J. Phys. Chem. 1982, 86, 2262-2267
2262
Electron Density Theory of Atoms and Molecules N. H. March Theoretlcel chemisby Deparfment, Unh.ersity of Oxford, Oxford OX1 3T0, England (Received: Ju& 20, 1981; I n Final Form: December 4, 198 I)
The ground-state energy E[p] of a molecule is a functional of the electron density p(r). The energy E is usefully separated into three parts: (i) single-particlekinetic energy, (ii) classical electrostatic energy, and (iii) exchange and correlation energy. Minimization of E[p] with respect to the electron density, subject to the normalization constraint, leads to the Euler equation of the variation problem. This, in turn, expresses the constancy of the chemical potential through the entire electronic charge cloud. Gross “physical”regularities are then discussed by use of the simplest Thomas-Fermi density function(al)for E[p]. These regularities are (a) energy relations in many molecules and (b) scaling properties in tetrahedral and octahedral molecules and molecular ions. Deviations from these gross regularities involve (i) density gradient “corrections”to the Thomas-Fermi theory and (ii) accurate evaluation of the chemical potential. Both of these points will be referred to. In connection with point (ii), the chemical potential will be related to the ionization potential and to electron correlation.
1. Introduction
In this paper, we shall review some consequences of the density functional theory as applied to atoms and to molecules at equilibrium. Although many of the results we focus on are derived from the simplest density description, namely, that afforded by the Thomas-Fermi statistical theory, we shall begin by summarizing the essential framework of the density functional theory. As assumed by Thomas,’ Fermi: and Dirac? and as was formally proved by Hohenberg and Kohn,4 the groundstate energy E of a many-electron system is a unique functional of the number of electrons per unit volume at r,p(r), i.e., E E E [ p ] . It will prove convenient to start from a decomposition of this energy into three parts: (i) single-particle kinetic energy, (ii) classical electrostatic potential energy, and (iii) exchange plus correlation energy. This is effected by writing the ground-state energy as
since U,, by definition does not depend on p. This important equation expresses the constant chemical potential p as a sum of terms which separately vary from point to point in the electronic charge cloud of the molecule. The constancy of p throughout the entire electron distribution’ is a statement that charge flow is complete and that the electron distribution has reached its ground-state equilibrium form. (The writer knows of only one case, the elementary example of N independent particles filling the lowest N levels of a one-dimensional harmonic oscillator well in which Euler eq 1.4 has been so far expressed as an explicit differential equation to determine the density. This example is summarized in the Appendix.) At this point, let us multiply eq 1.4 by p(r) and integrate through the whole of space to obtain 6t Np = Jpsp dr
On the right-hand side of eq 1.1, the single-particle kinetic energy has density t = t [ p ] , the exchange plus correlation energy has energy density e&] which includes correlation kinetic energy, while the remaining three terms are (i) the interaction energy of the charge cloud p with the nuclear potential VN, (ii) the self-Coulomb energy of the charge cloud p , written in terms of its own potential energy V,, and (iii) the bare nuclear-nuclear interaction energy U,. One then forms the Euler equation of the variation problem 6[E - p I v ] / 6 p = 0
(1.2)
where p is introduced as the Lagrange multiplier taking care of the normalization condition
where N is the total number of electrons. This Euler equation can be written (1)L.H.Thomas, h o c . Cambridge Phil. Soc., 23, 542 (1926). (2)E.Fermi, 2.Phys., 48, 73 (1928). Cambridge Phil. Soc., 26, 376 (1930). (3)P.A. M.Dirac, BOC. (4)P. Hohenberg and W. Kohn, Phys. Reo. B, 136,864(1964). 0022-3654/82/2086-2262$01.25/0
+ JpVN
dr
+ SpV, dr + J
p s dr 6P
(1.5) Denoting the electron-nuclear potential energy .fp VN d r by U,, and the electron-electron self-energy 1 / 2 j pV , d r by U,,, we can evidently write eq 1.5 as Np = l p -6 t d r U,, 2U,, + s p 5 d r (1.6)
+
6P
+
6P
A further point we wish to utilize here, before proceeding to the statistical limit of the theory, is that eq 1.4 can be regarded as of the form of a one-electron problem, since t by definition is a single-electron kinetic energy, with potential energyb V(r) = V d r ) + V,(r) + 6tXc/6p (1.7) We can, of course, ask for the one-electron wave functions +i(r)and their corresponding eigenvalues ti of this potential energy (1.7), which consists evidently of a Hartree part, VN + V,, and a part 6exc/6p from exchange and correlation interactions. We can form the density p ( r ) and the eigenvalue sum E, as5 p ( r ) = E. $i*(r) + i h ) (1.8) occupied states
(5) W. Kohn and L. J. Sham, Phys. Reo. A , 140,1133 (1965).
0 1982 American Chemical Society
The Journal of Physical Chemistty, Vol. 86,No. 12, 1982 2263
Electron Density Theory of Atoms and Molecules
E, =
occupied
(1.9)
ci
states
It is clear that an alternative form for E, to that in eq 1.9 can be written as
+ s p V ( r ) dr = st dr + Uen+ 2U, + s
E, = st d r
6%
p s p
d r (1.10)
where, in the second line of eq 1-10, we have used the explicit form (1.7) of the one-electron potential. Clearly, we can use eq 1.10 in eq 1.6 to write immediately Np = s p E d r 6P
+ E, - st d r
(1.11)
Equation 1.11 evidently relates E, - Np to the singleparticle kinetic energy t. Equations 1.1,1.4, and their consequences eq 1.10 and 1.11are the basic equations of formal density functional theory. However, to give physical and chemical content to these equations, it is necessary to specify approximations to the functionals. We turn immediately to the simplest electron density description, from which modern density functional theory has emerged, namely, the Thomas-Fermi statistical theory. 2. Energy Relations in Thomas-Fermi Statistical Theory For reasons which will become clear below, it is useful to separate the single-center atomic problem from the multicenter molecular case. 2.1. Energies in Heavy Atomic Ions. We focus here on the ground-state energy E(ZJV) of a heavy positive atomic ion with atomic number Z and N (52)electrons. In the Thomas-Fermi theory, one omits exchange and correlation, and finds t [ p ] by using Fermi gas theory locally as
(2.1) Evidently the Euler eq 1.4 becomes (2.2) PTF = 5/33~k(p(r))~/~ + V d r )+ Veh) Since it is known that for the neutral Thomas-Fermi atom the charge distribution p(r) is of infinite extent, and since p(r) 0, VN(r) Ve(r) 0 as r 00, it is clear from eq 2.2 that the constant chemical potential for the neutral atom must be zero. For positive ions, on the other hand, we have a finite classically allowed region of radius ro say, and since p(ro) = 0 eq 2.2 yields (Z - N)e2 /.Q&VO = Vdro) + Ve(r)= (2.3) r0 the last step following from Gauss’ theorem for a positive ion. Evidently, eq 2.3 is consistent with pm(Z,Z) = 0 for the neutral atom since in this limit (2- N) 0 and ro m. Equation 1.6 becomes, for heavy positive ions
-
+
- -
-
NP~(ZN =%~,1b(r)1~ d r/ ~+ = 5/3T + Uen+ 2Uee
uen
-+
+ 2uee (2.4)
where we have written T for the total kinetic energy ck$p513 d r in the Thomas-Fermi limit given by eq 2.1. But it was first shown by Focke that in the ThomasFermi theory the virial theorem is precisely obeyed, which (6)V. Fock, Phys. 2.Sowjetunion., 1, 747 (1932).
for atomic ions takes the form T = -E
(2.5)
Hence from eq 2.4 and 2.5 we may write
5/&
=
uen
+ 2uee - N P T ~ Z N
(2.6)
But from the Thomas-Fermi limit of eq 1.11we can write Npm(ZJV) = 5/3T E, - T = -73E +E, (2.7)
+
the last step in eq 2.7 invoking the virial result (2.5). Rewriting eq 2.7 we find7 immediately The limit N
-
E = 3/2[Es- NPTF(ZNI (2.8) Z of eq 2.8, yielding with pTF(Z,Z) = 0
EG,Z) = ?2E,(Z,Z)
(2.9)
was given by March and Plaskett.8 This result, that for heavy neutral atoms the total energy E is 312 of the eigenvalue sum E,, shows, of course, that the “correction” from self-energy of the electron cloud which is included twice in the eigenvalue sum is itself directly related to the total energy E in heavy neutral atoms. Furthermore, for the same limit N = 2, we can write eq 2.4 as (uen
+ 2Uee)/T
5
= -3
(2.10)
and since, from eq 2.5 and E = T + U it follows that (2.11) ( u e n + UeJ / T = -2 the combination of eq 2.10 and 2.11 yields (2.12) The analogue of eq 2.9-2.12 for molecules at equilibrium will be stressed below. To obtain relations 2.9-2.12 for neutral atoms, we have utilized the following: (i) Zpm(Z,Z)
