INHOMOGENEOUS ELECTRO B867 th 8丌(0lpqn)(n|p-4|0) Behavior Next we express the change of energy in terms of a(a). larizabil second-order perturbation theory we have A2(4r)2|a(q)12(0p|n)(lpg E=E0+ 入2ra(q) where kr is the Thomas-Fermi screening constant 入2r|b1(q) (33)and 2 q a(g) S(q)= pkx q2\,|9+2k On the other hand, combining Eqs.(15),(24),(25), 4kp2/9 gives This gives for K(), by (35) 1 / n(r)n(r) +G[n] q→0:K(q)=2r-c2+(c2-c4)y2+…];(44) q→2p:aK/q→+∞ 24x|b1(q)|22r|b1(q)2 q→∞:K(q)→ const×q (q)q?2 The power-series expansion of K(g),(43), leads to ∑Kq)|b1(q)|2.(34) -c2+(c2-c4)V2+…]6(r),(47) Comparison of Eqs.(33)and(34)gives hich in turn gives K(q)= 652-/mrr Equivalently, in terms of the dielectric constant +(c2-c)/n(r)dr+…|,(4 1-a(q) (36)i.., a gradient expansion. we may write K(q) o 0. At this point an important remark must be made e of the most significant features of K(o)is its singularity at g=2k p. This is responsible for the long- r→∞:K(r)~ const cos(2kr+b)/r3.(49) 3. The Nature of the Kernel K These obviously lie outside the framework of the The polarizability has the following properties, power-seric pansion(44)of k(@ and hence outside as function of q(see Fig. 1) the gradient expansion (49)of G[n]. This explains why neither the original Thomas-Fermi method [which (38) for the present system reduces to keeping only the first 2kp: da/ dq (39) term in(44), nor its generalizations by the addition of gradient terms, have correctly yielded wave-mechanical → (q)→ const/g (40) density oscillations, such as the density oscillations atoms which correspond to shell structure, or the Friedel These general properties are exemplified by the random- oscillations in alloys which are of the same general origin. phase approximation in which 1J. S. Langer and S. H. Vosko, Phys, Chem. Solids 12, 196 a(q)=[1+(q3/k2)S(q)]1 (41)(1090I~ HOMOGE~TI:OUS El I C I Ro ib GAS so that. (32) Next we express the change of energy in terms of ct q . By second-order perturbation theory we have li'(4z.)' Ia(q) I' (0I p, le)(~el p sl0) jj=ji,s+- n ~ q4 li'2 I (q)l' =~''()— —2— ~(q), 0 ~ q' li'2 Ib (q)l' Fr.o. 1. Behavior of the electronic po￾larizability n(q), as function of q (elec￾tronic density =4 &(10"cm '). 1.0 Rtq) 0.5 0 0 I I 1 2 q/qF ~ kg =—(4k p)'i' wheie h'y is e ' th Thomas-Fermi screening constan, (42) =~o— (33) and 0 & a(q)q' m(r)e(r—) drdr'+ G[e7 Ir—r'I li'4' Ib (q) I' V27r lb (q) I' + fl n (q) q' (i q' 1 E= n(r)ri(r)+— 2 +—2 K(q) I & (q) I' (34) Comparison of Eqs. (33) and (34) gives On the other hand, combining Eqs. (15), (24), (25), and (28) gives kr q' ) q+2kF 5(q)—= —, '+—1——I ln 2q 4k p2/ q—2k r (43) q —+~: K(q) —+ constXq . (See Fig. 2.) The power-series expansion of K q, , ea s o K(r) =27r[—cs+ (css—c4)V+ 76(r), 4 which in turn gives This gives for E(q), by (35), q—& 0: K(q) =2z.[—cs+ (css—c4)q'+ . 7; (44) q —+ 2kp. dK/dq —++~; (45) 2 (46) 2' E(q) =- q' n(q) G[N7 =G[rrs7+ 27r —cs 8(r)'dr (35) Equivaen y, l tl in' terrors of the dielectric constant, +(css—c4) I V'n(r) I'dr+, (48) we may write e(q) = 2~ 1 E(q) =- q' e(q)—1 (36) (37) i.e, , a gradient expansion. At this point an important remark must be made. One of the most significant features of K(q) is its singu larity at q= . ' t =2k . This is responsible for the long￾range Friedel oscillations" in E(r), ' r~~: E(r) const cos(2krr+ 8)/r'. (49 3. The Nature of the Kernel K Q(q) = 1+csq'+c4q'+ . . (38) (39) (4o) q—&0: q ~2k' '. de/dq ~ —oo I q —+~: n(q) ~const/q . These general properties are exemp lified by the random￾phase approximation in which The polarizability u(q) has the following properties, as function of q (see Fig. 1) These obviously lie outside the framewor r of the power-series expansion (44) of E(q) and hence outside e gr whyneieitherer the original Thomas-Fermi met od which for the present system reduces to keepingin onl y thee first ~44~~~ nor its eneralizations by the addition of gradient terms, have correctly yielded wave-mec anica density osci ations, suc atoms which correspond to shell structure, or the ne e oscillations in alloys which are of the same general origin. n(q) =[1+(q'/kp')S(q)7 —' . S. Langer and S. H, Vosko, Phys. Chem. Solids 12, 196 (41) (1960}