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)<业,Hv)=(v,(H+V-V)业 >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 realizedI 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 ground￾state 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'. As￾sume 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'+ $v (r)—v' (r)]ti (r)dr. Addition of (6) and (7) leads to the inconsistency E+E ~ &E+E~ Thus v (r) is (to within a constant) a unique functional of e(r); since, in turn, v(r) fixes H we see that the full many-particle ground state is a unique functional of rs(r). Ãfm] —= n (r)dr =cV. It is v ell known that for a system of Eparticles, the energy functional of 4' (12) has a minimum at the correct ground state 4, relative to arbitrary variations of 0' in which the number of particles is kept constant. In particular, let 4' be the ground state associated with a diferent external po￾tential v'(r). Then, by (12) and (9) B„L@']= v (r)I'(r) dr+Fc ri'], )8,$+]= v(r)e(r)dr+FLri]. Thus the minimal property of (10) is established rela￾tive to all density functions I'(r) associated with some other external potential v'(r). " If F(1) were a known and sufFiciently simple func￾tional 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 func￾tion. The major part of the complexities of the many￾electron 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 distri￾bution 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