Editing Crystal structure (section)

== Prediction of structure ==
{{Main| Crystal structure prediction}}
The difficulty of predicting stable crystal structures based on the knowledge of only the chemical composition has long been a stumbling block on the way to fully computational materials design. Now, with more powerful algorithms and high-performance computing, structures of medium complexity can be predicted using such approaches as [[evolutionary algorithms]], random sampling, or metadynamics.

The crystal structures of simple ionic solids (e.g., NaCl or table salt) have long been rationalized in terms of [[Pauling's rules]], first set out in 1929 by [[Linus Pauling]], referred to by many since as the "father of the chemical bond".<ref name="Paulingsrules">{{cite journal|author= L. Pauling|year = 1929|title = The principles determining the structure of complex ionic crystals|journal = [[J. Am. Chem. Soc.]]|volume = 51|issue = 4|pages= 1010–1026|doi= 10.1021/ja01379a006| bibcode=1929JAChS..51.1010P |author-link = Linus Pauling}}</ref> Pauling also considered the nature of the interatomic forces in metals, and concluded that about half of the five d-orbitals in the transition metals are involved in bonding, with the remaining nonbonding d-orbitals being responsible for the magnetic properties. Pauling was therefore able to correlate the number of d-orbitals in bond formation with the bond length, as well as with many of the physical properties of the substance. He subsequently introduced the metallic orbital, an extra orbital necessary to permit uninhibited resonance of valence bonds among various electronic structures.<ref>{{cite journal|doi=10.1103/PhysRev.54.899|title=The Nature of the Interatomic Forces in Metals|year=1938|last1=Pauling|first1=Linus|journal=Physical Review|volume=54|issue=11|pages=899–904|bibcode = 1938PhRv...54..899P }}</ref>

In the [[resonating valence bond theory]], the factors that determine the choice of one from among alternative crystal structures of a metal or intermetallic compound revolve around the energy of resonance of bonds among interatomic positions. It is clear that some modes of resonance would make larger contributions (be more mechanically stable than others), and that in particular a simple ratio of number of bonds to number of positions would be exceptional. The resulting principle is that a special stability is associated with the simplest ratios or "bond numbers": {{frac|1|2}}, {{frac|1|3}}, {{frac|2|3}}, {{frac|1|4}}, {{frac|3|4}}, etc. The choice of structure and the value of the [[axial ratio]] (which determines the relative bond lengths) are thus a result of the effort of an atom to use its valency in the formation of stable bonds with simple fractional bond numbers.<ref>{{cite journal|doi=10.1021/ja01195a024|year=1947|last1=Pauling|first1=Linus|title=Atomic Radii and Interatomic Distances in Metals|journal=Journal of the American Chemical Society|volume=69|issue=3|pages=542–553|bibcode=1947JAChS..69..542P }}</ref><ref>{{cite journal|doi=10.1098/rspa.1949.0032|title=A Resonating-Valence-Bond Theory of Metals and Intermetallic Compounds|year=1949|last1=Pauling|first1=L.|journal=[[Proceedings of the Royal Society A]]|volume=196|issue=1046|pages=343–362|bibcode = 1949RSPSA.196..343P |doi-access=free}}</ref>

After postulating a direct correlation between electron concentration and crystal structure in beta-phase alloys, [[Hume-Rothery]] analyzed the trends in melting points, compressibilities and bond lengths as a function of group number in the periodic table in order to establish a system of valencies of the transition elements in the metallic state. This treatment thus emphasized the increasing bond strength as a function of group number.<ref>{{cite journal|doi=10.1098/rspa.1951.0172|title=The Valencies of the Transition Elements in the Metallic State|year=1951|last1=Hume-rothery|first1=W.|first2=H. M.|first3=R. J. P.|journal=[[Proceedings of the Royal Society A]]|volume=208|issue=1095|pages=431|last2=Irving|last3=Williams|bibcode = 1951RSPSA.208..431H |s2cid=95981632}}</ref> The operation of directional forces were emphasized in one article on the relation between bond hybrids and the metallic structures. The resulting correlation between electronic and crystalline structures is summarized by a single parameter, the weight of the d-electrons per hybridized metallic orbital. The "d-weight" calculates out to 0.5, 0.7 and 0.9 for the fcc, hcp and bcc structures respectively. The relationship between d-electrons and crystal structure thus becomes apparent.<ref>{{cite journal|doi=10.1098/rspa.1957.0073|title=On the Relation between Bond Hybrids and the Metallic Structures|year=1957|last1=Altmann|first1=S. L.|first2=C. A.|first3=W.|journal=[[Proceedings of the Royal Society A]]| volume=240| issue=1221| pages=145| last2=Coulson| last3=Hume-Rothery| bibcode = 1957RSPSA.240..145A |s2cid=94113118}}</ref>

In crystal structure predictions/simulations, the periodicity is usually applied, since the system is imagined as being unlimited in all directions. Starting from a triclinic structure with no further symmetry property assumed, the system may be driven to show some additional symmetry properties by applying Newton's second law on particles in the unit cell and a recently developed dynamical equation for the system period vectors
<ref>{{cite journal|doi=10.1139/cjp-2014-0518|title=Dynamical equations for the period vectors in a periodic system under constant external stress|year=2015|last1=Liu|first1=Gang|journal=[[Can. J. Phys.]]| volume=93| issue=9| pages=974–978|bibcode = 2015CaJPh..93..974L |arxiv=cond-mat/0209372|s2cid=54966950}}</ref> (lattice parameters including angles), even if the system is subject to external stress.