Metal surface properties in the linear potential approximation

PHYSICAL REVIE% 8 15, VOLUME 15 FEBRUARY 1977 NUMBER 4 Metal surface properties in the linear potential approximation V. Sahni* The New School of Liberal Arts, Brooklyn College of the City University of J. B. Krieger* Department of Physics, Brooklyn College of the and ¹w York, Brooklyn, ¹wYork 11210 J. Gruenebaum City University of ¹wYork, Brooklyn, ¹wYork 11210 (Received 29 June 1976) Jellium metal surface properties including the dipole barrier, work function, and surface energy are obtained in the linear-potential approximation to the effective potential at the surface. The metal surface position and field strength are determined, respectively, by the requirement of overall charge neutrality and the constraint theorem. The surface energies are obtained both set on the electrostatic potential by the Budd-Vannimenus within the local density approximation and by application of a sum rule due to Vannimenus and Budd, and the two methods compared. The calculations are primarily analytic and all properties, with the exception of the exchange-correlation energy, are given in terms of universal functions of the field strength, The effects of for the correlation on the various properties are studied by employing three different approximations correlation energy per particle. The results obtained employing the Wigner expression for the correlation energy closely approximate those of Lang and Kohn. The use of different correlation functions, however, leads to only small differences in the results for the dipole barrier, work function, and the exchange-correlation contribution to the energy, but the results for the total surface energy are significantly different. I. INTRODUCTION In this paper we present the results of a model potential calculation of jellium metal' surface properties. The principal advantage of such model calculations" is the elimination of the requirement of a numerical solution of the Schrodinger equation for particles moving in a self-consistently obtained effective potential which is intrinsic Together to other more complex formalisms. with the application of certain theoretical constraints, which help to fully define the model. effective potential, it is then possible to determine various properties of interest, such as work functions and surface energies. Examples of typical constraints applicable to model metal surface calculations are the requirement of selfconsistency of the surface dipole barrier, ' the condition on the electrostatic potential at the metal surface set by the Budd-Vannimenus theorem' (BVT), and the Bayleigh-Ritz' energy minimum criteria, in addition to the charge neutrality condition. ' The choice and number of these constraints to be satisfied mould, of course, depend upon the complexity of the model potential employed. %e consider here the linear potential model. approximation to the effective potential at a metal surface (see Fig. l). For a given value of the field strength, the metal surface position is fixed The field by requiring overall charge neutrality. strength may then be determined by application of the BVT. The choice of the BVT criteria as the constraint is governed by the fact that its " '" application in our previous mork ' consistently l.ed to good results for both the work function and surface energy rather than, for example, just the latter, as is the case' on application of the variational principle for the energy. In Sec. II we give a discussion of the calculations, and definitions of properties and theorems employed. The results for the surface dipole barrier, work function, and surface energies are presented in Sec. III. The surface energies are determined both within the local-density approximation (LDA), which is meaningful provided both exchange and correlation are treated locally ii. u and by ap plication of a sum rule due to Vannimenus and Budd. The use of the Vannimenus-Budd theorem" (VBT) appears particularly attractive, as it completely eliminates the requirement of determining the individual components of the energy and any questions regarding the accuracy of the I DA "'i4'5 and the effects of inclusion of the gradient terms. These sets of results thus afford a comparison' ' Y eft FERMI P+ I = Fx efx) EVEL k3 =— F —k 3TT2 2 F t o a X FIG. 1. Schematic representation of the linear potential model of a metal surface indicating all relevant energies. The hatched region represents the uniform positive background beginning at the metal surface. J. B. V. SAHNI, 1942 KRIEGER, of the two methods for obtaining metal surface energies. In order to study the effects of correlation on the various surface properties, we have also employed three different correlation functions due to signer, Pines and Nozieres, and Vashista and Singwi, '" in our calculations. The correlation energy per particle as given by these expressions can differ by as much as twenty percent for low-density metals. Finally, we compare our results obtained by empl. oying the Wigner approximation for correlation with those of the self-consistent calculations of Lang and Kohn' (LK). " " II. CALCULATION OF METAL SURFACE PROPERTIES In this model calculation, tial V, «(x) the effective poten- " strength defined in terms of 'k~z is x» as E=kz/2xz, —, and e(x) is the step function. function g» (x) for x- 0 is given as (t, (x) = —(2/L)'~' sin[ kx+ 6(k)]. " In the region x~ 0, the Schrodinger the Airy differential equation' (2) equation is xpr(x) dx. &Q =4m The electrostatic potential V„(x) required for application of the BVT, VBT, and the electrostatic contribution to the surface energy is obtained by solution of Poisson's equation " d2V„ = — 4vpr(x), , (10) x' t'x V„(x) = A(t) —4v dx' ~co dx"pr(x"). 00 The surface energy of a metal E, which is the energy required to split the crystal in two along a plane may be obtained from the sum of the kinetic, electrostatic, exhange and correlation contributions. The kinetic energy E~ is given (3) (12) where whose solution is ll»(g) = C»Ai(&) &»i" = (k~»/160m)e, (4) where Ai(f) is the Airy function, C» is a normalization factor, g = (x-E/E)(2E)'(', and E is the energy. The factor C, and the phase shift 6(k) are determined by the requirement of the continuity of the wave function and its logarithmic derivative at the origin. Thus, C» = —(2/L)'(' sin6(k, xz)[Ai(- 0=1k, 'k' -, Jl )~, Ai'(- g, ) . '), (6) ( where go = (k»/k~+)(kzxz}'~', and where and A ff P8i ff p (x)= ( 0 (k» —k')[g )»dk pe (14) gy E„ is to the surface ener- and the sum of the exchange and correlation contributions E„, which must be taken together within the LDA is given as' wc= ky Pe i p&xV xdx Ai'(1) is the derivative of the Airy function. The fundamental quantities from which all other surface properties may be obtained are the electronic and total charge densities defined as t k'k(k)kk) 0 The electrostatic contribution 1 kg kll(k)dk — (6) f»}] and ( (- ()) = 0. 0) = V,', with the boundary conditions V, Applying the charge neutrality condition, the electrostatic potential may thus be written as" —fg» — —0, cot6(k, x~) = (8) (- where is the field the slope parameter the Fermi energy, The electronic wave d g» 15 respectively. Implicit in the definition of the total charge density is the assumption that the positive ions of the metal are smeared out and replaced by a uniform charge background of den-kz/3)(' ending abruptly at the metal sursity p+ — face position at & =a. The surface dipole barrier contribution to the work function is given by the expression sumed to be I GRUENEBAUM pr(x) = p, (x) —(k~z/3v')e(-x+a), at a metal surface (see Fig. 1} is as- V„~x) =Exe(x), J. AND ~ac pe & —&gp pe pg & &&p where &„, = &„+&~, &„, and &, are the average exchange and correlation energies per particle (16) METAL SURFACE PROPERTIES IN THE LINEAR POTENTIAL. . . for a uniform electron gas, and where p, =k'„/ 3m' is the mean interior electronic density. The expression for the derivative of the surface energy with respect to the signer-Seitz radius r, which according to the VBT is given as dE 9 4~r4 dr, .„ [ V., (- ~) —V„(x)]dx, 1948 1.5 1.0 0.5 (17) to obtain the surface enerdirectly by integration over r, , together with a suitable choice for the constant of integration. In this model the two variable parameters are the metal surface position a and the field strength E. For a given E the metal surface position may be determined either by the charge neutrality may also be employed —0.5 gy ~e3' I'IG. 2. Plot of the universal function for the metal surface position y, vs the slope parameter yz. condition pr(x) dx=0, (18) till. it satisfies the BVT. The work function of the metal or by application of the phase-shift rule of Sugiyama, ' according to which —, 8k„k g =— 3~ I k6(k) dk. (19) The field strength, or equivalently, the slope parameter x~ is adjusted so as to satisfy the requirement set on the electrostatic potential by the BVT such that' ~V= V„(~) —V„(-")=p. dc~ d (20) &~ is the sum of the kinetic, exchange, and correlation energies per particle for a uniform electron gas. —q, With the transformation y =kzx, and k/kz — such that the metal. surface position is now defined to be at y, =k~a, it can be shown that the quantities y, , p, /k3z, AP/kz, V„/kz, E~/kz~, E„/ kz', and r~dE, /dr, are all universal functions of the slope parameter y~ = k~x~. Furthermore, all the spatial integrals in the expressions for the metal surface position, the surface dipole barrier, the electrostatic potential, and the kinetic Thus, together energy can be done analytically. with the electronic density, the determination of these properties reduces to a simple numerical calculation of k-space integrals ranging from 0 in the to 1. The explicit expressions'employed present calculations are given in the Appendix. Plots of the universal functions y, , aP/k~, 160vE~/kz', E„/k~z, and (9v)r4dE, /dh, are given in Figs. 2-6. The metal surface position, surface dipole barrier, and surface energies are then easily determined for a specific metal [defined by its Fermi momentum kz --1/ar, ; o. = (-, w)' '] by adjusting the slope parameter x~ where C is then given as 4 =ay ——,'k2~ —p, „,, (21) where p. „, is the exchange and correlation part of the chemical potential. of a uniform electron gas defined as Wxc = d(p, e„,) (22) The exchange energy" per particle for the uniform electron gas is e„= —0 458/r, . . For the correlation energy per particle valid at metallic densities we employ the correlation functions" due to Wigner" (W), Pines and Nozihres" (PN), and Vashista and Singwi" (VS), e, = — .044/(r, +7.8), c~" = 0.0155 in/, —0.0575, = 0.016 t5 lnx, —0.056. 0, 3 0.2 0, 1 I I I I 1.0 2.0 3.0 4.0 FIG. 3. Plot of the universal function Aft)/kz, where is the surface dipole barrier, vs the slope parameter yp. 4Q V. SAHNI, 1944 J. B. KRIEGER, J. AND Sx1Q GRUENEBAUM -2 4TT 9 4x10 3x1Q 15 4 dEx dr ( a. u. -2 -2 -2 2x10 -2 1x10 -1 x10 FIG. 4. Plot of the universal function (160m/kz)E» where E~ is the kinetic energy, vs the slope parameter pp ~ FIG. 6. Plot of the universal function (T4 x)r ~ dE +dr, where E~ is the surface energy, vs the slope parameter pp the VS and PN expressions are similar, their respective values for &„. can differ by as much as 15/p for r, =6, for which value of r, the VS expression differs by 20% from that due to Wigner. The results for the various properties within the linear potential model employing these different correlation functions are presented in -2 ~ Although Sec. III. III. RESULTS A. Surface dipole barrier and work functions In Table I we present the results for the surface dipole barrier and work function for the three different correlation functions. For r, = 2-4, both the BVT and charge neutrality are exactly satisfied, whereas for r, ~ 4. 5, no choice of parameters in the linear potential model satisfy both these Thus for r, ~ 4. 5, the present work requirements. corresponds to satisfying the BVT as closely as 11 x10 -4 9 x10 7x10 5x10 3x 10 1x10 YF FIG. 5. Plot of the universal function E,~/&z, where energy, vs the slope parameter yz. E,~ is the electrostatic possible in the l. imit of this model, i. e. , by a potential of infinite slope, and by the charge neutr ality condition. A comparison of the Wigner (W) results with those of LK indicate that the model reproduces the large dipole moments required for high-density metals, differing by 0.23, 0. 04, and 0. 1S eV for r, =2. 0, 2. 5, and 3.0, respectively. The reason for the accuracy of these results is that by adjusting ~ U so as to satisfy the BVT, one has already obtained approximately 40% of the dipole barrier, exactly. The remaining contribution to ~Q from charge outside the metal is determined accurately since the model permits a large electronic spillover. In addition, satisfaction of the BVT also leads to very accurate electronic densities in this range. For r, =3.5 and 4. 0, the model. becomes progressively more reflecting as the potential rises more steeply leading to underestimates for the dipole barrier. Also, ~U in this range comprises only about 25% of the LK value. For r, ~ 4. 5, for which the contribution of the dipole barrier to the work function. is small, the results for AP are within 0.28 eV of those obtained by LK. In this range, the LK results' also do not satisfy the BVT, although their results more closely satisfy this condition than do the results of the infinite barrier model. ' Thus over the entire metallic range the results of this model. calculation for the dipole barrier and work function are within 0.32 eV of those due to LK. The use of the PN and VS correlation functions lead to results for &Q and 4 which are consistently I.ower than those obtained employing the W function. The PN and VS results for ~Q are within 0.03 eV of each other, the VS result dif- ' METAL SURFACE PROPERTIES IN THE LINEAR POTENTIAL. . . 15 1945 TABLE I. Results for the surface dipole barrier b, p and work function C for the correlation functions due to Wigner (W), Pines and Nozieres (PN) and Vashista and Singwi (VS). The values for the metal surface position y, and slope parameter y& quoted are those obtained using the Wigner expression for the correlation energy. 2.0 2.5 3.0 3.5 4, 0 4.5 5.0 5.5 6.0 Surface dipole barrier AP (eV) Lang-Kohn Present work g =kgb pp =kpxp 3.760 2.801 1.967 1.214 Present work PN VS 3.68 3.31 3.01 4.09 3.63 3.26 2.95 2.74 2.59 2.63 2.61 2. 58 2.68 2.57 2.60 2.58 2.54 4.02 3, 55 3.17 2.85 2.57 2.49 W PN VS 1.488 1.108 7.03 6.90 3.69 6.88 3.66 6.80 4.12 3.83 0.763 0.410 2.05 2.02 2. 32 1.18 0.53 1.09 0.50 0.28 0.25 0.23 0.21 1.43 0.91 0.56 0.35 0.16 3.79 2.13 -0.0370 0.507 0.000 0.000 0.000 0.000 Work function 4(eV) Lang-Kohn " -1.178 —1.178 -1.178 —1.178 0.59 0.28 0.25 0.23 0.21 1.12 0, 28 0.25 0.23 0.21 W W 0.04 2.51 2.49 2.45 ' (eV) W W 3.89 3.72 3.50 3.26 3.06 0.23 0.04 0.19 0.25 0.32 0.28 0.10 0.07 0.17 2.87 2.73 2.54 2.41 ~See Ref. 5. In Table II we present the results for the surface energy for the three different correlation functions determined in the LDA. The parameters employed are the same as those of Table I ob- conditions, and the results given only for those values of r, for which both these conditions are exactly satisfied, viz, r, =2-4. The use of the infinite barrier model for r, ~ 4. 5 leads to unrealistic results within the LDA, due primarily to the sensitivity of this method of computing energies to the vanishing of the el. ectronic density at the artificial barrier. ' Other methods however, such as the VBT employing the same model do lead to meaningful results for these low densities, as discussed in Ref. 3. We have also included in Table II the results for the infinite barrier model. for r, =2-4. A study of Table II indicates that the results for the total energy of the W column closely approximate those due to LK (see also Fig. 7). For r, =2-3, the results for the individual componthe accuracy ents are also close approximations, of the electronic densities being particularly well reflected in the results for the kinetic and ex- tained by satisfying the BVT and charge neutrality change-correlation fering by at most 0. 15 eV from the W result. On the other hand, it is the PN and W values for 4 which are more similar, being within 0.06 eV of each other, the VS and W results differing by a maximum of 0.17 eV. Thus these different correlation functions give rise to only small differences in the results for the dipole barrier and work function. Since our results are close to those of LK using the W correlation function, we expect that a completely self-consistent calculation empl. oying the other correlation functions would differ from LK by similarly small amounts, i. e. , of the order of 0. 1 or 0.2 eV. B. Surface energies-local density approximation energies. Since for TABLE II. Surface energies in ergs/cm2 as obtained in the local density approximation for the different correlation functions due to Wigner (W), Pines and Nozieres (PN), and Vashista and Singwi (US). The individual component values correspond to the Wigner expression for the correlation energy. $ Surface energy components (Wigner correlation function) Present work Surface energies Infinite &xc 2.0 2.5 3.0 3.5 4.0 Present work Barrier potential —5897 -1861 -666 —236 -51 See Ref. 3. See Ref. 5. 3386 1458 718 383 213 PN 1567 4077 -944 447 1832 964 565 359 44 205 206 59 23 Lang-Kohn" ~ &es 153 E, 185 —1008 -863 -843 80 223 218 201 92 36 231 199 194 158 224 208 r, =3.5 y. ~R1EgER, , SAHN& GRUENEBA™ 15 I.DA is q uite ' 'insensitive to cha n g es in the ffective potenti ial in sharp conntrast ra to the kinIn fact it can be s h ownn by expanding EgC about the exacct density th a tth th calculated E„, is prroportional to thee error in the calculated diip ol. e moment whic, in 'tt the infinite barb ermitting me a ramp, the resu itss for the total b obrastically altere,d a energy are dra o the infinite served by comp m aring the resu itss of r otentials given in a e The surface ene r g ies obtaine bta e S correlation functions m ' ' si nificantl. rent from those y di ff eren the signer expressi ress on These rise primarily due too the numerica differences in the values for th e k'i t' lesser degree, from thee differences energy. The exchange-corgV for both the PN an in the 200 LD E at g/cm -200 -400— -600— + Lang -1000 —+ —Kohn . ' on of the surface energie s obtained nsity approximation (-A), imenus-Budd dth orem (VBT), emp oyt core P lC i ner expression for corre a lO . ing the Wig resent the results o an repre Wl AN& J l -db. "- and 4. 0, the efffective poten e t t iaaal rises too steeply, or E„are larger an the results for n the corresp an sponding values . , as was obtained by I.K. However, ow e c 'nfinite barrier mo e s, p change-corre lation a i energies for a v s are considera, bl y more accura t e l.y obtained ine i t tic contribua t xam le, for r, =2. , „, i s E and nd are within q o lumn. Thus, althoug e 'on functions affect cor relation ct thee total energy im&V and thus E~, their ei explicit contribution ri taken togeether with exc an mately the same. - udd theorem -Use oof thee Vannimenus-Budd C. Surface energies-Use tiou we present results (see see Tab Tablee 'th tive of the sur face ) o m" r as obtainedd bby the sum rule of VanThe parame meters are again t the BVT is ei'th er tl the case for r, & d as p ossible, as y h ""Budd. &„di er 4% respective p, from those o I K. These result s to g ether wi th those of the other models s tu died i indicate that E as obtained „a " ppo sion for the derivative of e en of th fa th respect to the Wi gn er-Seitz radius dE /dy gy and surface eneer gi es in ergs cm, m as obtained o aine by a licat'on o employing the Wiigneer approxima t'ionn for correlation. dE, S Present work 2.0 2.5 3.0 3.5 4.0 45 5.0 5.5 6.0 ~See Hef. 13. S S Lang- Kohn ~ 3365 669 4447 755 119 —9 —36 —34 —23 —15 —ll 79 —54 —71 -67 —34 —37 -33 Present work —868 —41 123 143 130 111 97.5 Lang-Kohn -980 49 197 191 157 122 97.5 88 80 82 62 METAL SURFACE PROPERTIES IN THE LINEAR POTENTIAL. finite barrier model obtained from Eq. 17 is given in Ref. 3, and a plot of dE, /dr, over the entire metallic range is given in Fig. 8. The crosses in the figure represent the values obtained by Vannimenus and Budd employing the electrostatic potential. of LK. The surface energy is determined by integration over r, with the constant of integration" chosen such that the energy matches the LK value at r, =5. Although we could determine this constant by a physical criterion such as the t', ~ limit for which case the surface energy vanishes, we restrict ourselves to the above choice in order to enable comparisons with the work of Vannimenus and Budd. A graph of the surface energies thus obtained is al. so plotted in Fig. 7. We note that for high densities, in contrasts to both the step and infinite potential models, the present model does permit dE, /dr, to become positive and large. However, although the graph for the derivative appears very similar to the LK results, a comparison of the explicit values given in Table III indicate them to be quite different. This is interesting because use of the LDA within this model. with the same choice of parameters leads to very accurate surface energies. Since the LDA method depends primarily on the electronic density, and the VBT method on the total charge density inside the metal, the reason for the differences in the results of the two methods becomes evident on a comparison 1000— + Lang-Kohn 600— dES drs 400— 200— 2.5 I -)00- 30 3.5 4.0 I I 4, 5 5.0 5.5 I I I I + + + 6.0 + FIG. 8. Plot of the derivative of the surface energy E~ with respect to the signer-Seitx radius &~ vs ~~ for the Wigner expression for correlation. The crosses represent the results obtained by Vannimenus and Budd employing the electrostatic potentials of Lallg and Kohn. .. 1947 of the electronic and total charge densities and electrostatic potentials obtained with those of LK. Inside the metal, for example, the electronic density normalized with respect to the bulk value for r, =2 is within 4% of the LK density, differing 1.2 to by less than a percent in the range from —0.275 Fermi wavelengths. On the other hand, the total charge density near and at the metal surface can differ by as much as 30%. The major contribution to the integral of Eq. 1V for dE, /dr, arises from a region extending from the surface to approximately a third of a Fermi wavelength inside the metal. . In this range, the results for the electrostatic potential differs by (6-9)% from those of LK. Furthermore, inside the metal, however, the differences in V„are comparable to those of the electronic density. Thus it is these differences in total density, and hence V„ inside the metal which lead to the differences ln the derivative of the surface energy and to its integral, i.e. , the surface energy. The adjustment of the electrostatic potential such that it corresponds to the exact value at one point, viz. , at the metal. surface, cannot be expected to lead to precise values elsewhere. However, the results for the surface energy, though not as accurate as those of the LDA calculation, are fair approximations to the LK values, and meaningful over the entire metallic range (see Fig. 7). Thus a study of this more accurate model reaffirms the conclusion' that the VBT method for obtaining surface energies is particularly sensitive to the choice of effective potential. We observe, in conclusion, that the linear potential model together with the constraint of the sum rule due to Budd and Vannimenus leads to results for all surface properties comparable to those obtained by Lang and Kohn for jellium metal. The fact that the effective potential does not become constant but increases indefinitely is unimportant, since the effective potential is in substantial error only in the region far from the metal. surface where the electron density has exponentially decayed to a small fraction of its value at the surface. The majority of the calculations, as shown in the Appendix, are primarily analytic„and the universal curves permit a direct determination of surface properties on application of any theoretical, constraint. Furthermore, due to the accuracy of the results and semianalytic nature of the density, it is reasonable to employ this model in order to study the effects of the gradient and higher-order terms on the exchange and correlation energies, and such an investigation is in progress. We also observe that although the LDA and VBT methods for obtaining energies lead to good results in compari- - V. SAHXI, J. 8. J. KRIKGER, AND son with those of Lang and Kohn, the former method proves to be superior, and it would thus be of interest to determine how the results of the two methods compare on inclusion of the gradient term contribution to the existing LDA value. Finally, we note that use of different correlation functions does not affect in any significant manner either the work function or the exchange-correlation contribution to the surface energy, as obtained via the LDA, but do affect considerably the results for the total energy. This as explained earlier is because the component of the energy most sensitive to small variations in the effective potential is the kinetic energy, and its contribution to the total energy for high and medium densities is always significant. GRUENEBAUM erties were obtained by use of the foll. owing integral expressions. Ai2 df g &Ai' =fbi' f -Ai" — g d&= ', g'Ai' g (A7) i" —&A. & +xi (g) A~'(g)], A. i" dg=-,' g -g'Ai'g +gAz" + 2xf g (g)Af''(g)]. (AS) Metal surface position Application of the charge neutrality of Eq. j.8 leads to the expression With the transformation y =kzx, 0/k~ =q, the slope parameter is y„=k~x~, the metal surface is at y, = k„a, and the variable g is ~=(y f, 2) (1 sin25(q, condition yp ) (A10) (Al) q »1 )yZ present below expressions for the phase shift, electronic density, metal surface position, surface dipole barrier, electrostatic potential, and the kinetic energy in terms of universal functions of the slope parameter y~. 'tI)%t'e alternate expression is obtained from the Sugiyama phase shift rule of Eq. 19: An — Sm y, = — —3 dq q5(q, yz). D Surface dipole barrier &0/&p = (4/3&)[d(yz)+&(yz)], %lith the definition where 1yS»(-q'4") «(Syz)=ye ,, «(0) =yg' Aji( 2 ags) ~ »(0)— =-1.37172»g', &. —, &(q, y~) =cot '[1/q~( (A2) 2 3 1 ——y',2 J(y„) = — 4 3 y (A4) o dq(1 —q') cos2~[qy+5(q, yz)]. y ~V, (y) = 4 3, 3s d(y~)+ 8 + dq(l —q') sin'5(q, y~) ~ &f'{{y-q'y gg2( + )y q2»2/3) The universal. functions for the remaining dq(1-q')qsin20(q, y&). «0, 2 p~ 1 3, y~- 2' + 2' v. (y) Q' (A14) For j. o Electrostatic potential «0 =1 ——,' 2 (1 —q')«' dq— 1+K {A13) @y~) = yq~)]. g + —«(0)+ (A3) Electronic density For (A12) prop- 3 (, e{y.-y)+-'»{2». -y)e(y-y. i 3 8 y ) 2, cos26{q, y~) sin qy dq(l —q ) . -g , s' dg(l ~ 25(q, y )Hi 2qy) Q' (A15) METAL SURFACE PROPERTIES IN THE LINEAR POTENTIAL. . . For y~ 0 1949 Kinetic energy V„(X) hQ k~ k~ + g(i) 2 (7 —Vo) 8(X —S) 3w k~4 = (1/160s)8, (A1'7) where 80 8 = 1+ — dq q6(q, y~) —, 1 x [ 2g'Ai'(t;) —21. Ai "(g ) 0 A-i (g) Ai '(g)l . of York Faculty Research Program. N. D. Lang, in Solid State Physics, Advances in Research and Applications, edited by H. Ehrenreich, F. Seitz, and D. Turnbull (Academic, New York, 1973), Vol. 28, p. 243. V. Sahni, J. B. Krieger, and J. Gruenebaum, Phys. Bev. B 12, 3503 (1975). V. Sahni and J. Gruenebaum, Phys. Rev. B 15, 1929 New (1977). J. W. Kohn and L. Sham, Phys. Rev. 140, A1133 (1965). 5N. D. Lang and W. Kohn, Phys. Rev. B 1, 4555 (1970); 3, 1215 (1971). The self-consistency procedure has been numerically refined by Lang, and the improved results for the surface energy and dipole moments have been quoted in the text. H. F. Budd and J. Van». menus, Phys. Rev. Lett. 31, 1218 (1973); 31, 1430(E) (1973). Variational Principles (lnterscience, New York, 1966), p. 153. J. Bardeen, Phys. Bev. 49, 653 (1936). W. J. Swiatecki, Proc. Phys. Soc. Lond. A 64, 236 (1951). A. Sugj, yama, J. Phys. Soc. Jpn. 15, 965 (1960). B. L. Moiseiwitsch, E~"'l&z =(1/«')[&(Xz)h~l (A16) Supported in part by a grant from the City University des &(ax )), (A18) (A19) D. Lang and L. J. Sham, Solid State Commun. 17, 581 (1975). D. C. Langreth and P. Perdew, Solid State Commun. 17, 1425 {1975). ~3J. Vannimenus and H. F. Budd, Solid State Commun. ~~N. J. ' 15, 1739 (1974). J. Harris and R. O. Jones, J. Phys. F 4, 1170 (1974). '5E. Wikborg and J. E. Inglesfield, Solid State Commun. 16, 335 (1975). D. Pines, Elementary Excitations in Solids (Benjamin, New York, 1963), p. 94. D. Pines and P. Nozieres, The Theory of Quantum Liquids {Benjamin, New York, 1966), p. 330. P. Vashista and K. S. Singwi, Phys. Rev. B 6, 875 (1972). ~~Atomic units are used: ~e = K= m =1. The unit of energy is 27.21 eV. ~ L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1965), p. 71. M. Abramowitz and I. A. Stegun, Handbook' of Mathematical Functions (Dover, New York, 1965), p. 446. S. Raimes, 8'ave Mechanics of Electrons in Metals (North-Holland, Amsterdam, 1961), p. 177.