TDDFT method

A number of types of calculations can be performed. These include optimization of geometry, transition structure optimization, frequency calculation, and IRC calculation. It is also possible to compute electronic excited states using the TDDFT method. Solvation effects can be included using the COSMO method. Electric fields and point charges may be included in the calculation. Relativistic density functional calculations can be run using the ZORA method or the Pauli Hamiltonian. The program authors recommend using the ZORA method. 

A number of molecular properties can be computed. These include ESR and NMR simulations. Hyperpolarizabilities and Raman intensities are computed using the TDDFT method. The population analysis algorithm breaks down the wave function by molecular fragments. IR intensities can be computed along with frequency calculations. 

Solvatochromic shifts for cytosine have also been calculated with a variety of methods (see Table 11-7). Shukla and Lesczynski [215] studied clusters of cytosine and three water molecules with CIS and TDDFT methods to obtain solvatochromic shifts. More sophisticated calculations have appeared recently. Valiev and Kowalski used a coupled cluster and classical molecular dynamics approach to calculate the solvatochromic shifts of the excited states of cytosine in the native DNA environment. Blancafort and coworkers [216] used a CASPT2 approach combined with the conductor version of the polarizable continuous (CPCM) model. All of these methods predict that the first three excited states are blue-shifted. S i, which is a nn state, is blue-shifted by 0.1-0.2 eV in water and 0.25 eV in native DNA. S2 and S3 are both rnt states and, as expected, the shift is bigger, 0.4-0.6eV for S2 and 0.3-0.8 eV for S3. S2 is predicted to be blue-shifted by 0.54 eV in native DNA. 

This brief analysis allows to conclude that the fact that the superiority of the TDDFT method. .. has not been unequivocally established. .. in particular for d —> d transitions [116] is not an unfortunate accident but a logical consequence of deeply rooted deficiencies inherent to the underlying single-determinant nature of the TDDFT method and the announced proof of superiority will hardly whenever take place. 

In recent years, the first applications of DFT to excited electronic states of molecules have been reported. In the so-called time-dependent DFT (TDDFT) method, the excitation energies are obtained as the poles of the frequency-dependent polarizability tensor [29], Several applications of TDDFT with standard exchange correlation functionals have shown that this method can provide a qualitatively correct description of the electronic excitation spectrum, although errors of the order of 0.5 eV have to be expected for the vertical excitation energies. TDDFT generally fails for electronic states with pronounced charge transfer character. 

Several types of Werner complexes have been investigated over the last few years by TDDFT methods. They include metal oxide, metal halide, metal oxyhalide compounds, and transition metal complexes with bidentate ligands such as ethylenediamine and acetylacetonato. 

Table 9 Excitation energies (eV) of Mn04 computed by TDDFT methods compared to ASCF-DFT, SAC-CI, EOM-CCSD, and experimental values... 

For the description of the linear and nonlinear optical properties of metallotetrapyrroles, TDDFT methods have proven [133-148] to be an excellent alternative to conventional highly correlated ab initio methods, such as SAC-CI, STEOM-CC, and CASPT2, for which these systems still represent a severe computational challenge, especially when transition metals, lanthanides or actinides are involved. The few highly correlated ab initio calculations dealing with the excited state properties of metallotetrapyrroles that have appeared to date only concern magnesium and zinc porphyrins and porphyrazines [149-151]. Application of TDDFT methods to the electronic spectroscopy of a variety of metallotetrapyrroles, including homoleptic and heteroleptic sandwiches, will be illustrated in this section. 

Table 15 Excitation energies (eV) and oscillator strengths (in parentheses) of the lowest allowed excited states of MgP, ZnP, and NiP (calculated by TDDFT methods), compared to available CASPT2 and MRMP values, and to experimental data...

In view of the potential application of TDDFT methods to the spectroscopy of bioinorganic systems, the dependence of the excitation energies on the exchange-correlation functional and the basis set was carefully tested. In Table 16, the TDDFT results are compared to the known experimental data for the Q and B bands of CO-ligated mioglobin (Mb) [163], the SAC-CI calcula-... 

TDDFT methods have also been applied successfully to the description of the linear and nonlinear optical properties of heteroleptic sandwich complexes. The optical spectrum and the hyperpolarizability of Zr(OEP)(OEPz,) for which large first hyperpolarizabilities, /JSHG (SHG=second-harmonic generation) were measured in an electric field induced second-harmonic generation (EFISH) experiment [182], have been investigated by TDDFT methods [134]. The excitation energies and oscillator strengths calculated... 

The results discussed in this review indicate that the agreement of TDDFT with experiment is not as good for TM complexes as has been observed for small benchmark molecules such as N2, CO, CH20, and furan, where an accuracy of 0.1 - 0.2 eV has often been obtained. The best TDDFT methods for... 

Chirality is an important topic in chemistry and biochemistry, due to the natural occurrence of chiral molecules in living organisms. In circular dichroism (CD) one measures the differential absorption of left- and right-handed circularly polarized light, which for chiral species are different. Therefore, CD has turned out to be a powerful tool which provides information on the electronic and geometric structure of chiral molecules. Since most CD spectra are measured in solution we extended our DRF/TDDFT method to also calculate such properties. As a first example we studied... 

Regarding TDDFT benchmark studies of chiroptical properties prior to 2005, the reader is referred to some of the initial reports of TDDFT implementations and early benchmark studies for OR [15,42,47,53,98-100], ECD [92,101-103], ROA [81-84], and (where applicable) older work mainly employing Hartree-Fock theory [52,55, 85,104-111], Often, implementations of a new quantum chemistry method are verified by comparing computations to experimental data for relatively small molecules, and papers reporting new implementations typically also feature comparisons between different functionals and basis sets. The papers on TDDFT methods for chiroptical properties cited above are no exception in this regard. In the following, we discuss some of the more recent benchmark studies. One of the central themes will be the performance of TDDFT computations when compared to wavefunction based correlated ab initio methods. Various acronyms will be used throughout this section and the remainder of this chapter. Some of the most frequently used acronyms are collected in Table 1. 

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. 

In their recent papers, Tretiak et al. proposed the technique for calculations of TPA properties which is to some extent the combination of the methods described above [45, 74, 108, 109]. The method proposed by Tretiak takes an advantage of the quantities that can be calculated within the linear response theory framework. Tire remaining quantities that appear in the expressions for the two-photon absorption cross section can be evaluated as the functional derivatives based on the time-dependent density functional (TDDFT) method. Although the response theory is involved in their evaluation, it is important to note that the TPA cross section is calculated via SOS formulae. 

The master equations of the TDDFT method for open-shells follow (see [13] for a derivation) ... 

Figure 4.2 Potential-energy curves for the Sq and Si electronic states of 90° twisted ethylene as a function of CH2 monopyramidal-ization. TDDFT and ab initio [SA-2-CAS(2/2)] results are compared. The 6-3IG basis set was used with both methods. The zero of energy is chosen as in Figure 4.1. All coordinates not involved in pyrami-dalization were fixed at the values obtained by optimizing the geometry on Si at the respective level of theory subject to the constraint that it be twisted by 90° with no pyramidalization. The Sq/Si energy gap is unaffected by pyramidalization in the TDDFT method, in contrast to CAS(2/2), where pyramidalization is a dominant coordinate in tuning the energy gap to reach a conical intersection.

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