4.2 ADC(2) excitation energies and transition moments
5.1.2 Results
expressions with solely the right-hand eigenvectors. Another unfavorable consequence of our approximation is that it may slightly break the spatial symmetry, hence non-zero oscillator strengths may show up for forbidden transitions, or achiral molecules may acquire rotator strengths. We again expect that this effect can also be restrained by a properly chosen truncation parameter. These problems will be addressed in the following subsection.
To assess the errors originating from the LDF approximation, and to determine the TLDF parameter we endeavored to consider all important types of excitation and typical electronic structures. We paid particular attention to also select systems with Rydberg and charge transfer excitations, as well as conjugated or delocalized electronic structures, which are difficult to treat from the theoretical point of view, especially with local methods. The list of the considered excited states can be found in Tables5.2 to5.5.
The errors of the LDF approach for excitation energies and the corresponding oscillator strengths with respect to the canonical values are presented in Figs. 5.1 and 5.2 using various methods and basis sets.
0.0000 0.0025 0.0050 0.0075 0.0100
0.90 0.92 0.94 0.96 0.98 1.00
CIS calculations
Error in the exc. energy / eV
Wave function completeness
0.0000 0.0025 0.0050 0.0075 0.0100
0.90 0.92 0.94 0.96 0.98 1.00
TDHF calculations
Error in the exc. energy / eV
Wave function completeness
0.0000 0.0025 0.0050 0.0075 0.0100
0.90 0.92 0.94 0.96 0.98 1.00
TDA−TDDFT calculations
Error in the exc. energy / eV
Wave function completeness
0.0000 0.0025 0.0050 0.0075 0.0100
0.90 0.92 0.94 0.96 0.98 1.00
TDDFT calculations
Error in the exc. energy / eV
Wave function completeness
Figure 5.1: Error of excitation energies (eV) as a function of the TLDF parameter, which describes the completeness of the wave function. Dashed (solid) line: maximum (mean) errors. Blue (orange) line: results obtained with the aug-cc-pVDZ
(aug-cc-pVTZ) basis set.
Inspecting the results we can observe that the error introduced by the LDF approach decreases continuously in the range of 0.900 ≤ TLDF ≤ 1.000 for all four methods and for both bases with increasing TLDF parameter. The approximation is practically error-free for (TDA-)TDDFT calculations in the entire interval. The mean (maximum) error is less than 1 (2.5) meV in the least accurate case, which is more than two orders of magnitude smaller than the intrinsic error of the PBE0 functional for excitation energies.
0.000 0.005 0.010 0.015 0.020
0.90 0.92 0.94 0.96 0.98 1.00
CIS calculations
Error in the osci. strength
Wave function completeness
0.000 0.005 0.010 0.015 0.020
0.90 0.92 0.94 0.96 0.98 1.00
TDHF calculations
Error in the osci. strength
Wave function completeness
0.000 0.005 0.010 0.015 0.020
0.90 0.92 0.94 0.96 0.98 1.00
TDA−TDDFT calculations
Error in the osci. strength
Wave function completeness
0.000 0.005 0.010 0.015 0.020
0.90 0.92 0.94 0.96 0.98 1.00
TDDFT calculations
Error in the osci. strength
Wave function completeness
Figure 5.2: Error of oscillator strengths as a function of the TLDF parameter, which describes the completeness of the wave function. Dashed (solid) line: maximum (mean) errors. Blue (orange) line: results obtained with the aug-cc-pVDZ (aug-cc-pVTZ) basis
set.
As expected, if we choose the HF solution as the ground state reference, that is,αX= 1.0, the errors are more significant. In the case of the CIS method, the average error decreases below 5 meV for both bases at TLDF = 0.940; precisely, below 4 meV for the double-ζ and 1.5 meV for the triple-ζ basis. However, the maximum error is approximately 20 meV up to TLDF = 0.980 using the aug-cc-pVDZ basis set. Although both the average and maximum errors obtained in this range are also an order of magnitude smaller than the intrinsic error of the CIS method, it is still recommended to further increase the TLDF parameter. The maximum error at TLDF = 0.985 decreases dramatically, which guarantees the quantitatively and qualitatively adequate description of the problematic excitation (triad molecule, S1 state). At this value, the maximum error of the excitation energy does not exceed 2.5 meV for either basis set. Similar observations can be made for the TDHF calculations. The results obtained with the aug-cc-pVTZ basis are satisfactory in the entire range, the mean (maximum) error at TLDF = 0.950 is 2 (5) meV similarly to the CIS method. The results calculated with the aug-cc-pVDZ basis sets are less satisfactory as well, although the sudden improvement appears earlier in this case, at
TLDF = 0.970. In general, we can state that no significant difference can be observed between the aug-cc-pVDZ and aug-cc-pVTZ results for the DFT calculations, while, for the HF reference-based methods, the errors with the aug-cc-pVTZ basis set are slightly more favorable.
Similar results were obtained for the errors of the oscillator strengths as for the excitation energies using the corresponding methods and bases (see Fig. 5.2). The error introduced by the approximation, except for a few narrow ranges, decreases with increas-ing TLDF parameter in the interval studied. We can say that the oscillator strengths are somewhat more sensitive to the LDF approximation than the excitation energies. Similar observations were reported by Kjærgaard and co-workers for the ADMM method [63], while the COSX method of Neese et al. was not tested for transition moments [62]. For the TDA-TDDFT calculations, except for the dyad molecule, the approach is practically exact in the entire interval. The average error is less than 0.001 with both basis sets, while the maximum error is lower than 0.01 in every case, and no sudden and steep changes can be observed. Concerning the full TDDFT calculations, the average error is negligible as well, while the maximum error (D21L6 molecule) is a bit worse than for the TDA-TDDFT results. Nevertheless, it does not exceed 0.015 and it decreases below 0.005 at TLDF = 0.940 with both basis sets. Similarly to the excitation energies, the errors in the oscillator strengths are more pronounced for the CIS method. As it can be seen, the mean and maximum errors decrease dramatically at TLDF = 0.920 for the oscillator strengths as well with the triple-ζ basis. Beyond this value, the average error is approximately 0.001, while the maximum error decreases continuously from 0.006. In the case of the aug-cc-pVDZ basis, similarly to the excitation energies, the maximum and average errors drop rapidly atTLDF = 0.985. Using this threshold, the average (maximum) error is 0.002 (0.009), which is negligible compared to the intrinsic error of the method. For the TDHF method steep changes in the results do not occur using the aug-cc-pVTZ basis. The av-erage error decreases monotonically from 0.002 to 0.001 with increasing TLDF parameter, while the maximum error stays below 0.01. Akin to the excitation energies, the sudden improvement with the aug-cc-pVDZ basis appears earlier compared to the CIS method, at TLDF = 0.970. The maximum error is further reduced by increasing the threshold.
Relying on the above results and taking into account that the LDF approximation should be transferable to other DFT functionals, we have chosen TLDF = 0.985 as the default value of the truncation parameter. As we have seen, the errors are practically negligible even with TLDF < 0.985 for the PBE0 functional, but for the latter the exact exchange contribution is significantly scaled down (αX = 0.25). So that the LDF approxi-mation can be applied in a black box manner to other functionals with arbitraryαXvalues and also to the CIS and TDHF methods, we recommend the use of this default threshold.
With this value, the errors in the excitation energies or the oscillator strengths are two
orders of magnitude smaller than the intrinsic error of the methods, even if αX = 1.0.
For even more accurate calculations we propose TLDF = 0.990 as a tight threshold, which approximately halves the error.
To assess the performance of our approach using the default cutoff parameter, first we analyzed the consequences of the symmetry violation mentioned in the previous subsec-tion for the CIS method, where the most pronounced effect is foreseen. Unfortunately, to check the deviation of matrixAfrom hermiticity the entire matrix should be constructed.
It is only possible for smaller molecules, where the fitting domains almost include the whole molecule, and the approach practically reduces to the canonical method. Thus, instead we considered the error in the sigma vector introduced by our approximation for the bigger molecules of our test set. The canonical equations were solved, and the sigma vectors were evaluated with the resulting canonical excitation amplitudes using the local fitting approach. We found that the norm of the residual vector is about 10−4 a.u., thus the deviation between the symmetry-related elements of matrix A is presumably even smaller. The effect of the breaking of the spatial symmetry on transition moments was analyzed for the symmetric bisimide derivative and for other bigger symmetric molecules.
Our results show that the errors in the computed transition electric dipole moments for dark states, either in the length or in the velocity gauge, are on the 10−6 a.u. scale, while the corresponding elements of transition magnetic dipole moment vectors deviate by 10−4 a.u. or less from zero. This implies that forbidden transitions acquire oscillator strengths of 10−12, whereas the rotator strengths of achiral molecules are less then 10−10 a.u. This inaccuracy is very small, and we do not think that it causes any problem in practical applications.
Second, we computed the excitation energies, transition probabilities, and speedups for our test set. The detailed results obtained with the default threshold for each method and basis set are presented in Tables 5.2 to5.5.
Inspecting the CIS excitation energies (see Table 5.2) the average (maximum) un-signed error of the LDF approach is 1 (2) meV with the aug-cc-pVDZ basis set, whereas the average (maximum) unsigned error in the oscillator strengths is 0.002 (0.009). As we can see, significant speedups can be achieved in the calculations with the selected TLDF threshold. First of all, we can state that the magnitude of the speedups is consistent with the size and the electronic structure of the molecules. The lowest values, which are still close to a factor of two, were attained for the two smallest molecules. Speedups somewhat lower than what would be expected on the basis of the size of the molecule can be measured in some cases, such as the bisimide derivative and the P700 tetramer.
These discrepancies occur due to the extensive delocalized electronic structure of the rings, but these speedups are still about three-times as well. The most favorable result was obtained for the leupeptin molecule, for which the CIS calculations can be performed
Table 5.2: Canonical CIS excitation energies (ω, in eV) and oscillator strengths (f), the error of excitation energies (δω, in eV) and oscillator strengths (δf) with the present approach, and the speedups (S.u.) using the default threshold with the aug-cc-pVDZ
and aug-cc-pVTZ basis sets.
aug-cc-pVDZ aug-cc-pVTZ
Molecule Character ω δω f δf S.u. ω δω f δf S.u.
Lauric acid n→π∗ 6.847 0.001 0.001 0.000 1.96 6.868 0.001 0.001 0.000 2.17
Rydberg 9.060 0.002 0.083 0.000 9.063 0.001 0.081 0.000
Rydberg 9.148 0.001 0.002 0.000 9.135 0.000 0.004 0.000
Dyad π→π∗ 4.302 0.002 0.563 −0.002 1.94 4.301 0.000 0.563 0.000 3.34
π→π∗,CT 4.522 0.001 0.198 0.000 4.505 0.001 0.179 −0.002
σ→π∗ 4.634 0.001 0.002 0.000 4.611 0.000 0.002 0.000
Bisimide der. π→π∗ 3.108 0.000 1.272 −0.001 2.69 3.105 0.000 1.266 0.000 3.08
π→π∗ 4.365 0.000 0.000 0.000 4.363 0.000 0.000 0.000
π→π∗ 4.482 0.000 0.000 0.000 4.478 0.000 0.000 0.000
Leupeptin σ→π∗ 4.839 0.001 0.000 0.000 6.08 4.861 0.001 0.000 0.000 6.57
Rydberg 6.376 0.001 0.002 0.000 6.394 0.001 0.002 0.000
Rydberg 6.401 0.001 0.002 0.000 6.419 0.001 0.002 0.000
D21L6 CT 3.189 0.000 1.756 0.000 3.55 3.184 0.000 1.735 0.000 3.98
π→π∗,CT 4.360 0.001 0.017 0.000 4.355 0.001 0.016 0.000
Rydberg 4.676 0.000 0.037 0.000 4.668 0.000 0.033 0.000
Triad π→π∗,CT 4.151 0.000 0.386 0.005 4.68 4.147 0.000 0.364 0.001 3.76
π→π∗ 4.213 0.000 0.682 −0.006 4.207 0.000 0.701 −0.001
π→π∗ 4.417 0.000 0.000 0.000 4.414 0.000 0.000 0.000
P700 π→π∗ 2.323 0.000 0.531 −0.001 3.20
π→π∗ 2.356 0.000 0.493 −0.007
π→π∗ 2.371 0.000 0.172 0.009
π→π∗ 2.421 0.000 0.066 0.006
MAE/Average 0.001 0.002 3.44 0.000 0.000 3.82
Maximum 0.002 0.009 6.08 0.001 0.002 6.57
Minimum 0.000 0.000 1.94 0.000 0.000 2.17
six-times faster. Although the size of the systems studied varies widely, we can say that approximately 3.5-times faster calculations can be expected for systems containing 40 to 200 atoms. Similar conclusions can be drawn for the aug-cc-pVTZ basis set. The errors are slightly lower compared to the double-ζ basis set, both for the excitation energies and oscillator strengths. In this case, the average error of the excitation energies (oscillator strengths) is less than 1 meV (0.001), while the maximum error is 1 meV (0.002). The speedups gained are somewhat more favorable.
If we consider the fact that for the TDHF method four HF exchange contributions are approximated with LDF, it is not surprising that the errors are slightly larger, while the speedups are considerably better than those for the CIS method (see Table5.3). With the aug-cc-pVDZ basis set, the average (maximum) error of the excitation energy is 1 (4) meV, while it is 0.001 (0.008) in the oscillator strengths. The minimum (maximum) speedup factor is approximately 2.5 (8.0). Again, the average and maximum errors are smaller with the aug-cc-pVTZ basis, while the speedups are on average higher.
As it was discussed, the LDF approach is practically error-free in the entire range of 0.900 ≤ TLDF ≤ 1.000 for the TDA-TDDFT and TDDFT calculations, and it is, of
Table 5.3: Canonical TDHF excitation energies (ω, in eV) and oscillator strengths (f), the error of excitation energies (δω, in eV) and oscillator strengths (δf) with the present approach, and the speedups (S.u.) using the default threshold with the aug-cc-pVDZ
and aug-cc-pVTZ basis sets.
aug-cc-pVDZ aug-cc-pVTZ
Molecule Character ω δω f δf S.u. ω δω f δf S.u.
Lauric acid n→π∗ 6.676 0.001 0.000 0.000 2.45 6.691 0.001 0.000 0.000 2.44
Rydberg 9.027 0.002 0.091 0.000 9.030 0.001 0.090 0.000
Rydberg 9.138 0.001 0.002 0.000 9.124 0.000 0.002 0.000
Dyad π→π∗ 4.044 0.004 0.447 −0.003 3.29 4.040 0.000 0.447 −0.001 4.17
π→π∗,CT 4.320 0.001 0.203 0.000 4.300 0.001 0.186 0.000
σ→π∗ 4.456 0.003 0.001 0.000 4.427 0.000 0.001 0.000
Bisimide der. π→π∗ 2.809 0.004 1.053 0.000 3.36 2.802 0.002 1.050 0.000 3.30
π→π∗ 4.167 0.000 0.000 0.000 4.161 0.000 0.000 0.000
π→π∗ 4.296 0.002 0.000 0.000 4.288 0.000 0.000 0.000
Leupeptin σ→π∗ 4.672 0.002 0.000 0.000 7.99 4.687 0.001 0.000 0.000 7.97
Rydberg 6.217 0.001 0.002 0.000 6.228 0.001 0.002 0.000
Rydberg 6.242 0.001 0.002 0.000 6.255 0.001 0.002 0.000
D21L6 CT 2.955 0.001 1.591 −0.001 4.61 2.946 0.000 1.573 −0.001 5.14
π→π∗,CT 4.103 0.000 0.020 0.006 4.096 0.000 0.018 0.006
Rydberg 4.538 0.000 0.037 0.000 4.537 0.000 0.037 0.000
Triad π→π∗ 3.829 0.000 0.269 0.004 5.42 3.816 0.000 0.273 0.000 5.44
π→π∗,CT 3.911 0.001 0.615 −0.003 3.906 0.000 0.608 0.002
π→π∗ 4.174 0.000 0.000 0.000 4.169 0.000 0.000 0.000
P700 π→π∗ 1.886 0.000 0.359 0.008 4.79
π→π∗ 1.927 0.000 0.391 −0.007 π→π∗ 1.939 0.000 0.105 −0.002
π→π∗ 1.999 0.000 0.078 0.002
MAE/Average 0.001 0.002 4.56 0.001 0.001 4.75
Maximum 0.004 0.008 7.99 0.002 0.006 7.97
Minimum 0.000 0.000 2.45 0.000 0.000 2.44
course, also true for the default threshold (see Tables 5.4 and 5.5). The average error of the excitation energies is negligible, less than 1 meV for both basis sets, and similar statements can be made about the error in the oscillator strengths. In the case of the TDA-TDDFT calculations, the maximum error of the excitation energy is also less than 1 meV, while the maximum error of the oscillator strength is 0.003 (0.002) with the aug-cc-pVDZ (aug-cc-pVTZ) basis. The errors for full TDDFT calculations are very similar.
It is not surprising that the speedups are somewhat lower than those measured for the HF-based methods. The LDF approximation can only reduce the time required for the exact exchange part but not that for the DFT exchange-correlation contribution, which is comparable to the computation time of the Coulomb term. Moreover, it is also noteworthy that, according to our experience, the (TDA-)TDDFT iterations converge significantly faster than those for CIS and TDHF, so the overhead due to the preoptimization in our algorithm is relatively larger for the former methods. Despite all that, the average speedup with the aug-cc-pVDZ basis set is approximately three-times for TDA-TDDFT calculations, while in the case of the full TDDFT calculations it is somewhat higher, approximately four-times, but for particular systems speedup factors of 7 to 9 can be
Table 5.4: Canonical TDA-TDDFT excitation energies (ω, in eV) and oscillator strengths (f), the error of excitation energies (δω, in eV) and oscillator strengths (δf) with the present approach, and the speedups (S.u.) using the default threshold with
the aug-cc-pVDZ and aug-cc-pVTZ basis sets.
aug-cc-pVDZ aug-cc-pVTZ
Molecule Character ω δω f δf S.u. ω δω f δf S.u.
Lauric acid n→π∗ 5.889 0.000 0.000 0.000 1.05 5.904 0.000 0.000 0.000 1.48
Rydberg 6.913 0.000 0.057 0.000 6.930 0.000 0.055 0.000
Rydberg 7.362 0.000 0.003 0.000 7.357 0.000 0.002 0.000
Dyad CT 2.142 0.000 0.002 0.000 1.17 2.134 0.000 0.002 0.000 1.18
CT 3.208 0.000 0.030 −0.001 3.196 0.000 0.024 0.000
σ→π∗ 3.254 0.000 0.066 −0.003 3.249 0.000 0.058 −0.002
Bisimide der. π→π∗ 2.591 0.000 1.190 0.000 1.50 2.587 0.000 1.185 0.000 1.81
π→π∗ 3.070 0.000 0.000 0.000 3.072 0.000 0.000 0.000
π→π∗ 3.074 0.000 0.000 0.000 3.076 0.000 0.000 0.000
Leupeptin σ→π∗ 4.104 0.000 0.001 0.000 2.08 4.122 0.000 0.001 0.000 2.51
CT 4.893 0.000 0.000 0.000 4.920 0.000 0.000 0.000
CT 4.929 0.000 0.000 0.000 4.956 0.000 0.000 0.000
D21L6 CT 2.326 0.000 1.330 0.000 1.86 2.332 0.000 1.337 0.000 2.13
π→π∗,CT 3.068 0.000 1.016 0.000 3.069 0.000 0.992 0.000
π→π∗,CT 3.364 0.000 0.003 0.000 3.368 0.000 0.004 0.000
Triad CT 1.735 0.000 0.000 0.000 7.30 1.726 0.000 0.000 0.000 6.85
CT 1.933 0.000 0.005 0.000 1.929 0.000 0.005 0.000
CT 2.664 0.000 0.000 0.000 2.654 0.000 0.000 0.000
P700 π→π∗ 1.480 0.000 0.002 0.000 3.92
π→π∗ 1.489 0.000 0.004 0.000
π→π∗ 1.509 0.000 0.005 0.000
π→π∗ 1.533 0.000 0.001 0.000
MAE/Average 0.000 0.000 2.70 0.000 0.000 2.66
Maximum 0.000 0.003 7.30 0.000 0.002 6.85
Minimum 0.000 0.000 1.05 0.000 0.000 1.18
achieved. The errors obtained seem to be independent of which basis set is used, while, in contrast to the HF-based methods, the speedups are somewhat lower with the triple-ζ basis set. Of course, for functionals like PBE0, where the exact exchange contribution is relatively small, the computation time could be further reduced by decreasing the TLDF parameter. Anyway, we prefer the conservative default value forTLDF determined herein.
Though the comparison of the parent methods is not the subject of this study, it is instructive to realize that the character and the order of the lowest excited states are different for the various methods. The characters of the considered excited states with the CIS method are in accordance with the expectations and the results of more accurate ab initio methods [239,256]. The characters of the excitations obtained at the TDHF level are the same as for the CIS method, only two states of the triad molecule are swapped. If the KS solution is chosen as the ground state reference, the characters of the excitations differ in many cases from the aforementioned ones at both the TDA and the full TDDFT levels. The most remarkable difference can be observed for the triad molecule, where the three lowest excited states are CT excitations with unusually low excitation energies. Taking into consideration the known deficiencies of TDDFT
Table 5.5: Canonical TDDFT excitation energies (ω, in eV) and oscillator strengths (f), the error of excitation energies (δω, in eV) and oscillator strengths (δf) with the present approach, and the speedups (S.u.) using the default threshold with the
aug-cc-pVDZ and aug-cc-pVTZ basis sets.
aug-cc-pVDZ aug-cc-pVTZ
Molecule Character ω δω f δf S.u. ω δω f δf S.u.
Lauric acid n→π∗ 5.866 0.000 0.000 0.000 1.34 5.878 0.000 0.000 0.000 1.68
Rydberg 6.906 0.000 0.056 0.000 6.923 0.000 0.055 0.000
Rydberg 7.359 0.000 0.003 0.000 7.354 0.000 0.002 0.000
Dyad CT 2.141 0.000 0.002 0.000 2.12 2.134 0.000 0.002 0.000 1.88
π→π∗ 3.120 0.000 0.197 −0.001 3.117 0.000 0.194 0.000
σ→π∗ 3.198 0.000 0.008 −0.001 3.186 0.000 0.011 −0.001
Bisimide der. π→π∗ 2.358 0.000 0.792 0.000 2.32 2.354 0.000 0.790 0.000 2.88
π→π∗ 3.067 0.000 0.000 0.000 3.069 0.000 0.000 0.000
π→π∗ 3.071 0.000 0.000 0.000 3.073 0.000 0.000 0.000
Leupeptin σ→π∗ 4.077 0.000 0.001 0.000 3.15 4.091 0.000 0.001 0.000 3.17
CT 4.893 0.000 0.000 0.000 4.920 0.000 0.000 0.000
CT 4.929 0.000 0.000 0.000 4.956 0.000 0.000 0.000
D21L6 CT 2.249 0.000 1.184 0.000 3.33 2.254 0.000 1.190 0.000 3.33
π→π∗,CT 2.965 0.002 0.560 −0.002 2.966 0.002 0.541 −0.001
π→π∗,CT 3.260 0.000 0.051 0.002 3.264 0.000 0.052 0.003
Triad CT 1.735 0.000 0.000 0.000 9.20 1.726 0.000 0.000 0.000 7.10
CT 1.932 0.000 0.005 0.000 1.928 0.000 0.005 0.000
CT 2.664 0.000 0.000 0.000 2.654 0.000 0.000 0.000
P700 π→π∗ 1.479 0.000 0.002 0.000 5.74
π→π∗ 1.489 0.000 0.004 0.000
π→π∗ 1.508 0.000 0.004 0.000
π→π∗ 1.533 0.000 0.001 0.000
MAE/Average 0.000 0.000 3.89 0.000 0.000 3.34
Maximum 0.002 0.002 9.20 0.002 0.003 7.10
Minimum 0.000 0.000 1.34 0.000 0.000 1.68
methods [5,68–70] one suspects that these states are TDDFT artifacts. Furthermore, in the case of the P700 tetramer molecule, the four lowest excitations are local excitations of the individual units with the HF-based methods, which is in line with our expectations.
In contrast, at the DFT level the lowest excited states are CT excitations between two units. Nonetheless, it is important to note that the LDF approach does alter neither the character nor the order of the excited states.