HK Theorem I
For any system of interacting particles in V
ext
, the potential is
deﬁned uniquely, except for a constant, by the ground state
particle density n
0
(
r
)
.
Corollary
: Since the
H
is thus fully determined, it follows that
manybody wavefunctions for all states are determined. Thus
all properties of the system are fully determined by n
0
(
r
)
.
HK Theorem II
A universal functional for the energy E
[
n
]
in terms of the
density n
(
r
)
can be deﬁned, valid for
any
external potential
V
ext
(
r
)
. The exact ground state energy of the system is the
global minimum value of this functional, and the density
which minimizes the functional is the exact ground state
density n
0
(
r
)
.
Corollary
: The functional
E
[
n
]
alone is suﬃcient to determine
the exact
ground state
energy and density.
Notes
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V
ext
(
r
)
Ψ
i
({
r
})
Ψ
0
({
r
})
n
0
(
r
)
HK
I
Usually, the Schrödinger equation is solved by following
the small arrows above, where the potential
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 Winter '12
 Kotakoski
 ground state, Vext, ground state density, HK Theorem II

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