Excited-state singularities

Our assmnption that cq(P) be nonzero at P = 1 is not generally valid. In fact, one can prove [14] that the assumption will hold only for the ground state cq(1) = 0 for any excited state, which then implies that E is regular at P = 1. We can locate the excited-state singularities by determining the value of P at which the matrix element (V) diverges, using Eq. (27) as before. Suppose that co = 0... 

A different analysis applies to the LR approach (in either Tamm-Dancoff, Random Phase Approximation, or Time-dependent DFT version) where the excitation energies are directly determined as singularities of the frequency-dependent linear response functions of the solvated molecule in the ground state, and thus avoiding explicit calculation of the excited state wave function. In this case, the iterative scheme of the SS approaches is no longer necessary, and the whole spectrum of excitation energies can be obtained in a single run as for isolated systems. 

When written with the help of the Tl matrix as in (19), from (20) the OR parameter and other linear response properties are seen to afford singularities where co = coj, just like in the SOS equation (2). Therefore, at and near resonances the solutions of the TDDFT response equations (and response equations derived for other quantum chemical methods) yield diverging results that cannot be compared directly to experimental data. In reality, the excited states are broadened, which may be incorporated in the formalism by introducing dephasing constants 1 such that o, —> ooj — iT j for the excitation frequencies. This would lead to a nonsingular behavior of (20) near the coj where the real and the imaginary part of the response function varies smoothly, as in the broadened scenario at the top of Fig. 1. 

The deduction of the HPHF equations for excited states gives rise to a new challenge, since the excited orbital 6( is usually orthogonal to the occupied one Ofc. As a result, one of the overlap Aj in equation (14) could be zero, (or close to zero) which will give rise to singularities. 

Finally, we note that excited states can have a dimensional singularity structure that appears to be qualitatively different from that found for the ground state. In Chapter 8.1 the 6 expansions of three excited states of helium axe analyzed. The ls2s state appears to have the same structure as the groimd state, while the ls2s 5 and 2p states seem to have a rather different structure. 

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