Interatomic potential

Interatomic potentials are mathematical functions for calculating the potential energy of a system of atoms with given positions in space. Interatomic potentials are widely used as the physical basis of molecular mechanics and molecular dynamics simulations in chemistry, molecular physics and materials physics, sometimes in connection with such effects as cohesion, thermal expansion and elastic properties of materials. Interatomic potentials are mathematical functions for calculating the potential energy of a system of atoms with given positions in space. Interatomic potentials are widely used as the physical basis of molecular mechanics and molecular dynamics simulations in chemistry, molecular physics and materials physics, sometimes in connection with such effects as cohesion, thermal expansion and elastic properties of materials. Interatomic potentials can be written as a series expansion offunctional terms that depend on the position of one, two, three, etc.atoms at a time. Then the total of the system V canbe written as Here V 1 {displaystyle extstyle V_{1}} is the one-body term, V 2 {displaystyle extstyle V_{2}} the two-body term, V 3 {displaystyle extstyle V_{3}} thethree body term, N {displaystyle extstyle N} the number of atoms in the system, r → i {displaystyle {vec {r}}_{i}} the position of atom i, etc. i, j and k are indicesthat loop over atom positions. Note that in case the pair potential is given per atom pair, in the two-bodyterm the potential should be multiplied by 1/2 as otherwise each bond is countedtwice, and similarly the three-body term by 1/6. Alternatively,the summation of the pair term can be restricted to cases i < j {displaystyle extstyle i<j} and similarly for the three-body term i < j < k {displaystyle extstyle i<j<k} , ifthe potential form is such that it is symmetric with respect to exchangeof the j and k indices (this may not be the case for potentialsfor multielemental systems). The one-body term is only meaningful if the atoms are in an externalfield (e.g. an electric field). In the absence of external fields,the potential V should not depend on the absolute position ofatoms, but only on the relative positions. This meansthat the functional form can be rewritten as a functionof interatomic distances r i j = | r → i − r → j | {displaystyle extstyle r_{ij}=|{vec {r}}_{i}-{vec {r}}_{j}|} and angles between the bonds(vectors to neighbours) θ i j k {displaystyle extstyle heta _{ijk}} .Then, in the absence of external forces, the generalform becomes In the three-body term V 3 {displaystyle extstyle V_{3}} theinteratomic distance r j k {displaystyle extstyle r_{jk}} is not neededsince the three terms r i j , r i k , θ i j k {displaystyle extstyle r_{ij},r_{ik}, heta _{ijk}} are sufficient to give the relative positions of three atomsi,j,k in three-dimensional space. Any terms of order higher than2 are also called many-body potentials.In some interatomic potentials the manybody interactions are embedded into the terms of a pair potential (see discussion onEAM-like and bond order potentials below). In principle the sums in the expressions run over all N atoms.However, if the range of the interatomic potential is finite,i.e. the potentials V ( r ) ≡ 0 {displaystyle extstyle V(r)equiv 0} abovesome cutoff distance r c u t {displaystyle extstyle r_{cut}} ,the summing can be restricted to atoms within the cutoffdistance of each other. By also using a cellular methodfor finding the neighbours, the MD algorithm can bean O(N) algorithm. Potentials with an infiniterange can be summed up efficiently by Ewald summationand its further developments.