Refined transition densities

If the dipole operator in equation (4.12) is replaced by the density operator n, oneˆ gets, after similar considerations, an expression for the imaginary part of the Fourier transformed density fluctuation fδn=F[n−n^GS](forω >0) At a certain transition frequency ω_0n, this reads

=n

Transition densities of excitations that are energetically close to~ω0npenetrate into

=n

fδn(r, ω_0n)o

similar to the spectral lines in figures 4.1 (a3)+(b3). The proportion-ality in equation (4.8) corresponds to the approximation

=n

δnf(r, ω0n) o

≈ −T(k·r0n)ρ0n(r) . (4.29) Hence, it is only valid for |ω_0n−ω0j|T 1 for allj 6=n.

However, the real transition densities can be calculated by inverting equation (4.28) for a closed set ofM interfering transitions0→n_1...M. For this finite subset equation (4.28) can be written as a matrix-vector equation with a M×M correlation matrix C and a diagonal normalization matrix A

=n C contains the correlation factors C_ij = ^sin

(ω0ni−ω0nj)T

(ω0ni−ω_0nj)T . Therefore, it is symmetric and one on its diagonal. Moreover,C only depends on the excitation energies and A only on their strengths. The approximation (4.29) corresponds to setting C to unity.

The inversion of equation (4.28), which is used for decoupling the transition densities in the following, finally reads

ρ=A⁻¹C⁻¹=n δnfo

. (4.31)

A is diagonal and is easily inverted while C can be inverted by standard algorithms.

Boost ~yz

Y1

Z2

Boost ~yz

Z1 Y2

Figure 4.2: Transition densities of the Z1, Z2, Y1, and Y2 transitions ofNa4. The Z2 and Y1 transition densities are displayed for different propagation times and boost directions before and after the correction described in the text.

The gray arrows indicate the correction. (iso-surface at±0.0001 a₀⁻³) The approximated and corrected transition densities are discussed using figure 4.2 by means of theNa₄cluster and its spectrum in figures 4.1 (a3)+(a4) for different prop-agation times and boost directions. The latter are chosen parallel to the z-coordinate (∼z), parallel to the y-coordinate (∼y), or along the diagonal in the yz-plane (∼yz).

Thus, the excitations with transition dipoles parallel to the x-axis (X1-X3) are always switched off. The Z1 and Y2 excitations, whose transition densities are shown in figure 4.2 (top left block), are the dominant ones and hardly affected by the correction.

According to the spectrum in figure 4.1 (a3) transition densities of the Z1 and, if excited, the Y2 excitations notably penetrate into the Z2 transition density, which is shown in figure 4.2 (right block). The approximated transition densities according to equation (4.29) are shown on the left-hand side inside this block. For a boost parallel to the z-direction and a propagation time of 50 fs the transition density is already well described such that the decoupling (right-hand side inside the block) just adds a small correction. If the propagation time is reduced to10 fs, the approximated transition density is highly polluted by the Z1 transition density but corrected by the

decoupling. Finally, a boost along the diagonal of the yz-plane additionally excites the Y2 excitation, which breaks the z-symmetry of the approximated Z2 transition density. The correction, however, also removes this effect such that all corrected transition densities match perfectly.

This works the same for the other excitations seen in the spectrum. Still, a difficult benchmark case is the Y1 excitation, which is completely hidden by the Z1 excitation (if it is excited). If the Z1 excitation is switched off by a boost in y-direction, the Y1 transition density is accessible and, after T = 50 fs, the correction is rather small (figure 4.2, bottom left block). For a boost in yz-direction the Y1 transition density is completely hidden behind the Z1 transition density. The decoupling scheme can remove most of its influence and restore Y1 at least qualitatively.

50 fs

Figure 4.3: Transition densities of the Qy and Qx transitions as well as the CT1 and CT2 transitions before and after the correction. The gray arrows indicate the correction. (iso-surface at ±0.0002 a₀⁻³)

The same scheme works as well for the BChla with the spectrum in figures 4.1 (b3)+(b4). The respective transition densities, which all stem from a single 50 fs propagation, are shown in figure 4.3. The stronger Q_y and Q_x transitions are hardly influenced by the correction with this propagation time. However, the states CT1 an CT2, which are highly polluted especially by theQytransition, are decoupled success-fully. The corrected transition densities of these two states allow for the comparison with the transition densities from Q-Chem TDLDA calculations and their identifica-tion. They show increased intensity in the top left (CT1) and bottom right (CT2) corners inside the figure.

Finally, to measure the quality of the corrected transition densities, one can check how well the transition density reproduces the transition dipole from the spectral evaluation, i.e.,

r_0j =^! Z

rρ_0j(r)d³r . (4.32)

If the left-hand side and the right-hand side of equation (4.32) result in two vectors

with different directions, the correction did not work well as for the Y1 excitation of Na435. If the left-hand side and the right-hand side result in two vectors with almost identical directions but amplitudes that differ by some factor C, probably only the scaling with the matrixA⁻¹did not work well since the oscillator strength used within Awas not accurate enough.

The latter can be used to refine the strength and therefore the amplitude of the transition density of this excitation. If, e.g.,r_0j =CR

ρ_0jrd³r, the improved transi-tion dipole is ^√1

Cr0j, which leads, after renormalization with the improved A⁻¹, to a consistently improved transition density√

Cρ0j.

As final remark in this section, I want to emphasize that many of the excitations of the presented test systems cannot be identified with the traditional evaluation of real-time spectra and typical propagation real-times. The techniques presented in this section allow for a quantitative evaluation with even shorter propagation times.

35For the Y1 excitation ofNa4 the right-hand side of equation (4.32) from the transition density in figure 4.2 (Y1, boost ∼yz, corrected) still has a significant contribution in z-direction whereas the left-hand side of equation (4.32) from the spectral evaluation only has a y-component, as expected.