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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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