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Multimer Embedding for Molecular Crystals Utilizing up to Tetramer Interactions

Alexander List, A. Daniel Boese, Johannes Hoja

TL;DR

This work tackles the computational cost of accurate crystal-energy calculations for molecular crystals by extending a subtractive multimer embedding method to include up to tetramer corrections for lattice energies and up to trimer corrections for forces, stress, and harmonic vibrational properties. By embedding PBE0+MBD multimers into periodic PBE+MBD calculations and benchmarking against explicit periodic PBE0+MBD on the X23 set, the study maps the convergence behavior of multimers and identifies the most effective strategies (notably ME3(4Å)) for balancing accuracy and cost. The results show that tetramer corrections improve lattice energies with diminishing returns beyond closed and diamond tetramer types, while trimer interactions are crucial for accurate stress, cell volumes, and vibrational properties. The findings offer practical guidance for efficient, high-accuracy crystal-energy computations, including mixed-cutoff and energy-threshold approaches to reduce high-level work without compromising accuracy significantly.

Abstract

Molecular crystals possess a highly complex crystallographic landscape which in many cases results in the experimental observation of multiple crystal structures for the same compound. Accurate results can often be obtained for such systems by employing periodic density functional theory using hybrid functionals; however, this is not always computationally feasible. One possibility to circumvent these expensive periodic calculations is the utilization of multimer embedding methods. Therein, the fully periodic crystal is described at a lower level of theory, and subsequently monomer energies, dimer interaction energies, etc. are corrected via high-level calculations. In this paper, we further extend such a multimer embedding approach by one multimer order for all investigated properties, allowing us to compute lattice energies up to the tetramer embedding level, and atomic forces, the stress tensor, and harmonic phonons up to the trimer level. We test the significance of including these higher-order multimers by embedding PBE0+MBD multimers into periodic PBE+MBD calculations utilizing the X23 benchmark set of molecular crystals and comparing the results to explicit periodic PBE0+MBD calculations. We show that tetramer interactions systematically improve the lattice energy approximation and explore multiple possibilities for multimer selection. Furthermore, we confirm that trimer interactions are crucial for the description of the stress tensor, yielding cell volumes within 1 % of those of PBE0+MBD. Subsequently, this also results in an improvement of the description of vibrational properties, giving on average gamma point frequencies within 1.3 wave numbers and vibrational free energies within 0.3 kJ/mol of the PBE0+MBD results.

Multimer Embedding for Molecular Crystals Utilizing up to Tetramer Interactions

TL;DR

This work tackles the computational cost of accurate crystal-energy calculations for molecular crystals by extending a subtractive multimer embedding method to include up to tetramer corrections for lattice energies and up to trimer corrections for forces, stress, and harmonic vibrational properties. By embedding PBE0+MBD multimers into periodic PBE+MBD calculations and benchmarking against explicit periodic PBE0+MBD on the X23 set, the study maps the convergence behavior of multimers and identifies the most effective strategies (notably ME3(4Å)) for balancing accuracy and cost. The results show that tetramer corrections improve lattice energies with diminishing returns beyond closed and diamond tetramer types, while trimer interactions are crucial for accurate stress, cell volumes, and vibrational properties. The findings offer practical guidance for efficient, high-accuracy crystal-energy computations, including mixed-cutoff and energy-threshold approaches to reduce high-level work without compromising accuracy significantly.

Abstract

Molecular crystals possess a highly complex crystallographic landscape which in many cases results in the experimental observation of multiple crystal structures for the same compound. Accurate results can often be obtained for such systems by employing periodic density functional theory using hybrid functionals; however, this is not always computationally feasible. One possibility to circumvent these expensive periodic calculations is the utilization of multimer embedding methods. Therein, the fully periodic crystal is described at a lower level of theory, and subsequently monomer energies, dimer interaction energies, etc. are corrected via high-level calculations. In this paper, we further extend such a multimer embedding approach by one multimer order for all investigated properties, allowing us to compute lattice energies up to the tetramer embedding level, and atomic forces, the stress tensor, and harmonic phonons up to the trimer level. We test the significance of including these higher-order multimers by embedding PBE0+MBD multimers into periodic PBE+MBD calculations utilizing the X23 benchmark set of molecular crystals and comparing the results to explicit periodic PBE0+MBD calculations. We show that tetramer interactions systematically improve the lattice energy approximation and explore multiple possibilities for multimer selection. Furthermore, we confirm that trimer interactions are crucial for the description of the stress tensor, yielding cell volumes within 1 % of those of PBE0+MBD. Subsequently, this also results in an improvement of the description of vibrational properties, giving on average gamma point frequencies within 1.3 wave numbers and vibrational free energies within 0.3 kJ/mol of the PBE0+MBD results.

Paper Structure

This paper contains 13 sections, 7 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Computational Methods
  3. Energy
  4. Forces
  5. Stress
  6. Harmonic Vibrational Properties
  7. Computational Details
  8. Results and Discussion
  9. Lattice Energies for up to Tetramer Embedding
  10. Mixed Cutoffs and Energy Cutoffs for Multimer Selection
  11. Trimer Interactions for Forces and Stress
  12. Trimer Interactions for Harmonic Vibrational Properties
  13. Conclusion

Figures (5)

  • Figure 1: Different types of a) trimers and b) tetramers. Each vertex represents a monomer, and edges indicate whether or not two monomers are within cutoff distance.
  • Figure 2: Relative lattice energies in kJ/mol of the X23 set calculated with several approaches w.r.t PBE0+MBD. All calculations were carried out on top of the PBE0+MBD/light optimized structures.
  • Figure 3: Mean absolute errors (MAEs) of lattice energies of the X23 set as a function of the employed multimer cutoff distance for several approaches w.r.t. PBE0+MBD in kJ/mol.
  • Figure 4: Average tetramer interaction energy correction terms in kJ/mol for the X23 crystal structures calculated with ME4(a4Å) using light species default settings on top of the PBE0+MBD/light-optimized geometries, broken down to the individual tetramer types. Top left: Average of energy correction terms. Top right: Average of absolutes of energy correction terms. Bottom left: Average of energy correction terms, normalized per multimer. Bottom right: Average of absolutes of energy correction terms, normalized per multimer.
  • Figure 5: Relative volume errors for the optimized X23 cells with different levels of multimer embedding compared to the PBE0+MBD optimized cell volumes in %.