.Chem.Phys. 128, 134110 (2008).] We find the choice of functional used for the A-TDDFT step to be critical for positioning the 1h1p states with respect to the 2h2p states. We observe that D-TDDFT without HF exchange increases the error in excitations already underestimated by A-TDDFT. This problem is largely remedied by implementation of D- TDDFT including Hartree-Fock exchange.

In their famous paper Kohn and Sham formulated a formally exact density-functional theory (DFT) for the ground-state energy and density of a system of $N$ interacting electrons, albeit limited at the time by certain troubling representability questions. As no practical exact form of the exchange-correlation (xc) energy functional was known, the xc-functional had to be approximated, ideally by a local or semilocal functional. Nowadays however the realization that Nature is not always so nearsighted has driven us up Perdew's Jacob's ladder to find increasingly nonlocal density/wavefunction hybrid functionals. Time-dependent (TD-) DFT is a younger development which allows DFT concepts to be used to describe the temporal evolution of the density in the presence of a perturbing field. Linear response (LR) theory then allows spectra and other information about excited states to be extracted from TD-DFT. Once again the exact TD-DFT xc-functional must be approximated in practical calculations and this has historically been done using the TD-DFT adiabatic approximation (AA) which is to TD-DFT very much like what the local density approximation (LDA) is to conventional ground-state DFT. While some of the recent advances in TD-DFT focus on what can be done within the AA, others explore ways around the AA. After giving an overview of DFT, TD-DFT, and LR-TD-DFT, this article will focus on many-body corrections to LR-TD-DFT as one way to building hybrid density-functional/wavefunction methodology for incorporating aspects of nonlocality in time not present in the AA., Comment: 56 pages, 17 figures

The development of analytic-gradient methodology for excited states within conventional time-dependent density-functional theory (TDDFT) would seem to offer a relatively inexpensive alternative to better established quantum-chemical approaches for the modeling of photochemical reactions. However, even though TDDFT is formally exact, practical calculations involve the use of approximate functionals, in particular the TDDFT adiabatic approximation, whose use in photochemical applications must be further validated. Here, we investigate the prototypical case of the symmetric CC ring opening of oxirane. We demonstrate by direct comparison with the results of high-quality quantum Monte Carlo calculations that, far from being an approximation on TDDFT, the Tamm-Dancoff approximation (TDA) is a practical necessity for avoiding triplet instabilities and singlet near instabilities, thus helping maintain energetically reasonable excited-state potential energy surfaces during bond breaking. Other difficulties one would encounter in modeling oxirane photodynamics are pointed out but none of these is likely to prevent a qualitatively correct TDDFT/TDA description of photochemistry in this prototypical molecule., Comment: 19 pages, 17 figures, submitted to the Journal of Chemical Physics