4.2 ADC(2) excitation energies and transition moments
4.2.2 Results
Table 4.7: Canonical ADC(2) singlet excitation energies (ω, in eV), oscillator strengths (f), the error of excitation energies (δω, in eV) and oscillator strengths (δf) with the present approach, and the percentage of VNOs and NAFs dropped using the default thresholds with the aug-cc-pVTZ basis set for small molecules. Oscillator
strengths for symmetry-forbidden (s.f.) transitions are not displayed.
Molecule State Character ω f δω δf CS-NAFs VNOs RS-NAFs
Acetamide S1 n→π∗ 5.356 0.000 −0.006 0.000 59.3 54.1 82.6
S2 Rydberg 5.869 0.018 −0.018 0.000 59.3 54.1 82.1
S3 Rydberg 6.450 0.012 −0.028 0.000 59.3 51.6 81.7
Acetone S1 n→π∗ 4.252 s.f. −0.017 s.f. 60.1 56.2 83.1
S14 σ→π∗ 8.959 0.000 −0.012 0.000 60.1 55.6 83.0
S16 π→π∗ 9.102 0.000 −0.015 0.005 60.1 56.2 83.4
Benzene S3 π→π∗ 6.507 s.f. −0.026 s.f. 61.7 55.7 83.9
S5 π→π∗ 7.041 0.069 −0.033 0.000 61.7 57.0 84.0
Butadiene S1 π→π∗ 6.095 0.707 0.009 0.000 61.4 55.7 83.9
S5 π→π∗ 7.112 s.f. −0.008 s.f. 61.4 52.1 83.4
Formaldehyde S1 n→π∗ 3.834 s.f. −0.036 s.f. 59.5 50.0 80.3
S6 σ→π∗ 9.030 0.001 −0.020 0.000 59.5 49.2 80.3
S12 π→π∗ 10.557 0.038 −0.025 0.000 59.5 46.9 79.9
Formamide S3 Rydberg 6.688 0.015 −0.027 0.000 59.0 49.7 80.0
C2F4 →C2H4 S17 CT 9.577 0.000 −0.041 0.000 58.1 53.0 79.8
C2H4 →C2F4 S34 CT 10.915 0.000 −0.008 0.000 58.1 46.5 79.3
MAE/Average 0.021 0.000 59.9 52.7 81.9
Maximum 0.041 0.005 61.7 57.0 84.0
Minimum 0.006 0.000 58.1 46.5 79.3
and 81.9% of the RS-NAFs are dropped on the average. These mean values are quite representative, the fluctuation of the ratio of dropped VNOs and RS-NAFs for this test set is below 12% and 5%, respectively. The MAE (MAX) of the excitation energies is equal to (less than) that reported for singlet CC2 excitations, however, the number of the VNOs and RS-NAFs is slightly larger since the originally selected VNOs are augmented with the corresponding canonical orbitals. The errors for the corresponding ADC(2) oscillator strengths are also highly acceptable being lower than 0.001 in average and below 0.005 in every case. The benefit of using the new algorithm is that about 60% of the CS-NAFs can also be dropped without introducing any significant inaccuracy on top of the algorithm without the CS-NAF approximation (more details in Ref. 256). Consequently, most of the steps preceding the ADC(2) calculation in the reduced VNO/RS-NAF basis can be performed about 60% faster with the presented algorithm.
To demonstrate the robustness of our approach we also studied the triplet exci-tations of the molecules of the same test set. The triplet exciexci-tations considered were selected so that their character and excited-state wave function (in terms of its domi-nant configurations) will be as close to the singlet excitations included in the test set as possible. Note that the oscillator strength is zero for the spin-forbidden transition from the singlet ground state to the triplet excited state, hence oscillator strengths are not reported for the triplet excited states. The numerical values for the triplet excita-tion energies are collected in Table 4.8. The MAE and MAX errors are somewhat larger
Table 4.8: Canonical ADC(2) triplet excitation energies (ω, in eV), the error of excitation energies (δω, in eV) with the present approach, and the percentage of VNOs and NAFs dropped using the default thresholds with the aug-cc-pVTZ basis set for
small molecules.
Molecule State Character ω δω CS-NAFs VNOs RS-NAFs
Acetamide T1 n→π∗ 5.099 −0.019 59.3 50.5 81.3
T2 Rydberg 5.831 −0.030 59.3 55.8 82.6
T4 Rydberg 6.298 −0.034 59.3 54.1 81.7
Acetone T1 n→π∗ 3.887 −0.026 60.1 56.2 82.9
T8 σ →π∗ 8.345 −0.023 60.1 55.9 83.0
T9 π →π∗ 8.999 −0.034 60.1 54.6 82.7
Benzene T5 π →π∗ 6.468 −0.032 61.7 56.0 83.7
T7 π →π∗ 6.986 −0.038 61.7 57.0 83.8
Butadiene T1 π →π∗ 3.434 −0.027 61.4 55.7 83.3
T8 π →π∗ 7.483 −0.013 61.4 56.7 84.0
Formaldehyde T1 n→π∗ 3.375 −0.047 59.5 50.0 79.6
T7 σ →π∗ 8.260 −0.037 59.5 49.2 79.3
T14 π →π∗ 10.506 −0.022 59.5 46.2 79.6
Formamide T3 Rydberg 6.159 −0.038 59.0 52.8 81.1
C2F4 → C2H4 T19 CT 9.573 −0.041 58.1 53.0 79.7 C2H4 → C2F4 T37 CT 10.913 −0.005 58.1 50.9 79.4
MAE/Average 0.029 59.9 53.4 81.7
Maximum 0.047 61.7 57.0 84.0
Minimum 0.005 58.1 46.2 79.3
than the corresponding values obtained for the singlet excitations, but both values are more than appropriate. The comparable accuracy can be explained by the fact that the VNO and NAF approximations function almost identically in the two spin cases if the default thresholds are applied. In addition, we find the CS-NAF approximation similarly accurate for the triplet states as well.
We also investigated the AO basis-set dependence of our approximations by com-paring the results calculated with the aug-cc-pVTZ basis set to those obtained with aug-cc-pVDZ and aug-cc-pVQZ. The statistical measures for the corresponding errors are collected in Table 4.9. The MAE and MAX errors of the excitation energies with the aug-cc-pVDZ and aug-cc-pVQZ bases are about 2-3 times larger than the aug-cc-pVTZ errors. We find this acceptable, because the errors are still significantly lower than the intrinsic error of the ADC(2) method. Moreover, usually at least a triple-ζ basis set augmented with diffuse functions is necessary to bring down the AO basis-set error to the level comparable to the intrinsic error of ADC(2). In other words, double-ζ-quality basis sets are less relevant in production level calculations. On the other hand, the use
Table 4.9: Error measures for the ADC(2) excitation energies (δω, in eV) and oscilla-tor strengths, (δf) and the average, maximum, and minimum percentage of VNOs and NAFs dropped using the default thresholds with various basis sets for small molecules.
aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ
Singlet Triplet Singlet Triplet Singlet Triplet
δω MAE 0.045 0.039 0.021 0.029 0.064 0.080
MAX 0.086 0.079 0.041 0.047 0.125 0.124
δf MAE 0.001 − 0.000 − 0.001 −
MAX 0.008 − 0.006 − 0.008 −
Dropped CS-NAFs Avg. 76.5 76.5 59.9 59.9 45.8 45.8
Max. 77.5 77.5 61.7 61.7 47.5 47.5
Min. 75.9 75.9 58.1 58.1 44.7 44.7
Dropped VNOs Avg. 18.3 17.2 52.7 53.4 73.1 73.2
Max. 25.1 25.1 57.0 57.0 76.2 76.8
Min. 9.6 7.1 46.5 46.2 70.5 70.5
Dropped RS-NAFs Avg. 80.5 80.3 81.9 81.7 87.8 87.9
Max. 82.5 82.5 84.0 84.0 89.1 89.8
Min. 77.9 77.4 79.3 79.3 86.3 86.3
of the quadruple-ζ-quality sets are often not economical or even impossible in practice.
Additionally, the error of our approximations for the transition moments is fairly basis set independent.
Considering the basis-set dependence, the portion of the neglectable VNOs increases with the basis set size from 18% through 53% up to 73% for aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ, respectively. As expected, larger basis sets can be compressed more when describing the same correlation effect. An opposite trend is present for the CS-NAFs, which is explained by the fact that the ratio of the functions in the AO and the corresponding auxiliary bases increases with the cardinal number, but the number of the AO product densities also increases, and a larger portion of the original auxiliary basis is required to be retained for their accurate fitting. Finally, after introducing the system-dependently compressed VNO and CS-NAF spaces, the percentage of the discarded RS-NAFs becomes nearly basis set independent. All in all, the operation count and data size reduction is sizeable with all the basis sets. For instance, the 53% VNO and the 82% RS-NAF reduction yield a 25-fold cut in the operation count in the rate-determining steps of the ADC(2) iteration as well as a 25-times smaller three-center integral list with the aug-cc-pVTZ basis. This gain is even larger, almost 120-fold with the aug-cc-pVQZ basis set.
We have also tested the accuracy of our approximations on larger systems to demon-strate that the errors do not grow with the system size. For that purpose the seven
medium-sized dye molecules were collected from the literature, which were also useful in realistic applications. Using the default truncation thresholds determined above we com-puted the errors of singlet excitation energies and the corresponding oscillator strengths using the presented algorithm with respect to conventional ADC(2) (see Table 4.10).
Comparing the error measures with those obtained for the smaller molecules (see Ta-ble 4.7) we find a very good agreement. In fact, the MAE (MAX) of singlet excitation energies of the larger systems is even slightly better, being 0.015 (0.036) eV, than the corresponding value of 0.021 (0.041) eV presented in Table 4.7. We note that the os-cillator strengths are more representative from the practical point of view than for the smaller molecules, because there is at least one large value (around 0.2) for each molecule.
However, the errors of the individual oscillator strengths are only moderately larger than those for the smaller test molecules; the quality of the approximate oscillator strengths and the simulated spectra is excellent. The MAE of the oscillator strengths is 0.002, but two salient errors can be found in the test set. For the S1 state of diphenylamine and the S7 state of the benzcarbaxole derivative the errors are−0.023 and 0.013, respectively. In both cases, the CIS wave function is a poor approximation of the corresponding ADC(2) wave function, nevertheless, the improved version of the reduced space construction algo-rithm results in acceptable errors. The average portion of the dropped CS-NAFs, VNOs, and RS-NAFs, being about 60%, 56%, and 82%, respectively, does not decrease either, hence similarly good or better speedups are expected in practice for these larger systems.
The above analysis is also performed for the seven lowest-lying triplet excitations for each medium-sized dye molecules on the same set. If we first compare the errors obtained for the triplet excitations compiled in Table 4.11 to the corresponding singlet excitation energy errors in Table 4.10, we find that the triplet MAE is slightly smaller (0.009 vs. 0.015 eV), while the triplet MAX is a bit worse (0.042 vs. 0.036 eV), but both values are more than acceptable. A more noticeable difference, 0.009 vs. 0.029 eV, appears if the average triplet excitation energy error is compared to that for the smaller test molecules (see Table 4.8). However, this is at least partly explained by the fact that the smaller-molecule test set is less balanced in terms of low- and high-lying excited states. Finally, looking at the portion of the omitted VNOs and NAFs we find the former to be around 57% and the latter to be around 82% in average, which are very close to the corresponding values calculated for the smaller test systems.
So far we have characterized the efficiency of our approximations by reporting the ratio of the omitted VNOs and NAFs and determining the expected operation count re-duction at certain steps of the algorithm. An even better measure for that purpose is to compare the actual wall-clock times measured for the conventional and the reduced-cost ADC(2) variants. The total wall-clock times will be further partitioned into contributions required for the major steps of the computation. For the conventional approach these
Table 4.10: Canonical ADC(2) singlet excitation energies (ω, in eV), oscillator strengths (f), the error of excitation energies (δω, in eV) and oscillator strengths (δf),
as well as the percentage of VNOs and NAFs dropped.
Molecule State Character ω δω f δf CS-NAFs VNOs RS-NAFs
Hydrazone dye S1 π→π∗ 3.475 0.002 0.111 0.001 58.7 54.1 80.4
S2 CT 3.643 −0.004 s.f. s.f. 58.7 55.3 80.9
S3 n, σ→π∗ 3.670 −0.005 s.f. s.f. 58.7 54.5 80.7
S4 π→π∗ 4.073 0.007 0.459 0.004 58.7 53.4 80.3
S5 n, σ→π∗ 4.268 −0.006 0.000 0.000 58.7 54.5 80.7
S6 n, σ→π∗ 4.287 −0.006 0.001 0.000 58.7 55.2 80.9
S7 π→π∗ 5.007 0.026 0.050 0.006 58.7 54.0 80.5
Azobenzene S1 n, σ→π∗ 2.795 −0.023 s.f. s.f. 61.1 58.6 83.5
S2 π→π∗ 4.064 0.009 0.718 0.005 61.1 57.1 83.0
S3 π→π∗ 4.448 0.019 0.020 0.001 61.1 56.2 82.9
S4 π→π∗ 4.455 0.019 s.f. s.f. 61.1 57.4 83.0
S5 π→π∗ 5.182 0.024 s.f. s.f. 61.1 56.9 82.9
S6 Rydberg 6.127 −0.027 0.001 0.000 61.1 56.2 83.3
S7 Rydberg 6.315 −0.028 s.f. s.f. 61.1 57.5 83.3
Diphenylamine S1 π→π∗ 4.335 0.011 0.104 0.023 60.8 56.9 83.5
S2 π→π∗, Ryd. 4.418 0.010 0.194 −0.002 60.8 56.7 83.3
S3 π→π∗ 4.460 0.010 0.021 0.000 60.8 56.7 83.3
S4 Rydberg 4.649 −0.015 0.158 0.008 60.8 56.6 83.3
S5 Rydberg 5.045 −0.027 0.049 0.004 60.8 58.3 83.8
S6 Rydberg 5.144 −0.036 0.000 0.000 60.8 58.6 83.9
S7 π→π∗ 5.275 −0.027 0.003 0.000 60.8 58.1 83.8
6,6’-difluoro- S1 n, σ→π∗ 1.965 −0.006 0.000 0.000 59.9 54.7 81.0
indigo S2 n, σ→π∗ 2.479 −0.004 s.f. s.f. 59.9 54.8 81.1
S3 π→π∗ 2.933 0.009 0.173 0.003 59.9 55.8 81.2
S4 π→π∗ 3.431 0.010 s.f. s.f. 59.9 54.6 81.0
S5 π→π∗ 3.678 0.018 s.f. s.f. 59.9 54.4 80.9
S6 π→π∗ 3.697 0.017 0.140 0.000 59.9 54.4 80.9
S7 n, σ→π∗ 3.748 0.003 s.f. s.f. 59.9 55.9 81.2
Bithiophene S1 π→π∗ 3.803 0.017 0.588 0.000 60.7 55.3 82.3
derivative S2 π→π∗ 4.555 0.016 0.012 0.001 60.7 53.3 82.0
S3 Rydberg 4.844 0.004 0.000 0.000 60.7 55.3 82.2
S4 π→π∗ 4.886 0.022 0.041 0.004 60.7 55.1 82.2
S5 Rydberg 5.034 −0.005 0.001 0.000 60.7 52.4 81.9
S6 n, σ→π∗ 5.497 0.003 0.001 0.000 60.7 52.4 81.9
S7 π→π∗ 5.582 −0.018 0.021 0.006 60.7 55.0 82.3
N-methyl-2,3- S1 π→π∗ 3.304 0.005 0.031 0.000 61.6 58.2 83.4
benzcarbaxole S2 π→π∗ 4.057 0.016 0.036 0.000 61.6 58.3 83.4
S3 Rydberg 4.487 0.015 0.310 0.007 61.6 59.1 83.7
S4 Rydberg 4.516 −0.033 0.007 0.000 61.6 56.9 83.2
S5 Rydberg 4.950 −0.031 0.000 0.000 61.6 59.1 83.7
S6 π→π∗ 5.027 −0.031 0.105 0.000 61.6 59.1 83.7
S7 Rydberg 5.271 0.013 0.005 −0.013 61.6 57.6 83.4
Flavone S1 CT 3.280 0.009 0.616 0.005 59.7 58.6 83.0
derivative S2 π→π∗ 3.922 0.002 0.000 0.000 59.7 58.7 83.1
S3 n, σ→π∗ 4.040 0.016 0.065 0.000 59.7 58.7 83.1
S4 π→π∗ 4.197 0.004 0.014 0.000 59.7 56.9 82.9
S5 Rydberg 4.398 −0.030 0.015 0.000 59.7 59.1 83.2
S6 π→π∗ 4.730 0.020 0.030 −0.009 59.7 58.2 82.9
S7 Rydberg 4.869 −0.024 0.001 0.000 59.7 59.0 83.2
MAE/Average 0.015 0.002 60.4 56.4 82.4
Maximum 0.036 0.023 61.6 59.1 83.9
Minimum 0.002 0.000 58.7 52.4 80.3
Table 4.11: Canonical ADC(2) triplet excitation energies (ω, in eV), the error of excitation energies (δω, in eV), as well as the percentage of VNOs and NAFs dropped.
Molecule State Character ω δω CS-NAFs VNOs RS-NAFs
Hydrazone dye T1 π→π∗ 3.161 0.009 58.7 54.4 80.5
T2 CT 3.427 0.011 58.7 55.3 80.9
T3 π→π∗ 3.447 0.012 58.7 54.5 80.7
T4 π→π∗ 3.751 −0.002 58.7 54.1 80.5
T5 π→π∗ 3.792 0.004 58.7 54.5 80.7
T6 CT 4.189 0.004 58.7 54.8 80.7
T7 π→π∗ 4.279 0.013 58.7 54.5 80.7
Azobenzene T1 n, σ→π∗ 2.189 0.036 61.1 53.8 82.7
T2 π→π∗ 2.879 0.014 61.1 58.0 83.2
T3 π→π∗ 4.004 0.011 61.1 57.7 83.1
T4 π→π∗ 4.241 −0.004 61.1 57.5 83.1
T5 π→π∗ 4.302 −0.002 61.1 57.5 83.1
T6 π→π∗ 4.712 0.011 61.1 56.4 83.0
T7 π→π∗ 4.827 0.004 61.1 56.2 82.9
Diphenylamine T1 π→π∗ 3.634 0.012 60.8 57.6 83.4
T2 π→π∗ 4.103 0.008 60.8 57.1 83.3
T3 π→π∗ 4.110 0.003 60.8 59.2 83.7
T4 π→π∗ 4.263 0.009 60.8 57.2 83.4
T5 Rydberg 4.531 0.042 60.8 58.8 83.8
T6 π→π∗ 4.729 0.005 60.8 56.9 83.2
T7 π→π∗ 4.817 0.002 60.8 57.3 83.4
6,6’-difluoro- T1 π→π∗ 2.087 0.011 59.9 56.0 81.2
indigo T2 π→π∗ 2.786 0.006 59.9 54.6 80.9
T3 π→π∗ 3.286 −0.003 59.9 54.5 80.9
T4 π→π∗ 3.448 0.001 59.9 54.6 81.0
T5 π→π∗ 4.212 0.002 59.9 52.9 80.8
T6 π→π∗ 4.292 −0.010 59.9 55.7 81.1
T7 π→π∗ 4.357 0.002 59.9 52.9 80.7
Bithiophene T1 π→π∗ 2.697 0.011 61.6 55.4 82.2
derivative T2 π→π∗ 3.863 0.011 61.6 53.6 82.0
T3 π→π∗ 4.013 0.009 61.6 55.5 82.3
T4 π→π∗ 4.301 0.007 61.6 53.7 82.1
T5 Rydberg 4.784 0.004 61.6 55.4 81.9
T6 π→π∗ 5.262 −0.001 61.6 52.1 82.1
T7 n, σ→π∗ 5.380 −0.005 61.6 55.2 82.2
N-methyl-2,3- T1 π→π∗ 2.703 0.011 60.7 58.7 83.5
benzcarbaxole T2 π→π∗ 3.487 0.004 60.7 58.5 83.5
T3 π→π∗ 3.764 0.003 60.7 58.4 83.4
T4 π→π∗ 4.100 0.006 60.7 58.4 83.4
T5 π→π∗ 4.379 −0.020 60.7 59.5 83.5
T6 π→π∗ 4.489 0.008 60.7 59.1 83.7
T7 π→π∗ 4.604 −0.009 60.7 59.6 83.7
Flavone T1 CT 2.585 0.010 59.7 59.0 83.1
derivative T2 n, σ→π∗ 3.537 0.002 59.7 59.0 83.1
T3 π→π∗ 3.940 0.011 59.7 59.0 83.1
T4 n, σ→π∗ 4.007 0.004 59.7 58.8 82.9
T5 π→π∗ 4.277 −0.008 59.7 57.1 82.8
T6 π→π∗ 4.650 0.001 59.7 57.2 82.9
T7 π→π∗ 4.738 0.035 59.7 59.0 83.1
MAE/Average 0.009 60.4 56.5 82.4
Maximum 0.042 61.6 59.6 83.8
Minimum 0.001 58.7 52.1 80.5
parts include the time of the transformation of the three-center integrals from AO to canonical MO basis (tJ), and the time of the CIS (tCIS) and the ADC(2) (tADC(2)) itera-tions. In the case of the reduced-cost algorithm the overhead required for the evaluation and diagonalization of the necessary state-averaged density matrices and theWmatrices, and the integral transformation to the VNO/NAF bases are accumulated intotJ. On top of this, in the case of the presented algorithm, the time required for the construction of the CS-NAF basis (tCS−NAF) and the corresponding integral transformation is added to tJ leading to tˆJ=tJ+tCS−NAF.
Our wall-clock time measurements for the lowest-lying seven singlet and seven triplet excitation energies, and the corresponding singlet transition moments of the seven medium-sized test molecules discussed previously are presented in Table 4.12. Most
im-Table 4.12: The wall-clock times required for the various steps of the calculations in minutes for all the states.
Canonical Reduced-cost algorithm
Molecule tJ tCIS tADC(2) tTotal tˆJ tCIS tJ tADC(2) tTotal Speedup
Azobenzene 6.2 4.4 352.6 363.1 6.5 1.7 6.7 19.7 34.6 10.5
Hydrazone dye 4.9 3.8 492.8 501.5 5.1 1.5 7.6 35.9 50.1 10.0
Diphenylamine 5.6 3.2 511.8 520.5 6.0 1.3 6.2 20.4 33.9 15.4
Benzcarbaxole derivative 16.9 10.2 2218.1 2245.2 18.0 4.2 20.3 97.7 140.2 16.0
Indigo derivative 17.3 10.9 2032.4 2060.5 18.2 4.2 31.9 125.3 179.6 11.5
Bithiophene derivative 16.8 13.7 2894.5 2925.0 17.9 5.1 30.8 185.7 239.6 12.2
Flavone derivative 29.8 19.1 4926.7 4975.6 31.7 7.5 47.9 201.8 288.9 17.2
Average 13.3
Maximum 17.2
Minimum 10.0
portantly, the average overall speedup factor compared to the conventional algorithm is about 13 using the same hardware. This significant efficiency increase occurs quite reli-ably considering that the maximum (minimum) overall speedup is 17.2 (10.0). The time required for the CIS iterations as well as for the construction of VNOs and RS-NAFs can be halved compared to the algorithm without the CS-NAF approximations (see in Ref.
256), while the additional computational cost needed for the construction of the CS-NAF basis with respect to the evaluation of the canonical integral list J is only about 5%.
Consequently, the total calculation can be performed about 25% faster (and with con-siderably reduced storage requirement) via the presented algorithm. We expect similar benefits from the introduction of CS-NAFs in our reduced-cost CC2 method.
Chapter 5
Reduced-scaling approximations for excited states
5.1 Local density fitting approach for TDDFT and CIS methods
In this section an efficient scheme is presented for calculating excitation energies and transition moments at the CIS and TDDFT levels, as well as for the related TD-HF and TDA-TDDFT methods. We have successfully extended the local DF approximation to excited states, which was originally developed for ground-state self-consistent field (SCF) calculations. The construction of the subsets of auxiliary functions is discussed, and the details of the implementation are presented. To assess the efficiency of the algorithm, an extensive test set was compiled incorporating well-known examples relevant in the field of photochemistry [265].