D. LANG AND KOHN He r)=2mx erfc(xx)-2m12y-le-C g(的)+2|卖-式 Note that this cancellation simply reflects the fact 21-F·e(7-|-F) ①D5) that a plane of ions together with a slab of the negative charge distribution extending a distance with the standard assumption of pseudopotential 2a on either side of the plane is neutral. theory that the cores(the spherical regions of ra It is then easily seen that dius re about each ion)are nonoverlapping, and 60o1=-2zn242Fa, m, n,r) under the condition 2d>re, 6v2(x)is easily seen to e a periodic function also, that tends to cancel bv1(x).For-d≤x≤0, 1EG0,m,n,y)+H(, 62(x)=2m(-|x+1)(r。-|x+如l).⑩D6) The choice y"R t leads to equally rapid conver AEAH圣 ANE EFFECT ON SURFACE ENERGY T gence for all terms. With 80e=aZn, numerical computation shows that a=-0. 00395[(100)face of For purposes of the present work, attention was simple cubic lattice].(Results for fcc and bcc focused on Al, in which the lattice terms 6.1 and lattices are positive. 60s are large The first plane of ions at the The same procedure is employed for other lat (111)face of the semi-infinite (x<o)fcc structure tice types and other crystal faces. The somewhat was allowed to shift its position from x=-dd to more generalized x=(-2+a)d=x,(cf. Fig. 5), changing the surfac energy by an amount 4o. It is convenient to write cos mnc of Jacobis transformation must be employed in with the two terms corresponding to steps 1 and 2 these treatments Fig. 5. Zero point motion of APPENDIX D: ALGEBRAIC EXPRESSION FOR Sy( neglected in computing these two contributions It may be seen using the analysis of Appendix D Let vps '(F)be the pseudopotential at position F that Ao=R()-R(o),with due to the ion at lattice site y(which is at position FR). Further, let (x)be the potential at x due to R()=4nd7*"o(xx-x)n(x)-n]dx a semi-infinite uniform positive background filling the half-space x <0. Then +2ndm∫x”e(x+6-x)x(x)-ndx,(E1) 5(x)=(∑t()-q、x), (D1) The pseudopotential cores are assumed here al where the angular brackets denote an average over ways to be to the left of the point x=o(i. e, xx y and z directions, and where Evlrco, denotes a sum- mation over all sites of the semi-infinite lattice in the x<o half-space. It is convenient to write For the(111)face of the icc lattice, use of tech niques similar to those of Appendix C (withy-oo 6v(x)=6v1(x)+bv2(x), (D2) for simplicity)leads to the result Ade=s()-s(o where6n(x)=(∑(-z|F-式1)-9,(x)①D3a) s0)2mx222’z(m2+2y1 and0n2(x)=2(o2()+Z1卖-式) X cos(2mb)cos(2nc) The indicated average over y and z directions in Eg. (D3a)means that the charges at positions Fy Xexp[-2n(6)(+aO)(m2+n21/2],(E2) be regarded as smeared out uniformly over here the prime on ymn indicates omission of the each lattice plane. The evaluation of &u,(x)is ther term m==0 and that on Eu indicates omission of a simple one-dimensional electrostatic problem, the terms u=1,2 when I-0. Here a(a)=au+a whose result is shown as a dashed line in Fig. 6 with a1=a2=0, a3=a4-3, and as=ae=3: b2n41=0 x≤0(cf b2n=b; and can=如+b,can1=如. The first term in(E2)gives the energy change due to movemen 6n(x)=-27[x+de(-x-bd)]2 (D4) of the first lattice plane in the potential of the slab of negative charge between Use of Ashcroft's form for the pseudopotential nd the second term gives the effect of the re [Eq.(3. 1)]implies that ainder of the crystal (which is neutral