Post on 28-Oct-2021
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Physical meaning of naturalorbitals and naturaloccupation numbers
13.04.2016 N. Helbig Forschungszentrum Jülich
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Outline
1 Introduction
2 General properties
Non-interacting electrons
Interacting electrons
Correlation entropy
3 A toy model
Natural orbitals and occupation numbers
Description of excitations
4 Conclusions and Outlook
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Reduced density-matrix functional theory
One-body reduced density matrix
γgs(r, r′) = N
∫
d3r2...d3rNΨ
∗
gs(r′...rN)Ψgs(r...rN)
=∞∑
j=1
njϕ∗
j (r′)ϕj(r)
Ground-state energy
E [γ] = Ekin[γ]+Eext[γ]+EH [γ]+Exc[γ] = E [{nj}, {ϕj(r)}]
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Reduced density-matrix functional theory
Minimize total energy with respect to occupation numbers and
natural orbitals
N-representability conditions
0 ≤ nj ≤ 1,∞∑
j=1
nj = N,
∫
d3rϕ∗
j (r)ϕk(r) = δjk
Ensemble N-representability
∑
j
wj |Ψj〉〈Ψj |
instead of a pure state |Ψ〉 (see talks tomorrow)
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Reduced density-matrix functional theory
Problem
The exact Exc[γ] is unknown
From energy minimization one obtains approximate natural
orbitals and occupation numbers
For exact natural orbitals and occupation numbers one needs
to calculate Ψ(r1, · · · rN)
→ Introduce a one-dimensional model system
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Non-interacting electrons
Slater determinant
Ψ(r1...rN) =1√N!
∣∣∣∣∣∣∣
ϕ1(r1) · · · ϕ1(rN)...
...
ϕN(r1) · · · ϕN(rN)
∣∣∣∣∣∣∣
Density matrix
γ(r, r′) =N∑
j=1
ϕ∗
j (r′)ϕj(r)
Natural orbitals are single-particle orbitals
Occupation numbers are either zero or one
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Non-interacting electrons
Single-particle orbitals satisfy
[
−∇2
2+ vext(r)
]
ϕj(r) = ǫjϕj(r)
Lowest energy states are occupied
ǫ1 ≤ ǫ2 ≤ ... ⇒ nj = 1 for 1 ≤ j ≤ N
nj = 0 for j > N
The same holds for Hartree-Fock theory
(except that vext is replaced by the HF potential)
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Interacting electrons
Many-body wave function
Ψ(r1...rN) =∑
j
cjΦj(r1...rN)
with Slater determinants Φj(r1...rN) and∑
j |cj |2 = 1.
Density matrix
γ(r, r′) =∞∑
j=1
njϕ∗
j (r′)ϕj(r)
No single-particle equation associated to the natural orbitals
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Interacting electrons
Use M natural orbitals to set up the Slater determinants
Minimizes∫
d3r1 · · · d3rN |Ψ(r1 · · · rN)−ΨM(r1 · · · rN)|2
compared to any other set of M orbitals.
Relation between coefficients and occupation numbers
nj =∑
k ,ϕj∈Φk
|ck |2
If nj = 1(0) the corresponding natural orbital appears in all
(none) of the Slater determinants.
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Correlation entropy
Measure for correlation s = −∑∞
j=1 nj log nj
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 0.2 0.4 0.6 0.8 1
-x*log(x)
For non-interacting electrons s = 0.
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Small toy model
One-dimensional system with two electrons
vext(x) = − v
cosh2(x)
For non-interacting electrons
ǫj = −1
8
[√1 + 8v − 1 − 2(j − 1)
]
︸ ︷︷ ︸
>0
2
For v = 0.9: only one bound state
For v = 2.0: two bound states
Interaction
vint(x1, x2) =b
cosh2(x1 − x2)
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Natural orbitals
0
0,2
0,4
0,61s
t nat
. orb
. b=0.0b=0.5b=0.9b=1.0
-0,6-0,4-0,2
00,20,40,6
2nd
nat.
orb.
b=0.01b=1.3
-10 0 10x (a.u.)
-0,4-0,2
00,20,4
3rd
nat.
orb. b=1.5
b=3.0
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Occupation numbers
0 0.5 1 1.5 2 2.5 3Interaction strength (a.u.)
0
0.5
1
1.5
2O
cc. n
umbe
r/en
trop
y
n1
n2
n3
s
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Excited state
-15 -10 -5 0 5 10 15x (a.u.)
-0.4
-0.2
0
0.2
0.4
0.6
1st n
at. o
rb.
b=0.01b=0.5b=0.9
-15 -10 -5 0 5 10 15x (a.u.)
-0.4
-0.2
0
0.2
0.4
0.6
2nd
nat.
orb.
b=1.0b=1.3b=1.5
One natural orbital always unbound.
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Excitations
For b < 1.0 the natural orbitals of the ground state are
localized.
Excited state always has one unbound natural orbital.
First excited state of this system is ionized.
Excitations cannot be described by just changing the
occupation of the ground-state natural orbitals.
This is however what we always do for non-interacting
electrons, even for ionization.
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More bound states, v = 2.0
0
0.2
0.4
0.6
0.8
1st n
at. o
rb.
b=0.0
0
0.2
0.4
0.6
0.8
-15 -10 -5 0 5 10 15x (a.u.)
-0.6
-0.3
0
0.3
0.6
2nd
nat.
orb.
b=0.01b=0.5b=0.9
-15 -10 -5 0 5 10 15x (a.u.)
-0.6
-0.3
0
0.3
0.6b=1.0b=1.3b=1.5
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Excitations
For b < 1.0 the first two natural orbitals of the ground state
and the excited state are localized.
Excitation can be approximately described by just changing
the occupations of the ground-state natural orbitals.
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Excitations
For b < 1.0 the first two natural orbitals of the ground state
and the excited state are localized.
Excitation can be approximately described by just changing
the occupations of the ground-state natural orbitals.
Can excitations be described from ground-state natural orbitals?
It depends, sometimes yes, sometimes no.
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Molecular dissociation
Two potential wells at distance d
vext(x) == − v
cosh2(x − d/2)− v
cosh2(x + d/2)
Interaction
vint(x1, x2) =1
cosh2(x1 − x2)
Interaction decays exponentially with distance.
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Molecular dissociation
-20 -15 -10 -5 0 5 10 15 20x
0
0.2
0.4
0.6
0.8
KS
orb. d=7.0
d=11.0d=13.0d=15.0
0
0.2
0.4
0.6
0.8
1st n
at. o
rb.
d=1.0d=3.0d=5.0
-0.6-0.4-0.2
00.20.40.6
2nd
nat.
orb.
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Molecular dissociation
0 2 4 6 8 10 12 14Distance (a.u.)
0
0.5
1
1.5
2
Cor
rela
tion
entr
opy
n1
n2
s
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Conclusions
Natural orbitals change dramatically from non-interacting to
interacting particles.
Excitations can be described by a change in the occupation
numbers if the two states are similar in their localization.
Occupation numbers provide a measure for correlation.
nj = 0 and nj = 1 give more information on wave function.
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Work done together with...
I.V. Tokatly UPV/EHU, San Sebastián (Spain)
A. Rubio UPV/EHU, San Sebastián (Spain), MPI, Hamburg
(Germany)
References:
Phys. Rev. A 81, 022504 (2010)
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