PHYSICAL REVIEW VOLUME 136 NUMBER 3B 9 NOVEMEBR 1964 Inhomogeneous Electron Gas" P. HoHENBERGt Ecole Normale Superieure, Paris, france Ecole Normale Superieure, Paris, France and Faculte des Sciences, Orsay, france University of California at San Diego, La Jolla, California This paper deals with the ground state of n external potential u(r). It is proved that there exists a universal functional of the density, FLr(r)], independent of o(r), such that the ex- (2)n(r)=p(r/ro)with e arbitrary an elation energy and linear and higher order electronic polarizabilities of a uniform electron gas. This approac also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new extensions of these methods are presented heoretical considerations is a description of this D URING the last decade there has been considerable functional. Once known, it is relatively easy to deter- progress in understanding the properties of a mine the ground-state energy in a given external homogeneous interacting electron gas. I The point of potential view has been general, to regard the electrons as In Part II, we obtain an expression for F[n] when n ction of noninteracting particles deviates only slightly from uniformity, i.e, n(r)=no additional concept of collective +n(r), with A/no-0. In this case FLn] is entirely excitations ole in terms of On the other hand. there has been in existence since and the exact electronic polarizability a(q) of a uniform the 1920,s a different approach, represented by the electron gas. This procedure will describe correctl Thomas-Fermi method? and its refinements, in which the long-range Friedel charge oscillations set up by the electronic density n(r)plays a central role and in a localized perturbation. All previous refinements of the which the system of electrons is pictured more like a Thomas-Fermi method have failed to include these classical liquid. This approach has been useful, up In Part III we consider the case of a slowly val now, for simple though crude descriptions of inhomo- but not necessarily almost constant density, n(r) geneous systems like atoms and impurities in metals. =p(r/ro), fo-c0 For this case we derive an expansion Lately there have been also some important advances of FLn] in successive orders of ro or, equivalently of along this second line of approach, such as the work of the gradient operator v acting on n(r). The expansion and Borowitz, Baraf, 7and Du Bois and Kivelson. s The ground-state energy and the exact linear, quadratic present paper represents a contribution in the same area. etc, electric response functions of a uniform electron In Part I, we develop an exact formal variational gas to an external potential o(r). In this way we recover, Sitv ciple for the ground-state energy, in which the den- quite simply, all previously developed refinements of enters a universal functional F[n(r)], which applies to somewhat further. Comparison of this case with the all electronic systems in their ground state no matter nearly uniform one, discussed in Part II, also reveals what the external potential is. The main objective of why the gradient expansion is intrinsically incapable Supported in part by the U. S. Office of Naval Research. of properly describing the Friedel oscilations or adial oscillations of the electronic density in an atom tfor a review see, iof oin Info New s s Elementary Excitations summation of the gradient expansion can be carried which reflect the electronic shell structure. A partial Phy. 6, 1 rev95) of work up to 1956, see N. H. March, Advan. out(Sec. I4 ) but its usefulness has not yet been neets and E. S. Pavlovskii, Zh, Eksperir 31, 427(1956)[English transl. Soviet Phys.-JETP I EXACT GENERAL FORMULATION 4 D. A. Kirzhnits, Zh. Eksperim i. Teor 13 32,115(1957) The Density as Basic Variable H. W. Lewis. P We shall be consider number of electrons, enclosed in a large box and movin 8 D. F. Du Bois and M. G. Kivelson, Phys. Rev. 127, 1182 J. Friedel, Phil. Mag. 43, 153(1952). B864
PHYSICAL REVIEW VOLUM E 136, NUM B ER 3 8 9 NOVEMEBR 1964 InhOmOgeIIeouS EleCtrOn Gaa* P. HOHENBERGt Ecole Xornzale Superzeure, I'aris, France AND W. KonNt Ecole Xonnale Superieure, I'aris, Prance and I'aculte des Sciences, Orsay, France and University of Calzfo&nia at San Diego, La Jolla, Calzfornia (Received 18 June 1964) This paper deals with the ground state of an interacting electron gas in an external potential v(r). It is proved that there exists a universal functional of the density, FtI(r) g, independent of v(r), such that the expression E—=fs(r)n (r)dr+Ft I(r)j has as its minimum value the correct ground-state energy associated with s(r). The functional FLn(r)j is then discussed for two situations: (1) n(r) @san(r), 8/ao((1, and (2) a(r) =q (r/ra) with p arbitrary and 1'p ~~.In both cases Fcan be expressed entirely in terms of the correlation energy and linear and higher order electronic polarizabilities of a uniform electron gas. This approach also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new extensions of these methods are presented. INTRODUCTION ' ' &~IJRING the last decade there has been considerable progress in understanding the properties of a homogeneous interacting electron gas. ' The point of view has been, in general, to regard the electrons as similar to a collection of noninteracting particles with the important additional concept of collective excitations. On the other hand, there has been in existence since the 7920's a different approach, represented by the Thomas-Fermi method' and its re6nements, in which the electronic density n(r) plays a central role and in which the system of electrons is pictured more like a classical liquid. This approach has been useful, up to now, for simple though crude descriptions of inhomogeneous systems like atoms and impurities in nietals. Lately there have been also some important advances along this second line of approach, such as the work of Kompaneets and Pavlovskii, ' Kirzhnits, ' Lewis, ' Baraff and Borowitz, ' Bara6, ' and DuBois and Kivelson. ' The present paper represents a contribution in the same area. In Part I, we develop an exact formal variational principle for the ground-state energy, in which the density tz(r) is the variable function. Into this principle enters a universal functional PLtr(r)), which applies to all electronic systems in their ground state no matter what the external potential is. The main objective of *Supported in part by the U. S. Once of Naval Research. f NATO Post Doctoral Fellow. f Guggenheim Fellow. ' For a review see, for example, D. Pines, Elementary E'.'xci tati ons in Solids (W. A. Benjamin Inc., New York, 1963). ' For a review of work up to 1956, see N. H. March, Advan. Phys. 6, 1 (1957). A. S. Kompaneets and E. S. Pavlovskii, Zh. Eksperim. i. Teor. Fiz. 51, 427 (1956) [English transl. : Soviet Phys.—JETP 4, 328 (1957)j. D. A. Kirzhnits, Zh. Eksperim. i. Teor. Fiz. 32, 115 (1957) I English transl. : Soviet Phys.—JETP 5, 64 (1957)j. ' H. W. Lewis, Phys. Rev. 111, 1554 (1958). ' G. A. 13araff and S. Borowitz, Phys. Rev. 121, 1704 (1961). 7 G. A. BaraG, Phys. Rev. 123, 2087 (1961). 'D. F. Du Bois and M. G. Kivelson, Phys. Rev. 127, 1182 (1962). theoretical considerations is a description of this functional. Once known, it is relatively easy to determine the ground-state energy in a given external potential. In Part II, we obtain an expression for FLnj when tr deviates only slightly from uniformity, i.e., n(r)=1'cp +ts(r), with ts/tss —& 0; In this case FLej is entirely expressible in terms of the exact ground-state energy and the exact electronic polarizability n(g) of a uniform electron gas. This procedure will describe correctly the long-range Friedel charge oscillations' set up by a localized perturbation. All previous refinements of the Thomas-Fermi method have failed to include these. In Part III we consider the case of a slowly varying, but +of necessarily almost constant density, tr (r) = p(r/rs), rs —&oo. For this case we derive an expansion of F)trj in successive orders of rs ' or, equivalently of the gradient operator V acting on e(r). The expansion coeKcients are again expressible in terms of the exact ground-state energy and the exact linear, quadratic, etc. , electric response functions of a uniform electron gas to an external potential w(r). In this way we recover, quite simply, all previously developed refinements of the Thomas-Fermi method and are able to carry them somewhat further. Comparison of this case with the nearly uniform one, discussed in Part II, ,also reveals why the gradient expansion is intrinsically incapable of properly describing the Friedel oscillations or the radial oscillations of the electronic density in an atom which reQect the electronic shell structure. A partial summation of the gradient expansion can be carried out (Sec. III.4), but its usefulness has not yet been tested. I. EXACT GENERAL FORMULATION I. The Density as Basic Variable Ke shall be considering a collection of an arbitrary number of electrons, enclosed in a large box and moving ' J. Friedel, Phil. Nag. 45, 155 (1952)
INHOMOGENEOUS ELECTRON GAS B865 under the influence of an external potential v(r)and where FIn] is a universal functional, valid for any the mutual Coulomb repulsion. The Hamiltonian has number of particles and any external potential. This functional plays a central role in the present paper. H=T+V+U, (1) With its aid we define, for a given potential v(r), the whereto energy functional 2丿wy(r)vy(r)dr, ]=/v(r)n(r)dr+FLn] V=/v(r)*(r)ψ(r)lr, (3)Clearly, for the correct n(r), EmLn] equals the ground We shall now show that Ew[n] assumes its minimum )drdr. (4) value for the correct n(), if the admissible functions We shall in all that follows assume for simplicity that we are only dealing with situations in which the ground 11) state is nondegenerate. We denote the electronic density the ground state业by It is well known that for a system of N particles, the n(r)≡(ψ*(r)(r)v) (5) energy functional of Y' is clearly a functional of o(r 8]=业亚,Vy)+(,(T+U)业)(12) shall now show that conversely v(r) is a unique sume that another potential v(r), with ground state particles is kept constant. In particular, let ybe the w gives rise to the same density n(r). Now clearly ground state associated with a different external pe Unless v(r)-o(r)=const]y' cannot be equal to y tential t(r).TH since they satisfy different Schrodinger equations Hence, if we denote the Hamiltonian and ground-state 8[V]-/D(r)n(r)dr+F[n] energies associated with y and y' by H, Hand E, E we have by the minimal property of the ground state, (13) E=(亚,my)8[v]=/(r)n(r)dr+P[n] so that E<E+[v(r)-v(r)]a(r)dr (6)tive to all density functions n(r)associated with some other external potential w(r). 12 Interchanging primed and unprimed quantities, we find If FLn] were a known and sufficiently simple func in exactly the same way that tional of n, the problem of determining the ground-state energy and density in a given external potential would E<E+ Co(r)-o(r)]n(r)dr be rather easy since it requires merely the minimization of a functional of the three-dimensional density func tion. The major part of the complexities of the many Addition of(6)and(7)leads to the inconsistency electron problems are associated with the determination E+e<E+E (8) of the universal functional FLnJ Thus v(r)is(to within a constant)a unique functional 3. Transformation of the Functional F[n] of n(r); since, in turn, o(r)fixes H we see that the full many-particle ground state is a unique functional of Because of the long range of the Coulomb interaction n(r) it is for most purposes convenient to separate out from 2. The Variational Principle 11 This is obvious since the number of particles is itself a simple Since y is a functional of n(r), so is evidently the h we cannot prove whes the coid v( cleardrsinteger kinetic and interaction energy. We therefore define F[n(r)]=(v,(T+U)v) le form n(r)=no+n(r) in fact all, except some patholog 10 Atomic units are used distributions, can be realized
I N HOMOGENEOUS ELECTRON GAS under the influence of an external potential v(r) and the mutual Coulomb repulsion. The Hamiltonian has the form H= T+V+U, where'0 where Pfn] is a universal functional, valid for any number of particles" and any external potential. This functional plays a central role in the present paper. With its aid we define, for a given potential v(r), the energy functional ~~i*(r)~~i (r)dr, 2 (2) E„gn]=— v (r)I(r)dr+ FLN]. (10) V= v(r)i(*(r)P(r)dr, P*(r)P*(r')f(r')P (r)drdr' Clearly, for the correct is(r), E„ge] equals the groundstate energy E. We shall now show that E,ge] assumes its minimum value for the correct n(r), if, the admissible functions are restricted by the condition We shall in all that follows assume for simplicity that we are only dealing with situations in which the ground state is nondegenerate. We denote the electronic density in the ground state 0' by which is clearly a functional of v(r). We shall now show that conversely v(r) is a unique functional of N(r), apart from a trivial additive constant. The proof proceeds by reductio ad absurdum'. Assume that another potential v'(r), with ground state 4' gives rise to the same density N(r). Now clearly (unless v'(r) —v(r)=const] 0' cannot be equal to 4 since they satisfy different Schrodinger equations. Hence, if we denote the Hamiltonian and ground-state energies associated with 0' and 0' by H, B' and E, E', we have by the minimal property of the ground state, E'= (@',H'+') & (+,H'+) = (+, (H+ V' V)%'), — so that E'&E+ $v'(r) —v(r)]e(r)dr. Interchanging primed and unprimed quantities, we find in exactly the same way that E&E'+ $v (r)—v' (r)]ti (r)dr. Addition of (6) and (7) leads to the inconsistency E+E ~ &E+E~ Thus v (r) is (to within a constant) a unique functional of e(r); since, in turn, v(r) fixes H we see that the full many-particle ground state is a unique functional of rs(r). Ãfm] —= n (r)dr =cV. It is v ell known that for a system of Eparticles, the energy functional of 4' (12) has a minimum at the correct ground state 4, relative to arbitrary variations of 0' in which the number of particles is kept constant. In particular, let 4' be the ground state associated with a diferent external potential v'(r). Then, by (12) and (9) B„L@']= v (r)I'(r) dr+Fc ri'], )8,$+]= v(r)e(r)dr+FLri]. Thus the minimal property of (10) is established relative to all density functions I'(r) associated with some other external potential v'(r). " If F(1) were a known and sufFiciently simple functional of n, the problem of determining the ground-state energy and density in a given external potential would be rather easy since it requires merely the minimization of a functional of the three-dimensional density function. The major part of the complexities of the manyelectron problems are associated with the determination of the universal functional FLn]. 3. Transformation of the Functional P/n] Because of the long range of the Coulomb interaction, it is for most purposes convenient to separate out from 2. The Variational Principle Since 4 is a functional of n(r), so is evidently the kinetic and interaction energy. We therefore define ro At, oDllc url'its are- use '~ This is obvious since the number of particles is itself a simple functional of n(r). ~ We cannot prove whether an arbitrary positive density distribution a'(r), which satisaes the condition J'e'(r)dr=integer, can be realized by some external potential v'(r}. Clearly, to first order in R(r), any distribution oi the form n'(r) =no+n(r) can be so realized and we believe that in fact g,ll, except some patbologicaf distributions, can be realized
B866 P. HOHENBERG AND W. KOHN FLn] the classical Coulomb energy and write and 1 n(r)(r) f(rdr=0 (23) FLn] drdr+GLnT (14) Here we clearly must have a formal expansion of the so that E[n becomes 1 n(r)n(r) E[n]=|(r)n()dr+ 2/r- drdr+GLn],(15)GLn]=GLno]+ K(r-r)n(r)(r)drdr here G[n] is a universal functional like FL. L(r,r, r")i(r)i(r)n(rdrdr'dr (24) Now from the definition of F[n], Eg.(9), and G[n1, g.(14), we see that In this equation there is no term linear in i(r)since 1/C2(r, r) by translational invariance the coefficient of i(r)would G[n]=- V,r1(r,r,)lrerdr+- drdr.(16) be independent of readi ding to zero, by(23). The kernel appearing in the quadratic term is a functional of r-r'l Here n(r, r)is the one-particle density matrix; and only and may therefore be written as (r, r) is the two-particle correlation function defined in terms of the one-and two-particle density matrices as K(r-r)=(1/9∑K(q)e,(rr).(25) C(r, r)=n2(r,r;r,r)-n(r, r)n(r, r).(17) The higher order terms will not be further discussed Of course n1(r, r)=n(r) One may also quite trivially introduce a density From(16)we see that we can define an energy-density function grLn]=2V,Vrn1(r,r) er gLn]=8(n0)+/K(r)(r+r)n(r-r)dr+…,(20) 1C2(r-r/2;r+r/2) dr'(18)where go(no) is the density function of a unifor (kinetic, exchange such that energy). GEn]=g[n]dr (19) 2. Expression of the Kernel K in Terms of the Electronic Polarizability The fact that gr n] is a functional of n follows of course We shall now see that the kernel K appearing from the fact that y and hence ni and ne are Eqs.(24)and(26)is completely and exactly expressible It should be remarked, that wl while G[n] is a unique in terms of the electronic polarizability a(q). The latter functional of n, gr[n] is of course not the only possible is defined as follows: Consider an electron gas of mean energy-density functional. Clearly the functional small additional positive external-charge density 10 Write the electronic density, to first order in A, where the h() are entirely arbitrary, give equivalen A2∑b1(q)e results when used in conjunction with(19) Then eal with GIn] and grLn] in Let us now define the operator II. THE GAS OF ALMOST CONSTANT DENSITY 1. Form of the Functionals G[n] and g[n] (21) operators. Then, by first-order perturbation heory, O We consider here a gas whose density has the form where ck", Gx are the usual creation and annihilate n(r)=#0+(r) with a(g)(0 pan)(n e-glo) b1(q)=-(8丌)∑ (31)
P. HOHEN BERG AND W. KOHN F[n] the classical Coulomb energy and write and 1 F[n]=— 2 so that E,.[n] becomes iz (r)n (r') drdr'+ G[n], l r—r'l (14) R(r)dr=0 (23) Here we clearly must have a formal expansion of the following sort: 1 n (r)n (r') E,„[n]= p (r)n(r) dr+ — drdr'+G[n], (15) G[n]=G[np]+ E(r—r')R (r)R (r') drdr' l r—r'l Cz(r,r')=nz(r, r'; r,r') —nz(r, r)nz(r', r'). (17) Of course nz(r, r)=—n(r). From (16) we see that we can define an energy-density functional g p[n] = z V~V~ 1zz(r)r ) l g—~ where G[n] is a universal functional like F[n]. Now from the definition of F[n], Eq. (9), and G[n], Eq. (14), we see that 1 C,(r,r') G[n]=— V,V,.n~(r, r') l, , dr+ — drdr'. (16) l r—r'l Here n, (r,r') is the one-particle density matrix; and Cz(r,r') is the two-particle correlation function defined in terms of the one- and two-particle density matrices as + I(r,r',r") R(r) R(r') R(r"} dr dr' dr"+ . . (24) ln this equation there is no term linear in R(r) since by translational invariance the coefficient of R(r) would be independent of r leading to zero, by (23). The kernel appearing in the quadratic term is a functional of l r—r l only and may therefore be written as It(r—r')=(1/~l)Z It(q)e "' "' (25) g,[n]=gp(np)+ IC(r')R(r+-,'r')R(r ——, 'r')dr'+, (26) The higher order terms will not be further discussed here. One may also quite trivially introduce a density function such that 1 C,(r—r'/2; r+r'/2) dr' (18) where gp(np) is the density function of a uniform gas of electron density np (kinetic, exchange, and correlation energy). G[n]= g,[n]dr. The fact that g,[n] is a functional of n follows of course from the fact that 4' and hence e~ and e2 are. It should be remarked, that while G[n] is a unique functional of n, g,[n] is of course not the only possible energy-density functional. Clearly the functionals 2. Expression of the Kernel Xin Terms of the Electronic Polarizability We shall now see that the kernel IC appearing in Eqs. (24) and (26) is completely and exactly expressible in terms of the electronic polarizability n(q). The latter is defined as follows: Consider an electron gas of mean density eo in a background of uniform charge plus a small additional positive external-charge density i9 g,[n]=g,[n]yP ts, &'&[n], (20) n.„,(r) = (X/Q)g a(q) e-'q'. Write the electronic density, to first order in X, as (27) where the h~') are entirely arbitrary, give equivalent results when used in conjunction with (19). The following sections deal with G[n] and g,[n] in some simple cases. n(r) =np+ ()/Q)p b,(q)e-'q'. ~(V)—=bz(a)/~(a). Let us now define the operator (28) (29) n(r) =np+R(r), (21) R(r)/n, «1 (22) II. THE GAS OF ALMOST CONSTANT DENSITY 1. Form of the Functionals G[n] and g„[n] We consider here a gas whose density has the form Pq=g Ck q Ck, k (3o) ~(v) (oI pql n)(nip-. l o) b~(q) =—(8~) (31) where c~*, c~ are the usual creation and annihilation operators. Then, by first-order perturbation theory
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)(1090
I~ 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 polarizability n(q), as function of q (electronic 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 longrange 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 randomphase 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}
B868 P. HOHENBERG AND W. KOHN the Thomas-Fermi equation 2. The Gradient Expansion It is well k of the Thomas-Fermi equation is that n(r)must be a slowly varying function of r. This suggests study of th functional G[n], where n has the form (r=p(rro with It is obvious that this is quite a different class of systems than that considered in Part II (n=no+h, i/no<<1) III. THE GAS OF SLOWLY VARYING DENSITY since now we shall allow to have substantial varia- 1. The Thomas-Fermi equa aton tions. On the other hand, whereas in Part Il, i could contain arbitrarily short wavelengths, these are here For a first orientation we shall derive, from our general ruled out as ro becomes large variational principle, the elementary Thomas-Fermi We now make the basic assumption that for large ro, equation. For this purpose, we use the functional (18) the partial energy density g [n] may be expanded in and in(16)we neglect exchange and correlation effects, the form thus setting C2=0. We approximate the kinetic-energy term by its form for a free-electron gas, i.e. gr[n]=go(n(r))+2 g(n(r).Vn(r) g-[n]=To[kr(n)Pn (50) where the Fermi momentum k is given by +2 Lgi. d, 1)(n(r). vn(r) n(r) (51) +g,2)(n(r)Vvm(r)]+…(61) This results Here successive terms correspond to successive negative n(r)n(r) powers of the scale parameter ro. Quant E[n]= s(r)(r)dr+- - drdr go(n(r)), gi(n(r))etc, are functions (not functionals r-r, of n(r). No general proof of the existence of such an expansion is known to us, although it can be formally +io(3r)ia/[o(r)]8dr. (52) verified in special cases, e g-, when G[n(r)] can be ex- To determine n(r)we now set we know that, for a finite ro, the series does not strictly converge(see the discussion at the end of Sec. I1.3) may expect it to be useful (in the sense 6{En[n]-4/n(r)lr}=0 (53)totic convergence)for sufficiently large values of ro Now a good deal of progress can be made, using only where u is a Lagrange parameter. This results in the the fact that gi[n] is a universal functional of n independent of v (r). This requires gr[n] to be invariant under rotations about r. The coefficients g v(raf mdr+1(3T)/Ln(r)/3-H=O.(54)under rotations. Hence one finds by elementary being functions of the scalar n, are of course invariant siderations that gr[n] must have the form If we now introduce the "internal"potential gr[n]=80(n)+[g2)(x)V2n+g2()(n)(Vnwn)] +terms of order VA.(62) r (55) A further simplification results from the fact that we may eliminate from gr[n] an arbitrary divergence (54)is equivalent to the pair of equations 2iVhr'ln(see the end of Sec. 1. 3). It is then elemer n(r)=(1/3m)(2[u-o(r)-0(r)J)/,(56)tary to show that grLn]may be replaced by vvi (r)=-4rn(r (57)grLn]=go(0 )+g2 2)(n)Vn.Vn +{g42)()(V2n)(V2n)+g4)(n)(Vn)(VnV rom(56)and(57)we can eliminate n(r)and arrive at +g4(n)(Vn·Vn)2}+O(V).(63)
P. HOHEN BERG AN D W. KOHN the Thomas-Fermi equation V'v;(r) = (—2 &'/37r)f p —v(r) —v, (r)5'& (58) 10- FxG. 2. Behavior of the kernel E'(q), as a function of q (electronic density =4 &10"cm 3). 2. The Gradient Expansion It is well known that one condition for the validity of the Thomas-Fermi equation is that ri(r) must be a slowly varying function of r. This suggests study of the functional Gfej, where e has the form 0 0 with N(r) = y(r/rp), ro~~ . (59) (60) III. THE GAS OF SLOWLY VARYING DENSITY 1. The Thomas-Fermi Equation For a erst orientation we shall derive, from our general variational principle, the elementary Thomas-Fermi equation. For this purpose, we use the functional (18) and in (16) we neglect exchange and correlation effects, thus setting C2=0. We approximate the kinetic-energy term by its form for a free-electron gas, i.e. , It is obvious that this is quite a di6erent class of systems than that considered in Part 11 (N=ep+n, 8/Np«1), since now we shall allow q to have substantial variations. On the other hand, whereas in Part II, rI, could contain arbitrarily short wavelengths, these are here ruled out as r0 becomes large. We now make the basic assumption that for large r0, the partial energy density g,fnj may be expanded in the form g,f&i]=gp(N(r))+g g,(n(r)) Vps(r) g f&5= i'oft~—(~)]'~, where the Fermi momentum kl: is given by k p(n) = (37r'e)'~'. (50) (51) +Z Lg ""(~(r)) V'~(r)V~(r) +g;,&'&(n(r)) V,V,&i(r)]+ . . (61) This results in 1 &i(r)&p(r') Z„fe]= v(r)e(r)dr+- drdr' 2 fr—r'f +r'p (3~')'" f~(r)]""«(52) To determine e(r) we now set 8 E„fe]—&i e(r)dr =0, (53) where p, is a Lagrange parameter. This results in the equation m(r') v(r)+ dr'+-', (3 r')7'"fm(r)]' ' &Ii=0 —(54). /r —r'f If we now introduce the "internal" potential n (r') v, (r)—= dr', (55) (56) (57) (54) is equivalent to the pair of equations N(r) = (1/3m ){2'—v(r) —v, (r)5)'&' VPv;(r) =—4v 0(r) . From (56) and (57) we can eliminate m(r) and arrive at Here successive terms correspond to successive negative powers of the scale parameter rp. Quantities like gp(e(r)), g;(n(r)) etc. , are functions (not functionals) of N(r). No general proof of the existence of such an expansion is known to us, although it can be formally verified in special cases, e.g., when Gfe(r) 5 can be expanded in powers of fe(r)—Npj. At. the same time, we know that, for a finite r0, the series does not strictly converge (see the discussion at the end of Sec. II.3), but we may expect it to be useful (in the sense of asymptotic convergence) for suKciently large values of rp. Now a good deal of progress can be made, using only the fact that g,flj is a universal functional of n, independent of v(r). This requires g,fej to be invariant under rotations about r. The coeKcients g, ;, (n(r)), being functions of the scalar e, are of course invariant under rotations. Hence one 6nds by elementary considerations that g,fnj must have the form g,fnj= gp(n)+ fgp&'(e) V'm+gpt &(n)(VN Vn)5 +terms of order V~4. (62) A further simpli6cation results from the fact that we may eliminate from g,fej an arbitrary divergence Q,V,h, 'fN] (see the end of Sec. I.3).It is then elementary to show that g,fnj may be replaced by g,f&r j=gp(e)+gp "&(N)Ve V&p +{g"&(e)(V'e)(V'I)+g i'&(n)(V'&i)(VN VN) +g4'4&(e) (VN Ve)')+O(V P). (63)
INHOMOGENEOUS ELECTRON GAS B869 Here the subscripts refer to the number of gradient Now let us set operators(or the order in 1/ro)and the superscripts to the number of times that n appears to the right of It may be worth recalling admissible density function n=#10+-∑DAb1(q)+入2b2(q)+…]eir,(70) it differs from the energy density function grLn], Eq. Collecting terms of order o, A, xa, we find (18), by a divergence. g0( (72) 3. Identification of the coeficients of the GradientE -a(q)+{+g+2g2 q2 We shall now express the coefficients g,( p)(n)ap- pearing in Eg.(63) in terms of the expansion coef +2gy4+…}b(q)=0,(3) ficients, in powers of a, of the electronic polarizability (), and similar higl q≠0, We do this by applying our general expression(63)to the case of a nearly uniform electron gas, considered b(g) +…}a(q) already in Sec. II. 2. We go, however, beyond(28)and 4 write Also clear n(r)=+-∑b1(q)e+-∑b2()e-g+…….(65) H1=0 imilarly, we obtain The linear- and second-, third-, etc. order response functions are then defined by the relations r2+…}a(q^a(q-q).(5) b1(g)=a(g)a(q a(qu gya(qa(g2) (66) If we now expand the response functions in powers of () =1+C292+cag (q,q)=∑∑cmn"q (77) obtains with the use of (63) we can identify the functions ga(). Thus on ee[n]-p/n(r)dr (67) 0/4 (78) This gives g42)/4丌=号(-c+2c4-c2) (r)+ dr+go'-g2@2)(vn)2 Similarly all other coefficients g 2)(n)can be expressed in terms of the expansion coefficients cn of the linear polarizability a(a) of an electron gas of density n. 2g2 2)Vn+3g42'(vn)2+2g4@2)"(Vn)vn In an analogous manner we can express all g (a) in +2g4)Vn·V(V2n)+2g4)(四vn) terms of a(qu) and a(g1, q2); and generally g(o)in terms ofa(q1),…a(q1,q;…q-1) +g43)"(Vn)4+2g4(3)Vn,V(Vn)2 On dimensional grounds we can see from(63)that (81) 3g4(4)(Vn)4-4g4(4)V2n(Vn)2-4g)Vn.V(Vn)2 d Vvn|/四n<kp(m)
I NHOMOGEN EOUS F LECT RON GAS X44r u(q) e—4$'I 0 ~ q' (69) 1 n= no+ —Q @bi(q)+X'bo(q)+ ]e-'4', (70) 0 ~ G[n]= g,[n]dr, (64) Here the subscripts refer to the number of gradient Now let us set operators (or the order in 1/rp) and the superscripts to the number of times that n appears to the right of g &"&(n). It may be worth recalling that while g„[n] is an admissible density function in the sense that IJ =go+ "Pi+~ Po+ ' ' ' it differs from the energy density function g,[n], Eq. Collecting terms of order go l(i $2 we find (18), by a divergence. go (no)—iio=0, (71) (72) 3. Identi6cation of the CoeKcients of the Gradient Exyansion —4x 4m ~(q)+ —+gp"+2go "&q' 2 2 +2g4&2&q4+ b((q) =0, (73) q&0, g q'+" ~(q) 2m. (74) Also clearly g g We shall now express the coefficients g„&"&(n) appearing in Eq. (63) in terms of the expansion coef- 6cients, in powers of q, of the electronic polarizability a(q), and similar higher order, nonlinear, response functions. giving We do this by applying our general expression (63) to ~ I ~ ~ )Q (o&- the case of a nearly uniform electron gas, considered b, (q) = 1+. j — ~qo+ already in Sec. II.2. We go, however, beyond (28) and k 4~) (4~1 write n(r)=np+ P b((q)e "'+—P b,(q)e-"'+ . (65) 0 0 Similarly, we obtain pg=o. The linear- and second-, third-, etc., order response functions are then de6ned by the relations bi(q) =~(q)~(q), bo (q)= 2 (i(q(,qo) ~(q()~(qp), f/I b (q) =E q'+" ~(q')~(q —q'). 8x (73) If we now expand the response functions in powers of q, 01+q.2=% etc. —F„[n]—p n(r)dr =0. 8e (67) This gives Now let us compare these expressions with what one obtains with the use of (63). We require that n(q) = 1+c,q'+c4q'+ ~(q, q') = 2 2 C-'q'"q ", "'j we can identify the functions g„&").Thus gp /4'&I =—co, go &"/4' =, ' ( c4+co'), —— g4 /4Ã= o (—co+2coc4 co ) . (76) (77) (78) (79) (80) n (r') v(r)+ dr'+go' go&'&'(Vn—)' f r—r'/ —2go &'& V'n+3g4&'&'(Von)'+2g4('&" (Vn)'V'n +2g4(o&~Vn ~ V (Von)+2g4(o& (VoVon) +g ( &"(V4n)4+2g "&Vn V'(V'n)' +g4('&(Vo(Vn)' —2Vn V(Von) —2(V'n)') —3g4(4)'(V'n)' —4g &'&V'n(Vn)' —4g &4&Vn V(Vn)' + .-~=0 (68) ~ Vn ~/n&&k, (n) i V,V;ni/i Vni«k, (n). (81) (82) Similarly all other coefficients g„&'&(n) can be expressed in terms of the expansion coe%cients c„of the linear polarizability o,(q) of an electron gas of density n. ln an analogous manner we can express all g„(3) in terms of c((qi) and n(qi, qo); and generally g„&"& in terms of n(qi), n(qi, qo, q„ i). On dimensional grounds we can see from (63) that the gradient expansion requires
B870 P, HOHENBERG AND W, KOHN Both of these conditions are necessary. For while( 81) where r, is the radius of igner-Seitz sphere defined would admit the case of a nearly uniform gas with a by small but short-wavelength nonuniformity similar cases are excluded by(82), as they must be. This expression is believed to be reasonably accurate 4. Partial Summation of Gradient Expansion only for ras1. At lower densities, such as occur in metals (25ras5), various approximate expressions In the preceding section we have expressed the coef- have been proposed. One is due to Wigner ficient gu(2)in terms of the expansion coefficient G, of the polarizability a(a), Eq(76). However, we may apply 2.210.9160.88 the expression(63)to the special case of the gas of almost constant density, discussed in Part II. This shows that the leading term go(n) and the subsequent sub- Other approximations are dr series involving coefficients gu/(n)may be summed to and Pines, 7 and Gaskell ls ue to Hubbard, 16Nozieres 点[n]=B()+/kn)Dm+)-m b.g2)(n) These coefficients are all determined in terms of the [n(r-r)-n(r)]dr+…(83) polarizability, a(a). For this latter quantity vailable, at present, a random- phase e ly from terms of the form of a di (41), which gives or of order in the superscript v of gn(). Here Kn(n(r)=-∑ d (2) The form(83)of g, has the merit of being exact in botl (89) limiting cases where either the density has everywhere 4丌24kr2k2 nearly the same value(see Part II)or is slowly varying 11 Its quantitative value for calculating the electronic structure of actual atomic, molecular, or solid-state 4180kx2k systems is at present uncertain but is being examined. Inclusion of the first of these in the energy expression it will, unlike the simple Thomas-Fermi theory, yield: agrees with a correction to the Thomas-Fermi energy (1)a finite density at the nucleus, and (2)oscillations An expression for a(g), allowing in an approximate in the charge density corresponding to shell structure manner for exchange effects has been proposed by Hubbard. 6 It is 5. Approximate Expressions for the Coefficients q2\.q2 In the previous section we have expressed the coef ficients gu) appearing in the gra udient expansion(63) in terms of properties of the uniform electron gas. We where S(a) is defined in Eq.(43). This form yields now collect some results of existing calculations refer- ring to the uniform electron gas which are useful for 6 ur present purposes. For typical metallic densities this has the opposite sign from the random-phase approximation expression(88 This is the sum of the kinetic +exchange+ce Thus we see that the lowest nonvanishing gradient ce nergy density of a uniform gas of density n ection to the Thomas-Fermi theory depends quite has available the high-density expansion of ge sensitively on refinements in the theory of the electronic polarizability, a(g) o(n)= 2.210916 +002m-006+0()}n,(5) 16 E P. Wigner, Phys. Rev. 40, 1002(19 ondon)A243,336(1957) 1: P. Nozieres and D. Pines, Phys. Rev. lll, 442(1958) 18 T. Gaskell, Proc. Phys. Soc. (London)77, 1182(1961);80, Gell-Mann and K. Brueckner, Phys. Rev. 106, 364(1957). 1091(1962
P. HOHEN BERG AND AV. KOHN Both of these conditions are necessary. For while (81) would admit the case of a nearly uniform gas with a small but short-wavelength nonuniformity, this and similar cases are excluded by (82), as they must be. 4. Partial Summation of Gradient Expansion In the preceding section we have expressed the coefficient g„(2& in terms of the expansion coefficient c, of the polarizabilityn(q), Eq. (76).However, we may apply the expression (63) to the special case of the gas of almost constant density, discussed in Part II.This shows that the leading term gp(n) and the subsequent subseries involving coeAicients g„("(n) may be summed to yield g,[n-t=gp(n(r))+ A. „(,i(r')[n(r+ —',r') —n(r) j where r, is the radius of the Wigncr-Seitz sphere defined by s4vrr,s= 1/n. (86) Other approximations are due to Hubbard, " Nozieres and Pines " and Gaskell " b. g„"'(n) This expression is believed to be reasonably accurate only for r,(1. At lower densities, such as occur in metals (2&r,&5), various approximate expressions have been proposed. One is due to Wigner" 2.21 0.916 0.88 gp(n)- — —— — n. r.' r, r,+7 8. These coefficients are all determined in terms of the electronic polarizability, n(q). For this latter quantity there is available, at present, a random-phase expression, Eq. (41), which gives y [n(r—-',r') —n(r)]dr'+ . (83) apart possibly from terms of the form of a divergence or of higher order in the superscript v of g ~'~. Here 27( n(q) = 1+ $(q) kg.' kp 2 (88) 1 2' 1 &.(.) (r') =—2 —, . ~—7q ~ r' en(r) (1) (84) and g ('& 1 1 7 4a 24 kc'kI;2 (89) g4(2) 1 1 (90) 4' 18O k„'kI, ' Inclusion of the erst of these in the energy expression agrees with a correction to the Thomas-Fermi energy functional derived by Kompaneets and Pavlovskii. ' An expression for n(q), allowing in an approximate manner for exchange effects has been proposed by Hubbard. " It is The form (83) of g„has the merit of being exact in both limiting cases where either the density has everywhere nearly the same value (see Part II) or is slowly varying. Its quantitative value for calculating the electronic structure of actual atomic, molecular, or solid-state systems is at present uncertain but. is being exaniined. However, it is already clear that if applied to an atom it will, unlike the simple Thomas-Fermi theory, yield: (1) a finite density at the nucleus, and (2) oscillations in the charge density corresponding to shell structure. S. Approximate Expressions for the CoefBcients of the Gradient Expansion In the previous section we have expressed the coefficients g„("i appearing in the gradient expansion (63) in terms of properties of the uniform electron gas. We now collect some results of existing calculations referring to the uniform electron gas which are useful for our present purposes. 2 q2 1 n(q) = 1+— + 5(q) 2 q'+kg' kz' where 5(q) is defined in Eq. (43). This form yields 4x 24 kg 2kp2 k p' (91) (92) gp(n) = 2.21 O.91.6 +0.062 lnr, —0.096+0(r,) n, (85) rs '4 M. Gell-Mann and K. Brueckner, Phys. Rev. 106, 364 (1957). a., gp(n) This is the sum of the kinetic+exchange+correlation energy density of a uniform gas of density m. Here one has available the high-density expansion of Gell-Mann and Brueckner'4; For typical metallic densities this has the opposite sign from the random-phase approximation expression (88). Thus we see that the lowest nonvanishing gradient correction to the Thomas-Fermi theory depends quite sensitively on refinements in the theory of the electronic polarizability, u(q). "E.P. Wigner, Phys Rev. 40, 10.02 (1934). "J.Hubbard, Proc. Roy. Soc. (London) A243, 336 (1957). z7 P. Nozieres and D. Pines, Phys. Rev. 111,442 (1958). "T.Gaskell, Proc. Phys. Soc. (London) 77, 1182 (1961); 80, 1091 (1962)
INHOMOGENEOUS ELECTRON GAS IV CONCLUDING REMARKS inhomogeneous electron gas and may suggest further In the preceding sections we have developed a theory of the electronic ground state which is exact in two ACKNOWLEDG MENTS limiting cases: The case of a nearly constant density This work was begun and, to a considerable extent (n=no+n(n), n(r)/nu<<1) and the case of a slowly varying density. Actual electronic systems do not belong authors(P. Hohenberg) acknowledges with thanks a to either of these two categories. The most promising NAto Postdoctoral Fellowship; the other author(w formulation of the theory at present appears to be that Kohn)a Guggenheim Fellowship. Both authors wish obtained by partial summation of the gradient expan- to thank the faculties of the Ecole Normale Super sion (Sec. III. 4). It has, however, not yet been tested Paris, and the Service de Physique des Solides, Orsay in actual physical problems. But regardless of the out- for their hospitality, and Professor A. Blandin, P come of this test, it is hoped that the considerations of J. Friedel, Dr. R. Balian, and Dr. C. De D this paper shed some new light on the problem of the for valuable discussions PHYSICAL REVIEW VOLUME 136. NUMBER 3B 9 NOVEMBER 1964 Scattering of a High-Intensity, Low-Frequency Electromagnetic Wave by an Unbound Electron ZOLTAN FRIEDt U.S. Noval ordnance Lo and Universily of California, Santa Barbara, california JoSEPH H. EBERLY U. S. Naval Ordnance Laboralory, Silver Spring, Maryland Received 15 June 1964) Thomsonscattering of a high-intensity, low-frequency, circularly-polarized electromagnetic wave by a free electron is considered. We find that by neglecting radiative corrections and pair effects, the Feynman Dyson perturbation expansion is summable, and the sum can be analytically continued in the form of a sum of continued fractions. By imposing the boundary conditions that at he photons and target electron opagate as free particles, we obtain results which differ from those reported by Brown and Kibble and by ldman. In particular our results differ in two aspects. The first difference is in the kinematics; nd Kibble and of Goldman using a mixed set of classical and quantum boundary valr I INTRODUCTION second area of concentration is the question of proper lE advent of masers and lasers has stimulated a description of the electromagnetic radiation emanating great deal of interest in the interaction of intense from a laser; i.e. questions of coherence and correla electromagnetic fields with matter. This activity has tion. And finally, the problem of interaction of laser been focused on three different aspects of the subject. light with matter has attracted considerable interest First, a great deal of attention has been devoted to the It is this latter question to which we are devoting our dynamics of production of high-intensity light. 1 A selves in this paper The particular problem of immediate interest is the A preliminary version of this work was presented at the effect of the presence of the high-intensity field on the Compton(Thomson) scattering amplitude. Recall that ological Instit Massachusetts; on leave from the U. S. Nava rell, the Thomson amplitude describes the scattering of a Council R Glauber, Phys. Rev. 130, 2529(1963); E C G, Sudarshan, 10,277(196 hysics, Varenna, 19(3(unpublished) 8(1963
I NHOMOGENEOUS ELECTRON GAS IV. CONCLUDING REMARKS In the preceding sections we have developed a theory of the electronic ground state which is exact in two limiting cases: The case of a nearly constant density (Is=np+rI(r), rI(r)/ep((1) and the case of a slowly varying density. Actual electronic systems do not belong to either of these two categories. The most promising formulation of the theory at present appears to be that obtained by partial summation of the gradient expansion (Sec. III.4). It has, however, not yet been tested in actual physical problems. But regardless of the outcome of this test, it is hoped that the considerations of this paper shed some new light on the problem of the inhomogeneous electron gas and may suggest further developments. ACKNOWLEDGMENTS This work was begun and, to a considerable extent, carried out at the University of Paris. One of the authors (P. Hohenberg) acknowledges with thanks a NATO Postdoctoral Fellowship; the other author (W. Kohn) a Guggenheim Fellowship. Both authors wish to thank the faculties of the Ecole Normale Superieure, Paris, and the Service de Physique des Solides, Orsay, for their hospitality, and Professor A. Blandin, Professor J. Friedel, Dr. R. Balian, and Dr. C. De Dominicis for valuable discussions. PHYSICAL REVIEW VOLUME 136, NUM BER 3 B 9 iVOVEM B ER 1964 Scattering of a High-Intensity, Low-Frequency Electromagnetic Wave by an Unbound Electron* ZOLTAN FRIED) U. S. iVaval Ordnance Laboratory, Silver Spring, Maryland und Univ' sity of California, Santa Barbara, California AND IosEPII H. EszRLvf. U. S. Naval Ordnance Laboratory, Silver Spring, Maryland (Received 15 June 1964) "Thomson" scattering of a high-intensity, low-frequency, circularly-polarized electromagnetic wave by a free electron is considered. We find that by neglecting radiative corrections and pair e6ects, the FeynmanDyson perturbation expansion is summable, and the sum can be analytically continued in the form of a sum of continued fractions. By imposing the boundary conditions that at t=&~ the photons and target electron propagate as free particles, we obtain results which differ from those reported by Brown and Kibble and by Goldman. In particular our results dier in two aspects. The 6rst difference is in the kinematics; namely, we find no intensity-dependent frequency shift in the scattered photon. The second difference is in the dynamics; that is, we obtain a different expression for the scattering amplitude. Both of these changes originate in the choice of boundary conditions. Instead of treating the asymptotic radiation 6eld classically, we choose our states as linear combinations of occupation-number states. Finally, contact is made with the results of Brown and Kibble and of Goldman using a mixed set of classical and quantum boundary values. I. INTRODUCTION 'HE advent of masers and lasers has stimulated a great deal of interest in the interaction of intense electromagnetic 6elds with matter. This activity has been focused on three different aspects of the subject. First, a great deal of attention has been devoted to the dynamics of production of high-intensity light. A *A preliminary version of this work was presented at the Pasadena Meeting of the American Physical Society, Bull. Am. Phys. Soc. 8, 615 (1963). f Present address: Lowell Technological Institute, Lowell, Massachusetts; on leave from the U. S. Naval Ordnance Laboratory. $ National Academy of Sciences—National Research Council Postdoctoral Research Associate, 1962-64. ' J. R. Singer, %users (John Wiley R Sons, Inc., New York, 1900); F. Schwabl and W. Thirring (to be published); W. E. Lamb, Jr., Lecture Notes, Enrico Fermi International School of Physics, Varenna, 1963 (unpublished). second area of concentration is the question of proper description of the electromagnetic radiation emanating from a laser; i.e., questions of coherence and correlation. ' And finally, the problem of interaction of laser light with matter has attracted considerable interest. ' It is this latter question to which we are devoting ourselves in this paper. The particular problem of immediate interest is the effect of the presence of the high-intensity field on the Compton (Thomson) scattering amplitude. Recall that the Thomson amplitude describes the scattering of a ' R. Glauber, Phys. Rev. 130, 2529 (1963);E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963); E. Wolf, Proc. Phys. Soc. (London) 80, 1269 (1962). ~ J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962); Z. Fried s.nd W. M. Frank, Nuovo Cimento 27, 218 (1963)
