3.4 Results and discussion
3.4.2 Accuracy: NMR shieldings
In this section we use some of the systems in Figure3.4to examine how much the DLPNO-MP2 isotropic shielding constants deviate from the RI-DLPNO-MP2 values, as we tighten the DLPNO thresholds. The mean absolute errors (MAEs) vs RI-MP2 for different values of TCutPNOare shown in Figure3.5a for the different nuclides in the systems ATP4−, caffeine, ebselen, and penicillin. We note that the errors diminish rapidly with decreasingTCutPNO and at the default value of 10−8for valence orbitals, they are much lower than the inherent error in MP2. For example, the MAE for C is 0.1 ppm, which is an order of magnitude smaller than the average deviation from CCSD(T) (see Section 2.3.3 and ref. 47). It is also noteworthy that the results are insensitive to core PNO truncation, except for the heavier elements, particularly P and Se. This may be in part because the pcSseg-2 basis set does not fully capture core correlation effects (see Section 2.3.6).
For simplicity in the discussion we want a single measure of the DLPNO error, however,
0.00
a 0.05 H
0.0 2.5 5.0 7.5
MAE / ppm
C N O P
0 10
20 S
Se
10
510
610
710
810
910
1010
1110
12T
CutPNOValence
0 1 2 3
MAWE / %
b
10
510
610
710
810
910
1010
1110
12T
CutPNOCore
Figure 3.5: Mean absolute error (a) and mean absolute weighted error (b) vs RI-MP2 in the shieldings of different nuclei in the systems ATP4−, caffeine, ebselen, and penicillin.
TCutPNO was varied for either the valence (left) or core (right) orbitals, with the other parameter set to 10−12 and 10−10, respectively. The line and marker styles in the legend apply to all subplots.
the shielding constants have very different magnitudes for different nuclides. Thus, we define an absolute weighted error (AWE), whereby we scale the error in the calculated isotropic shielding (e.g. vs RI-MP2) by a factor wA, derived from the experimental range of chemical shifts for nuclide A:
AWE =
σiDLPNO-MP2−σiRI-MP2
wA ×100%, i∈A (3.149)
wA= 1
2 δAmax−δminA
(3.150) We use the values for δminA and δAmax shown in Table 3.5. A reasonable target accuracy is a mean AWE (MAWE) below 0.5%, which corresponds to 0.5 ppm for 13C, 0.03 ppm for 1H, and about 1 ppm for 15N and 17O. As can be seen from Figure 3.5b, the MAWE
Table 3.5: Experimental chemical shift ranges (ppm) for different nuclides used for scaling in MAWE calculations.
A δminA δmaxA wA
1H -1 12 6.5
13C 0 200 100
15N 0 900 450
17O -40 1120 580
19F -300 400 350
33S -290 670 480
31P -180 250 215
77Se -1000 2000 1500
faithfully represents the trends in Figure3.5a, while also allowing us to combine the data for different nuclides, as they now have the same order of magnitude. The same data
10
510
610
710
810
910
1010
1110
12Valence T
CutPNO0.00 0.25 0.50 0.75 1.00 1.25
MAWE / %
ATP
4caffeine ebselen penicillin
10
510
610
710
810
910
1010
1110
12Core T
CutPNO0.02 0.04 0.06 0.08
Figure 3.6: Mean absolute weighted error vs RI-MP2 in the shieldings of the systems ATP4−, caffeine, ebselen, and penicillin. TCutPNO was varied for either the valence (left) or core (right) orbitals, with the other parameter set to 10−12 and 10−10, respectively.
The line and marker styles in the legend apply to both subplots.
are displayed in Figure 3.6, separating the different systems and averaging over all nuclei.
Here we can see that ATP4− is slightly more sensitive to core PNO truncation, probably because it contains all the phosphorus atoms in the dataset. The same system also suffers
from a PNO response instability at valence TCutPNO = 10−9 with MaxAE = 1.51 ppm for
15N vs 0.59 and 0.04 ppm for the same nucleus atTCutPNO = 10−8 and TCutPNO = 10−10, respectively. With the level shift-based adjustment (see Section 3.2.3), this error drops to 0.17 ppm at valence TCutPNO = 10−9.
10
310
210
1T
CutDO0.0 0.5 1.0 1.5
MAWE / %
T
S= 10
810
310
210
1T
CutDOT
S= 10
5ATP
4caffeine ebselen penicillin
Figure 3.7: Mean absolute weighted error vs RI-MP2 in the shieldings of the systems ATP4−, caffeine, ebselen, and penicillin. TCutDOwas varied andTS was set to either 10−8 (left) or 10−5 (right). The line and marker styles in the legend apply to both subplots.
The influence ofTCutDO (at a conservative value ofTCutPNO) is shown in Figure3.7 for the same set of molecules. In the left subplot the redundant PAO domains were truncated with the default eigenvalue threshold TS = 10−8, which leads to some numerical issues, as discussed in Section3.2.2, seen in the figure as large kinks in the curves for some molecules.
In the right subplot a value of TS = 10−5 was used which leads to smooth convergence of the results towards the RI-MP2 reference. At the default value of TCutDO= 10−2, the MAWE over the whole dataset is under 0.05%.
10
810
610
410
2F
Cut0.00 0.05 0.10 0.15 0.20 0.25 0.30
MAWE / %
ATP
4caffeine ebselen penicillin
Figure 3.8: Mean absolute weighted er-ror vs RI-MP2 in the shieldings of the sys-tems ATP4−, caffeine, ebselen, and peni-cillin with varyingFCut.
10
810
610
410
2T
CutPre0 2 4 6
MAWE / %
ATP
4caffeine ebselen penicillin no E
PreFigure 3.9: Mean absolute weighted error vs RI-MP2 in the shieldings of the systems ATP4−, caffeine, ebselen, and penicillin.
TCutPre was varied and dipole estimate-based contributions from the screened-out pairs were either included (solid lines) or excluded (dotted lines).
Setting TCutPNO to a conservative value, we examine the effect of FCut in Figure 3.8.
Apparently, it imparts negligible errors on the shieldings, especially at the default value of 10−5.
Turning to the pair prescreening threshold TCutPre, its effect on the shielding MAWE is shown in Figure 3.9. Despite the relatively small systems, using a large TDOij = 0.05 leads a significant number of pairs to be screened-out: 50–80% at TCutPre = 10−4 and 20–50% at TCutPre = 10−6. Even so, the effect on the shieldings is minor. Neglecting the contributions from the prescreening correction ∆EPre also does not worsen the results significantly for TCutPre ≤ 10−4, however, calculating these contributions takes a very small proportion of the computation time, so there are no savings to be made by skipping them. On the other hand, the default threshold values of TDOij = 10−5 andTCutPre = 10−6 might be too conservative for NMR shielding calculations.
H (133) 0.3
0.2 0.1 0.0 0.1 0.2
Shielding error vs RI-MP2 RIJONX / ppm C (169) N (19) O (21) F (6) P (3)
0.5 0.0 0.5 1.0 1.5 2.0 2.5
S (1) Se (1) 0
2 4 6 8 LoosePNO RIJONX
LoosePNO RIJCOSX NormalPNO RIJONX
NormalPNO RIJCOSX TightPNO RIJCOSX RI-MP2 RIJCOSX
Figure 3.10: Distribution of errors in the shieldings for the systems: (anth)2, ATP4−, caffeine, coronene, ebselen, penicillin, and tweezer complex. Calculations were per-formed (from left to right, respectively) using LoosePNO thresholds without and with the COSX approximation, using NormalPNO thresholds without and with COSX, and using TightPNO and RI-MP2 with COSX, and the errors were calculated against RI-MP2 without COSX for all datasets. The lines, boxes, and whiskers denote the median, in-terquartile range, and minimum/maximum values, respectively. The number of nuclei of each element are indicated in parentheses.
Having separately examined the effects of the most important DLPNO parameters on the shieldings, we now see what happens when all parameters are set to their default (NormalPNO) values. Figure 3.10 shows the error distributions for different nuclides over all 8 test systems in Figure 3.4. The NormalPNO/RIJONX distributions are fairly narrow and both the medians and the ranges (which are more relevant for chemical shift calculations) are much smaller than the inherent error in MP2. There is an overall bias towards overestimation of the shielding constants, so some error compensation can be expected in chemical shift calculations. The same figure also contains results with the LoosePNO threshold settings, which produce both larger systematic deviations and wider error distributions, at the borderline of what we may consider tolerable.
All calculations discussed so far were performed using the RIJONX approximation.
In practice, it is preferable to also approximate the exchange part of the Fock matrix, e.g. using the COSX approximation, as this significantly reduces the computation time.
However, the semi-numerical integration used in COSX introduces additional errors and numerical noise to the calculation, which may get amplified in DLPNO calculations.
Therefore, we have included in Figure 3.10 the error distributions from calculations em-ploying the RIJCOSX approximation with a rather large set of integration grids (corre-sponding to the DefGrid3keyword). Though the DLPNO and COSX errors are basically cumulative, the latter are fairly small so the conclusions from the previous paragraph still apply for the most part. One exception is the 1H shieldings, where the combined NormalPNO and COSX errors lead to a maximum deviation of 0.2 ppm and a spread of almost 0.4 ppm, which is larger than the method error and could in principle lead to a wrong assignment, compared to RI-MP2. This can be remedied by using larger COSX grids, in particular for the CPSCF and Z-CPSCF equations, which appear to be the main source of the deviations. As expected, tightening TCutPNO and TCutDO from LoosePNO to NormalPNO to TightPNO systematically reduces the errors towards the RI-MP2 reference.
0.00 0.02
MaxAE / ppm
H linear H -helix
2 4 6 8 10 12 14
Number of glycine monomers 0.0
0.2 0.4 0.6 0.8 1.0
MaxAE / ppm
C linear
C -helix N linear
N -helix O linear O -helix
Figure 3.11: Maximum absolute errors in the shieldings of different nuclides from Nor-malPNO calculations vs RI-MP2 for polyglycine chains in either linear (dotted lines) or α-helix conformation (solid lines). RIJCOSX was used throughout.
As the effect of the local approximations becomes more significant with increasing system size, the relative error in the properties can also be expected to increase, up to the point where all approximations are fully active: orbital domains have reached a constant size, etc. For an intrinsic (i.e. local) property like NMR shielding this means that absolute errors should also reach an upper limit for large enough systems. To examine this rela-tionship, we use the calculations on the glycine chain systems, discussed in Section 3.4.4.
Figure 3.11shows the maximum absolute errors in the shieldings of different nuclides, as a function of the chain length. Apparently, the errors do increase slightly with the system size and the effect is more pronounced for the less sparse α-helix conformation. However, the errors do not grow indefinitely, leveling off around (gly)10, and are within the ranges shown in Figure3.10. A notable outlier is an oxygen nucleus in linear (gly)3 with a rather large error of 0.97 ppm for 17O vs 0.25 and 0.41 ppm for (gly)2 and (gly)4, respectively.
Using the level shift-based adjustment, discussed in Section 3.2.3, the MaxAE for 17O in linear (gly)3 is reduced to 0.42 ppm, consistent with the slightly oscillating trend.