correlation is empirically improved. Finally, the new approaches can be interpreted as dressed TDDFT methods as well with semi-empirical non-adiabatic XC kernels. Note also that the DH-ADC(2) scheme is consistent with both TDA-TDDFT and ADC(2) in the sense that the TDA-TDDFT or the ADC(2) transition energies and moments are recovered in the αC = 0 and αC = αX = 1 limits, respectively. The similar statement holds for the spin-scaled DH-ADC(2) variants as well. The anticipated advantage of the new models over the previous CIS(D)-based DHs is twofold. First, concerning excitation energies, ADC(2) moderately but consistently outperforms CIS(D), thus, an improvement in the calculated transition energies is expected, especially when the weight of double excitations is relatively large in the excited-state wave function. Second, in contrast to the CIS(D)-based DHs, the present methods also allow us to evaluate the transition moments at a higher level taking into account the effect of double excitations, which should considerably raise the quality of the computed spectral intensities. Compared to the parent ADC(2) methods or the dressed TDDFT models, also the improvement of excitation energies and transition moments is expected due to the empirical parameters introduced. Furthermore, the double counting of electron correlation, which can come up in the case of the dressed TDDFT approaches, is avoided in the present method.
hexatriene, octatetraene, cyclopropene, cyclopentadiene, norbornadiene, benzene, naph-thalene, furan, pyrrole, imidazole, pyridine, pyrazine, pyrimidine, pyridazine, triazine, tetrazine, formaldehyde, acetone, benzoquinone, formamide, acetamide, propanamide) were considered. The popular benchmark set of Gordon and co-workers [282] includes 33 valence (π →π∗,n →π∗,σ →π∗, n→σ∗) and 30 Rydberg excitations for 14 molecules.
For this test set the reoptimized geometries and the composite CC3-CCSDR(3) aug-cc-pVTZ reference excitation energies of SG [132] were used. Finally, for testing the behavior of the various methods for CT excitations the ethylene-tetrafluoroethylene dimer of Dreuw et al. was employed [251].
The errors employed for the evaluation of the computed excitation energies and oscillator strengths are obtained by subtracting the reference from the computed value.
The statistical error measures presented in the tables are the ME, MAE, RMS deviation, and MAX. The reported error measures for the oscillator strengths are calculated from their absolute errors only for states with f >0.2 as small values would bias our results.
The timing benchmarks were performed for the flavone derivative dyad, which has 36 atoms and 106 valence electrons, as well as for the D21L6 dyad with 98 atoms and 262 valence electrons. The aug-cc-pVTZ basis set and the corresponding auxiliary basis were used in these calculations resulting in 1311 (2916) and 3412 (7590) AOs (fitting functions), respectively. The reported computation times are wall-clock times measured on a machine with 128 GB of main memory and an 8-core 1.7 GHz Intel Xeon E5-2609 v4 processor.
In this initial study we do not endeavor to find the best DH-ADC(2) form but our intention is to demonstrate the advantages of the new ansatz over the previous CIS(D)-based ones. Hence, we only selected two DHs, the PBE0-2 (DH CIS(D)-based on the PBE hybrid functional) method of Chai and Mao [95] and the DSD-PBEP86 (dispersion-corrected, spin-component scaled DH with PBE exchange and P86 correlation) approach of Kozuch and Martin [104]. The performance of these DHs is excellent not only for ground-state properties, but they are among the best-performing functionals for excited states accord-ing to the recent study of SG [132]. Note that DSD-PBEP86 also includes an empirical dispersion correction, which does not contribute to excitation energies and transition moments and is ignored here. In line with the original nomenclature the DSD-PBEP86 variant that only includes OS correlation will be referred to as DOD-PBEP86.
In general, we refrain from the complete reoptimization of the parameters and keep the original parametrization of the ground-state DHs as far as possible. On the one hand, we think that the ground- and excited-state state theories should be consistent. On the other hand, as it was proven in previous studies, the performance of the approaches can hardly be improved by the refitting the functional parameters. Inspired by Ref.
132 we tested the following possibilities. First, we did not alter any parameter in the
original DHs. The resulting schemes will be denoted by PBE0-2/ADC(2) and DSD-PBEP86/ADC(2). Second, spin-scaling was introduced into PBE0-2 by multiplyingαC of the original functional, 12, by the SCS parameters derived for SCS-MP2 [97] and employing the resulting 65αC = 35 and 13αC = 16 coefficients as αOSC and αSSC , respectively. The 2/SCS-ADC(2) acronym will stand for this approach. The corresponding PBE0-2/SOS-ADC(2) scheme was derived by setting αOSC =, 1.3αC = 0.65, and αSSC = 0. The DSD-PBEP86 method already includes spin-scaling, and here αOSC = 0.53 andαSSC = 0.25 were rescaled as 65αOSC and 13αSSC , respectively, and the latter two were used as the scaling parameters of the OS and SS contributions in the DSD-PBEP86/SCS-ADC(2) method.
The similar holds for the DOD-PBEP86/SOS-ADC(2) ansatz, where the coefficient αOSC of the original DH, 0.53, was multiplied by 1.3. Third, we reoptimized the spin-scaling parameters for the SCS- and SOS-ADC(2)-based ans¨atze. In the abbreviation of the refitted method prefix f will be added to the acronym of the correlation method, e.g., as PBE0-2/fSCS-ADC(2). The spin-scaling parameters were reoptimized on the SG test set, and we obtained αOSC = 0.624 and αSSC = 0.260 for PBE0-2/fSCS-ADC(2), αOSC = 0.770 for PBE0-2/fSOS-ADC(2), αOSC = 0.555 and αSSC = 0.075 for DSD-PBEP86/fSCS-ADC(2), andαOSC = 0.575 for DOD-PBEP86/fSOS-ADC(2). It is interesting to note that these parameters are very similar to those attained for the corresponding CIS(D)-based analogues [132].
To distinguish the genuine CIS(D)-based DHs similar conventions will apply as for the new approaches utilizing ADC(2). For instance, PBE0-2/CIS(D) will denote the PBE0-2 DH approach where the (D) correction scaled by the original αC is added to the excitation energies, while the DSD-PBEP86/SCS-CIS(D) abbreviation will refer to the approach that scales the OS and SS contributions of the (D) correction similar to DSD-PBEP86/SCS-ADC(2). Here, we do not adopt the more flexible functional form proposed in Ref. 132, and the “direct” and “indirect” terms in the (D) correction are scaled with the same coefficient. We also note that the TDA-based variants of these methods are considered since these are the direct non-iterative analogues of our approaches.
The various error measures for the excitation energies of the SG test set obtained with the CIS(D)- and ADC(2)-based approaches are compiled in Table6.1, where the er-rors of the CC2 and ADC(2) methods and their spin-scaled variants are also presented. As we can see, without the optimization of the parameters, the performance of the CIS(D)-and ADC(2)-based DHs is very similar. For PBE0-2, all the error measures are slightly but consistently better if the second-order terms are evaluated with ADC(2), whereas, for DSD-PBEP86, the use of ADC(2) does not seem to enhance the accuracy. As expected, the refitting of the scaling coefficients consistently improves the results, especially for PBE0-2, while, for DSD-PBEP86, the improvement only in the maximum error is consid-erable. Comparing the performance of PBE0-2 and DSD-PBEP86, the latter seems to be
Table 6.1: Error measures for the calculated excitation energies (in eV) for the SG test set.
Method ME MAE RMS MAX
CC2 -0.06 0.12 0.17 0.49
SCS-CC2 0.05 0.11 0.13 0.30
SOS-CC2 0.10 0.17 0.19 0.32
ADC(2) -0.10 0.16 0.20 0.54
SCS-ADC(2) 0.01 0.09 0.12 0.35
SOS-ADC(2) 0.06 0.14 0.16 0.30
CIS(D) 0.01 0.16 0.21 0.46
SCS-CIS(D) 0.10 0.17 0.20 0.44
SOS-CIS(D) 0.15 0.21 0.25 0.53
PBE0-2/CIS(D) 0.13 0.14 0.18 0.46
PBE0-2/SCS-CIS(D) 0.18 0.18 0.21 0.41
PBE0-2/SOS-CIS(D) 0.20 0.20 0.24 0.55
PBE0-2/ADC(2) 0.08 0.12 0.16 0.45
PBE0-2/SCS-ADC(2) 0.15 0.15 0.18 0.39
PBE0-2/SOS-ADC(2) 0.17 0.18 0.21 0.53
PBE0-2/fSCS-ADC(2) 0.00 0.09 0.12 0.32 PBE0-2/fSOS-ADC(2) 0.00 0.10 0.12 0.31 DSD-PBEP86/CIS(D) -0.01 0.10 0.15 0.48 DSD-PBEP86/SCS-CIS(D) -0.02 0.09 0.14 0.50 DOD-PBEP86/SOS-CIS(D) -0.03 0.09 0.15 0.51 DSD-PBEP86/ADC(2) -0.05 0.12 0.15 0.49 DSD-PBEP86/SCS-ADC(2) -0.06 0.10 0.14 0.50 DOD-PBEP86/SOS-ADC(2) -0.07 0.10 0.15 0.51 DSD-PBEP86/fSCS-ADC(2) 0.00 0.09 0.12 0.41 DOD-PBEP86/fSOS-ADC(2) 0.00 0.09 0.12 0.41
superior to the former, but the optimization of the parameters minimizes the differences.
As the new ADC(2)-based DHs are as expensive as the parent wave function methods, the new DHs should, of course, also be competitive with the latter. In this respect the results show that the DH-ADC(2) ans¨atze are consistently superior, however, if spin scaling is applied, the original wave function methods are rather accurate, and only DSD-PBEP86 with reoptimized parameters can compete with them.
The performance of the new models for excitation energies and transition probabil-ities was first tested for the test set of Thiel and co-workers. Unfortunately this set lacks Rydberg excitations, but it is a representative collection of valence excitations including a
bunch of states with considerable double excitation character. The results are presented in Table 6.2.
Table 6.2: Error measures for the calculated excitation energies (in eV, for all the 121 states) and oscillator strengths (for 58 states withf >0.2) for the Thiel test set using
TZVP basis set [249].
Excitation energies Oscillator strengths
Method ME MAE RMS MAX ME MAE RMS MAX
CCSD 0.29 0.29 0.38 1.27 0.03 0.04 0.07 0.38
CC2 0.11 0.17 0.26 0.95 0.04 0.04 0.08 0.41
ADC(2) 0.07 0.19 0.28 1.00 0.01 0.04 0.08 0.39
SCS-ADC(2) 0.19 0.24 0.35 1.22 0.03 0.06 0.10 0.39
SOS-ADC(2) 0.25 0.30 0.41 1.34 0.04 0.07 0.11 0.39
CIS(D) 0.24 0.32 0.48 1.65 0.17 0.21 0.29 0.85
SCS-CIS(D) 0.34 0.37 0.54 1.59 0.18 0.21 0.30 0.89
SOS-CIS(D) 0.39 0.42 0.59 1.79 0.18 0.21 0.30 0.90
PBE0-2/CIS(D) 0.29 0.33 0.47 1.68 0.17 0.19 0.28 0.84
PBE0-2/SCS-CIS(D) 0.35 0.38 0.52 1.68 0.17 0.20 0.28 0.86
PBE0-2/SOS-CIS(D) 0.38 0.41 0.55 1.68 0.18 0.20 0.29 0.87
PBE0-2/ADC(2) 0.24 0.30 0.44 2.01 0.01 0.06 0.12 0.50
PBE0-2/SCS-ADC(2) 0.31 0.36 0.48 1.99 0.03 0.07 0.13 0.51
PBE0-2/SOS-ADC(2) 0.34 0.39 0.51 1.98 0.02 0.06 0.10 0.40
PBE0-2/fSCS-ADC(2) 0.14 0.23 0.41 1.98 -0.01 0.07 0.11 0.36 PBE0-2/fSOS-ADC(2) 0.13 0.23 0.40 1.95 -0.01 0.07 0.11 0.35
DSD-PBEP86/CIS(D) 0.13 0.24 0.36 1.48 0.16 0.19 0.28 0.81
DSD-PBEP86/SCS-CIS(D) 0.11 0.22 0.34 1.41 0.16 0.19 0.28 0.82 DOD-PBEP86/SOS-CIS(D) 0.10 0.22 0.33 1.37 0.16 0.19 0.28 0.82
DSD-PBEP86/ADC(2) 0.07 0.19 0.29 0.96 0.02 0.08 0.13 0.41
DSD-PBEP86/SCS-ADC(2) 0.05 0.17 0.26 0.98 -0.01 0.06 0.10 0.37 DOD-PBEP86/SOS-ADC(2) 0.04 0.16 0.25 0.99 0.01 0.07 0.12 0.37 DSD-PBEP86/fSCS-ADC(2) 0.13 0.21 0.30 1.00 0.02 0.07 0.12 0.43 DOD-PBEP86/fSOS-ADC(2) 0.12 0.20 0.30 1.02 0.02 0.07 0.12 0.41
The conclusions are somewhat different from those for the SG test set. Concerning excitation energies, the ADC(2)-based models consistently outperform the CIS(D)-based ones, but the improvement is more pronounced for DSD-PBEP86. The trends regard-ing the relative performance of the two functionals are similar to those identified for the CIS(D)-based approaches [132]. The reoptimization of the parameters significantly im-proves the error measures for PBE0-2, while it does not help at all for DSD-PBEP86.
Again, DSD-PBEP86 is superior to PBE0-2 in all respects. It is interesting to realize that the spin-scaling worsens the results both for ADC(2) and for the ADC(2)-based PBE0-2
DH, but for DSD-PBEP86 the opposite tendency can be observed. The best overall per-formance is attained by the spin-scaled ADC(2)-based DSD-PBEP86 variants. Their ME, MAE, and RMS errors are the lowest, and their maximum error is just slightly higher than the best ones. These methods outperform not only the other DH models but also all the wave function approaches listed in the table including CC2 and the much more expensive CCSD. If we take a closer look at the excitation energies, it is also obvious that the results bear out our expectations, that is, for states with considerable double excitation character, such as the lowest transitions of conjugated systems, the excitation energies computed using the new methods are noticeably better. More detailed statistics can be found in the supplementary material of Ref. 277.
The advantage of the new DH schemes over the previous CIS(D)-based ones is even more conspicuous for oscillator strengths. The new methods have almost perfect MEs, which are an order of magnitude smaller than those for the corresponding CIS(D)-based DHs. The other error measures are reduced by at least a factor of two. Comparing the performance of the DH-ADC(2) schemes we can conclude that there is no considerable difference in the accuracy of the variants utilizing PBE0-2 and DSD-PBEP86, only the maximum errors are somewhat more favorable for the latter. The refitting of the coeffi-cients only slightly changes the results, and sizeable improvement can only be observed for the maximum error with PBE0-2. Similar to the excitation energies, the spin-scaling moderately improves the accuracy only for PBEP86. The performance of the DSD-PBEP86 variants using spin-scaled ADC(2) correlation, which are the most accurate methods for excitation energies, is also convincing for oscillator strengths. Their error statistics are comparable to those of the parent wave function methods and only a little worse than that of the other wave function approaches. All in all, the DSD-PBEP86/SCS-ADC(2) and the DOD-PBEP86/SOS-DSD-PBEP86/SCS-ADC(2) approaches seem to be most efficient for the Thiel test set. Taking into consideration that the latter is significantly less expensive, it is the preferred method.
The performance of the methods for excitation energies was also tested for the benchmark set of Gordon and co-workers, which includes a comparable number of va-lence and Rydberg excitations and thus allows for tracking the behavior of the various approaches for both types of states. The results can be seen in Table 6.3.
If the statistics over all the states are considered, the excitation energies follow similar trends as discussed for the Thiel test set. The replacement of the (D) correction by ADC(2) moderately but consistently increases the accuracy. The spin-scaling is not effective for PBE0-2 but somewhat improves the results for DSD-PBEP86. As to the refitting of the parameters, the situation is reversed: it has practically no effect in the case of DSD-PBEP86, whereas it sensibly decreases the error for PBE0-2. The performance
Table6.3:Errormeasuresforthecalculatedexcitationenergies(ineV)ofall,Rydberg,andvalencestatesoftheGordontestset(63, 30,and33states,respectively). AllexcitationsRydbergexcitationsValenceexcitations MethodMEMAERMSMAXMEMAERMSMAXMEMAERMSMAX CCSD0.100.110.150.490.040.040.050.140.170.170.200.49 CC2-0.090.190.260.77-0.220.240.310.640.030.140.200.77 ADC(2)-0.080.180.270.79-0.190.230.330.690.020.130.200.79 SCS-ADC(2)0.070.170.230.82-0.020.160.210.620.160.170.240.82 SOS-ADC(2)0.150.220.270.840.060.170.210.710.230.260.310.84 CIS(D)0.010.220.321.11-0.150.190.290.570.160.250.351.11 SCS-CIS(D)0.160.230.331.200.020.160.200.650.290.290.411.20 SOS-CIS(D)0.230.270.371.270.100.180.220.740.350.360.471.27 PBE0-2/CIS(D)0.140.190.291.140.020.100.150.550.260.280.381.14 PBE0-2/SCS-CIS(D)0.230.240.331.150.120.140.180.650.330.340.421.15 PBE0-2/SOS-CIS(D)0.270.280.361.150.170.180.220.710.360.370.461.15 PBE0-2/ADC(2)0.120.180.281.030.010.100.140.540.210.250.361.03 PBE0-2/SCS-ADC(2)0.200.220.311.040.110.130.170.650.290.310.401.04 PBE0-2/SOS-ADC(2)0.240.260.341.040.160.170.210.700.320.340.431.04 PBE0-2/fSCS-ADC(2)0.080.170.271.030.020.170.210.640.130.180.321.03 PBE0-2/fSOS-ADC(2)0.110.200.281.030.080.210.240.740.140.190.311.03 DSD-PBEP86/CIS(D)-0.020.180.260.95-0.180.210.260.590.130.150.250.95 DSD-PBEP86/SCS-CIS(D)0.000.160.250.95-0.150.180.250.600.130.140.250.95 DOD-PBEP86/SOS-CIS(D)0.000.150.250.94-0.130.170.250.600.130.140.250.94 DSD-PBEP86/ADC(2)-0.050.170.250.92-0.190.220.260.590.080.130.240.92 DSD-PBEP86/SCS-ADC(2)-0.030.150.240.91-0.160.190.250.590.080.110.230.91 DOD-PBEP86/SOS-ADC(2)-0.030.140.240.91-0.140.180.250.600.070.110.220.91 DSD-PBEP86/fSCS-ADC(2)0.020.150.230.92-0.120.150.200.500.140.150.250.92 DOD-PBEP86/fSOS-ADC(2)0.020.150.230.92-0.110.150.200.500.140.150.250.92
of DSD-PBEP86 is remarkably better than that of PBE0-2. The accuracy of DOD-PBEP86/SOS-ADC(2) is again very good. It has the smallest MAE among all the second-order methods, it is only preceded by CCSD. Its ME is excellent and even smaller than that of CCSD. In average, it is also more accurate than the more costly second-order methods, CC2, ADC(2), and SCS-ADC(2), though its maximum error is higher. It clearly outperforms the parent SOS-ADC(2) method, whose computational expenses are very similar.
If the valence and Rydberg states are separately considered, the picture is more complicated than it seems. For Rydberg excitations the ADC(2) correlation does not improve the accuracy, and without the reoptimization of the scaling coefficients PBE0-2 performs somewhat better than DSD-PBEP86. The spin-scaling worsens the results for PBE0-2 and is hardly effective for DSD-PBEP86. The spin-scaled DSD-PBEP86 variants underperform the parent wave function methods, which are anyway quite good for Rydberg states. For valence states, the conclusions are very similar to those that we have drawn for the entire test set, but the effects are more noticeable. Hence, these results suggest that, if excitation energies are considered, the advantage of the new methods is limited for Rydberg states, but their overall accuracy is still better because of their excellent performance for valence states. Of course, for transition probabilities, improved results are expected also for the Rydberg excitations.
We also tested the behavior of the new DHs for CT states, where common density functional approximations with low percentage of exact exchange badly fail. The potential energy curve for the lowest CT excited state along the intermolecular distance (R) of the ethylene-tetrafluoroethylene system [251] was recorded (see in Table B.3 and Figure B.1 in AppendixB). In this model system the excitation energy should asymptotically decay with −1/R, but certain DFT approaches yield more or less constant excitation energies.
The asymptotic behavior of the new models, just as that for the previous CIS(D)-based DHs, is correct, and the excitation energies of the ADC(2)-based DHs decay similar to those evaluated with ADC(2). Thus, these results suggest that the new methods are also suitable for CT excitations.
Finally, we inspected the basis set convergence of the new approaches in comparison to other DHs and the parent wave function methods (see supplementary material of277).
We carried out calculations with the aug-cc-pVDZ and aug-cc-pVQZ basis sets for the SG test set and evaluated the errors of the aug-cc-pVDZ and aug-cc-pVTZ excitation energies with respect to the aug-cc-pVQZ ones. If the average errors are considered, we can conclude that the convergence is fast, the errors are reduced by factors of 2 to 4 when going from the double- to the triple-ζ basis set. The average errors are rather satisfactory with the triple-ζ basis set being an order of magnitude smaller than the intrinsic error of the methods. As expected, the basis set error of the composite approaches is smaller
than that for the pure wave function methods. The tendencies are also similar for the maximum errors with the exception of the original and spin-scaled CIS(D) methods, for which the maximum error hardly changes if aug-cc-pVTZ is used instead of aug-cc-pVDZ.
Surprisingly this behavior is not inherited by the CIS(D)-based DHs. Taking into account these results we can state that the new ADC(2)-based approaches can safely be applied with triple-ζ quality basis sets.
The benchmark results for both the Thiel and the Gordon test set reveal the ex-cellent performance of the DOD-PBEP86/SOS-ADC(2) approach, especially if computa-tional expenses are also taken into account. However, there are numerous other DHs on the market, and a comparison to them would also be pertinent. Fortunately, the perfor-mance of several popular DH functional forms for excitation energies was analyzed in Ref.
132for the aforementioned two test sets with and without spin-scaling and reoptimization of the parameters. Concerning Thiel’s benchmark set the MAE of DOD-PBEP86/SOS-ADC(2) for excitation energies is 0.16 eV, and none of the methods studied in Ref. 132 surpasses our approach including also the DHs that use full TDDFT instead of TDA (see Table 5 in Ref. 132). For Gordon’s set the corresponding value is 0.14 eV, and there are several other methods that are comparable in accuracy to our approach (see Table 6 in Ref. 132). Among the methods relying on the SOS approximation better MAE can only be achieved with PBE0-DH, which is, however, less efficient for other test sets. We also note that, as discussed above, the tendencies for the relative performances of the various DH functional forms are similar with the CIS(D) and ADC(2) correlation. It means that probably one cannot get better excitation energies with the ADC(2)-based version of the other functionals considered in Ref. 132. Our most important results are visualized in Fig. 6.1.
Unfortunately, for transition probabilities no comprehensive benchmark exists in the literature. In fact, transition probabilities computed with DH approaches have hardly received attention so far. Nevertheless, it is unlikely that any CIS(D)-based DH could outperform our method.
The outcome of our benchmark studies presented demonstrates the increased accu-racy of the composite DFT-ADC(2) method. However, needless to say that the ADC(2)-based methods are more costly than their CIS(D) counterparts since one iteration step with ADC(2) is roughly as expensive as the evaluation of the (D) correction. Conse-quently, one could ask if the accuracy gain is worth the effort. Before answering this question we inspect if the computational costs of the DHs considered can be lowered by our cost-reduction techniques. We note that only the NAF technique is studied here, but further substantial speedup can be attained for the DH-ADC(2) and DH-SCS-ADC(2) ans¨atze using our NO-based cost-reduction approaches. However, as our preferred method
Figure 6.1: Performance of the most accurate ADC(2)-based DHs for the excitation energies of the SG, Thiel, and Gordon test sets and the oscillator strengths of the Thiel test set in comparison with the corresponding CIS(D)-based DHs and the parent
ADC(2) and SOS-ADC(2) methods.
utilizes SOS correlation, and for the latter the NO-based techniques are less efficient, we refrain from the detailed presentation of the corresponding results.
Table 6.4: Statistical measures for the errors due to the NAF approximation for the calculated excitation energies (in meV) of the Gordon test set using a truncation
threshold ofεNAF = 0.075 a.u.
Method ME MAE RMS MAX
CC2 -2 2 3 9
ADC(2) -2 2 3 6
SCS-ADC(2) -3 3 4 6
SOS-ADC(2) -4 4 5 7
CIS(D) -2 2 3 6
SCS-CIS(D) -4 4 4 6
SOS-CIS(D) -4 5 5 8
DSD-PBEP86/CIS(D) -2 2 2 5
DSD-PBEP86/SCS-CIS(D) -2 3 3 4
DOD-PBEP86/SOS-CIS(D) -3 3 3 4
DSD-PBEP86/ADC(2) -2 2 2 4
DSD-PBEP86/SCS-ADC(2) -2 2 3 4
DOD-PBEP86/SOS-ADC(2) -3 3 3 5
DSD-PBEP86/fSCS-ADC(2) -2 2 3 4
DOD-PBEP86/fSOS-ADC(2) -2 2 3 4
First, we recalculated the excitation energies for Gordon’s test set using the NAF approximation. A truncation threshold of εNAF = 0.075 a.u. was employed, which is a fairly conservative choice on the basis of our previous results. The error statistics for a couple of representative methods are shown in Table6.4. The results show that the errors are very small with this cutoff parameter. For the original wave function methods the errors are already minuscule, but for the DHs, where the second-order terms are scaled down, the error is even smaller, in average it is about 3 meV and does not exceed 5 meV.
This is close to two orders of magnitude smaller than the intrinsic error of these approaches and can thus be safely ignored. The average, maximum, and minimum number of NAFs kept is about 49, 52, and 47 % for both the KS- and HF-based methods. It means that about half of the auxiliary functions can be dropped in both cases.
Table 6.5: Computation times (in min) and number of iterations (Nit) for the lowest singlet excited state of the flavone derivative with (lower panel) and without (upper
panel) the NAF approximation.
Method NAFa PT2b Singlesc Doublesd Nit Totale
ADC(2) − 5.5 0.1 9.4 16 187.5
SOS-ADC(2) − 1.4 0.1 3.3 16 85.9
DSD-PBEP86/CIS(D) − 5.5 0.3 6.2 16 47.6
DOD-PBEP86/SOS-CIS(D) − 1.4 0.3 1.6 16 38.9
DSD-PBEP86/ADC(2) − 5.5 0.3 9.4 19 221.1
DOD-PBEP86/SOS-ADC(2) − 1.4 0.3 3.3 19 101.1
ADC(2) 1.6 2.6 0.1 3.5 16 92.2
SOS-ADC(2) 1.6 0.4 0.1 0.9 16 48.4
DSD-PBEP86/CIS(D) 1.6 2.6 0.3 2.9 16 42.6
DOD-PBEP86/SOS-CIS(D) 1.6 0.4 0.3 0.4 16 37.9
DSD-PBEP86/ADC(2) 1.6 2.6 0.3 3.5 19 107.2
DOD-PBEP86/SOS-ADC(2) 1.6 0.4 0.3 0.9 19 55.6
aWall-clock time for the construction of NAFs. 49.9% of the NAFs were kept for both the HF- and KS-based calculations. bWall-clock time for the PT2 (MP2) calculation.
cWall-clock time per iteration step for the first-order terms. dWall-clock time per iteration step for the second-order terms or the wall-clock time for the (D) correction.
eTotal wall-clock time including the time required for the integral transformation, which is uniformly 30.4 minutes.
Next, we measured the wall-clock times for a representative example, the flavone derivative, for the DSD-PBEP86 DHs and the relevant ADC(2) methods with and without the NAF approximation. The results are collected in Table 6.5. Note that the compu-tation times for ADC(2) and SCS-ADC(2) are practically identical, therefore timings for the latter are not reported in the table. As can be seen, the situation is not so bad even
without the cost-reduction. Even though an ADC(2) iteration step is 1.5-times more expensive than the evaluation of the (D) correction, and for this particular example more iterations are required for the ADC(2)-based DH, the DSD-PBEP86/ADC(2) calculation takes only about five-times longer than DSD-PBEP86/CIS(D) due to the overhead of the PT2 calculation and the processing of the integrals. This ratio is even better for the SOS approaches.
The NAF technique significantly accelerates the calculations. It has a moderate overhead, which only amounts to a couple of percents of the whole computation time, but speeds up the evaluation of the second-order contributions by a factor up to 4. This speedup might be surprising as only about 50 % of the fitting functions are dropped, and the scaling of the most expensive operations is linear with the size of the auxiliary basis. However, the reduction of the size of the auxiliary basis does not just explic-itly decreases the number of operations but reduces the size of the integral lists thereby enabling the improvement of the memory management and further speeding up the cal-culation. The overall speedup is much larger for the ADC(2)-based models, which treat the second-order terms iteratively, and their evaluation requires a much larger fraction of the computation time. For the CIS(D)-based approaches the gain due to the introduction of the NAF basis is almost lost because of the overhead of the NAF construction. Thanks to the NAF approximation, the ratio of the wall-times of the DSD-PBEP86/CIS(D) and DSD-PBEP86/ADC(2) calculations drops to only 2.4, while our preferred method, DOD-PBEP86/SOS-ADC(2) is just by a factor of 1.5 more expensive than the corresponding CIS(D)-based DH. Of course, we should not forget that the factor grows with the number of the roots since the time spent on the NAF construction, PT2, and integral calculation is relatively shorter. For example, for five excited states it would be about 2.5, but this is still acceptable. On the whole, also taking into account the excellent performance of the new schemes for transition probabilities, we think that the answer to the question raised above, that the ADC(2)-based ans¨atze are competitive due to increased computation time, is affirmative.
Finally, to demonstrate how far we can go with the new approaches we performed further benchmark calculations for the 98-atom D21L6 system. With the hardware em-ployed in our study ADC(2)-based DH calculations can be seamlessly carried out for molecules of about up to 60 atoms. For larger systems our NO-based cost-reduction tech-niques should be invoked. As mentioned above, these approximations only moderately speed up the SOS-ADC(2)-based methods, however, they efficiently reduce the memory requirements and thereby enable larger calculations. Utilizing these techniques the DOD-PBEP86/SOS-ADC(2) calculation for the lowest singlet state of the D21L6 molecule took 2 days. This promising result can be relatively more favorable when increasing the num-ber of roots targeted as the computation times for many expensive steps are independent
of the number of states. Significant speedup can also be gained for the fifth-order scaling DSD-PBEP86/SCS-ADC(2) method, with which the calculation required only about 2.5 days. These results suggest that the new methods can be routinely applied to molecules of up to 100 atoms.
Chapter 7 Summary
In this thesis numerous approaches have been presented for efficient quantum chem-ical calculations of excited-state properties of large molecules. In the first group of meth-ods, we have successfully reduced the number of variables required for the calculations using and combining various approximations. In the remainder an effective fourth-power-scaling double hybrid TDDFT approach was developed, with which the errors can be significantly reduced compared to methods with similar cost.
In Chapter 4 a reduced-cost implementation of the CC2 and ADC(2) methods is presented. We introduce approximations by restricting virtual natural orbitals and natu-ral auxiliary functions, which results, on the average, in more than an order of magnitude speedup compared to conventional algorithms. For the reduction of the size of the molecu-lar orbital basis state-specific natural orbitals are constructed for each excited state using the average of the approximate MP2 and the corresponding CIS(D) density matrices. The algorithmic considerations for the density construction are discussed in detail. The var-ious approximations are carefully benchmarked, and conservative truncation thresholds are selected which guarantee errors much smaller than the intrinsic error of the methods.
Using the canonical values as reference, we find that the mean absolute error for excita-tion energies is 0.02 eV for both methods, while that for oscillator strengths is 0.001 in the case of ADC(2). The rigorous cutoff parameters together with the significantly reduced operation count and storage requirements allow us to obtain accurate excitation energies and transition properties using triple-ζbasis sets for systems of up to one hundred atoms.
In Chapter 5reduced-scaling algorithms for further computational savings are pre-sented utilizing the locality of the molecular orbitals. First, an approximation is prepre-sented which can efficiently decrease the computational expenses of CIS and TDDFT methods, as well as of the related TDHF and TDA-TDDFT approaches. The approximation is the adaptation of the local density fitting scheme developed for Hartree–Fock calculations to excited states and reduces the quartic scaling of the methods to cubic. Our benchmark calculations show that, for molecules of 50 to 100 atoms, average speedups of 2 to 4 can be
achieved for CIS and TDDFT calculations at the expense of negligible errors in the cal-culated excitation energies and oscillator strengths, but for bigger systems or molecules of localized electronic structure significantly larger speedups can be gained. Then a framework for the reduced-scaling implementation of excited-state correlation methods is presented. An algorithm is introduced to construct excitation-specific local domains, which include all the important molecular orbitals for the excitation as well as for the electron correlation. The sizes of the resulting compact domains are further decreased utilizing our reduced-cost techniques based on the natural auxiliary function and local natural orbital approaches. Additional methodological improvements for the evaluation of density matrices are also discussed. The results of benchmark calculations performed at the ADC(2) level are presented, and it is demonstrated that the speedups achieved are significant even for systems of fewer than 100 atoms, while the errors introduced by our approximation are highly acceptable. Our results show that the new reduced-scaling algorithm allows us to carry out correlated excited-state calculations using triple-ζ basis sets with diffuse functions for systems of up to 400 atoms or 13000 atomic orbitals.
Finally, in Chapter 6 a new type of combined double hybrid TDDFT-ADC(2) method is proposed for the calculation of spectral properties of molecular systems. The new ansatz is an extension of the previous double hybrid methods replacing the non-iterative doubles correction of CIS(D) by non-iteratively evaluated ADC(2)-like second-order terms, reminiscent of dressed TDDFT approaches. Our results show that, compared to the existing double hybrids, the ADC(2)-based methods do not increase the average accu-racy of the computed excitation energies for Rydberg states but consistently improve the quality of the transition energies for valence states, especially for those of sizable double excitation character. For oscillator strengths, the performance of the new methods is sig-nificantly better. The proposed method with a mean absolute error of about 0.15 eV for excitation energies, outperforms other double hybrid approaches, ADC(2), its spin-scaled variants, and also the computationally more demanding CC2 model.
During my doctoral studies the following features were implemented in the Mrcc program package [240]:
• conventional and root-following Davidson algorithm for a general eigenvalue prob-lem
• incore, out-of-core, integral-direct CIS/TDHF/TDDFT/TDA-TDDFT sigma vec-tor construction
• oscillator strength calculation for the CIS/TDHF/TDDFT/TDA-TDDFT methods
• ground-state CC2 and MP2 methods