**Cite This:**

*J. Chem. Theory Comput.*2022, 18, 7, 4077-4081

# Symmetric Molecular Dynamics

- Sam CoxSam CoxDepartment of Chemical Engineering, University of Rochester, Rochester, New York 14627, United StatesMore by Sam Cox
- and
- Andrew D. White
*****Andrew D. WhiteEmail:Department of Chemical Engineering, University of Rochester, Rochester, New York 14627, United States*****[email protected] (he/him).More by Andrew D. White

## Abstract

We derive a formulation of molecular dynamics that generates only symmetric configurations. We implement it for all 2D planar and 3D space groups. An atlas of 2D Lennard-Jones crystals under all planar groups is created with symmetric molecular dynamics.

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### License Summary*

You are free to share (copy and redistribute) this article in any medium or format and to adapt (remix, transform, and build upon) the material for any purpose, even commercially within the parameters below:

Creative Commons (CC): This is a Creative Commons license.

Attribution (BY): Credit must be given to the creator.

*Disclaimer

This summary highlights only some of the key features and terms of the actual license. It is not a license and has no legal value. Carefully review the actual license before using these materials.

### License Summary*

You are free to share (copy and redistribute) this article in any medium or format and to adapt (remix, transform, and build upon) the material for any purpose, even commercially within the parameters below:

Creative Commons (CC): This is a Creative Commons license.

Attribution (BY): Credit must be given to the creator.

*Disclaimer

This summary highlights only some of the key features and terms of the actual license. It is not a license and has no legal value. Carefully review the actual license before using these materials.

*input*. There is a finite number of symmetry groups. We simply simulate under all symmetry groups to generate symmetric structures.

*restraints*. (8) These harmonic restraints generally keep the system close to symmetric, but unlike the method we propose here, no single configuration is actually symmetric. Symmetry has certainly been considered as a

*measure*of molecular configurations. For example, Zabrodsky et al. (9) proposed a continuous symmetry measure, which is used to quantify the symmetry of atoms. This has been used to directly optimize Lennard-Jones clusters with symmetry. (10) Of course, the direct use of symmetry for crystal structure prediction with Monte Carlo is common, (11,12) and generative models with explicitly included symmetry are common. (13) There are no molecular dynamics methods though that can directly sample space groups, which would be useful for crystal structure prediction and modeling biological assemblies. (14) Symmetric molecular dynamics may also be viewed as a special case of objective molecular dynamics, which is a general method that encompasses any infinite or finite periodic tiling of a simulation. (15,16) Similarly, others have explored generalizing periodic boundary conditions to other tilings. (17−20)

## I. Theory

### A. Equation of Motion

*N*indistinguishable particles in

*D*dimensions under a Hamiltonian

*H*(

**p**(

*t*),

**q**(

*t*)). We wish to constrain

*H*so that

**q**(

*t*) is symmetric at all times. Symmetry is a property of

**q**(

*t*) and a specific symmetry group of position transformations

*G*, like mirrors along the

*x*axis.

**q**(

*t*) is point group symmetric if applying any element of the group results in no change to the positions (ignoring ordering of particles)

*g*· means applying the group element to each particle individually, ∼ means row equivalence, and

*G*is a finite group. Group elements are represented as affine matrices in space and planar groups.

*I*is the identity transformation.

*t*= 0, the particles can be partitioned into

*N*/|

*G*| =

*n*group orbits. A group orbit is the set generated by applying all elements of group

*G*to positions

**q**

_{i}(

*t*)

**q**

_{i}(

*t*) itself, because

*G*contains the identity element. We can label the particles as

*q*

_{ij}(

*t*) where

*i*indicates the orbit and

*j*indicates the group element. In crystallography, the

**q**

_{i0}particles are called the asymmetric unit. We can satisfy eq 1 at all

*t*by specifying the following holonomic constraint:

*G*| – 1 of these constraints per group orbit, and each removes

*D*degrees of freedom. This means the degrees of freedom of the dynamics is

*D*× (

*n*– 1). We can simulate dynamics under the holonomic constraints by simply only modeling the asymmetric unit─they are the generalized coordinates. (22)

*N*–

*n*) particles only when computing forces. This is similar to Dayal et al. (15) Practically this is done by setting these constrained particles’ positions just before computing forces. Similar to work on periodic boundary conditions, these equations of motion may lead to linear momentum conservation problems. (23,24)

*G*-invariant, where

*G*is any planar, space, or permutation group:

*U*(

*g*·

**q**) =

*U*(

**q**). That makes the forces,

*F*(

**q**),

*G*-equivariant

*G*| factor accounts for intragroup orbit interactions that are not explicitly computed. This translates an algorithm of an outer loop over the asymmetric unit and an inner loop over all particles.

### B. Bravais Lattice

*D D*-dimensional unit cell vectors. Particles always remain in one cell among the lattice cells, which are called images. For example, we could simulate the “root” cell and its 26 neighboring cells in 3 dimensions. We follow the approach above and treat each image of the system with virtual particles while only integrating the root cell. This means all images of the system are explicit, and we can violate the minimum image convention. We were not signatories of the minimum image convention anyway. This approach allows the cell vectors to shrink well below the distance cutoff of the potential, provided we have enough virtual particles to populate past the cutoff of the asymmetric unit of the origin cell. You can simulate 3

^{aD}images to allow the cells to shrink to at least 1/

*a*the cutoff distance.

*B*, we can transform between the representations via

*s*(

*t*) is the fraction of each lattice vector (i.e., fractional coordinates). Wrapping is trivial with fractional coordinates:

*s*(

*t*) fmod 1.0 will wrap the coordinates. All point group transformations are applied in

**s**(

*t*); however, a

*B*

^{–1}term should be added to eq 3 so that it operates on fractional coordinates.

**L**of shape

*D*×

*D*×

*D*×

*D*that maps from a triclinic box vector to the proper Bravais lattice box vectors of the space group. For example,

*L*

_{2011}is the contribution to Bravais lattice vector 2’s

*x*component from triclinic box vector 1’s

*y*component. There are many choices that could be made for

**L**. For example, to make a cubic Bravais lattice from a triclinic box vector, we require a single parameter

*a*to define the three lattice vectors (

*a*, 0, 0),(0,

*a*, 0),(0, 0,

*a*). We could set

*a*by averaging all the vector lengths, averaging all vector components, or selecting

*a*to be the first element of the first vector. Each of these choices gives a different

**L**, and some have large null spaces. NPT is then accomplished via scaling Monte Carlo moves in the triclinic box vectors (

*B*′) following Frenkel and Eppenga, (27) and the proper Bravais lattice is computed via

*B*=

**L**

*B*′.

### C. Wyckoff Positions

*q*

_{i0}is in a special position called a Wyckoff position─like the origin. (21) To perform constrained molecular dynamics of particle

*q*

_{i0}(

*t*) occupying a Wyckoff position, we define a subgroup

*G*′ that contains the elements of

*G*which do not leave

*q*

_{i0}(

*t*) invariant plus an identity group element. The identity of this subgroup is not the identity transform but instead a transform that projects from a general position into the Wyckoff position. For example, the Wyckoff position may be the vertical line

*x*= 0, and the identity group element would be the transform

*x*′ = 0,

*y*′ =

*y*. We will denote this group element as

*P*to hint it is a projection.

*q*

_{i0}(

*t*) must stay in a Wyckoff position at all times to satisfy eq 1. This can be accomplished via traditional constrained molecular dynamics of Lagrange multipliers. (28) Omitting the indices on

**q**

_{i0}(

*t*), our holonomic constraint is

*J*[σ] is the Jacobian of σ with respect to constraint dimension and element of

**q**(

*t*). We can solve for λ by knowing that σ[

**q**(

*t*+ Δ

*t*)] =

**0**

*t*is the time step,

*m*is the mass of the particle, and

**q**′(

*t*+ Δ

*t*) is

**q**(

*t*) integrated without the constraint force by Δ

*t*. All terms are constant except σ[

**q**′(

*t*+ Δ

*t*)], which simplifies computation.

## II. Methods

^{2D}images are simulated where

*D*is the dimension. To generate starting configurations, points were randomly generated and filtered to fit into the space group asymmetric unit as specified by Aroyo. (31) Point group generators and Wyckoff sites were taken from the Bilbao crystallography server. (32−34)

*L*,

*m*, and ϵ are the fundamental units of length, mass, and energy, respectively.

## III. Results

*P*4/

*mbm*) has 80 particles in a unit cell when there are 5 in the asymmetric unit, meaning the interaction potential felt has more particles contributing to it.

*P*= 0.25,

*T*= 0.1) for 1 M steps. Next, we do a constrained equilibration under NVT for 100k steps at

*T*= 0.05. This structure is then the proposed crystal structure for the given symmetry group. Figure 2 shows the root mean square deviation (RMSD) if the resultant structure is simulated under no symmetry constraint in NVE for 5k steps. The assumption is that if the structure does not collapse (RMSD rise), it is metastable. We indeed find that this protocol under no symmetry constraints (p1) gives the correct hexagonal packing.

## IV. Discussion

## V. Conclusions

## Acknowledgments

We thank Prof. Glen Hocky and Dr. Charles Matthews for valuable discussions and feedback. This work was supported by the NSF under grant 1751471.

## References

This article references 36 other publications.

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Ryckaet, et al. (1977) for integrating the Cartesian equations of motion of a system with holonomic constraints, has been extended to allow the independent constraint of arbitrary internal coordinates. To illustrate this new methodol., and to investigate the effects of dihedral angle constraints on the equil. and dynamical properties of macromols., parallel sets of mol. dynamics simulations and normal mode analyses of a small dipeptide were carried out. One without constraints, and one with a single backbone dihedral angle constrained. The avs. and the fluctuations of the energies, and of the internal degrees of freedom are not significantly modified by the constraint. However, in the region between 100 and 1400 cm-1 of the normal mode spectrum, the constraint shifts the frequencies of the modes, and modifies their contributions to the spectra of the internal coordinates. 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Bellemans (1978). However, the present simulations were substantially more precise; the run lengths were typically ∼20 times longer than those of R. and B. The present result also agrees with the simulation given by P. A. Wielopolski and E. R. Smith (1986). Thermodn. and structural data from the present simulations also agree well with these other simulations.**7**Ciccotti, G.; Ryckaert, J.-P. Molecular dynamics simulation of rigid molecules.*Computer Physics Reports*1986,*4*, 346, DOI: 10.1016/0167-7977(86)90022-5Google ScholarThere is no corresponding record for this reference.**8**Anishkin, A.; Milac, A. L.; Guy, H. R. Symmetryrestrained molecular dynamics simulations improve homology models of potassium channels.*Proteins: Struct., Funct., Bioinf.*2010,*78*, 932, DOI: 10.1002/prot.22618Google Scholar8https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXhtVSlsb4%253D&md5=0394ee96764f1f68d55663c5b1d33cd5Symmetry-restrained molecular dynamics simulations improve homology models of potassium channelsAnishkin, Andriy; Milac, Adina L.; Guy, H. RobertProteins: Structure, Function, and Bioinformatics (2010), 78 (4), 932-949CODEN: PSFBAF ISSN:. (Wiley-Liss, Inc.)Most crystd. homo-oligomeric ion channels are highly sym., which dramatically decreases conformational space and facilitates building homol. models (HMs). However, in mol. dynamics (MD) simulations channels deviate from ideal symmetry and accumulate thermal defects, which complicate the refinement of HMs using MD. In this work the authors evaluate the ability of symmetry constrained MD simulations to improve HMs accuracy, using an approach conceptually similar to Crit. Assessment of techniques for protein Structure Prediction (CASP) competition: build HMs of channels with known structure and evaluate the efficiency of proposed methods in improving HMs accuracy (measured as deviation from exptl. structure). Results indicate that unrestrained MD does not improve the accuracy of HMs, instantaneous symmetrization improves accuracy but not stability of HMs during subsequent unrestrained MD, while gradually imposing symmetry constraints improves both accuracy (by 5-50%) and stability of HMs. Moreover, accuracy and stability are strongly correlated, making stability a reliable criterion in predicting the accuracy of new HMs. Proteins 2010. © 2009 Wiley-Liss, Inc.**9**Zabrodsky, H.; Peleg, S.; Avnir, D. Continuous symmetry measures.*J. Am. Chem. 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A versatile, simple tool is developed as a continuous symmetry measure. Its main property is the ability to quantify the distance of a given (distorted mol.) shape from any chosen element of symmetry. The generality of this symmetry measure allows one to compare the symmetry distance of several objects relative to a single symmetry element and to compare the symmetry distance of a single object relative to various symmetry elements. The continuous symmetry approach is presented for the case of cyclic mols., first in a practical way and then with a rigorous math. anal. The versatility of the approach is then further demonstrated with alkane conformations, with a vibrating ABA water-like mol., and with a three-dimensional anal. of the symmetry of a [2 + 2] reaction in which the double bonds are not ideally aligned.**10**Oakley, M. T.; Johnston, R. L.; Wales, D. J. Symmetrisation schemes for global optimization of atomic clusters.*Phys. Chem. Chem. 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We report results for 38-, 75-, and 98-atom Lennard-Jones clusters, which are all multiple-funnel systems. Exploiting approx. symmetry reduces the mean time taken to locate the global min. by up to two orders of magnitude, with smaller improvements in efficiency for LJ55 and LJ74, which correspond to simpler single-funnel energy landscapes.**11**van Eijck, B. P.; Kroon, J. Upack program package for crystal structure prediction: Force fields and crystal structure generation for small carbohydrate molecules.*Journal of computational chemistry*1999,*20*, 799, DOI: 10.1002/(SICI)1096-987X(199906)20:8<799::AID-JCC6>3.0.CO;2-ZGoogle Scholar11https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXjsFShsL4%253D&md5=073401b9191ccee6cefa88d726b36c5dUpack program package for crystal structure prediction: force fields and crystal structure generation for small carbohydrate moleculesVan Eijck, Bouke P.; Kroon, JanJournal of Computational Chemistry (1999), 20 (8), 799-812CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)The UPACK program package for crystal structure generation was used to build hypothetical crystal structures for 32 pyranoses and 24 polyalcs. A subset of these was used to compare six force fields in their ability to reproduce the exptl. obsd. structures, preferably with low energies and high rankings with respect to the structure with lowest energy. One of these force fields, UNITAT, was derived from GROMOS87 by adjusting some parameters to obtain a better geometric description of these crystal structures. This united-atom force field and the all-atom OPLS force field performed best, and were used in the crystal structure generation for the full set of compds. For these carbohydrates, hundreds of hypothetical polymorphic structures are generated in an energy window of ∼25 kJ/mol. On av., the exptl. structures had reasonable energies and rankings. These lists are good starting points for more sophisticated calcns., and can be used for structure detn. by powder diffraction methods. In a few cases, serious doubt was raised concerning the H-bond network reported in the literature, for which more plausible alternatives appear to exist.**12**Fredericks, S.; Parrish, K.; Sayre, D.; Zhu, Q. Pyxtal: A python library for crystal structure generation and symmetry analysis.*Comput. Phys. Commun.*2021,*261*, 107810, DOI: 10.1016/j.cpc.2020.107810Google Scholar12https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXhtlWhsLs%253D&md5=6e93cdb95622515c90baf45b7a9d04fePyXtal: A Python library for crystal structure generation and symmetry analysisFredericks, Scott; Parrish, Kevin; Sayre, Dean; Zhu, QiangComputer Physics Communications (2021), 261 (), 107810CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)We present PyXtal, a new package based on the Python programming language, used to generate structures with specific symmetry and chem. compns. for both at. and mol. systems. This software provides support for various systems described by point, rod, layer, and space group symmetries. With only the inputs of chem. compn. and symmetry group information, PyXtal can automatically find a suitable combination of Wyckoff positions with a step-wise merging scheme. Further, when the mol. geometry is given, PyXtal can generate different dimensional org. crystals with mols. occupying both general and special Wyckoff positions. Optionally, PyXtal also accepts user-defined parameters (e.g., cell parameters, min. distances and Wyckoff positions). In general, PyXtal serves three purposes: (1) to generate custom structures, (2) to modulate the structure by symmetry relations, (3) to interface the existing structure prediction codes that require the generation of random sym. structures. In addn., we provide several utilities that facilitate the anal. of structures, including symmetry anal., geometry optimization, and simulations of powder X-ray diffraction (XRD). Full documentation of PyXtal is available at https://pyxtal.readthedocs.io. Program Title: PyXtalCPC Library link to program files:http://dx.doi.org/10.17632/wfyxyhjzwx.12Licensing provisions: MIT [1]Programming language: Python 3Nature of problem: Knowledge of structure at the at. level is the key to understanding materials' properties. Typically, the structure of a material can be detd. either from expt. (such as X-ray diffraction, spectroscopy, microscopy) or from theory (e.g., enhanced sampling, structure prediction). In many cases, the structure needs to be solved iteratively by generating a no. of trial structure models satisfying some constraints (e.g., chem. compn., symmetry, and unit cell parameters). Therefore, it is desirable to have a computational code that is able to generate such trial structures in an automated manner. The PyXtal package is able to generate many possible random structures for both at. and mol. systems with all possible symmetries. To generate the trial structure, the algorithm can either start with picking the symmetry sites randomly from high to low multiplicities, or use sites that are predefined by the user. For mols., the algorithm can automatically detect the mols.' symmetry and place them into special Wyckoff positions while satisfying their compatible site symmetry. With the support of symmetry operations for point, rod, layer and space groups, PyXtal is suitable for the computational modeling of systems from zero, one, two, and three dimensional bulk crystals.**13**Xie, T.; Fu, X.; Ganea, O.-E.; Barzilay, R.; Jaakkola, T. Crystal diffusion variational autoencoder for periodic material generation. 2021, arXiv:2110.06197.*arXiv preprint*. https://arxiv.org/abs/2110.06197 (accessed 2022-06-10).Google ScholarThere is no corresponding record for this reference.**14**Cannon, K. A.; Ochoa, J. M.; Yeates, T. O. High-symmetry protein assemblies: patterns and emerging applications.*Curr. Opin. Struct. Biol.*2019,*55*, 77, DOI: 10.1016/j.sbi.2019.03.008Google Scholar14https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXms1Ggtrk%253D&md5=55277a1232c59feb420d94b8957294d9High-symmetry protein assemblies: patterns and emerging applicationsCannon, Kevin A.; Ochoa, Jessica M.; Yeates, Todd O.Current Opinion in Structural Biology (2019), 55 (), 77-84CODEN: COSBEF; ISSN:0959-440X. (Elsevier Ltd.)The accelerated elucidation of three-dimensional structures of protein complexes, both natural and designed, is providing new examples of large supramol. assemblies with intriguing shapes. Those with high symmetry - based on the geometries of the Platonic solids - are particularly notable as their innately closed forms create interior spaces with varying degrees of enclosure. We survey known protein assemblies of this type and discuss their geometric features. The results bear on issues of protein function and evolution, while also guiding novel bioengineering applications. Recent successes using high-symmetry protein assemblies for applications in interior encapsulation and exterior display are highlighted.**15**Dayal, K.; James, R. D. Nonequilibrium molecular dynamics for bulk materials and nanostructures.*Journal of the Mechanics and Physics of Solids*2010,*58*, 145, DOI: 10.1016/j.jmps.2009.10.008Google Scholar15https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXksVentg%253D%253D&md5=78c01085107887eea5c179309937328fNonequilibrium molecular dynamics for bulk materials and nanostructuresDayal, Kaushik; James, Richard D.Journal of the Mechanics and Physics of Solids (2010), 58 (2), 145-163CODEN: JMPSA8; ISSN:0022-5096. (Elsevier Ltd.)We describe a method of constructing exact solns. of the equations of mol. dynamics in non-equil. settings. These solns. correspond to some viscometric flows, and to certain analogs of viscometric flows for fibers and membranes that have one or more dimensions of at. scale. This work generalizes the method of objective mol. dynamics (OMD) (). It allows us to calc. viscometric properties from a mol.-level simulation in the absence of a constitutive equation, and to relate viscometric properties directly to mol. properties. The form of the solns. is partly independent of the form of the force laws between atoms, and therefore these solns. have implications for coarse-grained theories. We show that there is an exact redn. of the Boltzmann equation corresponding to one family of OMD solns. This redn. includes most known exact solns. of the equations of the moments for special kinds of mols. and gives the form of the mol. d. function corresponding to such flows. This and other consequences leads us to propose an addn. to the principle of material frame indifference, a cornerstone of nonlinear continuum mechanics. The method is applied to the failure of carbon nanotubes at an imposed strain rate, using the Tersoff potential for carbon. A large set of simulations with various strain rates, initial conditions and two choices of fundamental domain (unit cell) give the following unexpected results: Stone-Wales defects play no role in the failure (though Stone-Wales partials are sometimes seen just prior to failure), a variety of failure mechanisms is obsd., and most simulations give a strain at failure of 15-20%, except those done with initial temp. above about 1200 K and at the lower strain rates. The latter have a strain at failure of 1-2%.**16**Xu, H.; Drozdov, G.; Hourahine, B.; Park, J. G.; Sweat, R.; Frauenheim, T.; Dumitrică, T. Collapsed carbon nanotubes: From nano to mesoscale via density functional theory-based tight-binding objective molecular modeling.*Carbon*2019,*143*, 786, DOI: 10.1016/j.carbon.2018.11.068Google Scholar16https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXisVOiur%252FO&md5=8cce8e84f8015d7f29b56ef027769870Collapsed carbon nanotubes: From nano to mesoscale via density functional theory-based tight-binding objective molecular modelingXu, Hao; Drozdov, Grigorii; Hourahine, Ben; Park, Jin Gyu; Sweat, Rebekah; Frauenheim, Thomas; Dumitrica, TraianCarbon (2019), 143 (), 786-792CODEN: CRBNAH; ISSN:0008-6223. (Elsevier Ltd.)Due to the inherent spatial and temporal limitations of atomistic modeling and the lack of efficient mesoscopic models, mesoscale simulation methods for guiding the development of super strong lightwt. material systems comprising collapsed C nanotubes (CNTs) are currently missing. Here the authors establish a path for deriving ultra-coarse-grained mesoscopic distinct element method (mDEM) models directly from the quantum mech. representation of a collapsed CNT. Atomistic calcns. based on d. functional-based tight-binding (DFTB) extended with Lennard-Jones interactions allow for the identification of the cross-section and elastic consts. of an elastic beam idealization of a collapsed CNT. Application of the DFTB quantum treatment is possible due to the simplification in the no. of atoms introduced by accounting for the helical and angular symmetries exhibited by twisted and bent CNTs. The multiscale modeling chain established here is suitable for deriving ultra-coarse-grained mesoscopic models for a variety of microscopic filaments presenting complex interat. bondings.**17**Hansson, T.; Oostenbrink, C.; van Gunsteren, W. Molecular dynamics simulations.*Curr. Opin. Struct. Biol.*2002,*12*, 190, DOI: 10.1016/S0959-440X(02)00308-1Google Scholar17https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XivVals7o%253D&md5=6e9fb898569ce0cb983ecb807c9d978cMolecular dynamics simulationsHansson, Tomas; Oostenbrink, Chris; van Gunsteren, Wilfred F.Current Opinion in Structural Biology (2002), 12 (2), 190-196CODEN: COSBEF; ISSN:0959-440X. (Elsevier Science Ltd.)A review. Mol. dynamics simulations have become a std. tool for the investigation of biomols. Simulations are performed of ever bigger systems using more realistic boundary conditions and better sampling due to longer sampling times. Recently, realistic simulations of systems as complex as transmembrane channels have become feasible. Simulations aid our understanding of biochem. processes and give a dynamic dimension to structural data; for example, the transformation of harmless prion protein into the disease-causing agent has been modeled. The promise of the first protein dynamics simulation 25 yr ago has been realized, as realistic simulations of systems as complex as transmembrane channels have recently become feasible.**18**Denton, R.; Hu, Y. Symmetry boundary conditions.*J. Comput. Phys.*2009,*228*, 4823, DOI: 10.1016/j.jcp.2009.03.033Google Scholar18https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXmtlOhu70%253D&md5=0e436531fc2a277f4a487d2f24f80d57Symmetry boundary conditionsDenton, R. E.; Hu, Y.Journal of Computational Physics (2009), 228 (13), 4823-4835CODEN: JCTPAH; ISSN:0021-9991. (Elsevier B.V.)A simple approach to energy conserving boundary conditions using exact symmetries is described which is esp. useful for numerical simulations using the finite difference method. Each field in the simulation is normally either sym. (even) or antisym. (odd) with respect to the simulation boundary. Another possible boundary condition is an antisym. perturbation about a nonzero value. One of the most powerful aspects of this approach is that it can be easily implemented in curvilinear coordinates by making the scale factors of the coordinate transformation sym. about the boundaries. The method is demonstrated for MHDs (MHD), reduced MHD, and a hybrid code with particle ions and fluid electrons. These boundary conditions yield exact energy conservation in the limit of infinite time and space resoln. Also discussed is the interpretation that the particle charge reverses sign at a conducting boundary with boundary normal perpendicular to the background magnetic field.**19**Wagner, G. J.; Karpov, E. G.; Liu, W. K. Molecular dynamics boundary conditions for regular crystal lattices.*Computer Methods in Applied Mechanics and Engineering*2004,*193*, 1579, DOI: 10.1016/j.cma.2003.12.012Google ScholarThere is no corresponding record for this reference.**20**Roy, A.; Post, C. B. Microscopic symmetry imposed by rotational symmetry boundary conditions in molecular dynamics simulation.*J. Chem. Theory Comput.*2011,*7*, 3346, DOI: 10.1021/ct2000843Google Scholar20https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhtFSmt7bP&md5=be841a6c186f2abc7f685f14d8d2cddcMicroscopic Symmetry Imposed by Rotational Symmetry Boundary Conditions in Molecular Dynamics SimulationRoy, Amitava; Post, Carol BethJournal of Chemical Theory and Computation (2011), 7 (10), 3346-3353CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A large no. of viral capsids, as well as other macromol. assemblies, have an icosahedral structure or structures with other rotational symmetries. This symmetry can be exploited during mol. dynamics (MD) to model in effect the full viral capsid using only a subset of primary atoms plus copies of image atoms generated from rotational symmetry boundary conditions (RSBC). A pure rotational symmetry operation results in both primary and image atoms at short-range, and within nonbonded interaction distance of each other, so that nonbonded interactions cannot be specified by the min. image convention and explicit treatment of image atoms is required. As such, an unavoidable consequence of RSBC is that the enumeration of nonbonded interactions in regions surrounding certain rotational axes must include both a primary atom and its copied image atom, thereby imposing microscopic symmetry for some forces. We examd. the possibility of artifacts arising from this imposed microscopic symmetry of RSBC using two simulation systems: a water shell and human rhinovirus 14 (HRV14) capsid with explicit water. The primary unit was a pentamer of the icosahedron, which has the advantage of direct comparison of icosahedrally equiv. spatial regions, for example regions near a 2-fold symmetry axis with imposed symmetry and a 2-fold axis without imposed symmetry. An anal. of structural and dynamic properties of water mols. and protein atoms found similar behavior near symmetry axes with imposed symmetry and where the min. image convention fails compared with that in other regions in the simulation system, even though an excluded vol. effect was detected for water mols. near the axes with imposed symmetry. These results validate the use of RSBC for icosahedral viral capsids or other rotationally sym. systems.**21**Wyckoff, R. W. G.*The Analytical Expression of the Results of the Theory of Space-groups*; Carnegie Institution of Washington: 1922; Vol. 318.Google ScholarThere is no corresponding record for this reference.**22**You can also arrive at this solution by constraining the Cartesian coordinates and finding Lagrange multipliers.

There is no corresponding record for this reference.**23**Shirts, R. B.; Burt, S. R.; Johnson, A. M. Periodic boundary condition induced breakdown of the equipartition principle and other kinetic effects of finite sample size in classical hard-sphere molecular dynamics simulation.*J. Chem. Phys.*2006,*125*, 164102, DOI: 10.1063/1.2359432Google Scholar23https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XhtFKqsbzM&md5=5386ebdf0bc6644f39946f83ff9fd8b5Periodic boundary condition induced breakdown of the equipartition principle and other kinetic effects of finite sample size in classical hard-sphere molecular dynamics simulationShirts, Randall B.; Burt, Scott R.; Johnson, Aaron M.Journal of Chemical Physics (2006), 125 (16), 164102/1-164102/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We examine consequences of the non-Boltzmann nature of probability distributions for one-particle kinetic energy, momentum, and velocity for finite systems of classical hard spheres with const. total energy and nonidentical masses. By comparing two cases, reflecting walls (NVE or microcanonical ensemble) and periodic boundaries (NVEPG or mol. dynamics ensemble), we describe three consequences of the center-of-mass constraint in periodic boundary conditions: the equipartition theorem no longer holds for unequal masses, the ratio of the av. relative velocity to the av. velocity is increased by a factor of [N/(N-1)]1/2, and the ratio of av. collision energy to av. kinetic energy is increased by a factor of N/(N-1). Simulations in one, two, and three dimensions confirm the analytic results for arbitrary dimension.**24**Kuzkin, V. A. On angular momentum balance for particle systems with periodic boundary conditions.*ZAMM-Journal of Applied Mathematics and Mechanics/ Zeitschrift für Angewandte Mathematik und Mechanik*2015,*95*, 1290, DOI: 10.1002/zamm.201400045Google ScholarThere is no corresponding record for this reference.**25**Musil, F.; Grisafi, A.; Bart′ok, A. P.; Ortner, C.; Cs′anyi, G.; Ceriotti, M. Physics-inspired structural representations for molecules and materials.*Chem. Rev.*2021,*121*, 9759, DOI: 10.1021/acs.chemrev.1c00021Google Scholar25https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXhs1aisL3J&md5=b75f32ea20fa0d83fb028415f540e1f3Physics-inspired structural representations for molecules and materialsMusil, Felix; Grisafi, Andrea; Bartok, Albert P.; Ortner, Christoph; Csanyi, Gabor; Ceriotti, MicheleChemical Reviews (Washington, DC, United States) (2021), 121 (16), 9759-9815CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review. The first step in the construction of a regression model or a data-driven anal., aiming to predict or elucidate the relationship between the at. scale structure of matter and its properties, involves transforming the Cartesian coordinates of the atoms into a suitable representation. The development of at.-scale representations has played, and continues to play, a central role in the success of machine-learning methods for chem. and materials science. This review summarizes the current understanding of the nature and characteristics of the most commonly used structural and chem. descriptions of atomistic structures, highlighting the deep underlying connections between different frameworks, and the ideas that lead to computationally efficient and universally applicable models. It emphasizes the link between properties, structures, their phys. chem. and their math. description, provides examples of recent applications to a diverse set of chem. and materials science problems, and outlines the open questions and the most promising research directions in the field.**26**White, A.*Deep learning for molecules and materials*; 2021. https://whitead.github.io/dmol-book/ (accessed 2022-06-01).Google ScholarThere is no corresponding record for this reference.**27**Frenkel, D.; Eppenga, R. Monte carlo study of the isotropic-nematic transition in a fluid of thin hard disks.*Physical review letters*1982,*49*, 1089, DOI: 10.1103/PhysRevLett.49.1089Google Scholar27https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL38Xlslaqs7w%253D&md5=4df39126e79e336dd27f7186e9cbffb7Monte Carlo study of the isotropic-nematic transition in a fluid of thin hard disksFrenkel, Daan; Eppenga, RobPhysical Review Letters (1982), 49 (15), 1089-92CODEN: PRLTAO; ISSN:0031-9007.The 1st numerical detn. of the thermodn. isotropic-nematic transition in a simple 3-dimensional model fluid, viz., a system of infinitely thin hard platelets, is reported. Thermodn. properties were studied using the const.-pressure Monte Carlo method; Widom's particle-insertion method was used to measure the chem. potential. The phase diagram differs considerably from predictions of a 2nd-virial (Onsager) theory. Virial coeffs. up to the 5th were computed; b5 is neg.**28**Miyamoto, S.; Kollman, P. A. Settle: An analytical version of the shake and rattle algorithm for rigid water models.*Journal of computational chemistry*1992,*13*, 952, DOI: 10.1002/jcc.540130805Google Scholar28https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK38Xlslykt7o%253D&md5=65da9d55c7905abeaf7708d91a09e6e4SETTLE: an analytical version of the SHAKE and RATTLE algorithm for rigid water modelsMiyamoto, Shuichi; Kollman, Peter A.Journal of Computational Chemistry (1992), 13 (8), 952-62CODEN: JCCHDD; ISSN:0192-8651.An anal. algorithm, called SETTLE, for resetting the positions and velocities to satisfy the holonomic constraints on the rigid water model is presented. This method is based on the Cartesian coordinate system and can be used in place of SHAKE and RATTLE. The authors implemented this algorithm in the SPASMS package of mol. mechanics and dynamics. Several series of mol. dynamics simulations were carried out to examine the performance of the new algorithm in comparison with the original RATTLE method. SETTLE is of higher accuracy and is faster than RATTLE with reasonable tolerances by three to nine times on a scalar machine. The performance improvement ranged from factors of 26 to 98 on a vector machine since the method presented is not iterative.**29**Leimkuhler, B.; Matthews, C. Robust and efficient configurational molecular sampling via langevin dynamics, The.*J. Chem. Phys.*2013,*138*, 174102, DOI: 10.1063/1.4802990Google Scholar29https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXmvFyjt7w%253D&md5=01ca1309fa2e43af93c7bea142c66ac4Robust and efficient configurational molecular sampling via Langevin dynamicsLeimkuhler, Benedict; Matthews, CharlesJournal of Chemical Physics (2013), 138 (17), 174102/1-174102/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A wide variety of numerical methods are evaluated and compared for solving the stochastic differential equations encountered in mol. dynamics. The methods are based on the application of deterministic impulses, drifts, and Brownian motions in some combination. The Baker-Campbell-Hausdorff expansion is used to study sampling accuracy following recent work by the authors, which allows detn. of the stepsize-dependent bias in configurational averaging. For harmonic oscillators, configurational averaging is exact for certain schemes, which may result in improved performance in the modeling of biomols. where bond stretches play a prominent role. For general systems, an optimal method can be identified that has very low bias compared to alternatives. In simulations of the alanine dipeptide reported here (both solvated and unsolvated), higher accuracy is obtained without loss of computational efficiency, while allowing large timestep, and with no impairment of the conformational exploration rate (the effective diffusion rate obsd. in simulation). The optimal scheme is a uniformly better performing algorithm for mol. sampling, with overall efficiency improvements of 25% or more in practical timestep size achievable in vacuum, and with redns. in the error of configurational avs. of a factor of ten or more attainable in solvated simulations at large timestep. (c) 2013 American Institute of Physics.**30**Leimkuhler, B.; Matthews, C. Efficient molecular dynamics using geodesic integration and solvent-solute splitting.*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*2016,*472*, 20160138, DOI: 10.1098/rspa.2016.0138Google ScholarThere is no corresponding record for this reference.**31**Aroyo, M. I.*International Tables for Crystallography*; Wiley Online Library: 2013.Google ScholarThere is no corresponding record for this reference.**32**Aroyo, M. I.; Perez-Mato, J.; Orobengoa, D.; Tasci, E.; de la Flor, G.; Kirov, A. Crystallography online: Bilbao crystallographic server.*Bulg. Chem. Commun.*2011,*43*, 183Google Scholar32https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhtlaqs7rN&md5=d81dab069e4816ac6ec6184d8d960e1fCrystallography online: Bilbao crystallographic serverAroyo, M. I.; Perez-Mato, J. M.; Orobengoa, D.; Tasci, E.; de la Flor, G.; Kirov, A.Bulgarian Chemical Communications (2011), 43 (2), 183-197CODEN: BCHCE4; ISSN:0324-1130. (Bulgarian Academy of Sciences)The Bilbao Crystallog. Server is a web site with crystallog. databases and programs available online. It was operating for more than ten years and new applications are being added regularly. The programs available on the server do not need a local installation and can be used free of charge. The server gives access to general information related to crystallog. symmetry groups (generators, general and special positions, maximal subgroups, Brillouin zones, etc.). Apart from the simple tools for retrieving the stored data, there are programs for the anal. of group-subgroup relations between space groups. There are also software package studying specific problems of solid-state physics, structural chem. and crystallog.**33**Aroyo, M. I.; Perez-Mato, J. M.; Capillas, C.; Kroumova, E.; Ivantchev, S.; Madariaga, G.; Kirov, A.; Wondratschek, H. Bilbao crystallographic server: I. databases and crystallographic computing programs.*Zeitschrift für Kristallographie-Crystalline Materials*2006,*221*, 15, DOI: 10.1524/zkri.2006.221.1.15Google Scholar33https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XmtFWnuw%253D%253D&md5=a2b474332c2a0c85b8fb7cc92de49861Bilbao crystallographic server: I. Databases and crystallographic computing programsAroyo, Mois Ilia; Perez-Mato, Juan Manuel; Capillas, Cesar; Kroumova, Eli; Ivantchev, Svetoslav; Madariaga, Gotzon; Kirov, Asen; Wondratschek, HansZeitschrift fuer Kristallographie (2006), 221 (1), 15-27CODEN: ZEKRDZ; ISSN:0044-2968. (Oldenbourg Wissenschaftsverlag GmbH)The Bilbao Crystallog. Server is a web site with crystallog. databases and programs available online at www.cryst.ehu.es. It has been operating for about six years and new applications are being added regularly. The programs available on the server do not need a local installation and can be used free of charge. The only requirement is an Internet connection and a web browser. The server is built on a core of databases, and contains different shells. The innermost one is formed by simple retrieval tools which serve as an interface to the databases and permit to obtain the stored symmetry information for space groups and layer groups. The k-vector database includes the Brillouin zones and the wave-vector types for all space groups. As a part of the server one can find also the database of incommensurate structures. The second shell contains applications which are essential for problems involving group-subgroup relations between space groups (e.g. subgroups and supergroups of space groups, splittings of Wyckoff positions), while the third shell contains more sophisticated programs for the computation of space-group representations and their correlations for group-subgroup related space groups. There are also programs for calcns. focused on specific problems of solid-state physics. The aim of the article is to report on the current state of the server and to provide a brief description of the accessible databases and crystallog. computing programs. The use of the programs is demonstrated by illustrative examples.**34**Aroyo, M. I.; Kirov, A.; Capillas, C.; Perez-Mato, J.; Wondratschek, H. Bilbao crystallographic server. ii. representations of crystallographic point groups and space groups.*Acta Crystallographica Section A: Foundations of Crystallography*2006,*62*, 115, DOI: 10.1107/S0108767305040286Google Scholar34https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XhslWnur8%253D&md5=689fb23c14e777b10fed4390b438b9a9Bilbao Crystallographic Server. II. Representations of crystallographic point groups and space groupsAroyo, Mois I.; Kirov, Asen; Capillas, Cesar; Perez-Mato, J. M.; Wondratschek, HansActa Crystallographica, Section A: Foundations of Crystallography (2006), A62 (2), 115-128CODEN: ACACEQ; ISSN:0108-7673. (Blackwell Publishing Ltd.)The Bilbao Crystallog. Server is a web site with crystallog. programs and databases freely available online (http://www.cryst.ehu.es). The server gives access to general information related to crystallog. symmetry groups (generators, general and special positions, maximal subgroups, Brillouin zones etc.). Apart from the simple tools for retrieving the stored data, there are programs for the anal. of group-subgroup relations between space groups (subgroups and supergroups, Wyckoff-position splitting schemes etc.). There are also software packages studying specific problems of solid-state physics, structural chem. and crystallog. This article reports on the programs treating representations of point and space groups. There are tools for the construction of irreducible representations, for the study of the correlations between representations of group-subgroup pairs of space groups and for the decompns. of Kronecker products of representations.**35**Frenkel, D.; Smit, B.; Ratner, M. A.*Understanding molecular simulation: from algorithms to applications*; Academic Press: San Diego, 1996; Vol. 2.Google ScholarThere is no corresponding record for this reference.

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ARTICLE SECTIONSThis article references 36 other publications.

**1**Parrinello, M.; Rahman, A. Polymorphic transitions in single crystals: A new molecular dynamics method.*J. Appl. Phys.*1981,*52*, 7182, DOI: 10.1063/1.3286931https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL38XislSnuw%253D%253D&md5=a0a5617389f6cabbf2a405c649aadf03Polymorphic transitions in single crystals: A new molecular dynamics methodParrinello, M.; Rahman, A.Journal of Applied Physics (1981), 52 (12), 7182-90CODEN: JAPIAU; ISSN:0021-8979.A Lagrangian formulation is introduced; it can be used to make mol. dynamics (MD) calcns. on systems under the most general, externally applied, conditions of stress. In this formulation the MD cell shape and size can change according to dynamic equations given by this Lagrangian. This MD technique was used to the study of structural transitions of a Ni single crystal under uniform uniaxial compressive and tensile loads. Some results regarding the stress-strain relation obtained by static calcns. are invalid at finite temp. Under compressive loading, the model of Ni shows a bifurcation in its stress-strain relation; this bifurcation provides a link in configuration space between cubic and hexagonal close packing. Such a transition could perhaps be obsd. exptl. under extreme conditions of shock.**2**Ryckaert, J.-P.; Ciccotti, G.; Berendsen, H. J. Numerical integration of the cartesian equations of motion of a system with constraints: molecular dynamics of nalkanes.*J. Comput. Phys.*1977,*23*, 327, DOI: 10.1016/0021-9991(77)90098-52https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE2sXktVGhsL4%253D&md5=b4aecddfde149117813a5ea4f5353ce2Numerical integration of the Cartesian equations of motion of a system with constraints: molecular dynamics of n-alkanesRyckaert, Jean Paul; Ciccotti, Giovanni; Berendsen, Herman J. C.Journal of Computational Physics (1977), 23 (3), 327-41CODEN: JCTPAH; ISSN:0021-9991.A numerical algorithm integrating the 3N Cartesian equation of motion of a system of N points subject to holonomic constraints is applied to mol. dynamics simulation of a liq. of 64 butane mols.**3**Andersen, H. C. Rattle: A “oevelocity” version of the shake algorithm for molecular dynamics calculations.*J. Comput. Phys.*1983,*52*(1), 24– 34, DOI: 10.1016/0021-9991(83)90014-13https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL2cXjvFOntw%253D%253D&md5=770dfdc612edc5847839ca28ea3d6501RATTLE: a "velocity" version of the SHAKE algorithm for molecular dynamics calculationsAndersen, Hans C.Journal of Computational Physics (1983), 52 (1), 24-34CODEN: JCTPAH; ISSN:0021-9991.An algorithm, called RATTLE, for integrating the equations of motion in mol. dynamics calcns. for mol. models with internal constraints is presented. RATTLE calcs. the positions and velocities at the next time from the positions and velocities at the present time step, without requiring information about the earlier history. It is based on the Verlet algorithm and retains the simplicity of using Cartesian coordinates for each of the atoms to describe the configuration of a mol. with internal constraints. RATTLE guarantees that the coordinates and velocities of the atoms in a mol. satisfy the internal constraints at each time step.**4**Tobias, D. J.; Brooks, C. L., III Molecular dynamics with internal coordinate constraints.*J. Chem. Phys.*1988,*89*, 5115, DOI: 10.1063/1.4556544https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1cXmtlOhtLw%253D&md5=ce33ec54b3c728522a51be34b8e5c927Molecular dynamics with internal coordinate constraintsTobias, Douglas J.; Brooks, Charles L., IIIJournal of Chemical Physics (1988), 89 (8), 5115-27CODEN: JCPSA6; ISSN:0021-9606.The method of J.P. Ryckaet, et al. (1977) for integrating the Cartesian equations of motion of a system with holonomic constraints, has been extended to allow the independent constraint of arbitrary internal coordinates. To illustrate this new methodol., and to investigate the effects of dihedral angle constraints on the equil. and dynamical properties of macromols., parallel sets of mol. dynamics simulations and normal mode analyses of a small dipeptide were carried out. One without constraints, and one with a single backbone dihedral angle constrained. The avs. and the fluctuations of the energies, and of the internal degrees of freedom are not significantly modified by the constraint. However, in the region between 100 and 1400 cm-1 of the normal mode spectrum, the constraint shifts the frequencies of the modes, and modifies their contributions to the spectra of the internal coordinates. Except for the lowest frequency torsional modes, in which anharmonic effects are significant, the behavior of the mol. dynamics power spectra is similar to that of the normal mode spectra.**5**Ryckaert, J.-P. Special geometrical constraints in the molecular dynamics of chain molecules.*Mol. Phys.*1985,*55*, 549, DOI: 10.1080/002689785001015315https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL2MXlsVOnuro%253D&md5=4642e82cf33a67826ed8db68b2a885ccSpecial geometrical constraints in the molecular dynamics of chain moleculesRyckaert, J. P.Molecular Physics (1985), 55 (3), 549-56CODEN: MOPHAM; ISSN:0026-8976.The mol.-dynamics simulation of chain mols. with a full at. description was considered in the case of n-alkane mols. In order (a) to keep the time step in the numerical integration of the equations of motions to a reasonable value (10-15-10-14 s), and (b) to describe such flexible systems with a min. no. of degrees of freedom, it is useful to impose geom. constraints to freeze the fastest intramol. motions of the chain. Given the large no. and the nature of the geom. constraints involved in this model, the method of constraints used to solve the dynamics in terms of at. cartesian coordinates needs to be generalized to arbitrary constraints and solved in an iterative way. Such a generalized algorithm is given, and was used in calcns. on the intramal. dynamics of cyclotetradecane (C14H28).**6**Edberg, R.; Evans, D. J.; Morriss, G. Constrained molecular dynamics: Simulations of liquid alkanes with a new algorithm.*J. Chem. Phys.*1986,*84*, 6933, DOI: 10.1063/1.4506136https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL28XksV2ht7s%253D&md5=ab74e47e9e17653583a5ff1b58f4e78aConstrained molecular dynamics: simulations of liquid alkanes with a new algorithmEdberg, Roger; Evans, Denis J.; Morriss, G. P.Journal of Chemical Physics (1986), 84 (12), 6933-9CODEN: JCPSA6; ISSN:0021-9606.A new algorithm is given for mol.-dynamics simulation involving holonomic constraints. Constrained equations of motion were derived by using the Gauss principle of least constraint. The algorithm uses a fast, exact soln. for constraint forces, and a new procedure to correct for accumulating numerical errors. Several simulations for liq. butane and decane were done with the new algorithm. An av. trans population of 60.6 ± 1.5% in liq. butane was obtained at 291 K and 0.583 g/mL. This result essentially agreed with that from a simulation given by J. P. Ryckaert and A. Bellemans (1978). However, the present simulations were substantially more precise; the run lengths were typically ∼20 times longer than those of R. and B. The present result also agrees with the simulation given by P. A. Wielopolski and E. R. Smith (1986). Thermodn. and structural data from the present simulations also agree well with these other simulations.**7**Ciccotti, G.; Ryckaert, J.-P. Molecular dynamics simulation of rigid molecules.*Computer Physics Reports*1986,*4*, 346, DOI: 10.1016/0167-7977(86)90022-5There is no corresponding record for this reference.**8**Anishkin, A.; Milac, A. L.; Guy, H. R. Symmetryrestrained molecular dynamics simulations improve homology models of potassium channels.*Proteins: Struct., Funct., Bioinf.*2010,*78*, 932, DOI: 10.1002/prot.226188https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXhtVSlsb4%253D&md5=0394ee96764f1f68d55663c5b1d33cd5Symmetry-restrained molecular dynamics simulations improve homology models of potassium channelsAnishkin, Andriy; Milac, Adina L.; Guy, H. RobertProteins: Structure, Function, and Bioinformatics (2010), 78 (4), 932-949CODEN: PSFBAF ISSN:. (Wiley-Liss, Inc.)Most crystd. homo-oligomeric ion channels are highly sym., which dramatically decreases conformational space and facilitates building homol. models (HMs). However, in mol. dynamics (MD) simulations channels deviate from ideal symmetry and accumulate thermal defects, which complicate the refinement of HMs using MD. In this work the authors evaluate the ability of symmetry constrained MD simulations to improve HMs accuracy, using an approach conceptually similar to Crit. Assessment of techniques for protein Structure Prediction (CASP) competition: build HMs of channels with known structure and evaluate the efficiency of proposed methods in improving HMs accuracy (measured as deviation from exptl. structure). Results indicate that unrestrained MD does not improve the accuracy of HMs, instantaneous symmetrization improves accuracy but not stability of HMs during subsequent unrestrained MD, while gradually imposing symmetry constraints improves both accuracy (by 5-50%) and stability of HMs. Moreover, accuracy and stability are strongly correlated, making stability a reliable criterion in predicting the accuracy of new HMs. Proteins 2010. © 2009 Wiley-Liss, Inc.**9**Zabrodsky, H.; Peleg, S.; Avnir, D. Continuous symmetry measures.*J. Am. Chem. Soc.*1992,*114*, 7843, DOI: 10.1021/ja00046a0339https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK38XlsFSgsb8%253D&md5=deda198744536a9eb0d116dc7edc609cContinuous symmetry measuresZabrodsky, Hagit; Peleg, Shmuel; Avnir, DavidJournal of the American Chemical Society (1992), 114 (20), 7843-51CODEN: JACSAT; ISSN:0002-7863.The notion that for many realistic issues involving symmetry in chem., it is more natural to analyze symmetry properties in terms of a continuous scale rather than in terms of yes or no. Justification of that approach is dealt with using examples such as: symmetry distortions due to vibrations; changes in the allowedness of electronic transitions due to deviations from an ideal symmetry; continuous changes in environmental symmetry with ref. to crystal and ligand field effects; nonideal symmetry in concerted reactions; symmetry issues of polymers and large random objects. A versatile, simple tool is developed as a continuous symmetry measure. Its main property is the ability to quantify the distance of a given (distorted mol.) shape from any chosen element of symmetry. The generality of this symmetry measure allows one to compare the symmetry distance of several objects relative to a single symmetry element and to compare the symmetry distance of a single object relative to various symmetry elements. The continuous symmetry approach is presented for the case of cyclic mols., first in a practical way and then with a rigorous math. anal. The versatility of the approach is then further demonstrated with alkane conformations, with a vibrating ABA water-like mol., and with a three-dimensional anal. of the symmetry of a [2 + 2] reaction in which the double bonds are not ideally aligned.**10**Oakley, M. T.; Johnston, R. L.; Wales, D. J. Symmetrisation schemes for global optimization of atomic clusters.*Phys. Chem. Chem. Phys.*2013,*15*, 3965, DOI: 10.1039/c3cp44332a10https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXislyns7k%253D&md5=d4eea68fb6ca314b7d83ed01d5feae33Symmetrisation schemes for global optimisation of atomic clustersOakley, Mark T.; Johnston, Roy L.; Wales, David J.Physical Chemistry Chemical Physics (2013), 15 (11), 3965-3976CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Locating the global min. of at. and mol. clusters can be a difficult optimization problem. Here we report benchmarks for procedures that exploit approx. symmetry. This strategy was implemented in the GMIN program following a theor. anal., which explained why high-symmetry structures are more likely to have particularly high or particularly low energy. The anal., and the corresponding algorithms, allow for approx. point group symmetry, and can be combined with basin-hopping and genetic algorithms. We report results for 38-, 75-, and 98-atom Lennard-Jones clusters, which are all multiple-funnel systems. Exploiting approx. symmetry reduces the mean time taken to locate the global min. by up to two orders of magnitude, with smaller improvements in efficiency for LJ55 and LJ74, which correspond to simpler single-funnel energy landscapes.**11**van Eijck, B. P.; Kroon, J. Upack program package for crystal structure prediction: Force fields and crystal structure generation for small carbohydrate molecules.*Journal of computational chemistry*1999,*20*, 799, DOI: 10.1002/(SICI)1096-987X(199906)20:8<799::AID-JCC6>3.0.CO;2-Z11https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXjsFShsL4%253D&md5=073401b9191ccee6cefa88d726b36c5dUpack program package for crystal structure prediction: force fields and crystal structure generation for small carbohydrate moleculesVan Eijck, Bouke P.; Kroon, JanJournal of Computational Chemistry (1999), 20 (8), 799-812CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)The UPACK program package for crystal structure generation was used to build hypothetical crystal structures for 32 pyranoses and 24 polyalcs. A subset of these was used to compare six force fields in their ability to reproduce the exptl. obsd. structures, preferably with low energies and high rankings with respect to the structure with lowest energy. One of these force fields, UNITAT, was derived from GROMOS87 by adjusting some parameters to obtain a better geometric description of these crystal structures. This united-atom force field and the all-atom OPLS force field performed best, and were used in the crystal structure generation for the full set of compds. For these carbohydrates, hundreds of hypothetical polymorphic structures are generated in an energy window of ∼25 kJ/mol. On av., the exptl. structures had reasonable energies and rankings. These lists are good starting points for more sophisticated calcns., and can be used for structure detn. by powder diffraction methods. In a few cases, serious doubt was raised concerning the H-bond network reported in the literature, for which more plausible alternatives appear to exist.**12**Fredericks, S.; Parrish, K.; Sayre, D.; Zhu, Q. Pyxtal: A python library for crystal structure generation and symmetry analysis.*Comput. Phys. Commun.*2021,*261*, 107810, DOI: 10.1016/j.cpc.2020.10781012https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXhtlWhsLs%253D&md5=6e93cdb95622515c90baf45b7a9d04fePyXtal: A Python library for crystal structure generation and symmetry analysisFredericks, Scott; Parrish, Kevin; Sayre, Dean; Zhu, QiangComputer Physics Communications (2021), 261 (), 107810CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)We present PyXtal, a new package based on the Python programming language, used to generate structures with specific symmetry and chem. compns. for both at. and mol. systems. This software provides support for various systems described by point, rod, layer, and space group symmetries. With only the inputs of chem. compn. and symmetry group information, PyXtal can automatically find a suitable combination of Wyckoff positions with a step-wise merging scheme. Further, when the mol. geometry is given, PyXtal can generate different dimensional org. crystals with mols. occupying both general and special Wyckoff positions. Optionally, PyXtal also accepts user-defined parameters (e.g., cell parameters, min. distances and Wyckoff positions). In general, PyXtal serves three purposes: (1) to generate custom structures, (2) to modulate the structure by symmetry relations, (3) to interface the existing structure prediction codes that require the generation of random sym. structures. In addn., we provide several utilities that facilitate the anal. of structures, including symmetry anal., geometry optimization, and simulations of powder X-ray diffraction (XRD). Full documentation of PyXtal is available at https://pyxtal.readthedocs.io. Program Title: PyXtalCPC Library link to program files:http://dx.doi.org/10.17632/wfyxyhjzwx.12Licensing provisions: MIT [1]Programming language: Python 3Nature of problem: Knowledge of structure at the at. level is the key to understanding materials' properties. Typically, the structure of a material can be detd. either from expt. (such as X-ray diffraction, spectroscopy, microscopy) or from theory (e.g., enhanced sampling, structure prediction). In many cases, the structure needs to be solved iteratively by generating a no. of trial structure models satisfying some constraints (e.g., chem. compn., symmetry, and unit cell parameters). Therefore, it is desirable to have a computational code that is able to generate such trial structures in an automated manner. The PyXtal package is able to generate many possible random structures for both at. and mol. systems with all possible symmetries. To generate the trial structure, the algorithm can either start with picking the symmetry sites randomly from high to low multiplicities, or use sites that are predefined by the user. For mols., the algorithm can automatically detect the mols.' symmetry and place them into special Wyckoff positions while satisfying their compatible site symmetry. With the support of symmetry operations for point, rod, layer and space groups, PyXtal is suitable for the computational modeling of systems from zero, one, two, and three dimensional bulk crystals.**13**Xie, T.; Fu, X.; Ganea, O.-E.; Barzilay, R.; Jaakkola, T. Crystal diffusion variational autoencoder for periodic material generation. 2021, arXiv:2110.06197.*arXiv preprint*. https://arxiv.org/abs/2110.06197 (accessed 2022-06-10).There is no corresponding record for this reference.**14**Cannon, K. A.; Ochoa, J. M.; Yeates, T. O. High-symmetry protein assemblies: patterns and emerging applications.*Curr. Opin. Struct. Biol.*2019,*55*, 77, DOI: 10.1016/j.sbi.2019.03.00814https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXms1Ggtrk%253D&md5=55277a1232c59feb420d94b8957294d9High-symmetry protein assemblies: patterns and emerging applicationsCannon, Kevin A.; Ochoa, Jessica M.; Yeates, Todd O.Current Opinion in Structural Biology (2019), 55 (), 77-84CODEN: COSBEF; ISSN:0959-440X. (Elsevier Ltd.)The accelerated elucidation of three-dimensional structures of protein complexes, both natural and designed, is providing new examples of large supramol. assemblies with intriguing shapes. Those with high symmetry - based on the geometries of the Platonic solids - are particularly notable as their innately closed forms create interior spaces with varying degrees of enclosure. We survey known protein assemblies of this type and discuss their geometric features. The results bear on issues of protein function and evolution, while also guiding novel bioengineering applications. Recent successes using high-symmetry protein assemblies for applications in interior encapsulation and exterior display are highlighted.**15**Dayal, K.; James, R. D. Nonequilibrium molecular dynamics for bulk materials and nanostructures.*Journal of the Mechanics and Physics of Solids*2010,*58*, 145, DOI: 10.1016/j.jmps.2009.10.00815https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXksVentg%253D%253D&md5=78c01085107887eea5c179309937328fNonequilibrium molecular dynamics for bulk materials and nanostructuresDayal, Kaushik; James, Richard D.Journal of the Mechanics and Physics of Solids (2010), 58 (2), 145-163CODEN: JMPSA8; ISSN:0022-5096. (Elsevier Ltd.)We describe a method of constructing exact solns. of the equations of mol. dynamics in non-equil. settings. These solns. correspond to some viscometric flows, and to certain analogs of viscometric flows for fibers and membranes that have one or more dimensions of at. scale. This work generalizes the method of objective mol. dynamics (OMD) (). It allows us to calc. viscometric properties from a mol.-level simulation in the absence of a constitutive equation, and to relate viscometric properties directly to mol. properties. The form of the solns. is partly independent of the form of the force laws between atoms, and therefore these solns. have implications for coarse-grained theories. We show that there is an exact redn. of the Boltzmann equation corresponding to one family of OMD solns. This redn. includes most known exact solns. of the equations of the moments for special kinds of mols. and gives the form of the mol. d. function corresponding to such flows. This and other consequences leads us to propose an addn. to the principle of material frame indifference, a cornerstone of nonlinear continuum mechanics. The method is applied to the failure of carbon nanotubes at an imposed strain rate, using the Tersoff potential for carbon. A large set of simulations with various strain rates, initial conditions and two choices of fundamental domain (unit cell) give the following unexpected results: Stone-Wales defects play no role in the failure (though Stone-Wales partials are sometimes seen just prior to failure), a variety of failure mechanisms is obsd., and most simulations give a strain at failure of 15-20%, except those done with initial temp. above about 1200 K and at the lower strain rates. The latter have a strain at failure of 1-2%.**16**Xu, H.; Drozdov, G.; Hourahine, B.; Park, J. G.; Sweat, R.; Frauenheim, T.; Dumitrică, T. Collapsed carbon nanotubes: From nano to mesoscale via density functional theory-based tight-binding objective molecular modeling.*Carbon*2019,*143*, 786, DOI: 10.1016/j.carbon.2018.11.06816https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXisVOiur%252FO&md5=8cce8e84f8015d7f29b56ef027769870Collapsed carbon nanotubes: From nano to mesoscale via density functional theory-based tight-binding objective molecular modelingXu, Hao; Drozdov, Grigorii; Hourahine, Ben; Park, Jin Gyu; Sweat, Rebekah; Frauenheim, Thomas; Dumitrica, TraianCarbon (2019), 143 (), 786-792CODEN: CRBNAH; ISSN:0008-6223. (Elsevier Ltd.)Due to the inherent spatial and temporal limitations of atomistic modeling and the lack of efficient mesoscopic models, mesoscale simulation methods for guiding the development of super strong lightwt. material systems comprising collapsed C nanotubes (CNTs) are currently missing. Here the authors establish a path for deriving ultra-coarse-grained mesoscopic distinct element method (mDEM) models directly from the quantum mech. representation of a collapsed CNT. Atomistic calcns. based on d. functional-based tight-binding (DFTB) extended with Lennard-Jones interactions allow for the identification of the cross-section and elastic consts. of an elastic beam idealization of a collapsed CNT. Application of the DFTB quantum treatment is possible due to the simplification in the no. of atoms introduced by accounting for the helical and angular symmetries exhibited by twisted and bent CNTs. The multiscale modeling chain established here is suitable for deriving ultra-coarse-grained mesoscopic models for a variety of microscopic filaments presenting complex interat. bondings.**17**Hansson, T.; Oostenbrink, C.; van Gunsteren, W. Molecular dynamics simulations.*Curr. Opin. Struct. Biol.*2002,*12*, 190, DOI: 10.1016/S0959-440X(02)00308-117https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XivVals7o%253D&md5=6e9fb898569ce0cb983ecb807c9d978cMolecular dynamics simulationsHansson, Tomas; Oostenbrink, Chris; van Gunsteren, Wilfred F.Current Opinion in Structural Biology (2002), 12 (2), 190-196CODEN: COSBEF; ISSN:0959-440X. (Elsevier Science Ltd.)A review. Mol. dynamics simulations have become a std. tool for the investigation of biomols. Simulations are performed of ever bigger systems using more realistic boundary conditions and better sampling due to longer sampling times. Recently, realistic simulations of systems as complex as transmembrane channels have become feasible. Simulations aid our understanding of biochem. processes and give a dynamic dimension to structural data; for example, the transformation of harmless prion protein into the disease-causing agent has been modeled. The promise of the first protein dynamics simulation 25 yr ago has been realized, as realistic simulations of systems as complex as transmembrane channels have recently become feasible.**18**Denton, R.; Hu, Y. Symmetry boundary conditions.*J. Comput. Phys.*2009,*228*, 4823, DOI: 10.1016/j.jcp.2009.03.03318https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXmtlOhu70%253D&md5=0e436531fc2a277f4a487d2f24f80d57Symmetry boundary conditionsDenton, R. E.; Hu, Y.Journal of Computational Physics (2009), 228 (13), 4823-4835CODEN: JCTPAH; ISSN:0021-9991. (Elsevier B.V.)A simple approach to energy conserving boundary conditions using exact symmetries is described which is esp. useful for numerical simulations using the finite difference method. Each field in the simulation is normally either sym. (even) or antisym. (odd) with respect to the simulation boundary. Another possible boundary condition is an antisym. perturbation about a nonzero value. One of the most powerful aspects of this approach is that it can be easily implemented in curvilinear coordinates by making the scale factors of the coordinate transformation sym. about the boundaries. The method is demonstrated for MHDs (MHD), reduced MHD, and a hybrid code with particle ions and fluid electrons. These boundary conditions yield exact energy conservation in the limit of infinite time and space resoln. Also discussed is the interpretation that the particle charge reverses sign at a conducting boundary with boundary normal perpendicular to the background magnetic field.**19**Wagner, G. J.; Karpov, E. G.; Liu, W. K. Molecular dynamics boundary conditions for regular crystal lattices.*Computer Methods in Applied Mechanics and Engineering*2004,*193*, 1579, DOI: 10.1016/j.cma.2003.12.012There is no corresponding record for this reference.**20**Roy, A.; Post, C. B. Microscopic symmetry imposed by rotational symmetry boundary conditions in molecular dynamics simulation.*J. Chem. Theory Comput.*2011,*7*, 3346, DOI: 10.1021/ct200084320https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhtFSmt7bP&md5=be841a6c186f2abc7f685f14d8d2cddcMicroscopic Symmetry Imposed by Rotational Symmetry Boundary Conditions in Molecular Dynamics SimulationRoy, Amitava; Post, Carol BethJournal of Chemical Theory and Computation (2011), 7 (10), 3346-3353CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A large no. of viral capsids, as well as other macromol. assemblies, have an icosahedral structure or structures with other rotational symmetries. This symmetry can be exploited during mol. dynamics (MD) to model in effect the full viral capsid using only a subset of primary atoms plus copies of image atoms generated from rotational symmetry boundary conditions (RSBC). A pure rotational symmetry operation results in both primary and image atoms at short-range, and within nonbonded interaction distance of each other, so that nonbonded interactions cannot be specified by the min. image convention and explicit treatment of image atoms is required. As such, an unavoidable consequence of RSBC is that the enumeration of nonbonded interactions in regions surrounding certain rotational axes must include both a primary atom and its copied image atom, thereby imposing microscopic symmetry for some forces. We examd. the possibility of artifacts arising from this imposed microscopic symmetry of RSBC using two simulation systems: a water shell and human rhinovirus 14 (HRV14) capsid with explicit water. The primary unit was a pentamer of the icosahedron, which has the advantage of direct comparison of icosahedrally equiv. spatial regions, for example regions near a 2-fold symmetry axis with imposed symmetry and a 2-fold axis without imposed symmetry. An anal. of structural and dynamic properties of water mols. and protein atoms found similar behavior near symmetry axes with imposed symmetry and where the min. image convention fails compared with that in other regions in the simulation system, even though an excluded vol. effect was detected for water mols. near the axes with imposed symmetry. These results validate the use of RSBC for icosahedral viral capsids or other rotationally sym. systems.**21**Wyckoff, R. W. G.*The Analytical Expression of the Results of the Theory of Space-groups*; Carnegie Institution of Washington: 1922; Vol. 318.There is no corresponding record for this reference.**22**You can also arrive at this solution by constraining the Cartesian coordinates and finding Lagrange multipliers.

There is no corresponding record for this reference.**23**Shirts, R. B.; Burt, S. R.; Johnson, A. M. Periodic boundary condition induced breakdown of the equipartition principle and other kinetic effects of finite sample size in classical hard-sphere molecular dynamics simulation.*J. Chem. Phys.*2006,*125*, 164102, DOI: 10.1063/1.235943223https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XhtFKqsbzM&md5=5386ebdf0bc6644f39946f83ff9fd8b5Periodic boundary condition induced breakdown of the equipartition principle and other kinetic effects of finite sample size in classical hard-sphere molecular dynamics simulationShirts, Randall B.; Burt, Scott R.; Johnson, Aaron M.Journal of Chemical Physics (2006), 125 (16), 164102/1-164102/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We examine consequences of the non-Boltzmann nature of probability distributions for one-particle kinetic energy, momentum, and velocity for finite systems of classical hard spheres with const. total energy and nonidentical masses. By comparing two cases, reflecting walls (NVE or microcanonical ensemble) and periodic boundaries (NVEPG or mol. dynamics ensemble), we describe three consequences of the center-of-mass constraint in periodic boundary conditions: the equipartition theorem no longer holds for unequal masses, the ratio of the av. relative velocity to the av. velocity is increased by a factor of [N/(N-1)]1/2, and the ratio of av. collision energy to av. kinetic energy is increased by a factor of N/(N-1). Simulations in one, two, and three dimensions confirm the analytic results for arbitrary dimension.**24**Kuzkin, V. A. On angular momentum balance for particle systems with periodic boundary conditions.*ZAMM-Journal of Applied Mathematics and Mechanics/ Zeitschrift für Angewandte Mathematik und Mechanik*2015,*95*, 1290, DOI: 10.1002/zamm.201400045There is no corresponding record for this reference.**25**Musil, F.; Grisafi, A.; Bart′ok, A. P.; Ortner, C.; Cs′anyi, G.; Ceriotti, M. Physics-inspired structural representations for molecules and materials.*Chem. Rev.*2021,*121*, 9759, DOI: 10.1021/acs.chemrev.1c0002125https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXhs1aisL3J&md5=b75f32ea20fa0d83fb028415f540e1f3Physics-inspired structural representations for molecules and materialsMusil, Felix; Grisafi, Andrea; Bartok, Albert P.; Ortner, Christoph; Csanyi, Gabor; Ceriotti, MicheleChemical Reviews (Washington, DC, United States) (2021), 121 (16), 9759-9815CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review. The first step in the construction of a regression model or a data-driven anal., aiming to predict or elucidate the relationship between the at. scale structure of matter and its properties, involves transforming the Cartesian coordinates of the atoms into a suitable representation. The development of at.-scale representations has played, and continues to play, a central role in the success of machine-learning methods for chem. and materials science. This review summarizes the current understanding of the nature and characteristics of the most commonly used structural and chem. descriptions of atomistic structures, highlighting the deep underlying connections between different frameworks, and the ideas that lead to computationally efficient and universally applicable models. It emphasizes the link between properties, structures, their phys. chem. and their math. description, provides examples of recent applications to a diverse set of chem. and materials science problems, and outlines the open questions and the most promising research directions in the field.**26**White, A.*Deep learning for molecules and materials*; 2021. https://whitead.github.io/dmol-book/ (accessed 2022-06-01).There is no corresponding record for this reference.**27**Frenkel, D.; Eppenga, R. Monte carlo study of the isotropic-nematic transition in a fluid of thin hard disks.*Physical review letters*1982,*49*, 1089, DOI: 10.1103/PhysRevLett.49.108927https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL38Xlslaqs7w%253D&md5=4df39126e79e336dd27f7186e9cbffb7Monte Carlo study of the isotropic-nematic transition in a fluid of thin hard disksFrenkel, Daan; Eppenga, RobPhysical Review Letters (1982), 49 (15), 1089-92CODEN: PRLTAO; ISSN:0031-9007.The 1st numerical detn. of the thermodn. isotropic-nematic transition in a simple 3-dimensional model fluid, viz., a system of infinitely thin hard platelets, is reported. Thermodn. properties were studied using the const.-pressure Monte Carlo method; Widom's particle-insertion method was used to measure the chem. potential. The phase diagram differs considerably from predictions of a 2nd-virial (Onsager) theory. Virial coeffs. up to the 5th were computed; b5 is neg.**28**Miyamoto, S.; Kollman, P. A. Settle: An analytical version of the shake and rattle algorithm for rigid water models.*Journal of computational chemistry*1992,*13*, 952, DOI: 10.1002/jcc.54013080528https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK38Xlslykt7o%253D&md5=65da9d55c7905abeaf7708d91a09e6e4SETTLE: an analytical version of the SHAKE and RATTLE algorithm for rigid water modelsMiyamoto, Shuichi; Kollman, Peter A.Journal of Computational Chemistry (1992), 13 (8), 952-62CODEN: JCCHDD; ISSN:0192-8651.An anal. algorithm, called SETTLE, for resetting the positions and velocities to satisfy the holonomic constraints on the rigid water model is presented. This method is based on the Cartesian coordinate system and can be used in place of SHAKE and RATTLE. The authors implemented this algorithm in the SPASMS package of mol. mechanics and dynamics. Several series of mol. dynamics simulations were carried out to examine the performance of the new algorithm in comparison with the original RATTLE method. SETTLE is of higher accuracy and is faster than RATTLE with reasonable tolerances by three to nine times on a scalar machine. The performance improvement ranged from factors of 26 to 98 on a vector machine since the method presented is not iterative.**29**Leimkuhler, B.; Matthews, C. Robust and efficient configurational molecular sampling via langevin dynamics, The.*J. Chem. Phys.*2013,*138*, 174102, DOI: 10.1063/1.480299029https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXmvFyjt7w%253D&md5=01ca1309fa2e43af93c7bea142c66ac4Robust and efficient configurational molecular sampling via Langevin dynamicsLeimkuhler, Benedict; Matthews, CharlesJournal of Chemical Physics (2013), 138 (17), 174102/1-174102/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A wide variety of numerical methods are evaluated and compared for solving the stochastic differential equations encountered in mol. dynamics. The methods are based on the application of deterministic impulses, drifts, and Brownian motions in some combination. The Baker-Campbell-Hausdorff expansion is used to study sampling accuracy following recent work by the authors, which allows detn. of the stepsize-dependent bias in configurational averaging. For harmonic oscillators, configurational averaging is exact for certain schemes, which may result in improved performance in the modeling of biomols. where bond stretches play a prominent role. For general systems, an optimal method can be identified that has very low bias compared to alternatives. In simulations of the alanine dipeptide reported here (both solvated and unsolvated), higher accuracy is obtained without loss of computational efficiency, while allowing large timestep, and with no impairment of the conformational exploration rate (the effective diffusion rate obsd. in simulation). The optimal scheme is a uniformly better performing algorithm for mol. sampling, with overall efficiency improvements of 25% or more in practical timestep size achievable in vacuum, and with redns. in the error of configurational avs. of a factor of ten or more attainable in solvated simulations at large timestep. (c) 2013 American Institute of Physics.**30**Leimkuhler, B.; Matthews, C. Efficient molecular dynamics using geodesic integration and solvent-solute splitting.*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*2016,*472*, 20160138, DOI: 10.1098/rspa.2016.0138There is no corresponding record for this reference.**31**Aroyo, M. I.*International Tables for Crystallography*; Wiley Online Library: 2013.There is no corresponding record for this reference.**32**Aroyo, M. I.; Perez-Mato, J.; Orobengoa, D.; Tasci, E.; de la Flor, G.; Kirov, A. Crystallography online: Bilbao crystallographic server.*Bulg. Chem. Commun.*2011,*43*, 18332https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhtlaqs7rN&md5=d81dab069e4816ac6ec6184d8d960e1fCrystallography online: Bilbao crystallographic serverAroyo, M. I.; Perez-Mato, J. M.; Orobengoa, D.; Tasci, E.; de la Flor, G.; Kirov, A.Bulgarian Chemical Communications (2011), 43 (2), 183-197CODEN: BCHCE4; ISSN:0324-1130. (Bulgarian Academy of Sciences)The Bilbao Crystallog. Server is a web site with crystallog. databases and programs available online. It was operating for more than ten years and new applications are being added regularly. The programs available on the server do not need a local installation and can be used free of charge. The server gives access to general information related to crystallog. symmetry groups (generators, general and special positions, maximal subgroups, Brillouin zones, etc.). Apart from the simple tools for retrieving the stored data, there are programs for the anal. of group-subgroup relations between space groups. There are also software package studying specific problems of solid-state physics, structural chem. and crystallog.**33**Aroyo, M. I.; Perez-Mato, J. M.; Capillas, C.; Kroumova, E.; Ivantchev, S.; Madariaga, G.; Kirov, A.; Wondratschek, H. Bilbao crystallographic server: I. databases and crystallographic computing programs.*Zeitschrift für Kristallographie-Crystalline Materials*2006,*221*, 15, DOI: 10.1524/zkri.2006.221.1.1533https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XmtFWnuw%253D%253D&md5=a2b474332c2a0c85b8fb7cc92de49861Bilbao crystallographic server: I. Databases and crystallographic computing programsAroyo, Mois Ilia; Perez-Mato, Juan Manuel; Capillas, Cesar; Kroumova, Eli; Ivantchev, Svetoslav; Madariaga, Gotzon; Kirov, Asen; Wondratschek, HansZeitschrift fuer Kristallographie (2006), 221 (1), 15-27CODEN: ZEKRDZ; ISSN:0044-2968. (Oldenbourg Wissenschaftsverlag GmbH)The Bilbao Crystallog. Server is a web site with crystallog. databases and programs available online at www.cryst.ehu.es. It has been operating for about six years and new applications are being added regularly. The programs available on the server do not need a local installation and can be used free of charge. The only requirement is an Internet connection and a web browser. The server is built on a core of databases, and contains different shells. The innermost one is formed by simple retrieval tools which serve as an interface to the databases and permit to obtain the stored symmetry information for space groups and layer groups. The k-vector database includes the Brillouin zones and the wave-vector types for all space groups. As a part of the server one can find also the database of incommensurate structures. The second shell contains applications which are essential for problems involving group-subgroup relations between space groups (e.g. subgroups and supergroups of space groups, splittings of Wyckoff positions), while the third shell contains more sophisticated programs for the computation of space-group representations and their correlations for group-subgroup related space groups. There are also programs for calcns. focused on specific problems of solid-state physics. The aim of the article is to report on the current state of the server and to provide a brief description of the accessible databases and crystallog. computing programs. The use of the programs is demonstrated by illustrative examples.**34**Aroyo, M. I.; Kirov, A.; Capillas, C.; Perez-Mato, J.; Wondratschek, H. Bilbao crystallographic server. ii. representations of crystallographic point groups and space groups.*Acta Crystallographica Section A: Foundations of Crystallography*2006,*62*, 115, DOI: 10.1107/S010876730504028634https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XhslWnur8%253D&md5=689fb23c14e777b10fed4390b438b9a9Bilbao Crystallographic Server. II. Representations of crystallographic point groups and space groupsAroyo, Mois I.; Kirov, Asen; Capillas, Cesar; Perez-Mato, J. M.; Wondratschek, HansActa Crystallographica, Section A: Foundations of Crystallography (2006), A62 (2), 115-128CODEN: ACACEQ; ISSN:0108-7673. (Blackwell Publishing Ltd.)The Bilbao Crystallog. Server is a web site with crystallog. programs and databases freely available online (http://www.cryst.ehu.es). The server gives access to general information related to crystallog. symmetry groups (generators, general and special positions, maximal subgroups, Brillouin zones etc.). Apart from the simple tools for retrieving the stored data, there are programs for the anal. of group-subgroup relations between space groups (subgroups and supergroups, Wyckoff-position splitting schemes etc.). There are also software packages studying specific problems of solid-state physics, structural chem. and crystallog. This article reports on the programs treating representations of point and space groups. There are tools for the construction of irreducible representations, for the study of the correlations between representations of group-subgroup pairs of space groups and for the decompns. of Kronecker products of representations.**35**Frenkel, D.; Smit, B.; Ratner, M. A.*Understanding molecular simulation: from algorithms to applications*; Academic Press: San Diego, 1996; Vol. 2.There is no corresponding record for this reference.