Schreiner and Grimme have examined a few compounds (see these previous posts) with long C-C bonds that are found in congested systems where dispersion greatly aids in stabilizing the stretched bond. Their new paper¹ continues this theme by examining 1 (again) and 2, using computations, and x-ray crystallography and gas-phase rotational spectroscopy and electron diffraction to establish the long C-C bond.

The distance of the long central bond in 1 is 1.647 Å (x-ray) and 1.630 Å (electron diffraction). Similarly, this distance in 2 is 1.642 Å (x-ray) and 1.632 Å (ED). These experiments discount any role for crystal packing forces in leading to the long bond.

A very nice result from the computations is that most functionals that include some dispersion correction predict the C-C distance in the optimized structures with an error of no more than 0.01 Å. (PW6B95-D3/DEF2-QZVP structures are shown in Figure 1.) Not surprisingly, HF and B3LYP without a dispersion correction predict a bond that is too long.) MP2 predicts a distance that is too short, but SCS-MP2 does a very good job.

Figure 1. PW6B95-D3/DEF2-QZVP optimized structures of 1 and 2.

References

1) Fokin, A. A.; Zhuk, T. S.; Blomeyer, S.; Pérez, C.; Chernish, L. V.; Pashenko, A. E.; Antony, J.; Vishnevskiy, Y. V.; Berger, R. J. F.; Grimme, S.; Logemann, C.; Schnell, M.; Mitzel, N. W.; Schreiner, P. R., "Intramolecular London Dispersion Interaction Effects on Gas-Phase and Solid-State Structures of Diamondoid Dimers." J. Am. Chem. Soc. 2017, 139, 16696-16707, DOI: 10.1021/jacs.7b07884.

InChIs

1: InChI=1S/C28H38/c1-13-7-23-19-3-15-4-20(17(1)19)24(8-13)27(23,11-15)28-12-16-5-21-18-2-14(9-25(21)28)10-26(28)22(18)6-16/h13-26H,1-12H2
InChIKey=MMYAZLNWLGPULP-UHFFFAOYSA-N

2: InChI=1S/C26H34O2/c1-11-3-19-15-7-13-9-25(19,21(5-11)23(27-13)17(1)15)26-10-14-8-16-18-2-12(4-20(16)26)6-22(26)24(18)28-14/h11-24H,1-10H2
InChIKey=VPBJYHMTINJMAE-UHFFFAOYSA-N

The Grimme group has examined all 1812 C₆₀ isomers, in part to benchmark some computational methods.¹ They computed all of these structures at PW6B95-D3/def2-QZVP//PBE-D3/def2-TZVP. The lowest energy structure is the expected fullerene 1 and the highest energy structure is the nanorod 2 (see Figure 1).

Figure 1. Optimized structures of the lowest (1) and highest (2) energy C₆₀ isomers.

About 70% of the isomers like in the range of 150-250 kcal mol^-1 above the fullerene 1, and the highest energy isomer 2 lies 549.1 kcal mol^-1 above 1. To benchmark some computational methods, they selected the five lowest energy isomers and five other isomers with higher energy to serve as a new database (C60ISO), with energies computed at DLPNO-CCSD(T)/CBS*. The mean absolute deviation of the PBE-D3/def2-TZVP relative energies with the DLPNO-CCSD(T)/CBS* energies is relative large 10.7 kcal mol^-1. However, the PW6B95-D3/def2-QZVP//PBE-D3/def2-TZVP method is considerably better, with a MAD of only 1.7 kcal mol^-1. This is clearly a reasonable compromise method for fullerene-like systems, balancing accuracy with computational time.

They also compared the relative energies of all 1812 isomers computed at PW6B95-D3/def2-QZVP//PBE-D3/def2-TZVP with a number of semi-empirical methods. The best results are with the DFTB-D3 method, with an MAD of 5.3 kcal mol^-1.

References

1) Sure, R.; Hansen, A.; Schwerdtfeger, P.; Grimme, S., "Comprehensive theoretical study of all 1812 C₆₀ isomers." Phys. Chem. Chem. Phys. 2017, 19, 14296-14305, DOI: 10.1039/C7CP00735C.

InChIs

1: InChI=1S/C60/c1-2-5-6-3(1)8-12-10-4(1)9-11-7(2)17-21-13(5)23-24-14(6)22-18(8)28-20(12)30-26-16(10)15(9)25-29-19(11)27(17)37-41-31(21)33(23)43-44-34(24)32(22)42-38(28)48-40(30)46-36(26)35(25)45-39(29)47(37)55-49(41)51(43)57-52(44)50(42)56(48)59-54(46)53(45)58(55)60(57)59
InChIKey=XMWRBQBLMFGWIX-UHFFFAOYSA-N

2: InChI=1S/C60/c1-11-12-2-21(1)31-41-32-22(1)3-13(11)15-5-24(3)34-43(32)53-55-47-36-26-6-16-17-7(26)28-9-19(17)20-10-29-8(18(16)20)27(6)37-46(36)54(51(41)55)52-42(31)33-23(2)4(14(12)15)25(5)35-44(33)58-56(52)48(37)39(29)50-40(30(9)10)49(38(28)47)57(53)59(45(34)35)60(50)58
InChIKey=AGZHNPDQKMDYHI-UHFFFAOYSA-N

Computed NMR spectra have become a very useful tool in identifying chemical structures. I have blogged on this multiple times. A recent trend has been the development of computational procedures that lead to computed spectra (again, see that above link). Now, Grimme, Neese and coworkers have offered their approach to computed NMR spectra, including spin-spin splitting.¹

Their procedure involves four distinct steps.

1. Generation of the conformer and rotamer space. This is a critical distinctive element of their method in that they take a number of different tacks for sampling conformational space to insure that they have identified all low-energy structures. This involves a combination of normal mode following, genetic structure crossing (based on genetic algorithms for optimization), and molecular dynamics. Making this all work is their choice of using the computational efficient GFN-xTB² quantum mechanical method.
2. The low-energy structures are then subjected to re-optimization at PBEh-3c and then single-point energies obtained at DSD-BLYP-D3/def2-TZVPP including treatment of solvation by COSMO-RS. The low-energy structures that contribute 4% or more of the Boltzmann-weighted population are then carried forward.
3. Chemical shifts and spin-spin coupling constants are then computed with the PBE0 method and the pcS and pcJ basis sets developed by Jensen for computing NMR shifts.³
4. Lastly, the chemical shifts and coupling constants are averaged and the spin Hamiltonian is solved.

The paper provides a number of examples of the application of the methodology, all with quite good success. The computer codes to run this method are available for academic use from xtb@thch.uni-bonn.de.

References

1) Grimme, S.; Bannwarth, C.; Dohm, S.; Hansen, A.; Pisarek, J.; Pracht, P.; Seibert, J.; Neese, F., "Fully Automated Quantum-Chemistry-Based Computation of Spin–Spin-Coupled Nuclear Magnetic Resonance Spectra." Angew. Chem. Int. Ed. 2017, 56, 14763-14769, DOI: 10.1002/anie.201708266.

2) Grimme, S.; Bannwarth, C.; Shushkov, P., "A Robust and Accurate Tight-Binding Quantum Chemical Method for Structures, Vibrational Frequencies, and Noncovalent Interactions of Large Molecular Systems Parametrized for All spd-Block Elements (Z = 1–86)." J. Chem. Theory Comput. 2017, 13, 1989-2009, DOI: 10.1021/acs.jctc.7b00118.

3) Jensen, F., "Basis Set Convergence of Nuclear Magnetic Shielding Constants Calculated by Density Functional Methods." J. Chem. Theory Comput. 2008, 4, 719-727, DOI: 10.1021/ct800013z.

Following on previous work (see these posts on ladderane and hexaphenylethane), Schreiner, Grimme and co-workers have examined the structure of the all-meta tri(di-t-butylphenyl)methane dimer 1₂.¹ In the study of hexaphenylethane,² Schreiner and Grimme note that t-butyl groups stabilize highly congested structures through dispersion, identifying them as “dispersion energy donors”.³ The idea here is that the dimer of 1 will be stabilized by these many t-butyl groups. In fact, the neutron diffraction study of the crystal structure of 1₂ shows an extremely close approach of the two methane hydrogens of only 1.566 Å, the record holder for the closest approach of two formally non-bonding hydrogen atoms.

To understand the nature of this dimeric structure, they employed a variety of computational techniques. (Shown in Figure 1 is the B3LYPD3^ATM(BJ)/def2-TZVPP optimized geometry of 1₂.) The HSE-3c (a DFT composite method) optimized crystal structure predicts the H^…H distance is 1.555 Å. The computed gas phase structure lengthens the distance to 1.634 Å, indicating a small, but essential, role for packing forces. Energy decomposition analysis of 1₂ at B3LYP-D3^ATM(BJ)/def2-TZVPP indicates a dominant role for dispersion in holding the dimer together. While 1₂ is bound by about 8 kcal mol^-1, the analogue of 1₂ lacking all of the t-butyl groups (the dimer of triphenylmethane 2₂) is unbound by over 8 kcal mol^-1. Topological electron density analysis does show a bond critical point between the two formally unbound hydrogen atoms, and the noncovalent interaction plot shows an attractive region between these two atoms.

Figure 1. ATM(BJ)/def2-TZVPP optimized geometry of 1₂, with most of the hydrogens suppressed for clarity. (Selecting the molecule will launch Jmol with the full structure, including the hydrogens.)

References

1) Rösel, S.; Quanz, H.; Logemann, C.; Becker, J.; Mossou, E.; Cañadillas-Delgado, L.; Caldeweyher, E.; Grimme, S.; Schreiner, P. R., "London Dispersion Enables the Shortest Intermolecular Hydrocarbon H···H Contact." J. Am. Chem. Soc. 2017, 139, 7428–7431, DOI: 10.1021/jacs.7b01879.

2) Grimme, S.; Schreiner, P. R., "Steric Crowding Can Stabilize a Labile Molecule: Solving the Hexaphenylethane Riddle." Angew. Chem. Int. Ed. 2011, 50 (52), 12639-12642, DOI: 10.1002/anie.201103615.

3) Grimme, S.; Huenerbein, R.; Ehrlich, S., "On the Importance of the Dispersion Energy for the Thermodynamic Stability of Molecules." ChemPhysChem 2011, 12 (7), 1258-1261, DOI: 10.1002/cphc.201100127.

InChIs

1: InChI=1S/C43H64/c1-38(2,3)31-19-28(20-32(25-31)39(4,5)6)37(29-21-33(40(7,8)9)26-34(22-29)41(10,11)12)30-23-35(42(13,14)15)27-36(24-30)43(16,17)18/h19-27,37H,1-18H3
InChIKey=VFNQDWKFTWSJAU-UHFFFAOYSA-N

Assessing when a molecular system might be subject to sizable static (non-dynamic) electron correlation, necessitating a multi-reference quantum mechanical treatment, is perhaps more art than science. In general one suspects that static correlation will be important when the frontier MO energy gap is small, but is there a way to get more guidance?

Grimme reports the use of fractional occupancy density (FOD) as a visualization tool to identify regions within molecules that demonstrate significant static electron correlation.¹ The method is based on the use of finite temperature DFT.^2,3

The resulting plots of the FOD for a series of test cases follow our notions of static correlation. Molecules, such as alkanes, simple aromatics, and concerted transition states show essentially no fractional orbital density. On the other hand, the FOD plot for ozone shows significant density spread over the entire molecule; the transition state for the cleavage of the terminal C-C bond in octane shows FOD at C₁ and C₂ but not elsewhere; p-benzyne shows significant FOD at the two radical carbons, while the FOD is much smaller in m-benzyne and is negligible in o-benzyne.

This FOD method looks to be a simple tool for evaluating static correlation and is worth further testing.

References

(1) Grimme, S.; Hansen, A. "A Practicable Real-Space Measure and Visualization of Static Electron-Correlation Effects," Angew. Chem. Int. Ed. 2015, 54, 12308-12313, DOI: 10.1002/anie.201501887.

(2) Mermin, N. D. "Thermal Properties of the Inhomogeneous Electron Gas," Phys. Rev. 1965, 137, A1441-A1443, DOI: 10.1103/PhysRev.137.A1441.

(3) Chai, J.-D. "Density functional theory with fractional orbital occupations," J. Chem. Phys 2012, 136, 154104, DOI: doi: 10.1063/1.3703894.

Grimme and coworkers have a featured article on computing host-guest complexes in a recent ChemComm.¹ They review the techniques his group has pioneered, particularly dispersion corrections for DFT and ways to treat the thermodynamics in moving from electronic energy to free energy. they briefly review some studies done by other groups. They conclude with a new study of eight different host guest complexes, three of which are shown in Figure 1.

1	2
3

Figure 1. TPSS-D3(BJ)/def2-TZVP optimized structures of 1-3.

These eight host-guest complexes are fairly large systems, and the computational method employed means some fairly long computations. Geometries were optimized at TPSS-D3(BJ)/def2-TZVP, then single point energy determined at PW6B95-D3(BJ)/def2-QZVP. Solvent was included using COSMO-RS. The curcurbituril complex 2 includes a counterion (chloride) along with the guest adamantan-1-aminium. Overall agreement of the computed free energy of binding with the experimental values was very good, except for 3 and the related complex having a larger nanohoop around the fullerene. The error is due to problems in treating the solvent effect, which remains an area of real computational need.

An interesting result uncovered is that the binding energy due to dispersion is greater than the non-dispersion energy for all of these complexes, including the examples that are charged or where hydrogen bonding may be playing a role in the bonding. This points to the absolute necessity of including a dispersion correction when treating a host-guest complex with DFT.

As an aside, you’ll note one of the reasons I was interested in this paper: 3 is closely related to the structure that graces the cover of the second edition of my book.

References

(1) Antony, J.; Sure, R.; Grimme, S. "Using dispersion-corrected density functional theory to understand supramolecular binding thermodynamics," Chem. Commun. 2015, 51, 1764-1774, DOI: 10.1039/C4CC06722C.

The importance of dispersion in determining molecular structure, even the structure of a single medium-sized molecule, is now well recognized. This means that quantum methods that do not account for dispersion might give very poor structures.

Grimme¹ takes an interesting new twist towards assessing the geometries produced by computational methods by evaluating the structures based on their rotational constants B₀ obtained from microwave experiments. He uses nine different molecules in his test set, shown in Scheme 1. This yields 25 different rotational constants (only one rotational constant is available from the experiment on triethylamine). He evaluates a number of different computational methods, particularly DFT with and without a dispersion correction (either the D3 or the non-local correction). The fully optimized geometry of each compound with each method is located to then the rotational constants are computed. Since this provides B_e values, he has computed the vibrational correction to each rotational constant for each molecule, in order to get “experimental” B_e values for comparisons.

Scheme 1.

Grimme first examines the basis set effect for vitamin C and aspirin using B3LYP-D3. He concludes that def2-TZVP or lager basis sets are necessary for reliable structures. However, the errors in the rotational constant obtained at B3LYP-D3/6-31G* is at most 1.7%, and even with CBS the error can be as large as 1.1%, so to my eye even this very small basis set may be completely adequate for many purposes.

In terms of the different functionals (using the DZVP basis set), the best results are obtained with the double hybrid B2PLYP-D3 functional where the mean relative deviation is only 0.3%; omitting the dispersion correction only increases the mean error to 0.6%. Common functionals lacking the dispersion correction have mean errors of about 2-3%, but with the correction, the error is appreciably diminished. In fact B3LYP-D3 has a mean error of 0.9% and B3LYP-NL has an error of only 0.6%. In general, the performance follows the Jacob’s Ladder hierarchy.

References

(1) Grimme, S.; Steinmetz, M. "Effects of London dispersion correction in density functional theory on the structures of organic molecules in the gas phase," Phys. Chem. Chem. Phys. 2013, 15, 16031-16042, DOI: 10.1039/C3CP52293H.

The role of dispersion in large systems is increasingly recognized as critical towards understanding molecular geometry. An interesting example is this study of acene dimers by Grimme.¹ The heptacene and nonacene dimers (1 and 2) were investigated with an eye towards the separation between the “butterfly wings” – is there a “stacked” conformation where the wings are close together, along with the “open” conformer?

The LPNO-CEPA/CBS potential energy surface of 1 shows only a single local energy minima, corresponding to the open conformer. B3LYP-D3 and B3LYP-NL, two different variations of dealing with dispersion (see this post), do a reasonable job at mimicking the LPNO-CEPA results, while MP2 indicates the stacked conformer is lower in energy than the open conformer.

B3LYP-D3 predicts both conformers for the nonacene dimer 2, and the optimized structures are shown in Figure 2. The stacked conformer is slightly lower in energy than the open one, with a barrier of about 3.5 kcal mol^-1. However in benzene solution, the open conformer is expected to dominate due to favorable solvation with both the interior and exterior sides of the wings.

open

stacked

Figure 1. B3LYP-D3/ef2-TZVP optimized structures of the open and stacked conformations of 2.

References

(1) Ehrlich, S.; Bettinger, H. F.; Grimme, S. "Dispersion-Driven Conformational Isomerism in σ-Bonded Dimers of Larger Acenes," Angew. Chem. Int. Ed. 2013, 41, 10892–10895, DOI: 10.1002/anie.201304674.

InChIs

1: InChI=1S/C60H36/c1-2-10-34-18-42-26-50-49(25-41(42)17-33(34)9-1)57-51-27-43-19-35-11-3-4-12-36(35)20-44(43)28-52(51)58(50)60-55-31-47-23-39-15-7-5-13-37(39)21-45(47)29-53(55)59(57)54-30-46-22-38-14-6-8-16-40(38)24-48(46)32-56(54)60/h1-32,57-60H
InChIKey=OYVUURMCCBPDLI-UHFFFAOYSA-N

2: InChI=1S/C76H44/c1-2-10-42-18-50-26-58-34-66-65(33-57(58)25-49(50)17-41(42)9-1)73-67-35-59-27-51-19-43-11-3-4-12-44(43)20-52(51)28-60(59)36-68(67)74(66)76-71-39-63-31-55-23-47-15-7-5-13-45(47)21-53(55)29-61(63)37-69(71)75(73)70-38-62-30-54-22-46-14-6-8-16-48(46)24-56(54)32-64(62)40-72(70)76/h1-40,73-76H
InChIKey=FTSMWBVFVPAXOB-UHFFFAOYSA-N

Basis set superposition error plagues all practical computations. This error results from the use of incomplete basis sets (thus pretty much all computations will suffer from this problem). The primary example of this error is in the formation of a supermolecule AB from the monomers A and B. Superposition occurs when in the computation of the supermolecule, basis functions centered on B are used to supplement the basis set of A, not to describe the bonding or interaction between the two monomers, but simply to better the description of the monomer A itself. Thus, BSSE always serves to increase the binding in the supermolecule. Recently, this concept has been extended to intramolecular BSSE, as discussed in these posts (A and B).

The counterpoise correction proposed by Boys and Bernardi corrects for the superposition by computing the energy of each monomer using the basis sets centered on both monomers, often referred to as ghost orbitals because the functions are used but not the nuclei upon which they are centered. This can overcorrect for superposition but is the only widely utilized approach to treat the problem. A variation on this approach is what has been suggested for the intramolecular
BBSE problem.

A major discouragement for wider use of counterpoise correction is its computational cost. Kruse and Grimme offer a semi-empirical approach that is extremely cost effective and appears to strongly mimic the traditional counterpoise correction.¹

They define the geometric counterpoise scheme (gCP) that provides an energy correction E_gCP that can be added onto the electronic energy. This term is defined as

Eq. (1)

where σ is an empirically fitted scaling term. The atomic contributions are defined as

where e^miss are the errors in the energy of an atom with a particular target basis set, relative to the energy with some large basis set:

(On a technical matter, the atomic terms are computed in an electric field of 0.6a.u. in order to get some population into higher energy orbitals.) The f_dec term is a decay function that relates to the distance between the atoms (R_nm) and the overlap
(S_nm):

where N^virt is the number of virtual orbitals on atom m, and α and β are fit parameters. Lastly, the overlap term comes from the integral of a single Slater orbital with coefficient

where ξ_s and ξ_p are optimized Slater exponents from extended Huckel theory, and η is the last parameter that needs to be fit.

There are four parameters and these need to be fit for each specific combination of method (functional) and basis set. Kruse and Grimme provide parameters for a number of combinations, and suggest that the parameters devised for B3LYP are suitable for other functionals.

So what is this all good for? They demonstrate that for a broad range of benchmark systems involving weak bonds, the that gCP corrected method coupled with the DFT-D3 dispersion correction provides excellent results, even with B3LYP/6-31G*! This allows one to potentially run a computation on very large systems, like proteins, where large basis sets, like TZP or QZP, would be impossible. In a follow-up paper,² they show that the B3LYP/6-31G*-gCP-D3 computations of a few Diels-Alder reactions and computations of strain energies of fullerenes match up very well with computations performed at significantly higher levels.

Once this gCP method and the D3 correction are fully integrated within popular QM programs, this combined methodology should get some serious attention. Even in the absence of this integration, these energy corrections can be obtained using the web service provided by Grimme at http://www.thch.uni-bonn.de/tc/gcpd3.

References

(1) Kruse, H.; Grimme, S. "A geometrical correction for the inter- and intra-molecular basis set superposition error in Hartree-Fock and density functional theory calculations for large systems," J. Chem. Phys 2012, 136, 154101-154116, DOI: 10.1063/1.3700154

(2) Kruse, H.; Goerigk, L.; Grimme, S. "Why the Standard B3LYP/6-31G* Model Chemistry Should Not Be Used in DFT Calculations of Molecular Thermochemistry: Understanding and Correcting the Problem," J. Org. Chem. 2012, 77, 10824-10834, DOI: 10.1021/jo302156p

This is a short post mainly to bring to the reader’s attention a couple of recent JCTC papers.

The first is a benchmark study by Hujo and Grimme of the geometries produced by DFT computations that are corrected for dispersion.¹ They use the S22 and S66 test sets that span a range of compounds expressing weak interactions. Of particular note is that the B3LYP-D3 method provided the best geometries, suggesting that this much (and justly) maligned functional can be significantly improved with just the simple D3 fix.

The second paper entails the description of Truhlar and Cramer’s latest iteration on their solvation model, namely SM12.² The main change here is the use of Hirshfeld-based charges, which comprise their Charge Model 5 (CM5). The training set used to obtain the needed parameters is much larger than with previous versions and allows for treating a very broad set of solvents. Performance of the model is excellent.

References

(1) Hujo, W.; Grimme, S. "Performance of Non-Local and Atom-Pairwise Dispersion Corrections to DFT for Structural Parameters of Molecules with Noncovalent Interactions," J. Chem. Theor. Comput. 2013, 9, 308-315, DOI: 10.1021/ct300813c

(2) Marenich, A. V.; Cramer, C. J.; Truhlar, D. G. "Generalized Born Solvation Model SM12," J. Chem. Theor. Comput. 2013, 9, 609-620, DOI: 10.1021/ct300900e