# Research

Due to their relevance in pharmaceuticals, organic semiconductor materials, and many other applications, the interest in molecular crystals has grown in the past decade. Electronic structure methods are a mandatory tool to model those systems accurately on a quantum mechanical level. While formally exact solutions can be formulated, real live applications have to be sufficiently fast in order to provide computed results in a reasonable time frame.

Thus, my research concentrates on the development and application of fast electronic structure methods, with particular focus on organic crystals and their polymorph prediction. The following sections highlight some of my recent research in the areas.

### London dispersion interactions

London dispersion interactions are also known as attractive part of the van der Waals forces and are omnipresent in all electronic systems. They arise as zero-point vibration of coupled charge fluctuations and are a purely quantum mechanically interaction without classical analog. The dispersion interaction can be viewed as the long-range limit of the correlation energy. For this, an exact expression can be formulated from the adiabatic fluctuation dissipation theorem

$\displaystyle E_{\text{c}} = -\frac{1}{2\pi} \int_{0}^{1} \!\textrm{d} \lambda \int \!\textrm{d} \mathbf{r} \textrm{d} \mathbf{r}' \frac{1}{|\mathbf{r}-\mathbf{r}'|} \times \int_{0}^{\infty} \!\textrm{d} \omega \left[ \chi_{\lambda}(\mathbf{r}, \mathbf{r}', i\omega) - \chi_{0}(\mathbf{r}, \mathbf{r}', i\omega) \right] \, .$

The dynamical response function involved is way to complicated to be computed for our systems of interest with about 10³ interacting electrons. Thus, a typical approach is the usage of an effective mean field theory like density funcational theory (DFT) or Hartree-Fock (HF) combined with a dispersion correction that approximates the above expression.

Fig. 1: Overview about dispersion-correction strategies to mean-field quantum chemical methods (left) and three main approximations used in the long-range perturbation theory of interfragment interactions (right).

Typically, the response function is locally partitioned into fragments (usually atoms) and subsequently the interaction is described perturbatively while the perturbing field is expanded into multipoles. Modern state-of-the-art dispersion corrections differ in (1) the order in the many-body perturbation theory, (2) the truncation of the mulipole expansion, (3) the partitioning in which the atom-in-molecules (or solid) dispersion coefficients are obtained, and (4) the shape of the damping function that connects the correction scheme with the underlying mean field model.

My work mostly involves the semiclassical D3 dispersion correction mainly developed by Stefan Grimme. This model exhibits a number of appealing features. First, it has a very low computational complexity and can be combined with intrinsically extremely fast electronic structure methods. This is due to its dependency on the molecular geometry only without density based information that requires integration of properties on a real space grid. Second, it can be applied for practically the whole periodic table and provides the correct long-range limit for all finite gap systems. Finally, the accuracy is excellent and similar to more complex density based procedures. For instance mass densities of organic crystals can be computed with small residuum error of about 1-3% and electronic lattice energies are close to or within the chemical accuracy of 1 kcal/mol. I have compiled a new benchmark set of ice polymorphs, which mainly confirms the accuracies though the focus is on electrostatic and induction interactions.

• S. Grimme, A. Hansen, J. G. Brandenburg, and C. Bannwarth, “Dispersion-corrected mean-field electronic structure methods,” Chem. Rev., vol. 116, pp. 5105-5154, 2016. doi:10.1021/acs.chemrev.5b00533

Mean-field electronic structure methods like Hartree−Fock, semilocal density functional approximations, or semiempirical molecular orbital (MO) theories do not account for long-range electron correlations (London dispersion interaction). Inclusion of these effects is mandatory for realistic calculations on large or condensed chemical systems and for various intramolecular phenomena (thermochemistry). This Review describes the recent developments (including some historical aspects) of dispersion corrections with an emphasis on methods that can be employed routinely with reasonable accuracy in large-scale applications. The most prominent correction schemes were classified into three groups: (i) nonlocal, density-based functionals, (ii) semiclassical C6-based, and (iii) one-electron effective potentials. The properties as well as pros and cons of these methods are critically discussed, and typical examples and benchmarks on molecular complexes and crystals are provided. Although there are some areas for further improvement (robustness, many-body and short-range effects), the situation regarding the overall accuracy is clear. Various approaches yield long-range dispersion energy with a typical relative error of 5\%. For many chemical problems, this accuracy is higher compared to the underlying mean-field method (i.e., a typical semilocal (hybrid) functional like B3LYP).

@article {brandenburg_ref18,
author = {S. Grimme and A. Hansen and J. G. Brandenburg and C. Bannwarth},
title = {Dispersion-corrected mean-field electronic structure methods},
journal = {{Chem. Rev.}},
volume = {116},
pages = {5105-5154},
year = {2016},
doi = {10.1021/acs.chemrev.5b00533},
url = {../wp-content/papercite-data/pdf/brandenburg_ref18.pdf},
abstract = {Mean-field electronic structure methods like Hartree−Fock, semilocal density functional approximations, or semiempirical molecular orbital (MO) theories do not account for long-range electron correlations (London dispersion interaction). Inclusion of these effects is mandatory for realistic calculations on large or condensed chemical systems and for various intramolecular phenomena (thermochemistry). This Review describes the recent developments (including some historical aspects) of dispersion corrections with an emphasis on methods that can be employed routinely with reasonable accuracy in large-scale applications. The most prominent correction schemes were classified into three groups: (i) nonlocal, density-based functionals, (ii) semiclassical C6-based, and (iii) one-electron effective potentials. The properties as well as pros and cons of these methods are critically discussed, and typical examples and benchmarks on molecular complexes and crystals are provided. Although there are some
areas for further improvement (robustness, many-body and short-range effects), the situation regarding the overall accuracy is clear. Various approaches yield long-range dispersion energy with a typical relative error of 5\%. For many chemical problems, this accuracy is higher compared to the underlying mean-field method (i.e., a typical semilocal (hybrid) functional like B3LYP).}
}

• J. G. Brandenburg and S. Grimme, “Dispersion corrected hartree-fock and density functional theory for organic crystal structure prediction,” Top. Curr. Chem., vol. 345, p. 1–23, 2014. doi:10.1007/128_2013_488
[BibTeX] [Abstract]

We present and evaluate dispersion corrected Hartree-Fock (HF) and Density Functional Theory (DFT) based quantum chemical methods for organic crystal structure prediction. The necessity of correcting for missing long-range electron correlation, also known as van der Waals (vdW) interaction, is pointed out and some methodological issues such as inclusion of three-body dispersion terms are discussed. One of the most efficient and widely used methods is the semi-classical dispersion correction D3. Its applicability for the calculation of sublimation energies is investigated for the benchmark set X23 consisting of 23 small organic crystals. For PBE-D3 the mean absolute deviation (MAD) is below the estimated experimental uncertainty of 1.3 kcal/mol. For two larger π-systems, the equilibrium crystal geometry is investigated and very good agreement with experimental data is found. Since these calculations are carried out with huge plane-wave basis sets they are rather time consuming and routinely applicable only to systems with less than about 200 atoms in the unit cell. Aiming at crystal structure prediction, which involves screening of many structures, a pre-sorting with faster methods is mandatory. Small, atom-centered basis sets can speed up the computation significantly but they strongly suffer from basis set errors. We present the recently developed geometrical counterpoise correction gCP. It is a fast semi-empirical method, which corrects for most of the inter- and intramolecular basis set superposition error. For HF calculations with nearly minimal basis sets, we additionally correct for short-range basis incompleteness. We combine all three terms in the HF-3c denoted scheme which performs excellently for the X23 sublimation energies with an MAD of only 1.5 kcal/mol, which is close to the huge basis set DFT-D3 result.

@article{brandenburg_ref06,
author = {J. G. Brandenburg and S. Grimme},
title = {Dispersion corrected Hartree-Fock and density functional theory for organic crystal structure prediction},
journal = {{Top. Curr. Chem.}},
year = {2014},
volume = {345},
pages = {1--23},
doi = {10.1007/128_2013_488},
abstract = {We present and evaluate dispersion corrected Hartree-Fock (HF) and Density Functional Theory (DFT) based quantum chemical methods for organic crystal structure prediction. The necessity of correcting for missing long-range electron correlation, also known as van der Waals (vdW) interaction, is pointed out and some methodological issues such as inclusion of three-body dispersion terms are discussed. One of the most efficient and widely used methods is the semi-classical dispersion correction D3. Its applicability for the calculation of sublimation energies is investigated for the benchmark set X23 consisting of 23 small organic crystals. For PBE-D3 the mean absolute deviation (MAD) is below the estimated experimental uncertainty of 1.3 kcal/mol. For two larger π-systems, the equilibrium crystal geometry is investigated and very good agreement with experimental data is found. Since these calculations are carried out with huge plane-wave basis sets they are rather time consuming and routinely applicable only to systems with less than about 200 atoms in the unit cell. Aiming at crystal structure prediction, which involves screening of many structures, a pre-sorting with faster methods is mandatory. Small, atom-centered basis sets can speed up the computation significantly but they strongly suffer from basis set errors. We present the recently developed geometrical counterpoise correction gCP. It is a fast semi-empirical method, which corrects for most of the inter- and intramolecular basis set superposition error. For HF calculations with nearly minimal basis sets, we additionally correct for short-range basis incompleteness. We combine all three terms in the HF-3c denoted scheme which performs excellently for the X23 sublimation energies with an MAD of only 1.5 kcal/mol, which is close to the huge basis set DFT-D3 result.}
}

• J. G. Brandenburg, T. Maas, and S. Grimme, “Benchmarking dft and semiempirical methods on structures and lattice energies for ten ice polymorphs,” J. Chem. Phys., vol. 142, p. 124104, 2015. doi:10.1063/1.4916070

Water in different phases under various external conditions is very important in bio-chemical systems and for material science at surfaces. Density functional theory methods and approximations thereof have to be tested system specifically to benchmark their accuracy regarding computed structures and interaction energies. In this study, we present and test a set of ten ice polymorphs in comparison to experimental data with mass densities ranging from 0.9 to 1.5 g/cm$^3$ and including explicit corrections for zero-point vibrational and thermal effects. London dispersion inclusive density functionals at the generalized gradient approximation (GGA), meta-GGA, and hybrid level as well as alternative low-cost molecular orbital methods are considered. The widely used functional of Perdew, Burke and Ernzerhof (PBE) systematically overbinds and overall provides inconsistent results. All other tested methods yield reasonable to very good accuracy. BLYP-D3$^{atm}$ gives excellent results with mean absolute errors for the lattice energy below 1 kcal/mol (7\% relative deviation). The corresponding optimized structures are very accurate with mean absolute relative deviations (MARDs) from the reference unit cell volume below 1\%. The impact of Axilrod-Teller-Muto (atm) type three-body dispersion and of non-local Fock exchange is small but on average their inclusion improves the results. While the density functional tight-binding model DFTB3-D3 performs well for low density phases, it does not yield good high density structures. As low-cost alternative for structure related problems, we recommend the recently introduced minimal basis Hartree-Fock method HF-3c with a MARD of about 3\%.

@article{brandenburg_ref12,
author = {J. G. Brandenburg and T. Maas and S. Grimme},
title = {Benchmarking DFT and semiempirical methods on structures and lattice energies for ten ice polymorphs},
journal = {{J. Chem. Phys.}},
year = {2015},
volume = {142},
pages = {124104},
doi = {10.1063/1.4916070},
url = {../wp-content/papercite-data/pdf/brandenburg_ref12.pdf},
abstract = {Water in different phases under various external conditions is very important in bio-chemical systems and for material science at surfaces. Density functional theory methods and approximations thereof have to be tested system specifically to benchmark their accuracy regarding computed structures and interaction energies. In this study, we present and test a set of ten ice polymorphs in comparison to experimental data with mass densities ranging from 0.9 to 1.5 g/cm$^3$ and including explicit corrections for zero-point vibrational and thermal effects. London dispersion inclusive density functionals at the generalized gradient approximation (GGA), meta-GGA, and hybrid level as well as alternative low-cost molecular orbital methods are considered. The widely used functional of Perdew, Burke and Ernzerhof (PBE) systematically overbinds and overall provides inconsistent results. All other tested methods yield reasonable to very good accuracy. BLYP-D3$^{atm}$ gives excellent results with mean absolute errors for the lattice energy below 1 kcal/mol (7\% relative deviation). The corresponding optimized structures are very accurate with mean absolute relative deviations (MARDs) from the reference unit cell volume below 1\%. The impact of Axilrod-Teller-Muto (atm) type three-body dispersion and of non-local Fock exchange is small but on average their inclusion improves the results. While the density functional tight-binding model DFTB3-D3 performs well for low density phases, it does not yield good high density structures. As low-cost alternative for structure related problems, we recommend the recently introduced minimal basis Hartree-Fock method HF-3c with a MARD of about 3\%.}
}

### Cost-efficient methods for organic crystals

Dispersion-corrected density functional theory (DFT-D) is in principle applicable to many interesting targets. However, the computational demands, for example, to sample a huge number of polymorphs, are still too high. I have (co-)developed a number of cost-efficient schemes to model molecular complexes and organic crystals in particular.

Fig. 2: Highlight of the different intermolecular interactions (left) and the real-space convergence behavior of the London dispersion interaction (right).

These range from semiempirical tight-binding methods (DFTB3-D3) to density functional based methods (PBEh-3c). Due to their partially empirical character, a rigorous benchmarking is mandatory. This involves energetic and geometric properties of both gas phase and solid phase systems. Some typical benchmark results are shown below with  the dispersion corrected meta generalized gradient approximated (GGA) functional TPSS-D3 in converged projector augmented plane wave (PAW) basis sets, the popular uncorrected B3LYP hybrid functional in a small single particle basis set of double-zeta quality, and the semiempirical density functional tight binding DFTB3. The importance of a proper dispersion correction is apparent and my recommended method TPSS-D3 and the low-cost alternative DFTB3-D3 perform excellently.

Fig. 3: Performance of various methods for the computation of gas and solid phase interaction energies and geometries, respectively.

• J. G. Brandenburg and S. Grimme, “Accurate modeling of organic molecular crystals by dispersion-corrected density functional tight-binding (DFTB),” J. Phys. Chem. Lett., vol. 5, p. 1785–1789, 2014. doi:10.1021/jz500755u

The ambitious goal of organic crystal structure prediction challenges theoretical methods regarding their accuracy and efficiency. Dispersion-corrected density functional theory (DFT-D) in principle is applicable, but the computational demands, for example, to compute a huge number of polymorphs, are too high. Here, we demonstrate that this task can be carried out by a dispersion-corrected density functional tight binding (DFTB) method. The semiempirical Hamiltonian with the D3 correction can accurately and efficiently model both solid- and gas-phase inter- and intramolecular interactions at a speed up of 2 orders of magnitude compared to DFT-D. The mean absolute deviations for interaction (lattice) energies for various databases are typically 2-3 kcal/mol (10-20\%), that is, only about two times larger than those for DFT-D. For zero-point phonon energies, small deviations of $<$0.5 kcal/mol compared to DFT-D are obtained.

@article{brandenburg_ref07,
author = {J. G. Brandenburg and S. Grimme},
title = {Accurate modeling of organic molecular crystals by dispersion-corrected density functional tight-binding {(DFTB)}},
journal = {{J. Phys. Chem. Lett.}},
year = {2014},
volume = {5},
pages = {1785--1789},
doi = {10.1021/jz500755u},
url = {../wp-content/papercite-data/pdf/brandenburg_ref07.pdf},
abstract = {The ambitious goal of organic crystal structure prediction challenges theoretical methods regarding their accuracy and efficiency. Dispersion-corrected density functional theory (DFT-D) in principle is applicable, but the computational demands, for example, to compute a huge number of polymorphs, are too high. Here, we demonstrate that this task can be carried out by a dispersion-corrected density functional tight binding (DFTB) method. The semiempirical Hamiltonian with the D3 correction can accurately and efficiently model both solid- and gas-phase inter- and intramolecular interactions at a speed up of 2 orders of magnitude compared to DFT-D. The mean absolute deviations for interaction (lattice) energies for various databases are typically 2-3 kcal/mol (10-20\%), that is, only about two times larger than those for DFT-D. For zero-point phonon energies, small deviations of $<$0.5 kcal/mol compared to DFT-D are obtained.
}
}

• J. G. Brandenburg, M. Hochheim, T. Bredow, and S. Grimme, "Low-cost quantum chemical methods for non-covalent interactions," J. Phys. Chem. Lett., vol. 5, p. 4275–4284, 2014. doi:10.1021/jz5021313

The efficient and reasonably accurate description of non-covalent interactions is important for various areas of chemistry, ranging from supramolecular host−guest complexes and biomolecular applications to the challenging task of crystal structure prediction. While London dispersion inclusive density functional theory (DFT-D) can be applied, faster “low-cost” methods are required for large-scale applications. In this Perspective, we present the state-of-the-art of minimal basis set, semiempirical molecular- orbital-based methods. Various levels of approximations are discussed based either on canonical Hartree−Fock or on semilocal density functionals. The performance for intermolecular interactions is examined on various small to large molecular complexes and organic solids covering many different chemical groups and interaction types. We put the accuracy of low-cost methods into perspective by comparing with first-principle density functional theory results. The mean unsigned deviations of binding energies from reference data are typically 10-30\%, which is only two times larger than those of DFT-D. In particular, for neutral or moderately polar systems, many of the tested methods perform very well, while at the same time, computational savings of up to 2 orders of magnitude can be achieved.

@article{brandenburg_ref11,
author = {J. G. Brandenburg and M. Hochheim and T. Bredow and S. Grimme},
title = {Low-Cost quantum chemical methods for non-covalent interactions},
journal = {{J. Phys. Chem. Lett.}},
year = {2014},
volume = {5},
pages = {4275--4284},
doi = {10.1021/jz5021313},
url = {../wp-content/papercite-data/pdf/brandenburg_ref11.pdf},
abstract = {The efficient and reasonably accurate description of non-covalent interactions is important for various areas of chemistry, ranging from supramolecular host−guest complexes and biomolecular applications to the challenging task of crystal structure prediction. While London dispersion inclusive density functional theory (DFT-D) can be applied, faster “low-cost” methods are required for large-scale applications. In this Perspective, we present the state-of-the-art of minimal basis set, semiempirical molecular- orbital-based methods. Various levels of approximations are discussed based either on canonical Hartree−Fock or on semilocal density functionals. The performance for intermolecular interactions is examined on various small to large molecular complexes and organic solids covering many different chemical groups and interaction types. We put the accuracy of low-cost methods into perspective by comparing with first-principle density functional theory results. The mean unsigned deviations of binding energies from reference data are typically 10-30\%, which is only two times larger than those of DFT-D. In particular, for neutral or moderately polar systems, many of the tested methods perform very well, while at the same time, computational savings of up to 2 orders of magnitude can be achieved.}
}

• S. Grimme, J. G. Brandenburg, C. Bannwarth, and A. Hansen, "Consistent structures and interactions by density functional theory with small atomic orbital basis sets," J. Chem. Phys., vol. 143, p. 54107, 2015. doi:10.1063/1.4927476

A density functional theory (DFT) based composite electronic structure approach is proposed to efficiently compute structures and interaction energies in large chemical systems. It is based on the well-known and numerically robust Perdew-Burke-Ernzerhoff (PBE) generalized-gradient-approximation in a modified global hybrid functional with a relatively large amount of non-local Fock-exchange. The orbitals are expanded in Ahlrichs-type valence-double zeta atomic orbital (AO) Gaussian basis sets, which are available for many elements. In order to correct for the basis set superposition error (BSSE) and to account for the important long-range London dispersion effects, our well-established atom-pairwise potentials are used. In the design of the new method, particular attention has been paid to an accurate description of structural parameters in various covalent and non-covalent bonding situations as well as in periodic systems. Together with the recently proposed three-fold corrected (3c) Hartree-Fock method, the new composite scheme (termed PBEh-3c) represents the next member in a hierarchy of “low-cost” electronic structure approaches. They are mainly free of BSSE and account for most interactions in a physically sound and asymptotically correct manner. PBEh-3c yields good results for thermochemical properties in the huge GMTKN30 energy database. Furthermore, the method shows excellent performance for non-covalent interaction energies in small and large complexes. For evaluating its performance on equilibrium structures, a new compilation of standard test sets is suggested. These consist of small (light) molecules, partially flexible, medium-sized organic molecules, molecules comprising heavy main group elements, larger systems with long bonds, 3d-transition metal systems, non-covalently bound complexes (S22 and S66×8 sets), and peptide conformations. For these sets, overall deviations from accurate reference data are smaller than for various other tested DFT methods and reach that of triple-zeta AO basis set second-order perturbation theory (MP2/TZ) level at a tiny fraction of computational effort. Periodic calculations conducted for molecular crystals to test structures (including cell volumes) and sublimation enthalpies indicate very good accuracy competitive to computationally more involved plane-wave based calculations. PBEh-3c can be applied routinely to several hundreds of atoms on a single processor and it is suggested as a robust “high-speed” computational tool in theoretical chemistry and physics.

@article{brandenburg_ref14,
author = {S. Grimme and J. G. Brandenburg and C. Bannwarth and A. Hansen},
title = {Consistent structures and interactions by density functional theory with small atomic orbital basis sets},
journal = {{J. Chem. Phys.}},
year = {2015},
volume = {143},
pages = {054107},
doi = {10.1063/1.4927476},
url = {../wp-content/papercite-data/pdf/brandenburg_ref14.pdf},
abstract = {A density functional theory (DFT) based composite electronic structure approach is proposed to efficiently compute structures and interaction energies in large chemical systems. It is based on the well-known and numerically robust Perdew-Burke-Ernzerhoff (PBE) generalized-gradient-approximation in a modified global hybrid functional with a relatively large amount of non-local Fock-exchange. The orbitals are expanded in Ahlrichs-type valence-double zeta atomic orbital (AO) Gaussian basis sets, which are available for many elements. In order to correct for the basis set superposition error (BSSE) and to account for the important long-range London dispersion effects, our well-established atom-pairwise potentials are used. In the design of the new method, particular attention has been paid to an accurate description of structural parameters in various covalent and non-covalent bonding situations as well as in periodic systems. Together with the recently proposed three-fold corrected (3c) Hartree-Fock method, the new composite scheme (termed PBEh-3c) represents the next member in a hierarchy of “low-cost” electronic structure approaches. They are mainly free of BSSE and account for most interactions in a physically sound and asymptotically correct manner. PBEh-3c yields good results for thermochemical properties in the huge GMTKN30 energy database. Furthermore, the method shows excellent performance for non-covalent interaction energies in small and large complexes. For evaluating its performance on equilibrium structures, a new compilation of standard test sets is suggested. These consist of small (light) molecules, partially flexible, medium-sized organic molecules, molecules comprising heavy main group elements, larger systems with long bonds, 3d-transition metal systems, non-covalently bound complexes (S22 and S66×8 sets), and peptide conformations. For these sets, overall deviations from accurate reference data are smaller than for various other tested DFT methods and reach that of triple-zeta AO basis set second-order perturbation theory (MP2/TZ) level at a tiny fraction of computational effort. Periodic calculations conducted for molecular crystals to test structures (including cell volumes) and sublimation enthalpies indicate very good accuracy competitive to computationally more involved plane-wave based calculations. PBEh-3c can be applied routinely to several hundreds of atoms on a single processor and it is suggested as a robust “high-speed” computational tool in theoretical chemistry and physics.}
}

• R. Sure, J. G. Brandenburg, and S. Grimme, "Small atomic orbital basis set first-principles quantum chemical methods for large molecular and periodic systems," ChemistryOpen, vol. 5, pp. 94-109, 2016. doi:10.1002/open.201500192

In quantum chemical computations the combination of Hartree-Fock or a density functional theory (DFT) approximation with relatively small atomic orbital basis sets of double-zeta quality is still widely used, for example, in the popular B3LYP/6-31G* approach. In this Review, we critically analyze the two main sources of error in such computations, that is, the basis set superposition error on the one hand and the missing London dispersion interactions on the other. We review various strategies to correct those errors and present exemplary calculations on mainly noncovalently bound systems of widely varying size. Energies and geometries of small dimers, large supramolecular complexes, and molecular crystals are covered. We conclude that it is not justified to rely on fortunate error compensation, as the main inconsistencies can be cured by modern correction schemes which clearly outperform the plain mean-field methods.

@article {brandenburg_ref17,
author = {R. Sure and J. G. Brandenburg and S. Grimme},
title = {Small atomic orbital basis set first-principles quantum chemical methods for large molecular and periodic systems},
journal = {{ChemistryOpen}},
volume = {5},
pages = {94-109},
year = {2016},
doi = {10.1002/open.201500192},
url = {../wp-content/papercite-data/pdf/brandenburg_ref17.pdf},
abstract = {In quantum chemical computations the combination of Hartree-Fock or a density functional theory (DFT) approximation with relatively small atomic orbital basis sets of double-zeta quality is still widely used, for example, in the popular B3LYP/6-31G* approach. In this Review, we critically analyze the two main sources of error in such computations, that is, the basis set superposition error on the one hand and the missing London dispersion interactions on the other. We review various strategies to correct those errors and present exemplary calculations on mainly noncovalently bound systems of widely varying size. Energies and geometries of small dimers, large supramolecular complexes, and molecular crystals are covered. We conclude that it is not justified to rely on fortunate error compensation, as the main inconsistencies can be cured by modern correction schemes which clearly outperform the plain mean-field methods.}
}

### Organic crystal structure prediction

The polymorphism of molecular crystals is important in various areas of chemistry and physics with possible applications for pharmaceutical compounds, pigments, explosives, and metal-organic framework materials. Many well known generic drugs such as aspirin and paracetamol have multiple polymorphs. Because the polymorph might have different properties (e.g. solubility)  pharmaceutical companies have to screen the polymorph landscape. Simulation techniques can help and guide experimentalists by supplying them with computed crystal energy landscapes.

Fig. 4: An exemplary crystal structure prediction (CSP) task is sketched.

The most stable crystal structures shall be predicted from the knowledge of their molecular composition. The computed energy landscape can then be used to identify the measured structures and to suggest plausible alternative polymorphs.

Last year, I participated in the 6th blind challenge for organic crystal structure prediction organized by the Cambridge structural database. For more information, have a look at the CSP homepage and the press release concerning the latest predictions with "Best Results Ever!" As the target molecules are approaching realistic real live applications (hydrated salt, co-crystal, high flexibility), the outcome is indeed promising and I am happy to contribute to some extend.

Using Sally (L) Price generated polymorphs (1000 per target molecule), a refinement using some of the above mentioned methods (DFTB3-D3, HF-3c, TPSS-D3) is conducted. While a few polymorphs have been lost during the refinement stages, the final TPSS-D3 energy ranking is excellent and among the best ones provided so far for these targets.

Fig. 5: My results for the 6th CSP blind test show the excellent accuracy of the computed energy ranking and highlights problems in the refinement stages.

• A. M. Reilly, R. I. Cooper, C. S. Adjiman, S. Bhattacharya, D. A. Boese, J. G. Brandenburg, P. J. Bygrave, R. Bylsma, J. E. Campbell, R. Car, D. H. Case, R. Chadha, J. C. Cole, K. Cosburn, H. M. Cuppen, F. Curtis, G. M. Day, R. A. {DiStasio Jr}, A. Dzyabchenko, B. P. van Eijck, D. M. Elking, J. A. van den Ende, J. C. Facelli, M. B. Ferraro, L. Fusti-Molnar, C. Gatsiou, T. S. Gee, R. de Gelder, L. M. Ghiringhelli, H. Goto, S. Grimme, R. Guo, D. W. M. Hofmann, J. Hoja, R. K. Hylton, L. Iuzzolino, W. Jankiewicz, D. T. de Jong, J. Kendrick, N. J. J. de Klerk, H. Ko, L. N. Kuleshova, X. Li, S. Lohani, F. J. J. Leusen, A. M. Lund, J. Lv, Y. Ma, N. Marom, A. E. Masunov, P. McCabe, D. P. McMahon, H. Meekes, M. P. Metz, A. J. Misquitta, S. Mohamed, B. Monserrat, R. J. Needs, M. A. Neumann, J. Nyman, S. Obata, H. Oberhofer, A. R. Oganov, A. M. Orendt, G. I. Pagola, C. C. Pantelides, C. J. Pickard, R. Podeszwa, L. S. Price, S. L. Price, A. Pulido, M. G. Read, K. Reuter, E. Schneider, C. Schober, G. P. Shields, P. Singh, I. J. Sugden, K. Szalewicz, C. R. Taylor, A. Tkatchenko, M. E. Tuckerman, F. Vacarro, M. Vasileiadis, A. Vázquez-Mayagoitia, L. Vogt, Y. Wang, R. E. Watson, G. A. de Wijs, J. Yang, Q. Zhu, and C. R. Groom, "Report on the sixth blind test of organic crystal-structure prediction methods," Acta Cryst. B, vol. 72, pp. 439-459, 2016. doi:10.1107/S2052520616007447

The sixth blind test of organic crystal-structure prediction (CSP) methods has been held, with five target systems: a small nearly rigid molecule, a polymorphic former drug candidate, a chloride salt hydrate, a co-crystal, and a bulky flexible molecule. This blind test has seen substantial growth in the number of submissions, with the broad range of prediction methods giving a unique insight into the state of the art in the field. Significant progress has been seen in treating flexible molecules, usage of hierarchical approaches to ranking structures, the application of density-functional approximations, and the establishment of new workflows and “best practices” forperforming CSP calculations. All of the targets, apart from a single potentially disordered Z = 2 polymorph of the drug candidate, were predicted by at least one submission. Despite many remaining challenges, it is clear that CSP methods are becoming more applicable to a wider range of real systems, including salts, hydrates and larger flexible molecules. The results also highlight the potential for CSP calculations to complement and augment experimental studies of organic solid forms.

@article {brandenburg_ref19,
author = {Reilly, Anthony M. and Cooper, Richard I. and Adjiman, Claire S. and Bhattacharya, Saswata and Boese, A. Daniel and Brandenburg, Jan Gerit and Bygrave, Peter J. and Bylsma, Rita and Campbell, Josh E. and Car, Roberto and Case, David H. and Chadha, Renu and Cole, Jason C. and Cosburn, Katherine and Cuppen, Herma M. and Curtis, Farren and Day, Graeme M. and {DiStasio Jr}, Robert A. and Dzyabchenko, Alexander and van Eijck, Bouke P. and Elking, Dennis M. and van den Ende, Joost A. and Facelli, Julio C. and Ferraro, Marta B. and Fusti-Molnar, Laszlo and Gatsiou, Christina-Anna and Gee, Thomas S. and de Gelder, R{\'e}ne and Ghiringhelli, Luca M. and Goto, Hitoshi and Grimme, Stefan and Guo, Rui and Hofmann, Detlef W.M. and Hoja, Johannes and Hylton, Rebecca K. and Iuzzolino, Luca and Jankiewicz, Wojciech and de Jong, Dani{\"e}l T. and Kendrick, John and de Klerk, Niek J.J. and Ko, Hsin-Yu and Kuleshova, Liudmila N. and Li, Xiayue and Lohani, Sanjaya and Leusen, Frank J.J. and Lund, Albert M. and Lv, Jian and Ma, Yanming and Marom, Noa and Masunov, Art{\"e}m E. and McCabe, Patrick and McMahon, David P. and Meekes, Hugo and Metz, Michael P. and Misquitta, Alston J. and Mohamed, Sharmarke and Monserrat, Bartomeu and Needs, Richard J. and Neumann, Marcus A. and Nyman, Jonas and Obata, Shigeaki and Oberhofer, Harald and Oganov, Artem R. and Orendt, Anita M. and Pagola, Gabriel I. and Pantelides, Constantinos C. and Pickard, Chris J. and Podeszwa, Rafa\l{} and Price, Louise S. and Price, Sarah L. and Pulido, Angeles and Read, Murray G. and Reuter, Karsten and Schneider, Elia and Schober, Christoph and Shields, Gregory P. and Singh, Pawanpreet and Sugden, Isaac J. and Szalewicz, Krzysztof and Taylor, Christopher R. and Tkatchenko, Alexandre and Tuckerman, Mark E. and Vacarro, Francesca and Vasileiadis, Manolis and V{\'a}zquez-Mayagoitia, Alvaro and Vogt, Leslie and Wang, Yanchao and Watson, Rona E. and de Wijs, Gilles A. and Yang, Jack and Zhu, Qiang and Groom, Colin R.},
title = {Report on the sixth blind test of organic crystal-structure prediction methods},
journal = {{Acta Cryst. B}},
volume = {72},
pages = {439-459},
year = {2016},
doi = {10.1107/S2052520616007447},
url = {../wp-content/papercite-data/pdf/brandenburg_ref19.pdf},
abstract = {The sixth blind test of organic crystal-structure prediction (CSP) methods has been held, with five target systems: a small nearly rigid molecule, a polymorphic former drug candidate, a chloride salt hydrate, a co-crystal, and a bulky flexible molecule. This blind test has seen substantial growth in the number of submissions, with the broad range of prediction methods giving a unique insight into the state of the art in the field. Significant progress has been seen in treating flexible molecules, usage of hierarchical approaches to ranking structures, the application of density-functional approximations, and the establishment of new workflows and “best practices” forperforming CSP calculations. All of the targets, apart from a single potentially disordered Z = 2 polymorph of the drug candidate, were predicted by at least one submission. Despite many remaining challenges, it is clear that CSP methods are becoming more applicable to a wider range of real systems, including salts, hydrates and larger flexible molecules. The results also highlight the potential for CSP calculations to complement and augment experimental studies of organic solid forms.}
}

• J. G. Brandenburg and S. Grimme, "Organic crystal polymorphism: a benchmark for dispersion corrected mean field electronic structure methods," Acta Cryst. B, vol. 72, pp. 502-513, 2016. doi:10.1107/S2052520616007885

We analyze the energy landscape of the 6th crystal structure prediction blind test targets with various first principles and semi-empirical quantum chemical methodologies. A new benchmark set of 59 crystal structures (termed POLY59) for testing quantum chemical methods based on the blind test target crystals is presented. We focus on different means to include London dispersion interactions within the density functional theory (DFT) framework. We show the impact of pair-wise dispersion corrections like the semi-empirical D2 scheme, the Tkatchenko-Scheffler TS method, and the density dependent dispersion correction dDsC. Recent methodological progress includes higher order contributions in both the many-body and multipole expansions. We use the D3 correction with Axilrod-Teller-Muto type three-body contribution, the TS based many body dispersion MBD, and the nonlocal van der Waals density functional vdW-DF2. The density functionals with D3 and MBD correction provide an energy ranking of the blind test polymorphs in excellent agreement with the experimentally found structures. As computationally less demanding method, we test our recently presented minimal basis Hartree-Fock method (HF-3c) and a density functional tight-binding Hamiltonian (DFTB). Considering the speed-up of three to four orders of magnitudes, the energy ranking provided by the low-cost methods is very reasonable. We compare the computed geometries with the corresponding X-ray data where TPSS-D3 performs best. The importance of zero-point vibrational energy and thermal effects on crystal densities is highlighted.

@article {brandenburg_ref20,
author = {J. G. Brandenburg and S. Grimme},
title = {Organic crystal polymorphism: A benchmark for dispersion corrected mean field electronic structure methods},
journal = {{Acta Cryst. B}},
volume = {72},
pages = {502-513},
year = {2016},
doi = {10.1107/S2052520616007885},
url = {../wp-content/papercite-data/pdf/brandenburg_ref20.pdf},
abstract = {We analyze the energy landscape of the 6th crystal structure prediction blind test targets with various first principles and semi-empirical quantum chemical methodologies. A new benchmark set of 59 crystal structures (termed POLY59) for testing quantum chemical methods based on the blind test target crystals is presented. We focus on different means to include London dispersion interactions within the density functional theory (DFT) framework. We show the impact of pair-wise dispersion corrections like the semi-empirical D2 scheme, the Tkatchenko-Scheffler TS method, and the density dependent dispersion correction dDsC. Recent methodological progress includes higher order contributions in both the many-body and multipole expansions. We use the D3 correction with Axilrod-Teller-Muto type three-body contribution, the TS based many body dispersion MBD, and the nonlocal van der Waals density functional vdW-DF2. The density functionals with D3 and MBD correction provide an energy ranking of the blind test polymorphs in excellent agreement with the experimentally found structures. As computationally less demanding method, we test our recently presented minimal basis Hartree-Fock method (HF-3c) and a density functional tight-binding Hamiltonian (DFTB). Considering the speed-up of three to four orders of magnitudes, the energy ranking provided by the low-cost methods is very reasonable. We compare the computed geometries with the corresponding X-ray data where TPSS-D3 performs best. The importance of zero-point vibrational energy and thermal effects on crystal densities is highlighted.}
}

### Spin crossover and thermochemistry

As several properties can depend on the crystal packing, this also holds for the spin state of a material. Spin crossover compounds can be used as magnetic sensors and noncovalent interactions can influence their properties.  I contributed to studies analyzing the relative stability of the high- and low-spin state of a few spin crossover complexes.

Fig. 6: Dependence of the spin state of metal-organic complexes on its packing in a crystal environment.

In collaboration with the group of Gerhard Erker, I analyzed the thermochemistry and some crystal packing effects in frustrated lewis pair (FLP) compounds. FLPs can activate small molecules like H2 and my theoretical studies helped interpreting thermodynamical features.

• D. Schweinfurth, S. Demeshko, S. Hohloch, M. Steinmetz, J. G. Brandenburg, S. Dechert, F. Meyer, S. Grimme, and B. Sarkar, "Spin crossover in Fe(II) and Co(II) complexes with the same click-derived tripodal ligand," Inorg. Chem., vol. 53, p. 8203–8212, 2014. doi:10.1021/ic500264k
[BibTeX] [Abstract]

The complexes [Fe(tbta)2](BF4)2·2EtOH (1), [Fe(tbta)2](BF4)2·2CH3CN (2), [Fe(tbta) 2](BF4)2·2CHCl3 (3), and [Fe(tbta)2](BF4)2 (4) were synthesized from the respective metal salts and the click-derived tripodal ligand tris[(1-benzyl- 1H-1,2,3-triazol-4-yl)methyl]amine (tbta). Structural characterization of these complexes (at 100 or 133 K) revealed Fe-N bond lengths for the solvent containing compounds 1−3 that are typical of a high spin (HS) Fe(II) complex. In contrast, the solvent-free compound 4 show Fe−N bond lengths that are characteristic of a low spin (LS) Fe(II) state. The Fe center in all complexes is bound to two triazole and one amine N atom from each tbta ligand, with the third triazole arm remaining uncoordinated. The benzyl substituents of the uncoordinated triazole arms and the triazole rings engage in strong intermolecular and intramolecular noncovalent interactions. These interactions are missing in the solvent containing molecules 1, 2, and 3, where the solvent molecules occupy positions that hinder these noncovalent interactions. The solvent-free complex (4) displays spin crossover (SCO) with a spin transition temperature T1/2 near room temperature, as revealed by superconducting quantum interference device (SQUID) magnetometric and Moössbauer spectroscopic measurements. The complexes 1, 2, and 3 remain HS throughout the investigated temperature range. Different torsion angles at the metal centers, which are influenced by the noncovalent interactions, are likely responsible for the differences in the magnetic behavior of these complexes. The corresponding solvent-free Co(II) complex (6) is also LS at lower temperatures and displays SCO with a temperature T1/2 near room temperature. Theoretical calculations at molecular and periodic DFT-D3 levels for 1−4 qualitatively reproduce the experimental findings, and corroborate the importance of intermolecular and intramolecular noncovalent interactions for the magnetic properties of these complexes. The present work thus represents rare examples of SCO complexes where the use of identical ligand sets produces SCO in Fe(II) as well as Co(II) complexes.

@article{brandenburg_ref09,
author = {D. Schweinfurth and S. Demeshko and S. Hohloch and M. Steinmetz and J. G. Brandenburg and S. Dechert and F. Meyer and S. Grimme and B. Sarkar},
title = {Spin crossover in {Fe(II)} and {Co(II)} complexes with the same click-derived tripodal ligand},
journal = {{Inorg. Chem.}},
year = {2014},
volume = {53},
pages = {8203--8212},
doi = {10.1021/ic500264k},
abstract = {The complexes [Fe(tbta)2](BF4)2·2EtOH (1), [Fe(tbta)2](BF4)2·2CH3CN (2), [Fe(tbta) 2](BF4)2·2CHCl3 (3), and [Fe(tbta)2](BF4)2 (4) were synthesized from the respective metal salts and the click-derived tripodal ligand tris[(1-benzyl- 1H-1,2,3-triazol-4-yl)methyl]amine (tbta). Structural characterization of these complexes (at 100 or 133 K) revealed Fe-N bond lengths for the solvent containing compounds 1−3 that are typical of a high spin (HS) Fe(II) complex. In contrast, the solvent-free compound 4 show Fe−N bond lengths that are characteristic of a low spin (LS) Fe(II) state. The Fe center in all complexes is bound to two triazole and one amine N atom from each tbta ligand, with the third triazole arm remaining uncoordinated. The benzyl substituents of the uncoordinated triazole arms and the triazole rings engage in strong intermolecular and intramolecular noncovalent interactions. These interactions are missing in the solvent containing molecules 1, 2, and 3, where the solvent molecules occupy positions that hinder these noncovalent interactions. The solvent-free complex (4) displays spin crossover (SCO) with a spin transition temperature T1/2 near room temperature, as revealed by superconducting quantum interference device (SQUID) magnetometric and Mo\"{o}ssbauer spectroscopic measurements. The complexes 1, 2, and 3 remain HS throughout the investigated temperature range. Different torsion angles at the metal centers, which are influenced by the noncovalent interactions, are likely responsible for the differences in the magnetic behavior of these complexes. The corresponding solvent-free Co(II) complex (6) is also LS at lower temperatures and displays SCO with a temperature T1/2 near room temperature. Theoretical calculations at molecular and periodic DFT-D3 levels for 1−4 qualitatively reproduce the experimental findings, and corroborate the importance of intermolecular and intramolecular noncovalent interactions for the magnetic properties of these complexes. The present work thus represents rare examples of SCO complexes where the use of identical ligand sets produces SCO in Fe(II) as well as Co(II) complexes.}
}

• B. -H. Xu, K. Bussmann, R. Fröhlich, C. G. Daniliuc, J. G. Brandenburg, S. Grimme, G. Kehr, and G. Erker, "An enamine/HB(C$_6$F$_5$)$_2$ adduct as a dormant state in frustrated lewis pair chemistry," Organometallics, vol. 32, p. 6745–6725, 2013. doi:10.1021/om4004225
[BibTeX] [Abstract]

The enamine piperidinocyclopentene reacts with HB(C6F5)2 by formation of the C-Lewis base/B-Lewis acid adduct 10. It shows a zwitterionic iminium ion/hydridoborate structure. However, this adduct formation is apparently reversible and may generate the “invisible” frustrated Lewis pair 11 as a reactive intermediate by hydroboration of the enamine CC bond in an equilibrium situation at room temperature. Consequently, the FLP 11 was trapped by typical FLP reactions, namely by the reaction with dihydrogen to give the ammonium/hydridoborate 12, the acetylene deprotonation products 13 and 14, and simple borane adducts with pyridine (15) and with an isonitrile (17). The products 10 and 12−15 and the isonitrile adduct 17 were characterized by X-ray diffraction. A DFT study determined the thermodynamic features of the 10 ⇄ 11 equilibrium and of a previously discussed reference system (18 ⇄ 19) derived by reacting piperidinocyclohexene with HB(C6F5)2.

@article{brandenburg_ref03,
author = {B.-H. Xu and K. Bussmann and R. Fr\"ohlich and C. G. Daniliuc and J. G. Brandenburg and S. Grimme and G. Kehr and G. Erker},
title = {An enamine/{HB}({C}$_6${F}$_5$)$_2$ adduct as a dormant state in frustrated Lewis pair chemistry},
journal = {{Organometallics}},
year = {2013},
volume = {32},
pages = {6745--6725},
doi = {10.1021/om4004225},
abstract = {The enamine piperidinocyclopentene reacts with HB(C6F5)2 by formation of the C-Lewis base/B-Lewis acid adduct 10. It shows a zwitterionic iminium ion/hydridoborate structure. However, this adduct formation is apparently reversible and may generate the “invisible” frustrated Lewis pair 11 as a reactive intermediate by hydroboration of the enamine CC bond in an equilibrium situation at room temperature. Consequently, the FLP 11 was trapped by typical FLP reactions, namely by the reaction with dihydrogen to give the ammonium/hydridoborate 12, the acetylene deprotonation products 13 and 14, and simple borane adducts with pyridine (15) and with an isonitrile (17). The products 10 and 12−15 and the isonitrile adduct 17 were characterized by X-ray diffraction. A DFT study determined the thermodynamic features of the 10 ⇄ 11 equilibrium and of a previously discussed reference system (18 ⇄ 19) derived by reacting piperidinocyclohexene with HB(C6F5)2.}
}

Top