Unrestricted TDDFT

Often, TDDFT studies are targeted at closed-shell systems. However, many metal complexes are paramagnetic, and the open d- or /-shells may pose additional challenges for TDDFT spectral computations. Fan et al. [338] have studied the CD of high-spin trigonal dihedral chromium complexes, and developed a spin-unrestricted TDDFT method for the computations of the CD spectra. When possible, an analogous closed-shell cobalt(III) complex was calculated as well for comparison. 

For that purpose, electronic spectra are first classified according to the character of the states involved. A very basic distinction relies on the electronic structure of the corresponding initial state from which the transition occurs. This initial state is not necessarily the ground state of the system, shown schematically in Figure 4, which includes the three most common possibilities. For a more detailed discussion of this point in the context of restricted and unrestricted TDDFT approaches, see Ref. 22. 

The naphthalene molecule, which is used as an example here, has a lowest triplet state of Bxu symmetry and n n (La) character. The dipole allowed transitions that are polarized in the plane of the molecule thus belong to transitions to Ag and B g excited states. The results of the unrestricted TDDFT-B3LYP treatment are shown in comparison with the experimental spectrum in Figure 18. 

Re(dmpe)32+. Of interest is that both complexes make available extremely oxidizing excited states using a visible photon, since their absorption and emission energies are 528 nm (600 nm) for Re, and 590 nm (681 nm) for Tc, respectively. Unrestricted open shell TDDFT calculations confirm the nature of the transitions as LMCT and predict the proper relative energies of Re versus Tc. 

Most implementations of TDDFT to an open-shell system use an spin-unrestricted approach, because orbital-energy differences concerned with partially occupied orbitals are generally too small in a spin-restricted approach, and the orbital-energy difference in DFT is the leading term in the electron-excitation energy. 

Figure 4.1 Important features of ground- and excited-state PESs for ethylene photodynamics and demonstration of the inadequacy of TDDFT and CIS methods for this problem, (a) Sq and PESs for ethylene in the pyramidalization and torsion coordinates (defined in the inset) that dominate the photodynamics. This surface was calculated using multireference perturbation theory — CAS(2/2) PT2. The global minimum on Si occurs at twisted and pyramidalized geometries. (b-d) A quantitative comparison of the Si PES obtained with CAS(2/2) PT2, TDDFT/B3LYP, and CIS, respectively. All calculations use the 6-3IG basis set. The TDDFT and CIS calculations are performed in a spin-unrestricted formalism. Contour values are given in eV, and in all cases the energies are referenced to the Sq equilibrium geometry at the corresponding level of theory. Only the multireference calculation captures the Si minimum correctly.

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