# eoslecture - Finite-temperature equation of state F V T = U...

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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 μ , j " = # 2 U # r i μ # 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 μ , j " = # 2 U # r i μ # r j " " 2 U " r i μ " r j # = \$ " 2 U " r ij 2 r i μ \$ r j μ ( ) r i # \$ r j # ( ) r ij 2 + " U " r ij r i μ \$ r j μ ( ) r i # \$ r j # ( ) r ij 3 \$ % μ , # r ij & ’ ( ( ) * + + • For i,j referring to different atoms • Also need terms where i=j • Notice that for each pair of components μ, ν , we have • Can be easily tested in code! Also Hessian is symmetric " 2 U " r i μ " r i # = " 2 U " r ik 2 r i μ \$ r k μ ( ) r i #...
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eoslecture - Finite-temperature equation of state F V T = U...

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