Oshiyama-Kakenhi@Univ.Tokyo-U.2012

Non-additivity in metallic tri-wire binding
Quantum Monte Carlo study

Ryo Maezono

rmaezono@mac.com

School of Information Science,	Japan Advanced Institute of Science and Technology, 	Ishikawa, Japan.	

vdW in Nano Wire・Diﬃculty of conventional DFT
XC functionals for Dispersion force

・Metallic Wire？

(Diﬀerent Power law)

→ local polarization not well be deﬁned...
(Screening Length)

・Non-additivity
Additive modeling ... valid only for local polarizations
01

Previous Studies
1
Between Insulating wires ~ 5z
1
Metal wires/RPA ~ 232z ( ln z )Metal-Semiconductor/SAPT
HuckelJ.F. Dobson�et.al., PRL96, 073201 (2006)

1~ αz

α =2~5J. Spencer and A. Alavi, (thesis)�

SAPT（TB + Perturbation）

1
Metal/QMC ~ α32z ( ln z )A. Misquitta�et.al., Phys. Rev. B, 82 , 075312, (2010).�

α = α ( rs ) ~ 2
N.D. Drummond et.al., Phys. Rev. Lett. 99, 247401 (2009).�

Much earlier works for 1-dim.
- Coulson and Davies, Trans. Faraday Soc. (1952).�
- Longuet-Higgins and Salem, Proc. R. Soc. A. (1961).�

04Asymptotic Behavior
Density Matrix ;・Local Polarization →Power Decay; Metals

Exponential Decay ; Insulatorsz −6 → Inter-wire Interaction with z −5・Metals/ No local polarization deﬁned...
RPA treatment ; U ( z ) =Valid for Dense GasrS

16π z 2 ⎡⎣ log ( 2.39z b )⎤⎦32

J.F. Dobson�et.al., Phys. Rev. Lett. 96, 073201 (2006)

Dilute region ...

QMC treatment
02

'vdW' in Metallic systems
No local polarization well deﬁned.

1) Zero point motion of the Plasmon
interaction between induced charge by quantum ﬂuctuation
Frequency shift depending on distance (stabilization)δ ⎡⎣ ω p ( z )⎤⎦ ~RPA evaluation
U ( z) =rS

16π z 2 ⎡⎣ log ( 2.39z b )⎤⎦1zα

32

J.F. Dobson�et.al., Phys. Rev. Lett. 96, 073201 (2006)

2) 2nd order Perturbation… shown later …
03

Kink in the exponent
HuckelNote; correlation eﬀect makes kink broader [Misquetta/SAPT, 2010]22

Rs = 10.0

u3 = ( e3 − e1 ) − 2u2

u2 = ( e2 − e1 ) ≤ 0

Cu2,3 ( z ) = αz09

Exponents of DecayRSumming up interactions…
We expect it not diverging.

aρ ; Num. of wires per unit areadN ( r ) = ρ ·2π r·drRREM~ ∫ u ( r )·dN ( r ) = ρ ∫ u ( r )·2π r·dr ；Energy per wireaa
Mu (r ) ~1、R → ∞αr
α =3 ;

R⎡1 ⎤ur·2πr·dr~()∫a⎢⎣ r ⎥⎦a
R→ Finite

α = 2 ; ∫a u (r )·2π r·dr ~ [ ln r ]a → Diverge !
RR15

Exponents of Decay
Possible mechanisms making it ﬁnite.
１）Divergence of u2 cancelled by u3 and higher.
(c.f., Our equilateral triangle gives repulsive u3 )２）Exponent falls oﬀ more rapidly at larger distances.
(at which we cannot perform accurate calculations)

３）Relativistic retardation reducing the interaction
at larger distances.
16

Non-additivity
Interaction due to induced polarizations
Forces not superpose...

Equilateral Triangle Geometry+� +� +�

zx

More remarkable in Metallic wires-� -� -� F12 +� +�+�d-� -� -�

yPossible in Nano Tube systems
J.F. Dobson et.al., Surf. Sci. 601, 5667 (2007).�

＜What are clariﬁed in this work＞
・Non additive contribution well ﬁtted by power law.
・Long-ranged (α< 3), physically important.

05

Modeling
DMC (diﬀusion Monte Carlo) method
・Intra-Wire ; Anti-symetrized products of PW orbitals
1D HEG → No Nodal problem.
・Inter-Wire ; treated as Distinguishable particles
・Odd number of electrons ; Real WF.
N=55, 111

Equilateral Triangle Geometry+� +� +�

zx
Bi-wire interactionu2 = ( e2 − e1 ) ≤ 0

Non-additive contribution

u3 = ( e3 − e1 ) − 2u2-� -� -� F12 +� +�+�d-� -� -�

ye2,3 ; Energy/wire @ Bi (Tri)-wire system
06

Rs = 1.0

u3 = ( e3 − e1 ) − 2u2

u2 = ( e2 − e1 ) ≤ 0

Cu2,3 ( z ) = αz07

Rs = 3.0

u2 = ( e2 − e1 ) ≤ 0

u3 = ( e3 − e1 ) − 2u2Cu2,3 ( z ) = αz08

Rs = 10.0

u3 = ( e3 − e1 ) − 2u2

u2 = ( e2 − e1 ) ≤ 0

Cu2,3 ( z ) = αz09

Exponents

Cu2,3 ( z ) = αzLong-ranged interactions ; α< 310Exponents of Decay
Possible mechanisms making it ﬁnite.
１）Divergence of u2 cancelled by u3 and higher.
(c.f., Our equilateral triangle gives repulsive u3 )２）Exponent falls oﬀ more rapidly at larger distances.
(at which we cannot perform accurate calculations)

３）Relativistic retardation reducing the interaction
at larger distances.
16

Kink in the exponent
HuckelNote; correlation eﬀect makes kink broader [Misquetta/SAPT, 2010]22

2nd order perturbation
Edisp = − ∑(2)n≠0,m≠0Ψ Ψ VAB Ψ ΨAnBmA0BmA0B
02(E + E ) − (E + E )AnB
0inter-molecular interactionsρ (r ) ρ (r )VAB ~ ∫ d 3rA ∫ d 3rB · A  BrA − rBABintegral representation

(AB
1
12 ∞
ω n0
ω m0= A= ∫ dω ·B
AB
AB2
ω n0 + ω m0 π 0En + Em − E0 + E0
ω A +ω2 ωB) ()(( )n0) ((m0) +ω )2218

Longuet-Higgins Representationof dispersion energy  α ( rB′, rB ;iω )α ( rA′, rA ;iω )1(2)3333Edisp=−dω·drdrdrdr·′′   AA∫BB∫∫
02πrA − rB · rA′ − rB′A
B∞

H. C. Longuet-Higgins, Discuss. Faraday Soc. 40, 7 1965 .�

non-local polarizability
AAAAA
ω
Ψ
ρr
Ψ
Ψ
ρ
r
Ψ′()()
 n00AnnA0α ( rA′, rA ;iω ) = ∑2A 2n≠0
ω−iω()
( n0 )
multi-pole expansionltlu111⎛ rA ⎞ ⎛ rB ⎞AABB~τ=Q̂
·T·Q̂ ∑∑⎟⎜⎟t,u ⎜ttuu⎝ R⎠ ⎝ R⎠
rA − rB R t,uR1+lt +lu t,u19

Exponent of decay
Edisp = − ∑ ∑ ∑1(2)n≠0,m≠0 t,u t ′, u′R

AB AB ABTTαi
ωα()t ′u′tt ′uu′ (iω )2+lt +lu +lt ′ +lu′ tu α tAt ′ (iω ) = ∫ d 3rA′d 3rA ·Qˆ t′ Aα ( rA′, rA ;iω ) Q̂tA′AQ̂tA ~ r lt

Induced dipole-dipole int.
(2)
Rank of tensor (lt , lt ′ ) = (lu , lu ) = (1,1) --> Edisp~1R=2+1+1+1+1

1R6

lowest possible contribution(lt , lt′ ) = (lu, lu ) = ( 0, 0)(2)--> Edisp
~1
1= 2
2+0
R
R20

Inv.Sq. contribution(2)Edisp ~ −

1l
3l′
3⎡⎤⎡
⎤r×
ρrdr·r×
ρ
rd′′
()
()
A ⎦A ∫⎣ B
B ⎦ rB′2+2l+2 l ′ ∫ ⎣ ARAB Insulator (localized) ; α ( rA′, rA ;iω ) ~ exp ⎡⎣ −γ rA′ − rA ⎤⎦33drdrA ′A∫γ −1 ~ Lc ; localization lengthA --> ∫ d rA′ ·α ( rA′, rA ;iω ) = 03RA

inv-sq. contribution vanishes

R >> Lc; integral region valid for multi-pole expansion
gets larger --> integral gets disappear

21

Kink in the exponent

Note; correlation eﬀect makes kink broader [Misquetta/SAPT, 2010]22

Exponents of Decay
RSumming up interactions…
We expect it not diverging.

aρ ; Num. of wires per unit areadN ( r ) = ρ ·2π r·drRR
EM~ ∫ u ( r )·dN ( r ) = ρ ∫ u ( r )·2π r·dr ；Energy per wireaa
Mu (r ) ~1、R → ∞αr
α =3 ;

R⎡1 ⎤ur·2πr·dr~()∫a⎢⎣ r ⎥⎦a
R→ Finite

α = 2 ; ∫a u (r )·2π r·dr ~ [ ln r ]a → Diverge !
RR15

Exponents of Decay
Possible mechanisms making it ﬁnite.
１）Divergence of u2 cancelled by u3 and higher.
(c.f., Our equilateral triangle gives repulsive u3 )２）Exponent falls oﬀ more rapidly at larger distances.
(at which we cannot perform accurate calculations)

３）Relativistic retardation reducing the interaction
at larger distances.
16

Summary
Non-additivity of vdW investigatedon 1-dim. metallic tri-wire system.
１）Bi-wire interaction u2
as well as non-additive contribution u3２）Divergence of u2

1
show long-ranged exponent u ~ 2.5z

when we make a bunch of wire.

(not physical, should not occur)

cancelled by u3 and higher？
(importance of non-additive contribution in metallic system)

23