Electronic Excitation Energies and Transition Moments

Using the spectral representation of the polarization propagator, Eq. (3.110), we can alternatively write 

In the previous sections it was shown that frequency-dependent linear response prop -erties, such as frequency-dependent polarizabilities, can be obtained as the value of the polarization propagator for the appropriate operators. Furthermore, all static second-order properties discussed in Chapters 4 and 5 can be calculated as the value of a polarization propagator for zero frequency. 

In addition to these properties, which are all related to the value of a particular polarization propagator, we can get further information about a molecule by studying the poles and residues of the linear response function or polarization propagator. We can see from Eq. (3.110), that the polarization propagator has a singularity or pole, if the frequency oj of the perturbation takes one of the following values 

Finding the poles of the polarization propagator is thus a way of directly calculating the vertical electronic excitation energies of a system. 

Furthermore, the residuum corresponding to a pole, fuvno = — E defined 

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The time-dependent Hartree-Fock (TDHF) method has a lengthy history as a technique for calculating the electronic excitation energies and transition moments of molecular systems. There are two ways to formulate TDHF theory that look quite different but are in fact equivalent. The formulation used below... 

The popularity of the SOS methods in calculations of non-linear optical properties of molecules is due to the so-called few-states approximations. The sum-over-states formalism defines the response of a system in terms of the spectroscopic parameters, like excitations energies and transition moments between various excited states. Depending on the level of approximation, those states may be electronic or vibronic or electronic-vibrational-rotational ones. Under the assumption that there are few states which contribute more than others, the summation over the whole spectrum of the Hamiltonian can be reduced to those states. In a very special case, one may include only one excited state which is assumed to dominate the molecular response through the given order in perturbation expansion. The first applications of two-level model to calculations of j3 date from late 1970s [93, 94]. The two-states model for first-order hyperpolarizability with only one excited state included can be written as ... 

The calculation of excitation energies and transition moments for unsaturated organic molecules has been one of the more successful applications of multiconhgurational quantum chemistry since the hrst application to the benzene molecule in 1992 [34]. Many hundred molecules have been studied. The CASSCF method allows optimization of excited state energy surfaces and this has been used to compute vibrationally resolved electronic spectra [35,36]. The method is used by several research groups for studies of photochemical reactions, including the localization of conical intersections, etc. 

The central problem of the sum-over-states perturbation theory lies in the evaluation of the energy resolvant which includes the inverse of excitation energies and transition moments of the one-electron operators... 

A typical example for the application of the state-averaging procedure is a calculation of the potential energy and transition moment surfaces for the X and A electronic states of NHj °. These states have a conical intersection which is shown in Fig. 5. The optimization of the excited-state wavefunctions in the vicinity of the crossing point would not have been possible in a singlestate treatment. 

Quadratic response theory in combination with self-consistent field (SCF), MCSCF, and coupled-cluster electronic structure methods have been used for calculation of excitation energies and transition dipole moments between excited electronic states <2000PCP5357>. The excited state polarizabilities for r-tetrazine are given by the double residues of the cubic response functions <1997CPFl(224)201>. 

Sum Over States (SOS). This method computes molecular orbitals, from which values for transition fi equencies may be calculated. First the electronic ground state of the molecular system is determined, after which one may apply either the Hartree-Fock-Roothan method or the LCAO (hnear combination of atomic orbitals) approximation. Then one accoimts for correlations by configuration interaction calculations to form the lowest-energy excited states and transition dipole moments of the molecule. Finally transition frequencies and dipole moments are employed along with the formulas for the hyperpolarizabilities. The SOS method needs as input, energies and transition moments for excited states. It yields Pico) directly (eq. 1) identification of contributing excited states is important. 

Abstract This chapter considers the properties of the molecular solute in electronic excited states determined from the linear response functions descriljed in the previous Chap. 3. Transition energies and transition moments are determined from a generalized eigenvalue equations, and the first-order properties in electronic excited states are expressed as analytical gradients of the corresponding tfansition energies with respect to suitable perturbations. 

Our present focus is on correlated electronic structure methods for describing molecular systems interacting with a structured environment where the electronic wavefunction for the molecule is given by a multiconfigurational self-consistent field wavefunction. Using the MCSCF structured environment response method it is possible to determine molecular properties such as (i) frequency-dependent polarizabilities, (ii) excitation and deexcitation energies, (iii) transition moments, (iv) two-photon matrix elements, (v) frequency-dependent first hyperpolarizability tensors, (vi) frequency-dependent polarizabilities of excited states, (vii) frequency-dependent second hyperpolarizabilities (y), (viii) three-photon absorptions, and (ix) two-photon absorption between excited states. 

The electrostatic effects associated with the relative orientation of the chromophores show a specific connection with the corresponding electronic excited states and dipole transition moments that can, in turn, be related to changes in the hyperpolarizabilities via summation-over-states (SOS) expressions [1 3]. When the molecules interact, the electronic excited states split. In the case of a collinear arrangement, the intensities (oscillator strengths) are shifted to the red, that is, to the states of lower energies, whereas for a side by side and parallel configuration, the intensities are shifted to the blue. Furthermore, since the excited state polarizabilities are usually larger than the... 

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