Pressure regulation and immobilized atoms - Lammps

where L is the box size). I will assume that the forces contributing tothe virial are dominated by pairwise interactions between atoms (denotedu µν (|⃗r µ −⃗r ν |), where ⃗r µ and ⃗r ν are the positions of atoms µ and ν). (Iomitted a discussion of more general N-body interactions [2], because Idon’t use them in my current work. If someone asks, I’m happy to rewritethis document to include them.)The potential energy (per unit cell) is the sum of all pairwise energiesbetween atoms, and their images in other cells, and can be written as:U = 1 ∑ ∑u µν (|⃗r µ −(⃗r ν +L⃗m)|)2⃗m∈Z 3 µ≠ν(1)useful notation: ⃗r ν,⃗m ≡ ⃗r ν +L⃗m (2)= 1 ∑ ∑ (∣ ∣)u µν ∣ ⃗r µ −⃗r ν,⃗m 2⃗m∈Z 3 µ≠ν(3)where ∑ µ≠νis a sum over all pairs of atoms in the system (mobile orimmobile), and ∑ ⃗m∈Z 3 is a sum over an atom’s images in all surroundingunit cells (located at position ⃗r ν +L⃗m, where ⃗m, is a vector with integerx,y,zcomponents). Thefactorof 1 2 compensatesforredundancyofequivalentpairs (µ ↔ ν). This yields:W = −3V dUdV= −L dUdL(where V = L 3 ) (4)= − L 2(∑ ∑ d⃗r µdL · ∂u (∣ ∣)µν ∣ ⃗r µ −⃗r ν,⃗m + d⃗r ν,⃗m∂⃗r µ dL · ∂u (∣ ∣)µν ∣ ⃗r µ −⃗r ν,⃗m ∂⃗r⃗m∈Z 3 µ≠νν,⃗m} {{ } } {{ }= −f ⃗ ν,⃗m→µ= − ⃗ f µ→ν,⃗m)(5)One can recognize − ∂∂⃗r µu µν(∣ ∣ ⃗r µ −⃗r ν,⃗m∣ ∣)as the force acting on atom µ exertedby atom ν’s image in unit cell indicated by ⃗m (denoted ⃗ f ν,⃗m→µ ). Wenow split this sum into terms involving mobile and immobile atoms. Weuse lower-case indices (i,j) to refer to mobile atoms, and upper-case indices(I,J) to refer to immobile atoms, respectively. Expressed this way, W =L2∑⃗m∈Z 3 ( ∑i≠j+ ∑ i,J( d⃗ridL ⃗ f j,⃗m→i + d⃗r j,⃗mdL ⃗ f i→j,⃗m)+ ∑ I,j( d⃗ridL ⃗ f J,⃗m→i + d⃗r J,⃗mdL ⃗ f i→J,⃗m)+ ∑ I≠J( d⃗rIdL ⃗ f j,⃗m→I + d⃗r j,⃗mdL ⃗ f I→j,⃗m)( d⃗rIdL ⃗ f J,⃗m→I + d⃗r J,⃗mdL ⃗ f I→J,⃗m) ) (6)Weusethenotation: ⃗s i ≡⃗r i /Ltodenotethenormalizedcoordinatesofatomi. The x,y,z components of ⃗s i lie in the range from 0 and 1. Because mobile(unconstrained, lower-case) atom positions are rescaled during pressureequilibration, this means that ⃗r i and ⃗r j are proportional to L (⃗r j = L⃗s i ),and consequently, ⃗s i and ⃗s j are independent of L. Consequently:L d⃗r idL = ⃗r iand L d⃗r j.⃗mdL = ⃗r j,⃗m = ⃗r j +L⃗m (7)3