Hohenberg–Kohn theorem

Basic theorems of density functional theory
Hohenberg–Kohn theorem
The Hohenberg–Kohn theorem states that the particle density
uniquely determines the properties of a many–particle system
(Hohenberg and Kohn, Phys. Rev. 136, B864 (1964))
In particular, the ground state total energy is a functional of
the density n(r)
E = E [n]
A functional is a mapping from functions to real numbers
’A function of a function’
Example: Definite integral
(1)
Basic theorems of density functional theory
Proof of the Hohenberg–Kohn theorem
Assume: nondegenerate ground state
The proof can also be extended to degenerate ground states
The external potential Vext uniquely determines the ground
state wave function Ψ as a solution of the Schr¨odinger
equation
The electron density n(r) can in turn be calculated from Ψ
Therefore, n(r) = n[Vext (r)]
Next, let us prove the one–to–one correspondence, i.e. that
there is only one function Vext (r), which yields n(r)
Basic theorems of density functional theory
Proof of the Hohenberg–Kohn theorem
Reductio ad absurdum: Let us assume there is another
potential Vext (r) which yields n(r)
Let us denote
H = T + Uel + Vext ⇒ HΨ = E Ψ
(2)
H = T + Uel + Vext ⇒ H Ψ = E Ψ
(3)
and
According to the variational principle
E = Ψ |H |Ψ < Ψ|H |Ψ ,
because we are assuming E is the ground state energy
(4)
Basic theorems of density functional theory
Proof of the Hohenberg–Kohn theorem
Because
n(r) = N
dr3 · · · drN |Ψ(r1 , r2 , · · · , rN )|2
dr2
(5)
we get
N
Ψ|Vext |Ψ =
∗
···
dr1 dr2 · · · drN Ψ
Vext (r) Ψ =
i=1
N
dri Vext (ri )
···
dr1 · · · dri−1 dri+1 · · · drN Ψ∗ Ψ =
i=1
N
i=1
1
dri Vext (ri )n(ri ) =
N
drVext (r)n(r) (6)
Basic theorems of density functional theory
Proof of the Hohenberg–Kohn theorem
Therefore
Ψ|H |Ψ = Ψ|H − Vext + Vext |Ψ =
(7)
drn(r)[Vext (r) − Vext (r)]
(8)
E+
and
E <E+
drn(r)[Vext (r) − Vext (r)]
(9)
Also, starting from E we get
E = Ψ|H|Ψ < Ψ |H|Ψ = E +
drn (r)[Vext (r)−Vext (r)]
(10)
Basic theorems of density functional theory
Proof of the Hohenberg–Kohn theorem
Since, by assumption n(r) = n (r),
E <E +
drn(r)[Vext (r) − Vext (r)]
(11)
By summing equations (10) and (11) we get
E + E =< E + E ,
(12)
which is a contradiction
There is, therefore, only one Vext (r) which yields n(r)
If we know n(r), we can calculate Vext (r) which yields the
Hamiltonian H which in turn yields all ground state properties
Basic theorems of density functional theory
Variational principle for the particle density
We have proven that if n is the true ground state density, then
the energy functional is minimized:
E [n] = Ψ|H|Ψ < Ψ |H|Ψ = E [n ]
(13)
The variational principle states that a small change in the true
density, δE [n], does not change E , i.e.
δE [n] = E [n + δn] − E [n] = 0
(14)
Due to the conservation of particle number we must have
drδn(r) = 0
(15)
Basic theorems of density functional theory
Variational principle for the particle density
Using Lagrange multipliers µ, we can write the requirement
for the ground state density:
δ E [n] − µ
Define functional derivative
δE [n] ≡
drn(r)
δE [n]
δn(r)
dr
=0
(16)
so that
δE [n]
δn(r)
δn(r)
(17)
The functional derivative is a function of r, it yields the rate
of change of E [n] when n(r) is changed by δn(r) at r
Basic theorems of density functional theory
Variational principle for the particle density
Using (17), equation (16) can be written as
dr
δE [n]
− µ δn(r) = 0
δn(r)
(18)
The above is valid for an arbitrary δn(r) only if
δE [n]
−µ=0
δn(r)
(19)
The above equation is called the Euler equation
It can be used to calculate the electron density, provided the
functional E [n] is known
Basic theorems of density functional theory
Functional derivative, examples
Example1: F [n] =
δF [n] =
drf (n(r)), where f (n) is a function
dr {f (n(r) + δn(r)) − f (n(r)}
δF [n] =
∂f (n)
δn(r) + O(δn2 )
∂n
∂f (n)
δF [n]
⇒
=
|
δn(r)
∂n n=n(r)
dr
(20)
(21)
(22)
Functional derivative equals the partial derivative of function
f (n) with respect to n
Basic theorems of density functional theory
Functional derivative, examples
Example2: F [n] = dr1 dr2 K (n(r1 ), n(r2 )), where K is a
function of two variables. As above we get
δF [n]
=
δn(r)
dr
∂
K (n(r), n(r )) + K (n(r ), n(r))
∂n(r)
(23)
For example, the electrostatic self–energy
Ees =
e2
2
dr
dr
n(r)n(r )
|r − r |
(24)
yields
δEes
= e2
δn(r)
dr
n(r )
|r − r |
(25)
Basic theorems of density functional theory
The energy functional
Define auxiliary functional F [n] = Ψ|T + Uel |Ψ . Now
E [n] = Ψ|T + Uel + Vext |Ψ = F [n] +
drn(r)Vext (r) (26)
Note: F can be defined even if Vext does not exist
This definition provides another way of proving the existence of
the energy functional
Let us separate the average electrostatic energy of the
electrons from F [n] and define functional G [n]
F [n] ≡ G [n] +
e2
2
dr
dr
n(r)n(r )
|r − r |
(27)
Basic theorems of density functional theory
The energy functional
The functional derivative of F is
δF [n]
δG [n]
=
+ e2
δn(r)
δn(r)
dr
n(r )
,
|r − r |
(28)
where the last term is the electrostatic potential caused by the
electron density
The external potential is usually caused by the positive
charges of the nuclei:
Vext = −e 2
dr
n+ (r )
,
|r − r |
(29)
where
Z δ(r − R)
n+ (r) =
R
(30)
Basic theorems of density functional theory
The energy functional
In general, n+ (r) can be e.g. the compensating background
charge in the homogeneous electron gas (the jellium model)
As can be seen from Eqs. (28) and (29), we can define the
total electrostatic potential as
φ(r) = −e
dr
n(r ) − n+ (r )
,
|r − r |
(31)
The Euler equation can then be written as
δG [n]
+ eφ(r) = µ
δn(r)
(32)
Here, φ(r) depends on the density. We yet have to find (or
approximate) G [n]

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