Quantum entanglement between electronic and
vibrational degrees of freedom in molecules
Laura K. McKemmish
School of Chemistry, The University of Sydney, NSW, 2006 Australia
Ross H. McKenzie
School of Mathematics and Physics, The University of Queensland, QLD 4072 Australia
Noel S. Hush
School of Molecular Biosciences and School of Chemistry, The University of Sydney, NSW,
2006 Australia
Jeffrey R. Reimers*
School of Chemistry, The University of Sydney, NSW, 2006 Australia
* to whom correspondence should be sent, email reimers@chem.usyd.edu.au
phone +61(2)93514417 fax +61(2)93513329
SUPPLEMENTARY MATERIAL
S1. Derivation of Eq. (15) showing the origin of fragile entanglement
In the localized limit of 2 J /  1 , the model can be approximated using a basis set
consisting of only the single zero-point vibrational level for each localized oscillator. These
two vibrational levels have a Franck-Condon overlap given by

F00  exp
2 
This approximation ignores vibronic interactions between these levels and the overtone bands
of each localized oscillator. However, the effects of such terms are small in the localized
limit as for example the next interaction is

  
F01  
 exp
2 
 
1/2
but this term is a maximum at   2  and at this value the ratio of the depression of the
ground vibronic level coming from the excluded 0-1 term to that from the included 0-0 term
is
4 2J
e 
which vanishes in the limit of 2 J / 
1.
1
Using this approximation, the vibronic interaction matrix becomes simply
 

H  2
 JF
 00

JF00 


 E0 

2
for which the lowest-energy vibronic eigenfunction is
 0  cos  | 1 01   sin  | 2  01 
where
tan 2 
2 J F00
.
E0
The density matrices for this system are then

2
 cos 
0V
0E
  
 F00 sin 2
 2
F00

sin 2 
2

2
sin  

with eigenvalues of
0
11


1
1  00
2

and
0
 22


1
1  00
2

where
 0   cos 2 2  F00 sin 2 2 
1/2
1/2

 .
1  F00
 1 
 1  E / (2 J F ) 
0
00 

2
S2. Model parameters for ammonia and benzene.
The inversion reaction in ammonia involves vibronic coupling between the nonbonding HOMO orbital and the symmetric NH antibonding orbital.1-3 Two electrons are
involved with this coupling. The localized two-state hamiltonian, Eqn. (3) of the main text,
can be rewritten, with an energy shift added for subsequent simplicity, as
E




T
(Q   )2   J  0
J


2
4
2
H 2L  

E0 

2 

J
T
(Q   )   J 
2
4
2 

 d2
. After a  =45 rotation, an equivalent diabatic representation in terms
2 dQ 2
of delocalized electronic states at E0 =0 can be expressed as
where T  
 2


T
Q
 Q


2
HD Q   
,
 2
  Q
T
Q  2J 
2


4
(see for generalizations to say fully include E0 ). The two delocalized diabatic states are
identified as the ground electronic state and singly excited electronic state at the highsymmetry D3h geometry, and their forms are taken to be independent of distortion. However,
when two electrons are available, a doubly excited state can also be generated that to a zeroth
approximation can be considered as analogously interacting with the singly excited state as
does the ground state. Hence this hamiltonian should be expanded to include the doubly
excited state:
H 3D
 2

T

Q
 Q

2

 2
   Q
T
Q  2J

2


0
 Q




.
 Q


 2
T
Q  4J 
2

0
Rotating this back to the localized diabatic description and reinserting E0 yields
H 3L
E0


2 
T  2 (Q  2 )  2  2 J  2


 2J



0

 2J
T

2
Q2  2J
2J




2J


E0 

2 
T
(Q  2 )   2 J 
2
2
2 
0
When E0 =0 this hamiltonian can be transformed non-diabatically into the block-diagonal
form
3
H 2D

T 




 2


2
Q2
0
T
 Q

2




0


 2
T
Q  4J 
2

2  Q
0
Q2  2 J
0
so that the actual 3-state problem becomes represented using an effective 2-state model
involving renormalized parameters.
We fit the experimental Swalen and Ibers5 potential for ammonia inversion using a
two-state model and so deduce the parameters listed in Table I of the main text. Most
significantly, the critical ratio 2 J /  is invariant to the renormalization, but J and  appear
twice as large as they should be, meaning that  / E takes on half of its proper value.
Also, this last transformation is not diabatic and hence using the effective 2-state model
introduces an approximation to the entanglement that would be manifest in the full 3-state
model.
A similar situation arises for benzene6-10 except it's Kekule distortion mixes the
HOMO and LUMO orbitals, orbital that in this case are both doubly degenerate. Hence 4
electrons are available to create a ladder of excitations from the ground-state to the
quadrupally excited state. A further complexity is that the doubly excited state actually has
three independent components, and here we assume that these can be replaced by a single
component, producing the 5-state delocalized diabatic model:
H 5D
 2

 Q
T  2 Q

 2
  Q
T
Q  2J

2


0
 Q



0
0


0
0


T
0
0
 Q
0

2
 Q
Q2  4J
 Q
0
T

2
Q2  6J
 Q




0


.
0



 Q


 2
T
Q  8J 
2

0
Over the range of the parameter space of interest for benzene's Kekule distortion,
(  Q / J )4 1 and in this limit the adiabatic ground-state and quadrupally-excited-state
surfaces can be represented using the effective two-level hamiltonian
 2


T
Q
 Q


2
H5D  
,
 2
 2  Q
T
Q  8J 


2
with the other states decoupling completely to remain at vertical excitation energies of
energies of 2J, 4J, and 6J and vibration frequency  . However, for the ground state, the
parameters that appear in this effective two-state model are again renormalized versions of
4
the actual physical parameters, with 2 J /  remaining preserved but  / E takes on a
quarter of its proper value. The values given in Table I of the main text were obtained using
the observed single-excitation energy of 4.77 eV and the observed vibration frequencies of
1309 cm-1 in the ground state11 and 1563 cm-1 in the first excited state.6, 12. From the deduced
parameters, the displacement can be determined and mapped back onto deduced diabatic
C=C and C-C bond lengths of 1.31 Å and 1.53 Å in the Kekule structures, respectively.
S3. Comparing the entanglement arising from full 3-state and 5-state models to that
from the corresponding effective 2-state models
Fig. S1 shows the entanglement within the lowest-energy vibronic wavefunctions S0
well as the first-excited state vibronic wavefunctions S1 evaluated over the parameter space
for the two-state (1-electron), 3-state (2-electrons), and 5-state (4-electrons) models. The
results for the 1-electron model are the same as those shown in Fig. 4 of the main text.
Inspection of the figure indicates that the inclusion of additional states introduces no
qualitative changes to the nature of the entanglement across the parameter space, but that
quantitative changes do occur that could be significant in practical situations.
For E0=0, the number of coupled states has only a small influence on the
entanglement in the two lowest vibronic levels, the most significant difference being a
translation of the contours towards slightly higher values of  / E for 2 J /   1.
However, as the molecular parameter  / E for individual molecules is renormalized in a
similar fashion on introduction of an effective 2-state model, this shift has little practical
consequence. Alternatively, for 2 J /   1 only the molecular parameters and not the shape
of the entanglement contours are modified and so for benzene the entanglement coming from
the full 5-state model is actually considerably enhanced, from S0 = 0.011 (effective 2-state
model) to S0 = 0.033 (5-state model). For molecules such as ammonia with 2 J /  1 , the
entanglement is very sensitive to small changes in the parameters and this is found to also
increase considerably, from S0 = 0.44 (effective 2-state model) to S0 = 0.67 (3-state model).
The molecules with the most attractive features for use in some device that exploits
entanglement are those the ones for which extensive treatment of the interactions within the
excited-state manifold are most required.
The effect of very small energy asymmetries E0 on fragile entanglement are amplified
by the 3-state and 5-state models, with the region of persistent entanglement narrowing as the
number of coupled states increases. However, for large asymmetries, inclusion of additional
states had a small effect which enhances the entanglement,
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6
FIG. S1. Entanglement of the vibronic ground-state (S0) and the first-excited state (S1)
wavefunctions vs. 2 J /  and  / E at various values of E0 /  for reactions involving
in frontier orbitals one electron such as: blue- CT, purple- BNB, 3PYR, PRC; two electrons
such as red- ammonia; and four electrons such as green- benzene.
7