eoslecture - Finite-temperature equation of state F ( V , T...

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Unformatted text preview: Finite-temperature equation of state F ( V , T ) = U + k B T ln 2sinh h " # 2 k B T $ % & ( ) * + ,- . / # Compute vibrational modes, frequencies Evaluate at a given volume V Compute F at various temperatures T with above Need the Hessian! Start again with fcc lattice Atoms should be relaxed (zero force on atoms) Evaluate at a given volume V Compute F at various temperatures T and volumes V Generate non-primitive basis-- 4 atoms ! start with non-primitive fcc basis r1(1)=0.0d0 r2(1)=0.0d0 r3(1)=0.0d0 r1(2)=0.5d0 r2(2)=0.5d0 r3(2)=0.0d0 r1(3)=0.5d0 r2(3)=0.0d0 r3(3)=0.5d0 r1(4)=0.0d0 r2(4)=0.5d0 r3(4)=0.5d0 Some parameters for lattice INTEGER, PARAMETER :: nn=4 INTEGER, PARAMETER :: Prec14=SELECTED_REAL_KIND(14) INTEGER, PARAMETER :: natoms=4*nn**3 INTEGER, PARAMETER :: nx=nn,ny=nn,nz=nn REAL(KIND=Prec14), PARAMETER :: sigma=1.0d0,epsilon=1.0d0 4x4x4 lattice of non-primitive units 64x4=256 atoms System has 256x3=768 degrees of freedom! The Hessian matrix needs to be 768x768 Generate the full lattice n=0 do ix=1,nx do iy=1,ny do iz=1,nz do i=1,4 n=n+1 rx(n)=(r1(i)+dble(ix-1))*sigma ry(n)=(r2(i)+dble(iy-1))*sigma rz(n)=(r3(i)+dble(iz-1))*sigma enddo enddo enddo enddo ! scale lengths by box size rx=rx/nx ry=ry/ny rz=rz/nz We need more than forces this time! H i , j " = # 2 U # r i # r j " Definition of Hessian matrix elements For N atoms, this means a 3Nx3N matrix Many elements will be zero due to cutoff First derivative is just related to force on j: But need higher order derivatives! " U " r j # = " U " r kj r j # $ r k # ( ) r kj k % We need more than forces this time! H i , j " = # 2 U # r i # r j " " 2 U " r i " r j # = $ " 2 U " r ij 2 r i $ r j ( ) r i # $ r j # ( ) r ij 2 + " U " r ij r i $ r j ( ) r i # $ r j # ( ) r ij 3 $ % , # r ij & ( ( ) * + + For i,j referring to different atoms Also need terms where i=j Notice that for each pair of components , , we have Can be easily tested in code! Also Hessian is symmetric " 2 U " r i " r i # = " 2 U " r ik 2 r i $ r k ( ) r i #...
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eoslecture - Finite-temperature equation of state F ( V , T...

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