Download On the leading energy correction for the statistical model of the atom

On the leading energy correction for the statistical model
of the atom using phase space localization techniques
Konstantin Merz and Prof. Dr. Heinz Siedentop, Mathematisches Institut
1
Motivation
3
• We consider the ground state energy E Z of neutral atoms, whose Z electrons are confined to two spatial dimensions. They are described by a
VZ 2 2
Hamiltonian HZ on H := i=1 L (R : Cq).
E := inf hΨ, HZ Ψi
Z
kΨk=1
A popular example would be graphene.
• In the non-relativistic case, the Hamiltonian is given by
!
Z
X
X
Z
1
2
+
.
HZ :=
pi −
|x
|
|x
−
x
|
i
i
j
1≤i< j≤Z
i=1
The leading order of the ground state energy is given by the infimum
of the Thomas-Fermi functional
Z
Z
2π
Zρ(x)
Z
2
ETF[ρ] =
ρ (x)dx −
dx
q
|x|
R2Z Z
R2
1
ρ(x)ρ(y)
+
dx dy
2
|x − y|
Z
ETF
R2 R2
n
o
R R
R
−1
2 2
on I = ρ ∈ L (R ) : ρ ≥ 0 a.e. ,
ρ(x)ρ(y) |x − y| < ∞, ρ = Z .
• The next order is called Scott correction and originates from quantum
effects close to the Coulomb singularity on the length scale Z −1.
!
1
1
2
1
2
2−
Z
,
E = − Z log Z + ETF + cS Z + O Z
2
2
where cS = −3 log(2) − 2γE + 1 ≈ −2.2339 which is derived by using trial
density matrices.
• The leading order Z 2 log Z is semiclassical and originates from the fact,
that the Thomas-Fermi functional is not bounded from below on I. This
forces us to introduce a cut-off at a distance of order Z −1 of the ThomasFermi potential. The other part of the Z 2 correction is semiclassical as
well and comes from the bulk of electrons at distance Z 0 = 1.
2
Thomas-Fermi equation in two dimensions
• Minimizing the Thomas-Fermi functional leads to the Thomas-Fermi equation
4πρ = qΦ+
with the Thomas-Fermi potential Φ = Zr − ρ ∗ |·|1 and the minimizer ρ < I.
• To solve this equation one notes that |·|1 is the Green’s function of the
pseudo-differential operator |p| /2π in two dimensions.
• Applying |p| /2π on both sides of the equation, performing a Fourier transform, solving the algebraic equation and performing the inverse Fourier
transform, leads to the solution of the Thomas-Fermi equation
"
!
!!#
1 qπ
q |x|
q |x|
Φ=Z
−
H0
− Y0
.
2
2
|x| 4
Φ(r)
Z
Macke Orbitals & Hellmann-Weizsäcker functional
• In Hartree-Fock theory, one uses Slater determinants for the wave functions
and replaces the actual Coulomb interaction with a mean-field potential.
For large particle numbers, the trial wave functions are approximated so
that it is not clear that they are all orthogonal to each other and thus,
fulfilling the Pauli principle.
• We define the Macke orbitals with the radial density ρm in the m-th angular momentum channel, which is the actual density times 2πr, in the
following way:
Rr


r


ρ
(t)dt
m
ρm
0



exp (imϕ) exp i2πnr
ψnr,m(r, ϕ) :=
2πrNm
Nm 
R r
ρ (t)dt
2
1
0 m
The map R → S , (x, y) 7→
Nm , ϕ can be understood as a coordinate transformation. One can easily see that these wave functions
are automatically orthogonal to each other in the new coordinates thus,
fulfilling the Pauli principle. By explicitly specifying the wave functions,
one also ensures the quantum-mechanical foundation of the functional including the Weizsäcker correction which can only be introduced ad-hoc in
Thomas-Fermi theory.
• Using the new description of particles, one can compute the energy funcP
tional nr,mhψnr,m, HZ ψnr,mi. Minimizing the sum over nr leads to the
Hellmann-Weizsäcker functional
"
!
#
Z
2
2
X
m
π
Z
√
Z
3
02
EHW =
ρm + ρm + 2 − ρm dr
3
r
r
m
R
Z Z
X
1
ρm(x)ρm0(y)
dx dy.
+
2 m,m0
|x − y|
R2 R2
• Neglecting the Weizsäcker and the electron-self-interaction term, the
Hellmann-minimizer can be obtained to be
√
p
1
1
H
ρm =
r2Φ − m2+ ≡
Φm+.
πr
π
• The non-interacting Hellmann functional is, like the Thomas-Fermi functional, unbounded from below. This is why we regularize Φm in the same
way as we regularized the Thomas-Fermi potential Φ, by introducing a
−1
cut-off at a distance of order Z .
4
Scott correction
• To find an upper bound to the Hellmann-Weizsäcker functional, one uses
Lieb’s variational principle and splits the angular momentum channels. We
use Bohr orbitals for small and the Macke orbitals for large angular momentum channels for the energy functional. For large angular momenta we
split the density at two points x1,2(m) which are motivated by the ThomasFermi equation. After a Poisson summation for large angular momenta
one finds with the regularized Hellmann potential ΦR,m, that
!
2
qπ
1
1
1
1 2
Z
2
Z
E ≤ − Z log Z + ET F + Z −
+ −
−
+ const Z 7/4.
2
8
2 2M 4K
10
Here, M ∝ Z α (α < 21 ) denotes the smallest angular
momentum
chanh 1i
nel for which one uses the Macke orbitals, K = Z 2 the largest allowed
1
π2
principal quantum number and 2 − 4 ≈ −1.9674.
• For the lower bound one can neglect the self-interaction. Again, we split
the angular momentum channels and use the hydrogen-like Hamiltonian
for small m. Using the Lieb-Thirring inequality for the trace over the negd2
m2
ative part of the Hamiltonian
H̃
=
−
2 + r 2 − Φ for large m, one finds
m
dr
41 1
that there exists an ∈ 192
, 2 , such that
8
6
4
2
0.2
0.4
0.6
0.8
1.0
r
Fig. 1. Thomas-Fermi potential with q = 2 - near the nucleus it behaves like |x|−1 and far out like |x|−3
• This also shows the scaling behaviour of the electron density ρZ (r) =
ZρZ=1(r) and why the statistical description leads to the Z 2-correction.
∞
X
2
2
1
Z
qπ 2
1
2
+3
2
E ≥
q tr(H̃m)− − const Z ≥ − log Z + ET F Z −
Z −cZ .
2
8
m=0
Z
5
3