MATERIAL Y MÉTODO
5.1. Discusión de resultados
Technique
The results obtained with the test set of small molecules (Sec. 4.2.1) and the linear alkanes (Sec. 4.2.2) are very encouraging, however, due to the reasons discussed in Secs. 2.5 and 2.7, for large molecular systems the numerical errors introduced by the LEDO approximation can lead to problems with the SCF convergence. Here, the use of the projection technique (Sec. 2.7) for such critical systems shall be assessed. To analyze the effect of the projection technique we have performed a series of test calculations on naphthalene, anthracene, polyynes and three isomers of C20. These compounds have been chosen since the SCF for LEDO-DFT with
the present auxiliary orbitals (Table 4.1) fails to converge. Note that the polyynes have already been reported [225] to be critical with respect to the SCF conver- gence. The isomers of C20enable an analysis of the SCF convergence behavior for
systems of fixed size but differing numbers of significant overlap densities ρµAνB.
These are increasing from the ring over the bowl to the cage isomer. We there- fore expect the most critical behavior for the cage isomer. The ring isomer of C20, however, is a special case since it can be regarded as a cyclic polyyne. The
starting structures for the optimizations of the ring, bowl and cage isomer have been taken from the work of Grossman et al.2 [226] The calculations have been
performed as described at the beginning of Sec. 4.2. For the structure optimiza- tions of the C20 isomers, however, the change of the total energy between two
steps was only required to be smaller than 10−6au. A threshold for the maximum allowed condition number of cmax = 4000 was employed in all calculations since
LEDO-DFT calculations do not converge for polyynes with chain lengths from six carbon atoms onwards and the ring isomer of C20 for values of cmax ≥ 5000.
Restricting the maximum condition number to smaller values, on the other hand, leads to increasing errors. Of course, the condition number depends on the posi-
tion of the atoms, and it consequently changes during the course of a structure optimization. However, in general, these changes can be expected to be small, which means that no additional basis set superposition error (BSSE)-like effect should be observed. Indeed, we have observed that the number of omitted COOs at the beginning and at the end of a structure optimization is identical in most of the cases. In the following we should like to report how electronic energies and structural parameters are affected by the projection technique.
Table 4.7 summarizes the errors which are introduced by the projection tech- nique and by the LEDO approximation. Reference for the LEDO-DFT calcu- lations are conventional DFT calculations employing the projection technique. Let us first comment on the errors introduced by the projection technique. As expected, for similar systems the error in the total energy grows with increasing number M of eliminated COOs. The maximum deviation of the bond distances remains very small in all cases. At first sight, the errors in bond distances for the cage isomer of C20 seem large. However, the maximum deviation of bond
distances of 0.18 pm for a calculation with conventional density fitting [127] (RI- J ) is of the same order of magnitude.
The errors in the total energies due to the LEDO approximation are of the order of 10−3au. The errors in the bond distances are smaller than 0.1 pm for most molecules and only little larger for the bowl and cage isomers of C20 with
deviations of 0.23 pm and 0.19 pm, respectively. The errors introduced by the LEDO approximation are of the same order as the errors introduced by the pro- jection technique and we therefore conclude that the application of the projection technique is justified. None of the investigated molecules posed any difficulties for LEDO-DFT with a value of cmax= 4000 and we assume it to be reliable for other
calculations as well. We therefore suggest to generally employ the projection technique in combination with the LEDO approximation.
4.2.4
Some “Real Life” Examples
In quantum chemical methods, apart from structure determinations, it is fre- quently important to estimate energy differences with accuracy rather than the absolute quantities of reactant and product. We therefore have investigated the
Table 4.7: Application of the projection technique with cmax = 4000: Errors
of total energies (∆E1 = E proj DFT − EDFT and ∆E2 = E proj LEDO− E proj DFT in 10 −3au)
and maximum absolute deviations of bond distances (∆d1 = dprojDFT − dDFT and
∆d2 = dprojLEDO − d proj
DFT in pm) for conventional DFT and LEDO-DFT. M is the
number of canonically orthogonalized orbitals eliminated by the projection.
DFT LEDO-DFT M ∆E1 ∆d1 ∆E2 ∆d2 Naphthalene 3 2.11 0.06 −1.20 0.09 Anthracene 8 4.87 0.09 −2.98 0.05 C6H2 3 3.99 0.09 0.22 0.03 C8H2 5 6.65 0.14 0.34 0.03 C10H2 6 7.99 0.10 0.48 0.05 C12H2 8 10.73 0.13 0.83 0.05 C14H2 9 12.06 0.10 0.88 0.07 C16H2 11 14.80 0.12 1.40 0.05 C18H2 13 17.53 0.13 1.87 0.05 C20H2 14 18.87 0.11 2.00 0.06 C20 (ring) 13 10.65 0.16 −2.18 0.02 C20 (bowl) 14 4.19 0.12 −2.60 0.23 C20 (cage) 14 9.41 0.31 7.27 0.19
performance of LEDO-DFT for some representative energy differences of three different orders of magnitude. We have computed the dissociation energy of ben- zene into three acetylene molecules as the prototype for a reaction in which the size of the reactant and product molecules differ. The rotational barrier of ethane and the relative stabilities of different isomers of C2H6O, C4H4, and C6H6 serve
as test systems in which additional errors introduced by non-additivity are ex- cluded. All calculations have been performed as described at the beginning of Sec. 4.2 and the results are listed in Tables 4.8 and 4.9.
Table 4.8: Reaction energy for the dissociation of benzene into three acetylene molecules as calculated with conventional DFT and LEDO-DFT. Calculations in which the projection technique was employed are marked by an asterisk. All energies and energy differences are given in au.
EDFT ELEDO Error
C6H6 −232.088927 −232.087863∗ 0.001064
C2H2 −77.270689 −77.270657 0.000032
Dissociation 0.276860 0.275892 −0.000968
Table 4.9: Relative stabilities of isomers of C2H6, C2H6O, C4H4 and C6H6
computed using conventional DFT and LEDO-DFT. Calculations in which the projection technique was employed are marked by an asterisk. All energies and energy differences are given in au.
E and ∆E DFT LEDO-DFT Error C2H6 staggered ethane −79.766975 −79.767021 −0.000046 eclipsed ethane −79.762432 −79.762144 0.000288 0.004543 0.004877 0.000334 C2H6O trans-Ethanol −154.926738 −154.926771 −0.000033 cis-Ethanol −154.924647 −154.924783 −0.000136 0.002091 0.001988 −0.000103 Dimethylether −154.912564 −154.912513 0.000051 0.014174 0.014258 0.000084 C4H4 Cyclobutadiene −154.572950 −154.572859 0.000091
E and ∆E DFT LEDO-DFT Error Tetrahedrane −154.548994 −154.548757 0.000237 0.023956 0.024102 0.000146 C6H6 Benzene −232.088920 −232.087863∗ 0.001057 Benzvalene −231.971679 −231.968558∗ 0.003121 0.117241 0.119305 0.002064 Dewar benzene −231.961620 −231.960587∗ 0.001003 0.127300 0.127276 −0.000024 Prismane −231.912299 −231.912036∗ 0.000263 0.176621 0.175827 −0.000794 Bicyclopropenyl −231.894209 −231.893997∗ 0.000212 0.194711 0.193886 −0.000845
Although the relative errors do increase as the quantity being computed de- creases, the smallest energies — the rotational barrier of ethane and the relative stability of cis-ethanol — are in error by only 7.4 % and 4.9 %, respectively. All other relative errors range from the excellent values of 0.02 % for the relative stability of Dewar benzene to 1.8 % for the relative stability of Benzvalene. Ac- cording to these results and the data collected in the previous Secs., the relative error in energy differences calculated with LEDO-DFT can be expected to be in between 1 % and 10% for values of the order of 10−3au and to be 1% or smaller for larger energy differences. It should furthermore be taken into account that the error arising from the use of approximate XC functionals and incomplete AO basis sets can be expected to be much larger than the consistently small errors introduced by the LEDO approximation. Despite the fact that the LEDO-DFT energy is not a bound to the exact energy, which means that the error in total energies can be both positive and negative, we conclude from our results that the prediction of energy differences such as relative stabilities or reaction energies with LEDO-DFT does not suffer from serious drawbacks.
The molecular systems considered so far have been chosen on the basis of a systematical assessment of LEDO-DFT. Most of these are not of great interest in the present chemical or biochemical research. We therefore finally should like to present some test calculations on a set of molecules better reflecting the potential interests of, e.g., life sciences. The calculations have been performed as described at the beginning of Sec. 4.2, however, the change of the total energy between two structure optimization steps was only required to be smaller than 10−6au and the projection technique with cmax = 4000 as recommended in Sec. 4.2.3
was employed for LEDO-DFT. Results for the following molecules (in alphabetic order) are presented in Table 4.10:
• Acyclovir (C8H11N5O3, an antiviral agent for treating herpes),
• Adrenaline (C9H13NO3, a hormone and neurotransmitter),
• Caffeine (C8H10N4O2, an alkaloid found in coffee beans and other plants),
• Capillin (C12H8O, an acetylenic ketone with fungicidal activity),
• (+)-trans-chrysanthemic acid (C10H16O2, a terpene constituent of the flower
Chrysanthemum cinerariaefolium),
• S-Ibuprofene (C13H18O2, an anti-inflammatory and analgetic drug),
• LSD (C20H25N3O, a potent hallucinogen),
• Nicotine (C10H14N2, a carcinogenic alkaloid found, e.g., in tobacco),
• α-Pinene (C10H16, a major constituent of turpentine),
• Valium (C16H13ClN2O, a sedative of the benzodiazepine family) and
• Vitamin C (C6H8O6, an acid with antioxidant properties essential for life).
Caffeine, capillin and acyclovir possess Cs symmetry. All other molecules are not
symmetric, as is typical for most compounds of natural origin or of relevance for pharmaceutical or biochemical applications. While the maximum error in bond distances is as high as 0.59 pm in the case of LSD, the RMS deviations clearly show that the overall agreement of the structures is actually very good. For
Table 4.10: Accuracy of LEDO-DFT calculations: Errors of computed total en- ergies (∆E = ELEDO−EDFTin 10−3au) and dipole moments (∆µ = µLEDO−µDFT
in 10−3au) and maximum absolute and root-mean-square (RMS) deviations of bond distances (∆d in pm) and bond angles (∆γ in degrees) for some represen- tative molecules.
∆d ∆γ
Molecule ∆E max RMS max RMS ∆µ
Acyclovir −3.66 0.20 0.07 0.5 0.2 −10.2 Adrenaline −2.72 0.25 0.09 0.5 0.2 35.5 Caffeine −0.75 0.10 0.06 0.2 0.1 23.6 Capillin 3.61 0.49 0.17 0.6 0.2 −23.6 Chrys. acid −4.04 0.38 0.12 0.3 0.2 13.6 S-Ibuprofen −4.32 0.55 0.15 0.4 0.2 49.6 LSD −8.01 0.59 0.15 1.3 0.3 26.3 Nicotine −0.02 0.30 0.10 0.4 0.1 0.3 α-Pinene −3.36 0.31 0.13 0.5 0.2 8.1 Valium −5.00 0.25 0.10 0.6 0.2 6.1 Vitamin C −2.08 0.23 0.07 0.2 0.1 −23.1
the errors introduced by the LEDO approximation into the bond angles we can observe a similar tendency as for the linear alkanes (cf. Sec. 4.2.2). The maximum errors are around 0.5 degrees or smaller with exception of LSD, which shows a relatively large deviation of 1.3 degrees. The RMS values, however, are again encouragingly low. Taking into account the errors in the structure parameters, all dipole moments are reproduced with very good accuracy. The largest relative error of 11.7 % is actually observed for α-Pinene, because the dipole moment as calculated without LEDO approximation is as low as 71.7×10−3au. We conclude that LEDO-DFT indeed seems to be reliable for investigations of systems as complex as presented in this section.
0 5 10 15 20 25 30 Chain length 0 50 100 150 t / min conventional DFT (semi-direct) RI-DFT (incore) LEDO-DFT (incore) XC XC (LEDO) diagonalization 0 5 10 15 20 25 30 Chain length 0 10 20 30 t / min conventional DFT RI-DFT LEDO-DFT XC XC (LEDO)
Figure 4.8: CPU times (in minutes on a 2 GHz Intel Xeon processor) for DFT single point (left) and analytical gradient (right) calculations on linear alkanes with conventional DFT, conventional density fitting (RI-DFT) and LEDO ap- proximation.