On the Total Position Spread Tensor, bold0mu mumu , in some … · 2020. 9. 2. · 2) x Ground...

Introduction Total Position Spread Tensor, Λ Our Results/Contribution Outlook

On the Total Position Spread Tensor, Λ, insome model systems

M. El Khatib1 S. Evangelisti1 T. Leininger1

1Laboratoire de Chimique et Physique Quantiques - UMR5626Université Paul Sabatier. Toulouse, France.

Group seminars @ AG-PaulusFreie Universität Berlin

October 2013

On the Total Position Spread Tensor, Λ M. El Khatib


Outline

1 Introduction

2 Total Position Spread Tensor, ΛTheoryImplementation in MOLPRO

3 Our Results/ContributionDiatomic molecules (FCI level)Mixed-valence systems



Introduction

• Metal and Insulators. Distinction proposed by Bethe,Sommerfeld and Bloch (1928).

• What is a metal-insulator transition?The process to change from a metal state to an insulatorone.

• Why to study them?

* Yet one of the fundamentally least understood problems incondensed matter physics.

* Electronic components are getting into the low dimensionalregime.



Introduction

• Why is it important?- Many applications: eg.: microelectronics, molecular

electronics.- Physical and chemical properties changes dramatically in

such transitions.• Why is the MIT a difficult problem to study?

- A good metal and a good insulator are very differentphysical systems.



Theory

Outline

1 Introduction





Theory

• The “Localization Tensor” (LT) is a quantity introduced inthe context of the theory of Kohn (1964) to characterize theelectrical conductivity properties.

• Kohn realized that the electrical conductivity is morerelated to a properly delocalization of the wavefunction.

• Resta and coworkers (1999), with the introduction of thelocalization tensor, provided an important tool to give aquantitative formulation of this localization (a per electronquantity).

λ2βγ =

< r̂β r̂γ >c

n(1)



Theory

• In the molecular studies, the full quantity (not divided by thenumber of electrons) seems more appropriate since one isinterested in the analysis of the molecular wavefunction.(Ángyán, Int. J. Quant. Chem. 2009, 109, 2340)

Λ = < r̂β r̂γ >c (2)

• What is a cumulant?A combination of moments that vanishes when somesubset of the variables is independent of the others.(R. Kubo, J. Phys. Soc. Jpn., 17, 1100 (1962))



Theory

On the cumulants (1)

This mathematical object has many interesting characteristics:

• They can be explicitly represented only by moments oflower or equal order.

Order Moment Cumulant1 < r̂β > < r̂β >2 < r̂β r̂γ > < r̂β r̂γ > − < r̂β >< r̂γ >3 < r̂β r̂γ r̂δ > < r̂β r̂γ r̂δ > − < r̂δ >< r̂β r̂γ >

− < r̂β >< r̂δ r̂γ > − < r̂γ >< r̂β r̂δ >+2 < r̂β >< r̂γ >< r̂δ >

• They are a linear combination of the products of thesemoments of lower or equal order so they are additive (sizeconsistency).



Theory


• The trace of the tensor is rotational invariant, for cumulantsof second order.

• A diagonal element of the cumulant tensor is the variance,and it is always a positive value.

• They also are invariant under translation of the origin ofcumulants of order ≥ 2.



Implementation in MOLPRO

Outline

1 Introduction






The position spread (1)• The total position operator:

r̂β =n∑

p=1

β̂(p) (3)

• We also consider its tensorial square.

r̂β r̂γ =n∑

p,q=1

β̂(p)γ̂(q) (4)

• The cumulant of the quadratic fluctuation of the position is:

〈r̂β r̂γ〉c =< Ψ|r̂β r̂γ |Ψ > − < Ψ|r̂β|Ψ >< Ψ|r̂γ |Ψ > (5)




The position spread (2)

The implementation done in MOLPRO that calculates the TPSon a CASSCF wavefunction needed:

• The bielectronic operator, r̂β r̂γ , is evaluated with thesecond order reduced density matrix.

• The monoelectronic operators are evaluated with the firstorder reduced density matrix.

• We validated our implementation by comparing our resultswith the implementation of Λ already made in NEPTUNUS.




The position spread (3)

• What were our motivations?• Study bigger molecular systems.• At CASSCF level, MOLPRO treats very well situations

where state averaged calculations are needed.• What our implementation calculate is:

• The total position spread, Λ.• It prints the different contributions: core-core-, active-core,

active-active.



Diatomic molecules (FCI level)

Outline

1 Introduction






Covalent systems (1)

(a) H2(7s3p3d3f ) (b) N2(3s2p)

Figure 1 : Total Position Spread, Λ, for the covalent systems (Λ‖, redfull line; Λ⊥, green line/circles; Re equilibrium distance).




Weakly bounded systems

(a) Be2 (7s7p4d3f ) (b) BeH– (7s7p4d3f − 6s4p3d)

Figure 2 : Total Position Spread, Λ, for the molecular anions andweakly bonded systems (Λ‖, red full line; Λ⊥, green line/circles; Reequilibrium distance).




Ionic systems (1)

Figure 3 : Potential energy surface for LiF (GS, 11Σ: red full line; ES,21Σ: green line/circles) (3s2p).




Ionic systems (2)

(a) TPS, Λ⊥(GS, 11Σ: red full line; ES,21Σ: green line/circles

(b) TPS, Λ‖(GS, 11Σ: red full line; ES,21Σ: green line/circles)

Figure 4 : Total Position Spread, Λ, for the LiF molecule.




Take home messages about diatomic molecules:

• The TPS tensor is a powerful indicator of the electronicrearrangements in a molecular wavefunction.

• When a bond is stretched from its equilibrium distance, thelongitudinal component of the TPS tensor increasesrapidly. Once the bond is broken, the TPS decreasesabruptly to the atomic values.

• Possible application of this technique to the study ofmixed-valence systems (?).



Mixed-valence systems

Outline

1 Introduction






H+4 model system (1) (FCI level)

D

R2 R1

Figure 5 : The H+4 model structure (Submitted to JCC)

• The average geometry: R(ξ,D) ≡ {R1(ξ,D),R2(ξ,D)}

R1(ξ,D) = (1/2− ξ)R+(D) + (1/2 + ξ)R0(D) (6)

R2(ξ,D) = (1/2 + ξ)R+(D) + (1/2− ξ)R0(D) (7)





Figure 6 : The potential energy curves of the two lowest states as a functionof the mixing parameter ξ, for a fixed intermolecular distance D = 6.0 bohr.





0.95

1.00

1.05

1.10

1.15

1.20

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Λxx (

bohr2

)

ξ

Ground stateExcited state

Figure 7 : The xx component of the Localization-Spread Tensor for the twolowest states as a function of the mixing parameter ξ, for a fixedintermolecular distance D = 6.0 bohr.





0.65

0.70

0.75

0.80

0.85

0.90

0.95

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Λyy (

bohr2

)

ξ

Ground stateExcited state

Figure 8 : The yy component of the Localization-Spread Tensor for the twolowest states as a function of the mixing parameter ξ, for a fixedintermolecular distance D = 6.0 bohr.





0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

-1.0 -0.5 0.0 0.5 1.0

Λzz (

bohr2

)

ξ

Figure 9 : The zz component of the Localization-Spread Tensor for the twolowest states as a function of the mixing parameter ξ, for a fixedintermolecular distance D = 6.0 bohr.




Take home messages about H+4:

• It shows the character of mixed-valence compounds ofClass II.

• Even being a simple system, it presents many features ofrealistic and much larger systems.

• The TPS shows a marked maximum in the transition-stateregion where the charge is very delocalized.




The spiro molecule (1) (CASSCF level)

Figure 10 : The structure of the Spiro π-σ-π system5, 5′(4H, 4H ′)-spirobi[cyclopenta[c]-pyrrole]2, 2′, 6, 6′ tetrahydro molecule.CAS(11,12); state averaged calculation. (submitted to Chem. Phys. Lett.).

• The mixing parameter employed:

Q(ξ) = (ξ)QA + (1− ξ)QB (8)





-609.270

-609.268

-609.266

-609.264

-609.262

-609.260

-609.258

-609.256

-609.254

-609.252

-609.250

0.0 0.2 0.4 0.6 0.8 1.0

Energ

y (

hart

ree)

ξ

Ground state (Roos)Excited state (Roos)

Figure 11 : Potential energy surfaces of the Spiro cation for the groundstate and first lower excited state (3s2p1d) using a CAS(11,12).





56.4

56.4

56.4

56.5

56.5

56.6

56.6

56.7

0.0 0.2 0.4 0.6 0.8 1.0

Λ (

bohr2

)

ξ

Λxx RoosΛyy Roos

Figure 12 : The TPS Λxx and Λyy components represented in bohr2 as afunction of the chosen mixing parameter ξ (GS).





56.5

56.5

56.5

56.6

56.6

56.7

56.7

56.8

56.8

0.0 0.2 0.4 0.6 0.8 1.0

Λ (

bohr2

)

ξ

Λxx RoosΛyy Roos

Figure 13 : The TPS Λxx and Λyy components represented in bohr2 as afunction of the chosen mixing parameter ξ (ES).





60.0

65.0

70.0

75.0

80.0

85.0

90.0

95.0

100.0

0.0 0.2 0.4 0.6 0.8 1.0

Λzz (

bohr2

)

ξ

Ground state, RoosExited state, Roos

Figure 14 : The TPS Λzz component represented in bohr2 as a function ofthe chosen mixing parameter ξ (GS & ES).




Take home messages about the implementation of the TPSin MOLPRO:

• The evaluation of the TPS in bigger systems is nowpossible.

• Averaged orbitals required to describe correctly chargetransfer processes in mixed-valence systems can be doneusing state-avaraged calculations at CASSCF level.



Summary

• The TPS is very sensitive to the change of the WF.• The TPS is able to describe the delocalized nature of the

electrons.

• Outlook• The Implementation of the TPS to be computed on a MRCI

WF.• Application to graphenes.• Study of conducting polymers.• Definition of the Local Position Spread Tensor.

• Download these slides:http://muammar.me/files/chemistry/



Acknowledgment

• University of Toulouse and the Centre National de laRecherche Scientifique (CNRS).

• L’Agence Nationale de la Recherche (ANR) under theaction ANR-11-INTB-1009 MITLOW in collaboration withthe Deutsche Forschungsgemeinschaft (DFG, Germany)[PhD grant].

• Erasmus Mundus program “TCCM Euromaster onTheoretical Chemistry and Computational Modelling” of theEuropean Union (FPA 2010-0147).



Thanks for your attention

“Und wenn du lange in einen Abgrund blickst, blickt derAbgrund auch in dich hinein”

Friedrich Nietzsche



On the cumulants (1)• They are a linear combination of the products of these

lower moments so they are additive (size consistency).

Consider two fragments A and B (well separated and with theirown orthonormal basis set of spin-orbitals).

ΨAB = ΨAΨB (9)

Given that the total position is represented by:

R̂ =n∑

i=1

r̂i (10)

And written in the second quantization as:

R̂ = R̂A + R̂B =∑ijεA

< i |R̂|j > a†i aj +

∑ijεB

< i |R̂|j > a†i aj (11)




The moment generating function is defined by:

F (t) =< Ψ|exp(t ∗ R)|Ψ > (12)

Finally the exponential factorize:

F (t) = FA(t)FB(t); G(t) = GA(t) + GB(t) (13)


On the Total Position Spread Tensor, bold0mu mumu , in some … · 2020. 9. 2. · 2) x Ground...

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