Analytical Expression for Hessian

Transcript

1 Analytical Expession fo Hessian We deive the expession of Hessian fo a binay potential the coesponding expessions wee deived in [] fo a multibody potential. In what follows, we use the convention that paticle indices ae always epesented as supe-scipts whe components of tensoial quantities ae epesented as sub-scipts. The total enegy of the system is given by [] U = m n m φ mn m ζ ψ mn, n m = U Bin U EAM, whee the supe-scipts Bin and EAM fo Embedded-Atom Method epesent the binay and the multibody contibutions espectively and the functions ψ mn and φ mn ae functions of mn n m only. We only pesent the expessions fo Hessian fo the mutibody pat below. The expessions fo the binay pat can also be obtained fom the expessions below fo the special case ζ =. Plugging the expession fom equation, we obtain = = m = m = m ζ n mψ mn, m ζ n mψ mn, ζ l m ζ l m ζ ζ ψ mn x i, n m ψ mn mn mn x i, n m whee mn ν xn ν xm ν, mn x n xm mn and x = mn i δ in δ im. mn Using this we obtain x i = ζ ζ ψ mn mn δ in mn mn δ im. 3 m l m n m Due to the two Konecke delta constaints, the st tem in expession 3 is nonzeo only fo n = i and thus we set n = i and do not sum ove n. Saly in the second tem, we set m = i and do not sum ove m to obtain = ζ ζ ψ ζ ζ ψ in in in in. 4 l m l i n i

2 Noting that ψ in = ψ ni, in = ni and in = in,we can ewite expession 4 as = ζ ζ ψ + ζ ζ ψ ni ni ni ni. 5 l m l i n i The summation and the poduct can be intechanged in the second tem of expession 5 to obtain = ζ ζ ψ + ζ ζ ψ ni ni. 6 ni ni l m l i n i Expession 6 can be e-witten as a single sum ove m whee χ = ζ ζ + ζ ψ l m l i, }{{}}{{} sum ove neighbous of m sum ove neighbous of i = ζ χ χ, 7 l m ζ ζ + and χ ψ l i. Note that unlike χ χ is not a function of alone but depends on the configuation of the system. It can be checked that expession 7 educes to the expession C3 of [] fo the case of binay potential whee ζ =. The expession fo the Hessian due to the multi-body pat alone is H ij U EAM U EAM x j = xi x j x i, = ζ χ x j χ, = ζ [ ] χ χ x j + ζ [ ] χ χ x j, = + Bij. 8

3 Expession fo : in expession 8 can be futhe simplified using ψ + ψ χ x j = = = = whee ψ ψ final fom as = ζ χ x j ψ ψ ψ + ψ ψ ψ x j x j, + ψ x j x j, + ψ x j x i x m, δ ji δ jm + ψ δ ji δ δ jm δ, etc. Thus Aij in expession 8 can be e-witten to obtain the [ ψ ψ 3 δ ji δ jm] [ ψ + ζδ χ δ ji δ jm ]. 0 educes to the expession C4 of [] fo ζ =. Futhe, we can obtain the expession fo the off-diagonal tem fom equation 0, by setting i j as [ ] = ψ ji [ ] ζχji ji ψji ji 3 ji ji ζδ χ ji ψ ji ji, 9 which can also be witten as [ = ψ ij ζχij ij ψij ij 3 ij ij + δ ] ψ ij ij i j, which educes to the expession C5 in [] fo ζ =. The expession fo i = j can be obtained fom equation 0 A ii = ζ [ ] χ ψ ψ 3 + ζδ χ ψ, 3 which is the genealisation of expession C6 in []. 3

4 Expession fo : in expession 8 can be simplified futhe using χ x j = x j ζ ζ +, l m l i = ζ ψ mp = ζ ψ mp + ψ ip ml x j ml + ψ ip ml δ jm x j, δ ji. 4 Plugging expession 4 in 8, we obtain = ζ [ ] χ χ x j, = ζ ζ + ψ ip χ ψ mp ml ml δ jm δ ji. 5 Fo i j, equation 5 gives = ζ ζ + ψ ip χ ψ mp δjl ml ml δ jm, 6 4

5 which can be futhe simplified using the Konecke delta constaints as = ζ ζ χ ζ mj ψ mp ψ mj mj ζ ζ χ ji ψ jp jl ψ jl l j +ζ ζ p j χ ψ ip ζ ψ ij jl ij ij. 7 We can e-wite equation 7 in the second tem e-witing jl as lj and in the thid tem change the unning index fom m to n to obtain the final fom, = ζ ζ ψ ζ mj ψ mp ψ mj,j +ζ ζ mj }{{} +ζ ζ Sum ove neighbous common to both i and j ψ ji ji ji n i ψ ni p j ni ni ψ jp }{{} Sum ove neighbous of i ζ l j ψ lj lj lj }{{} Sum ove neighbous of j ψ ip ζ ψ ij ij ij. i j Note that fo a pai of paticles i and j, the fist tem of 8 can be non-zeo even when i and j ae not the neighbous of each othe. This implies that the Hessian matix fo a multi-body potential is less spase than that fo a binay potential. Fo i = j, we obtain the following fom equation 5 B ii = ζ ζ + ψ ip χ ψ mp ml δ ml δ im 8 δ δ ii. 9 In the the above expession the tem with δ im vanishes due to the constaint m i. One of the tems with δ the second one also vanishes because of the 5

6 constaint l i. Thus expession 9 simplifies to B ii = ζ ζ ψ +ζ ζ ψ im im im ψ mp ψ ip ζ ψ. 0 Symmety checks: Symmety of the Hessian implies that H ij x j U EAM = x j H ji. Symmety of A ij in expessions and 3 is easy to check. We show hee the symmety of expessions 8 and 0. Fo compaison, we epoduce expession 8 below, = ζ ζ ζ mj ψ mp,j +ζ ζ ψ ji ψ ji ji +ζ ζ ψ p j ψ jp ψ ip m j ψ mj ζ Fom which we obtain by intechanging i j and, B ji = ζ ζ mj ψ mj ζ ψ mp m j,i +ζ ζ ψ ij ij ij +ζ ζ ψ mj m j mj mj mj ψ ip ψ mj ψ ij ψ ψ jp p j ζ mj mj ψ ij ij ψ ji mj, i j. ji ji, j i. 3 6

7 Compaing equations and 3, it is clea that the fist tem of both expessions ae the same. The second tem of equation is the same as the thid tem of equation 3 and vice-vesa. This shows the symmety of fo i j. To show the symmety of B ii we epoduce expession 0 below. B ii = ζ ζ ψ ζ ψ mp ψ +ζ ζ ψ im im im ψ ip. 4 Fom which we obtain by intechanging, B ii = ζ ζ ψ +ζ ζ ψ im im im ψ mp ψ ip ζ ψ. 5 In the second tem in expession 5, since l and m ae unning indices, we can exchange them and this shows that expessions 4 and 5 ae the same thus completing ou check. 7

8 Bibliogaphy [] Smaajit Kamaka, Edan Lene and Itama Pocaccia, Phys. Rev. E, 8, 0605, 00. [] G. Duan, D-H Xu, Q. Zhang, G-Y Zhang, T. Cagin, W. L. Johnson, and W. A. Goddad, III, Phy. Rev. B, 7, 408, 005, 74, 0990,