Physical meaning of natural orbitals and natural occupation numbers - - PowerPoint PPT Presentation

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Physical meaning of natural

• rbitals and natural
• ccupation numbers

13.04.2016

• N. Helbig

Forschungszentrum Jülich

Member of the Helmholtz-Association

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|ΨjΨj| 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

• rbitals 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

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 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

0,2 0,4 0,6

1st nat. orb.

b=0.0 b=0.5 b=0.9 b=1.0

• 0,6
• 0,4
• 0,2

0,2 0,4 0,6

2nd nat. orb.

b=0.01 b=1.3

• 10

10

x (a.u.)

• 0,4
• 0,2

0,2 0,4

3rd nat. orb.

b=1.5 b=3.0 13.04.2016

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Occupation numbers

0.5 1 1.5 2 2.5 3

Interaction strength (a.u.)

0.5 1 1.5 2

• Occ. number/entropy

n1 n2 n3 s

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Excited state

• 15
• 10
• 5

5 10 15

x (a.u.)

• 0.4
• 0.2

0.2 0.4 0.6

1st nat. orb.

b=0.01 b=0.5 b=0.9

• 15
• 10
• 5

5 10 15

x (a.u.)

• 0.4
• 0.2

0.2 0.4 0.6

2nd nat. orb.

b=1.0 b=1.3 b=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

• ccupation 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

0.2 0.4 0.6 0.8

1st nat. orb.

b=0.0 0.2 0.4 0.6 0.8

• 15
• 10
• 5

5 10 15

x (a.u.)

• 0.6
• 0.3

0.3 0.6

2nd nat. orb.

b=0.01 b=0.5 b=0.9

• 15
• 10
• 5

5 10 15

x (a.u.)

• 0.6
• 0.3

0.3 0.6 b=1.0 b=1.3 b=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

5 10 15 20

x

0.2 0.4 0.6 0.8

KS orb. d=7.0 d=11.0 d=13.0 d=15.0

0.2 0.4 0.6 0.8

1st nat. orb. d=1.0 d=3.0 d=5.0

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6

2nd nat. orb.

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Molecular dissociation

2 4 6 8 10 12 14

Distance (a.u.)

0.5 1 1.5 2

Correlation entropy 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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