# Intermolecular Interactions in Molecular Organic Crystals upon Relaxation of Lattice Parameters

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}(in 80% of the cases) [6], which is in the same range as the differences in lattice energy between enantiopure and racemic crystals [7,8].

_{latt}, is defined as the energy difference between a static, perfect, infinite crystal (ideal static solid, iss) and its related ideal static gas (isg) of non-interacting molecules in their lowest energy conformation at 0 K (Equation (1) where Z is the number of molecules in the unit cell).

^{−6}term. The accuracy of calculating noncovalent interaction energies prompted us to assess the B97-D functional in our calculation of the lattice energies of periodic molecular crystals which are held together by the same type of interactions (vdW, hydrogen bonding, and a mix of both). For an extensive discussion of the development of exchange-correlation functionals in DFT and an assessment of their performance, see [30].

## 2. Materials and Methods

**k**points. In 3D periodic systems, each sampling point is defined by its components k

_{1}, k

_{2}, and k

_{3}along the reciprocal lattice vectors

**b1**,

**b2**, and

**b3**as Equation (2):

**k**= k

_{1}

**b**+ k

_{1}_{2}

**b**+k

_{2}_{3}

**b**

_{3}## 3. Results

#### 3.1. Calculated Lattice Energies for the X23 Benchmark Set

^{0}

_{sub}(298K) and vibrational contributions (E

_{vib}+ 4RT, calculated using supercell phonon calculations with experimental C

_{p}data, where available). [20] For benzene, naphthalene, and cytosine, revised recent values from [48] were used.

_{2}) to −162.8 kJ/mol (for cytosine), a relative deviation from the experiment is a more appropriate measure. The B97-D functional shows a NRMSE of 7.94% from experiment compared to 13.23% for PBE.

_{latt}

^{ref,exp}for the X23 benchmark set [49], we obtained a MSE of −7.91 kJ/mol for PBE-D3 and −0.75 for B97-D, which correspond to 11.8% and 7.7%, respectively. Thus, comparing our results with the new reference values only marginally affect the performance, and even slightly reduce the deviation from the experiment.

#### 3.2. Convergence of Lattice Energies with Basis Set

#### 3.3. Optimization of Unit Cell Parameters

**a**|, |

**b**|, and |

**c**| lengths of the cell vectors, and the α, β, and γ unit cell angles. Simultaneous optimization of atomic positions and unit cell parameters make it possible to adapt the unit cell to the minimum energy molecular packing. This provides, on one hand, an additional assessment criterion by which to evaluate the accuracy of computational approaches to reproduce unit cell parameters. Second, the full structural relaxation of the unit cell to an ideal system at 0 K removes differences in unit cell parameters originating from different experimental conditions. This yields an ideal unit cell with lattice parameters in the absence of any finite temperature effect such as thermal expansion. This is highly relevant for the unbiased prediction of possible crystal structures starting from a molecular compound only (CSP_0), after which thermal corrections are employed in a second step (CSP_thd). The sensitivity of the lattice energy to the unit cell parameters cannot be estimated a priori, and is presented here for the first time for molecular crystals using periodic DFT calculations in Turbomole.

#### 3.3.1. Changes of Unit Cell Volumes and Lattice Parameters

^{3}to 848 a.u.

^{3}, i.e., by −6.96%), and the largest increase in unit cell volume occurs for CO

_{2}(from 1220 a.u.

^{3}to 1268 a.u.

^{3}, by +5.63%). These two crystal structures were obtained at very low temperatures, i.e., 160 K for ammonia [53] and 150 K for CO

_{2}[54], and the gaseous systems are a challenge in terms of electronic and solid state structure determination. The overestimation of the CO

_{2}unit cell volume was already noted by [17,19,20], and was shown not to originate from a particular type of dispersion correction method.

^{ref}

_{el}, which may serve as a reference for benchmarking volumes obtained via the minimization of electronic energies. On average, the V

^{ref}

_{el}values are 5% smaller than the experimentally-determined volumes. The plane-wave VASP PBE-D3 unit cell volumes had a MSE of 4.45% and a MAD of 0.93% from V

^{ref}

_{el}, which is very close to our GTO PBE-D3 values of 5.4% and 1.04%, respectively. The results of the comparison are given in detail in the Supplementary Information. Thus, both plane-wave and GTO basis functions are equally well-suited to obtaining accurate lattice parameters for the crystals of organic molecules.

#### 3.3.2. Lattice Energies after Unit Cell Optimization

## 4. Conclusions

## Supplementary Materials

^{el}

_{ref}.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Scheme 1.**Molecules from benchmark set X23 of small organic molecular crystals sorted by dominating noncovalent interaction type (CSD codes are given in the Supplementary Information Table S1).

**Figure 1.**Deviation in kJ/mol of calculated lattice energies with the PBE-D3 and B97-D functionals from experiment.

**Figure 3.**Convergence of calculated lattice energies with basis set size. (

**Top**): PBE-D3, (

**bottom**): B97-D.

**Figure 4.**Relative unit cells volume changes for organic crystals of the X23 benchmark set upon optimization of unit cell parameters with PBE-D3 (

**top**) and B97-D (

**bottom**).

**Figure 5.**Changes in deviation of the calculated lattice energies from the experiment upon minimization of the unit cell parameters. A negative number refers to an increase in the deviation; a positive number indicates better agreement with experiment.

**Figure 6.**Deviations from experimental lattice energies in kJ/mol from energy minimization and the simultaneous optimization of structural and unit cell parameters for PBE-D3 (

**top**) and B97-D (

**bottom**).

**Table 1.**B97-D calculated lattice energies in kJ/mol for molecular crystals of the X23 benchmark set compared to experimental lattice energies. Absolute and relative deviations from the experiment are given.

Compound | Calc. E_{latt}(kJ/mol) | Exp. (kJ/mol) | Deviation (kJ/mol) | Rel. Deviation (%) |
---|---|---|---|---|

1,4-Cyclohexanedione | −93.55 | −88.6 | −4.95 | 5.59 |

Adamantane | −74.50 | −69.4 | −5.10 | 7.36 |

Anthracene | −118.88 | −112.7 | −6.18 | 5.48 |

Benzene | −59.08 | −55.3 | −3.78 | 6.84 |

CO_{2} | −23.04 | −28.4 | 5.36 | 18.86 |

Hexamine | −94.79 | −86.2 | −8.59 | 9.97 |

Naphtalene | −87.13 | −83.1 | −4.03 | 4.85 |

Pyrazine | −66.01 | −61.3 | −4.71 | 7.68 |

Pyrazole | −81.01 | −77.7 | −3.31 | 4.26 |

Triazine | −60.70 | −61.7 | 1.00 | 1.62 |

Trioxane | −63.34 | −66.4 | 3.06 | 4.61 |

Acetic Acid | −70.75 | −72.8 | 2.05 | 2.82 |

Cytosine | −159.39 | −162.8 | 3.41 | 2.09 |

Imidazole | −89.90 | −86.8 | −3.10 | 3.57 |

Uracil | −132.99 | −135.7 | 2.71 | 2.00 |

Ammonia | −45.36 | −37.2 | −8.16 | 21.94 |

Cyanamide | −86.98 | −79.7 | −7.28 | 9.14 |

Ethylcarbamate | −87.00 | −86.3 | −0.70 | 0.81 |

Formamide | −79.90 | −79.2 | −0.70 | 0.88 |

Oxalic Acid (alpha) | −89.50 | −96.3 | 6.80 | 7.06 |

Oxalic Acid (beta) | −92.03 | −96.1 | 4.07 | 4.24 |

Succinic Acid | −125.69 | −130.3 | 4.61 | 3.54 |

Urea | −108.21 | −102.5 | −5.71 | 5.57 |

Interaction Type | PBE-D3 | B97-D | ||||||
---|---|---|---|---|---|---|---|---|

NRMSE (%) | RMSE (kJ/mol) | MSE (kJ/mol) | MAE (kJ/mol) | NRMSE (%) | RMSE (kJ/mol) | MSE (kJ/mol) | MAE (kJ/mol) | |

vdW | 9.76 | 7.34 | −5.69 | 5.90 | 8.21 | 4.91 | −2.84 | 4.55 |

Mixed | 10.16 | 9.67 | −9.58 | 9.58 | 2.70 | 2.86 | 1.27 | 2.28 |

H-bonding | 17.91 | 12.54 | −12.12 | 12.12 | 9.20 | 5.45 | −0.88 | 4.75 |

total | 13.32 | 9.84 | −8.60 | 8.70 | 7.94 | 4.82 | −1.14 | 4.32 |

**Table 3.**Calculated lattice energies of the molecular crystals from the X23 benchmark set upon simultaneous optimization of the molecular structures and unit cell parameters.

PBE-D3 | B97-D | |||||
---|---|---|---|---|---|---|

E_{latt,calc.}(kJ/mol) | Δ(E_{calc-exp})(kJ/mol) | Δ(E_{calc-exp})(%) | E_{latt,calc.}(kJ/mol) | Δ(E_{calc-exp})(kJ/mol) | Δ(E_{calc-exp})(%) | |

1,4-Cyclohexanedione | −88.60 | −11.81 | 13.33 | −94.98 | −6.38 | 7.20 |

Adamantane | −69.40 | −10.72 | 15.44 | −77.90 | −8.50 | 12.25 |

Anthracene | −112.70 | 0.81 | −0.72 | −118.88 | −6.18 | 5.48 |

Benzene | −55.30 | −3.44 | 6.22 | −62.12 | −6.82 | 12.33 |

CO_{2} | −28.40 | 0.78 | −2.73 | −23.41 | 4.99 | −17.56 |

Hexamine | −86.20 | −14.61 | 16.95 | −96.14 | −9.94 | 11.54 |

Naphthalene | −83.10 | −1.02 | 1.23 | −89.15 | −6.05 | 7.28 |

Pyrazine | −61.30 | −8.86 | 14.45 | −67.53 | −6.23 | 10.16 |

Pyrazole | −77.70 | −8.73 | 11.23 | −81.24 | −3.54 | 4.55 |

Triazine | −61.70 | −3.29 | 5.33 | −62.66 | −0.96 | 1.56 |

Trioxane | −66.40 | −4.49 | 6.76 | −63.76 | 2.64 | −3.97 |

Acetic Acid | −72.80 | −9.75 | 13.40 | −70.79 | 2.01 | −2.77 |

Cytosine | −162.80 | −7.54 | 4.63 | −160.37 | 2.43 | −1.49 |

Imidazole | −86.80 | −10.82 | 12.46 | −90.45 | −3.65 | 4.20 |

Uracil | −135.70 | −10.89 | 8.02 | −133.74 | 1.96 | −1.45 |

Ammonia | −37.20 | −14.20 | 38.17 | −45.82 | −8.62 | 23.16 |

Cyanamide | −79.70 | −16.98 | 21.30 | −87.39 | −7.69 | 9.65 |

Ethylcarbamate | −86.30 | −12.01 | 13.92 | −88.50 | −2.20 | 2.55 |

Formamide | −79.20 | −10.37 | 13.09 | −80.33 | −1.13 | 1.43 |

Oxalic Acid α | −96.30 | −7.77 | 8.07 | −89.80 | 6.50 | −6.75 |

Oxalic Acid β | −96.10 | −10.31 | 10.73 | −92.04 | 4.06 | −4.23 |

Succinic Acid | −130.30 | −15.12 | 11.60 | −126.73 | 3.57 | −2.74 |

Urea | −102.50 | −15.09 | 14.72 | −108.31 | −5.81 | 5.66 |

**Table 4.**Performance of PBE-D3 and B97-D in calculating the lattice energies upon simultaneous optimization of the molecular structure and unit cell parameters.

PBE-D3 | B97-D | |||||||
---|---|---|---|---|---|---|---|---|

NRMSE (%) | RMSE (kJ/mol) | MAE (kJ/mol) | MSE (kJ/mol) | NRMSE (%) | RMSE (kJ/mol) | MSE (kJ/mol) | MAE (kJ/mol) | |

vdW | 10.27 | 7.79 | 6.23 | −5.94 | 9.63 | 6.16 | −4.27 | 5.66 |

Mixed | 10.26 | 9.84 | 9.75 | −9.75 | 2.72 | 2.60 | 0.69 | 2.51 |

H-bonding | 18.73 | 13.06 | 12.73 | −12.73 | 9.62 | 5.53 | −1.41 | 4.95 |

total | 13.81 | 10.26 | 9.10 | −8.97 | 8.82 | 5.47 | −2.41 | 4.86 |

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**MDPI and ACS Style**

Stein, M.; Heimsaat, M.
Intermolecular Interactions in Molecular Organic Crystals upon Relaxation of Lattice Parameters. *Crystals* **2019**, *9*, 665.
https://doi.org/10.3390/cryst9120665

**AMA Style**

Stein M, Heimsaat M.
Intermolecular Interactions in Molecular Organic Crystals upon Relaxation of Lattice Parameters. *Crystals*. 2019; 9(12):665.
https://doi.org/10.3390/cryst9120665

**Chicago/Turabian Style**

Stein, Matthias, and Madalen Heimsaat.
2019. "Intermolecular Interactions in Molecular Organic Crystals upon Relaxation of Lattice Parameters" *Crystals* 9, no. 12: 665.
https://doi.org/10.3390/cryst9120665