ELEMENTS OF RARE EARTH MAGNETISM - Rare Earth Magnetism: Structures and Excitations - page 9

the periodic lattice potential in the solid. Hohenberg and Kohn (1964)
established two important results. Firstly, they showed that the external
potential is a unique functional of the electron density n ( r ), and hence
that the ground-state wavefunction Φ and the energy functional

Φ > + v ext ( r ) n ( r ) d r

< Φ |H| Φ > = < Φ | ( T + U ) | (1 . 2 . 4)

are unique functionals of n ( r ). Secondly, they proved that the energy
functional (1.2.4) attains its minimum value, the ground-state energy,
for the correct ground-state density. Hence, if the universal functional
< Φ

Φ > were known, it would be straightforward to use this
variational principle to determine the ground-state energy for any speci-
fied external potential. However, the functional is not known, and the
complexity of the many-electron problem is associated with its approx-
imate determination.
Guided by the successes of the one-electron model, Kohn and Sham
(1965) considered a system of non-interacting electrons with the same
density as that of the real system, satisfying the single-particle Schro-
dinger equation

|

( T + U )

|

2 + v eff ( r ) ψ i ( r )= ε i ψ i ( r ) .

h 2
2 m ∇

− (1 . 2 . 5)

ThegroundstateΦ S of such a system is just the antisymmetrized prod-
uct, or Slater determinant ,formedfromthe Z lowest-lying one-electron
orbitals, so that the electron density is the sum over these orbitals:

Z

2 .

n ( r )= | ψ i ( r ) | (1 . 2 . 6)

i

The effective potential v eff ( r ) must therefore be determined so that n ( r )
is also the ground-state density of the real system. To accomplish this,
the energy functional (1.2.4) may be written in the form

< Φ |H| Φ > = < Φ S | T | Φ S >

+ 1
2

e 2 n ( r )
| r − r |

d r + v ext ( r ) n ( r ) d r + E xc {n ( r )

(1 . 2 . 7)

},

where the first contribution is the kinetic energy of the non-interacting
system, and the second is the Hartree energy of the charge cloud. The
last term is the difference between the true kinetic energy and that of the
non-interacting system, plus the difference between the true interaction
energy of the system and the Hartree energy. This exchange-correlation