Here’s another extensive benchmarking study – this time on the use of TD-DFT to predict excitation energies.¹ This study looks at the performance of 28 different functionals, and compares the TD-DFT excitation energies against a data set of (a) computed vertical energies and (b) experimental energies. The performance is generally about the same for both data sets, with many functionals (especially the hybrid functionals) giving errors of about 0.3 eV. Performance can be a bit better when examining subclasses of compounds. For example, PBE0 and mPW1PW91 have a mean unsigned error of only 0.14 eV for a set of organic dyes.

References

(1) Jacquemin, D.; Wathelet, V.; Perpete, E. A.; Adamo, C., "Extensive TD-DFT Benchmark: Singlet-Excited States of Organic Molecules," J. Chem. Theory Comput., 2009, 5, 2420-2435, DOI: 10.1021/ct900298e

An emerging theme in this blog is Möbius systems, ones that can be aromatic or antiaromatic. Rzepa has led the way here, especially in examining annulenes with a twisted structure. Along with Schleyer and Schaefer, they have now explored a series of Möbius annulenes.¹ The particularly novel aspect of this new work is the examination of higher-order Möbius systems. In the commonly held notion of the Möbius strip, the strip contains a single half twist. Rzepa points out that the notion of twist must be considered as two parts, a part due to torsions and a part due to writhe.² We can think of the Möbius strip as formed by a ladder where the ends are connect such that the left bottom post connects with the top right post and the bottom right post connects with the top left post. Let’s now consider the circle created by joining the midpoints of each rug of the ladder. If this circle lies in a plane, then the torsion is π/N where N is the number of rungs in the ladder. But, the collection of midpoints does not have to lie in a plane, and if these points distort out of plane, that’s writhe and allows for less torsion in the strip.The sum of these two parts is called L_k and it will be an integral multiple of π. So the common Möbius strip has L_k = 1.

An example of a molecular analogue of the common Möbius strip is the annulene C₉H₉⁺ (1) – see figure 1. But Möbius strips can have more than one twist. Rzepa, Schleyer, and Schaefer have found examples with L_k = 2, 3, or 4. Examples are C₁₄H₁₄ (2) with one full twist (L_k = 2, two half twists), C₁₆H₁₆^2- (3) with three half twists, and C₂₀H₂₀²⁺ (4) with four half twists.

1	2
3	4

Figure 1. Structures of annulenes 1-4.

These annulenes with higher-order twisting, namely 2-4, are aromatic, as determined by a variety of measures. For example, all express negative NICS values, all have positive diagmagnetic exaltations, and all express positive isomerization stabilization energies (which are a measure of aromatic stabilization energy).

References

(1) Wannere, C. S.; Rzepa, H. S.; Rinderspacher, B. C.; Paul, A.; Allan, C. S. M.; Schaefer Iii, H. F.; Schleyer, P. v. R., "The Geometry and Electronic Topology of Higher-Order Charged M&oml;bius Annulenes" J. Phys. Chem. A 2009, ASAP, DOI: 10.1021/jp902176a

(2) Fowler, P. W.; Rzepa, H. S., "Aromaticity rules for cycles with arbitrary numbers of half-twists," Phys. Chem. Chem. Phys. 2006, 8, 1775-1777, DOI: 10.1039/b601655c.

As mentioned in Chapter 2 of my book, many post-HF methods predict that planar benzene has an imaginary frequency, whereby out-of-plane bending leads to a lower energy structure.¹ This anomaly was suggested to result from intramolecular basis set incompleteness.

Asturiol, Duran and Salvador provide more evidence that the root cause is intramolecular basis set superposition error.² They propose an extension of the standard counterpoise correction, which has been widely applied to interacting molecules. They divide the molecule into small fragments and apply the counterpoise correction to these fragments. For benzene, they use C-H or (CH)₂ fragments. With this counterpoise correction, the imaginary frequency corresponding to an out-of-plane distortion is removed for all combinations of either MP2 or CISD with the 6-31+G*, 6-311G or 6-311++G basis sets. The planar indenyl anion, which is found to have 4 imaginary frequencies at MP2/6-311G, has no imaginary frequencies when the counterpoise correction is used.

These authors have now shown that nucleic acid bases suffer from the same intramolecular superposition error.³ Uracil, thymine and guanine suffer from spurious imaginary frequencies with certain combinations of MP2 and Pople basis sets. However, all of these out-of-plane imaginary frequencies become real when the counterpoise correction is applied. The take-home message is to carefully mate the post-HF method and basis set combination – or else make the counterpoise correction!

References

(1) Moran, D.; Simmonett, A. C.; Leach, F. E.; Allen, W. D.; Schleyer, P. v. R.; Schaefer, H. F., III, "Popular Theoretical Methods Predict Benzene and Arenes To Be Nonplanar," J. Am. Chem. Soc. 2006, 128, 9342-9343, DOI: 10.1021/ja0630285

(2) Asturiol, D.; Duran, M.; Salvador, P., "Intramolecular basis set superposition error effects on the planarity of benzene and other aromatic molecules: A solution to the problem," J. Chem. Phys. 2008, 128, 144108, DOI: 10.1063/1.2902974

(3) Asturiol, D.; Duran, M.; Salvador, P., "Intramolecular Basis Set Superposition Error Effects on the Planarity of DNA and RNA Nucleobases," J. Chem. Theory Comput. 2009, 5, 2574-2581, DOI: 10.1021/ct900056u

Once more into the benzene dimer (see these previous posts: “Benzene dimer again“, “Benzene dimer“, “π-π stacking (part 2)“, “π-π stacking“)! Sherrill has published a detailed and impressive benchmark study of the benzene dimer in its three most important configurations: the D_6h stacked arrangement (1), the T-shaped arrangement (2) and the parallel displaced arrangement (3). ¹

First, they performed a careful extrapolation study to obtain accurate binding energies based on CCSD(T) with large basis sets. Then they compared the potential energy curves of the three configurations of benzene dimer obtained with this accurate method with those obtained with less computationally expensive methods. These alternates include RI-MP2, SCS-MP2 and a variety of different density functional. Their results are summarized in Table 1. The upshot is that the SCS-MP2 results are very similar to the much more expensive CCDS(T) values. And while the errors are a bit larger with the DFT methods, their performance is really quite good, especially given their dramatically lower costs. (Note that the “-D” indicates inclusion of Grimme’s dispersion correction term.) Particularly worth mentioning is the very fine performance of the MO6-2X functional.

Table 1. Binding energies (kcal mol^-1) of the three benzene dimers with different computational methods.

References

(1) Sherrill, C. D.; Takatani, T.; Hohenstein, E. G., "An Assessment of Theoretical Methods for Nonbonded Interactions: Comparison to Complete Basis Set Limit Coupled-Cluster Potential Energy Curves for the Benzene Dimer, the Methane Dimer, Benzene-Methane, and Benzene-H₂S" J. Phys. Chem. A 2009, ASAP, DOI: 10.1021/jp9034375

Here’s another fine paper from the Alonso group employing laser ablation molecular beam Fourier transform microwave spectroscopy coupled with computation to discern molecular structure. In this work they examine the low-energy tautomers of guanine.1 The four lowest energy guanine tautomers are shown in Figure 1. (Unfortunately, Alonso does not include the optimized coordinates of these structures in the supporting information – we need to more vigorously police this during the review process!) These tautomers are predicted to be very close in energy (MP2/6-311++G(d,p), and so one might expect to see multiple signals in the microwave originating from all four tautomers. In fact, they discern all four, and the agreement between the computed and experimental rotational constants are excellent (Table 1), especially if one applies a scaling factor of 1.004. Once again, this group shows the power of combined experiment and computations!

Figure 1. Four lowest energy (kcal mol^-1, MP2/6-311++G(d,p)) tautomers of guanine.

Table 1. Experimental and computed rotational constants (MHz) of the four guanine tautomers.

References

(1) Alonso, J. L.; Peña, I.; López, J. C.; Vaquero, V., "Rotational Spectral Signatures of Four Tautomers of Guanine," Angew. Chem. Int. Ed. 2009, 48, 6141-6143, DOI: 10.1002/anie.200901462

InChIs

Guanine: InChI=1/C5H5N5O/c6-5-9-3-2(4(11)10-5)7-1-8-3/h1H,(H4,6,7,8,9,10,11)/f/h8,10H,6H2
InChIKey=UYTPUPDQBNUYGX-GSQBSFCVCX