TDDFT,

Surfaces for Excited States The Role of Conical Intersections for Reactivity 

The simplest approach to the potential energy surface for any reaction is the optimization of the critical points minima, transition structures, and surface crossings. With these elements and the reaction path calculations described in Section 

it is often possible to represent the potential energy surface schematically and to rationalize problems of chemical 

The potential energy snrfaces for excited-state reactions are extensions of the reaction profiles nsually fonnd in the ground state (two different minima connected by a transition structure). In photochemistry, several reaction profiles are connected by a state crossing. In Fignre 2.4 we ontline two reaction profiles to introduce some important concepts in the analysis of photochemical reactivity. We also give an overview of the conclusions that can be drawn from these calculations, together with the more difficult problems that must be addressed with dynamics. 

In this first approach we follow Kasha s rule, that is, we consider that photochemical reactivity starts from the minimum of the lowest excited-state surface and focus on how the products are formed from there. Later we will discuss how the excited-state minimum can be reached from the FC region. There the molecnle may go through a crossing between two excited states or a state switch along an avoided crossing. The concepts we introduce now will also be useful to discuss these issues. 

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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. 

Time-Dependent Density Functional theory (TDDFT) has been considered with increasing interest since the late 1970 s and many papers have been published on the subject. The treatments presented by Runge and Gross (36) and Gross and Kohn (37) are widely cited in the discussion of the evolution of pure states. The evolution of mixed states has been considered extensively by Rajagopal et al. (38), but that treatment differs in many aspects from the form given here. 

In essentially all of the prior formulations of TDDFT a complex Lagrangian is used, which would amount to using the full expectation value in Eq. (2.9), not just the real part as in our presentation. The form we use is natural for conservative systems and, if not invoked explicitly at the outset, emerges in some fashion when considering such systems. A discussion of the different forms of Frenkel s variational principle, although not in the context of DFT, can be found in (39). 

Another place where we diverge from other develq)ments of TDDFT is in the use of the metric term Eq. (2.9). These terms arise in a non-trivial manner as the paths <2 (p) are manifestly nonlinear functionals of p and thus have... 

The mixed state TDDFT of Rajagopal et al. (38) differs from our formulation in the aspects mentioned alx)ve and in the nature of the operator space where the supervectors reside. A particularly notable distinction is in the use of the factorization D = QQ of the state density operator that leads to unconstrained variation over the space of Hilbert-Schmidt operatOTS, rather than to a constrained variaticxi of the space of Trace-Class operators. 

For a review of TDDFT the reader should consult (36) and Gorling (41). In the latter work TDKS is developed and a fairly exhaustive list of TDDFT references is given. 

Note that in all current implementations of TDDFT the so-called adiabatic approximation is employed. Here, the time-dependent exchange-correlation potential that occurs in the corresponding time-dependent Kohn-Sham equations and which is rigorously defined as the functional derivative of the exchange-correlation action Axc[p] with respect to the time-dependent electron-density is approximated as the functional derivative of the standard, time-independent Exc with respect to the charge density at time t, i. e.,... 

Tables 11-6, 11-7, and 11-8 show calculated solvatochromic shifts for the nucle-obases. Solvation effects on uracil have been studied theoretically in the past using both explicit and implicit models [92, 94, 130, 149, 211-214] (see Table 11-6). Initial studies used clusters of uracil with a few water molecules. Marian et al. [130] calculated excited states of uracil and uracil-water clusters with two, four and six water molecules. Shukla and Lesczynski [122] studied uracil with three water molecules using CIS to calculate excitation energies. Improta et al. [213] used a cluster of four water molecules embedded into a PCM and TDDFT calculations to study the solvatochromic shifts on the absorption and emission of uracil and thymine. Zazza et al. [211] used the perturbed matrix method (PMM) in combination with TDDFT and CCSD to calculate the solvatochromic shifts. The shift for the Si state ranges between (+0.21) - (+0.54) eV and the shift for the S2 is calculated to be between (-0.07) - (-0.19) eV. Thymine shows very similar solvatochromic shifts as seen in Table 11-6 [92],...

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. 

Bases stacked rather than hydrogen bonded have also been studied with quantum chemical methods [182, 244-247]. The nature of excited states in these systems has been debated and theoretical calculations are called to decide on the degree of excited state localization or delocalization, as well as the presence and energy of charge transfer states. The experimentally observed hypochromism of DNA compared to its individual bases has been known for decades [248], Accurate quantum chemical calculations are limited in these systems because of their increased size. Many of the reported studies have used TDDFT to calculate excited states of bases stacked with other bases [182, 244, 246, 247], However, one has to be cautious when us-... 

Shukla MK, Leszczynski J (2004) TDDFT investigation on nucleic acid bases comparison with experiments and standard approach. J Comput Chem 25 768-778... 

Sobolewski AL, Domcke W (2002) On the mechanism of nonradiative decay of DNA bases ab initio and TDDFT results for the excited states of 9H-adenine. Eur Phys J D 20 369... 

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