Supplemental Material for
“Plasmon Hybridization Model Generalized to Conductively Bridged Nanoparticle Dimers”
Lifei Liu1,2, Yumin Wang1,3, Zheyu Fang1,3,4, and Ke Zhao1,3*
1. Laboratory for Nanophotonics, Rice University, Houston, Texas 77005, United States
2. Department of Physics and Astronomy, M.S. 61, Rice University, Houston, Texas 770051892, United States
3. Department of Electrical and Computer Engineering, M.S. 366, Rice University, Houston,
Texas 77251-1892, United States
4. School of Physics, State Key Lab for Mesoscopic Physics, Peking University, Beijing
100871, China
*Corresponding author: Ke Zhao
Email: kz4@rice.edu
In the spherical coordinate systems of nanoparticle i (i =1, 2) with the origin at its center Oi, we
use ri, θi, and φi to refer to the radial distance, polar angle, and azimuthal angle of a point in
space. In addition, when θi and θj (i ≠ j) refer to the same point P on the surface of nanoparticle j,
θi is a function of θj and written as θi = Θi(θj), and Xi(θj) is used to denote the distance between Oi
and point P.
I. Self-interacting Energy of Surface Charge
For small dimers with a negligible retardation effect, the interaction energy between surface
charges Vint obeys the instantaneous Coulomb law:


 i (r1 ) i (r2 )
1
1
2
2
Vint   
 a d1  a d 2 | r  r |
2 i 1, 2 4 0
1
2
,


1
 1 (r1 ) 2 (r2 )
2
2

 a d1  a d 2 | r  r |
4 0
1
2
(S2)
where ε0 is the vacuum dielectric constant. The explicit form of Vint is:
(n0 e) 2 a 5
l2
2l
2
Vint 
{  [
S l( i ) 
( yl( i )  Vl( i ) ) T S l( i ) ]
i 1, 2 l
2 0
2l  1
2l  1

1
4ll '
2
(  V ) T  
Vll ' S l(1) S l('2 ) }
ll ' 2l '1
2
,
(S3a)
1
where

yl( i )  2 0 K iYl 0 cos i sin i d i
a l 1 cos jYl 0 (i )
Kj
sin j d j (i  j ) ,
X il 1 ( j )

Vl  2 0
(i )
2
0
0
  2  0  '0  0
2

V  2   0 

(S3b)
2
cos cos ' sin sin '
dd ' d
2  2 sin sin ' cos  2 cos cos '
(S3c)
(S3d)
0
 d d d
   0  1  0
1
2
cos1 cos 2 sin 1 sin  2
2  2 sin 1 sin  2 cos   2 cos1 cos 2  2(2  D / a )(cos1  cos 2 )  (2  D / a ) 2
(S3e)
a l 1
(S3f)
Vll '  0 Yl 0 (1 ) l 1
Yl '0 ( 2 ) sin1d1
X 2 (1 )
b
K1 , K 2  1 when 0  1  0 ( 0  arcsin ) and    0   2   , and K1 , K 2  0
a

anywhere else.
II. Equations for Plasmon Modes Derivation
Defining Lmax as the maximum order of multipoles included in the calculation, the equations of
motion for plasmon modes can be derived from lagrangian

 d 2S  
I  2 V  S  0,
dt


where I is the (2Lmax  1)  (2Lmax  1) unit matrix, V
matrix with the following elements ( l , l '  Lmax ):


Vl ,l '  Vl  L

Vl ,l '  L
max

max
, l '  Lmax
  ll '
2l '
Vll ' p2
2l '1
l
 p2
2l  1
L  T  Vint :
(S6a)
is a
(2Lmax  1)  (2Lmax  1)
(S6b)
(S6c)
2

Vl  L
max , l '

Vl , 2 L

Vl  L

V2 L

V2 L

V2 L
and
max
max
max
max
max
1

2l '
Vl 'l  p2
2l '1
yl(1)  Vl(1) 2

p
2l  1
, 2 Lmax 1
1, l
yl( 2 )  Vl( 2 ) 2

p
2l  1
a 3 ( yl(1)  Vl(1) )l 2
 2

b D (2l  1) p
1, l  Lmax
1, l  Lmax
1
(S6f)
(S6g)
(S6h)
a 3 (  V ) 2

p
2 2b 2 D
(S6i)
max
max
(S6e)
a 3 ( yl( 2 )  Vl( 2 ) )l 2


b 2 D (2l  1) p

S is a 2Lmax  1 column array with elements:

S l  S l(1)

Sl  L  Sl( 2 )

S2 L
(S6d)
T
(S6j)
(S6k)
(S6l)
The corresponding eigenvalue problem is:
 
det[V  I  2 ]  0 ,
(S6m)
where the square root of eigenvalues are the plasmon frequencies and the eigenvectors composed
of
Sl(1) , S l( 2 )
and
 T give the charge distribution of plasmon modes through Eq. (8) in the main
paper.
III. Equations for Optical Response
L  T  (Vint  Vext ) and in addition, the equation of motion
includes the dissipation term derived from F (see Eqs. (13), (14) in the main paper), resulting in
To calculate the optical response,
the linear response equations:
3
where


 d 2 S  dS   
I  2  
V  S   ,
dt
dt

 

S , I , and V are the same as those in section II, 
(S7a)
is a
(2Lmax  1)  (2Lmax  1)
matrix with nonzero elements only on the diagonal:


 l ,l   l  L
max
, l  Lmax

 2L
and
max
1, 2 Lmax 1


0
gT
0
g0
 p2
(S7b)
 p2
(S7c)

 is a 2Lmax  1 column array with elements:


l  l  L

2L
max 1

max
  l1
e 4
E0 e it
me a 3
e
E0 e it
me a
Solving Eq. (S7a) for
Sl(1) , S l( 2 )
(S8a)
(S8b)
and  , the polarizability can be obtained through Eq. (16) in
T
the main paper.
In both the plasmon modes and the optical response calculations, the most time-consuming part
is to calculate the different components of
dimensional integrals.

V , because the calculation involves multi-
4