1
The
I
mpact of
L
igand
F
ield
S
ymmetry on
M
olecular
Q
ubit
C
oherence
Nathanael P. Kazmierczak, Ruben Mirzoyan, and Ryan G. Hadt*
Division of Chemistry and Chemical Engineering, Arthur Amos Noyes Laboratory of Chemical Physics,
California Institute of Technology, Pasadena, CA 91125, United States
*Corresponding Author:
rghadt@caltech.edu
Abstract
Developing quantum bits (qubits) exhibiting room
temperature
electron
spin coherence is a key
goal of molecular quantum information science.
Here
we
develop
a
simple and powerful model
for predicting relative T
1
coherence
times
in transition metal complexes
from dynamic ligand field
principles.
By consider
ing
the excited state origins of
ground state
spin
-
phonon
coupling
, we
derive group theory selection rules governing
which vibrational symmetries can
induce
decoherence
.
Thermal weighting of the
coupling terms
produces surprisingly good
predictions of
experim
ental T
1
trends as a function of temperature and explains previously
confounding
features
in spin
-
lattice relaxation data.
We use this model to evaluate
experimental
relaxation rates across
S =
½
transition metal qubit candidates with diverse
structures,
gaining
new insight
s
into
the
interplay between spin
-
phonon coupling and molecular
symmetry
.
This
methodology
elucidates
the specific vibrational modes
giving
rise to decoherence
, suggesting
symmetry
-
based
design
2
strategies
and
provid
ing
insight
into
the origin of room temperature coherence in transition metal
complexes.
1.
Introduction
The use of paramagnetic transition metal complexes as molecular electron spin quantum bits
(qubits) has generated considerable interest over the past
decade
(
Figure 1
A
)
.
1
–
6
When placed into
a magnetic field, the Zeeman effect splits the energies of the
M
S
sublevels into a quantum two
-
level system that can be leveraged for applications in computing, sensing, and communication
(
Figure 1
B
)
.
2,7
Among these, molecular
quantum sensing constitutes a particular
ly exciting
application,
2
as molecular
electron
spin qubits can be synthetically tuned and
located in a targeted
fashion with
in chemical microenvironments
and interfaces
to read out properties of relevance in
areas such as catalysis and medicine.
T
he microenvironments of interest often exist under ambie
nt
conditions. Thus, developing molecular qubits that operate at room temperature remains a key goal
in the field.
1,8,9
The utility of molecular electron spin qubits is limited by the phase coherence lifetime T
2
,
which describes how long phase relations
are retained
between members of the ensemble
.
10
As
temperature increases in spin
-
dilute environments
, T
2
becomes limited by T
1
, the spin
-
lattice
relaxation time
.
T
1
d
escribes how quickly spin energy is transferred to the vibrational bath.
11
In
solid lattices, this process is
controlled by
spin
-
phonon coupling.
12
Three mechanisms for spin
-
phonon coupling deteriorate the performance of molecular qubits at room temperature, known a
s
the direct, Raman, and Orbach processes (
Figure 1
B
).
10,13,14
The direct process dissipates spin
energy through acoustic phonon emission and exerts the gre
atest contribution at low temperatures
(
e.g.
,
< 10
K).
15
The Raman process dissipates spin energy through inelastic scattering of phonons
3
from a virtual state, with acoustic phonons contributing at intermediate temperatures and optical
phonons (i.e.
,
local modes
6
) dominating at elevated temperatures near ambient conditions.
12,16
In
S
=
½
systems, the Orbach mechanism
generally
does not contribute strongly.
16
R
oom
temperature
coherence lifetimes
of molecular electron spin qubits are controlled by spin
-
phonon coupling with
the molecular vibrational modes.
4,17
Figure 1.
Overview of molecular electron spin qubits. (
A
) V(IV) and Cu(II) qubits considered in
this study.
8,9,18,19
VOPc = vanadyl phthalocyanine; CuPc = copper phthalocyanine;
[Cu(bdt)
2
]
2
-
=
copper bis(1,2
-
benzenedithiolate);
[Cu(bd
s
)
2
]
2
-
= copper bis(1,2
-
benzenediselenlate)
;
[VO(dmit)
2
]
2
-
=
vanadyl bis(1,3
-
dithiole
-
2
-
thione
-
4,5
-
dithiolate);
[V(bdt)
3
]
2
-
= vanadium tris(1,2
-
4
benzenedithiolate); [V(bd
s
)
3
]
2
-
= vanadium tris(1,2
-
benzenediselenate); (
B
) Electronic structure
and relaxation mechanisms of molecular qubits. (Left)
E
lectronic states
(example:
VOPc
)
in
single
-
valued point groups and double groups (Bethe notation) inclusive of spin
-
orbit coupling.
Cha
rge
-
transfer states not shown.
(Right) T
1
relaxation mechanisms.
Atomic color scheme: C
(grey), N (blue), O (red), S (yellow), Se (orange), Cu (brown), V (pink). H atoms not shown for
clarity.
A natural question arises: which vibrational modes exhibit the
strongest spin
-
phonon
coupling?
Because
the Raman process requires thermal population of an existing phonon mode
,
10
vibrational modes higher in frequency than 400 cm
-
1
are not expected to contribute
sign
i
ficantly
to the spin
-
lattice
r
elaxation
. While the phonon density of states and dispersion relation below 400
cm
-
1
can be probed using terahertz spectroscopy
20
and four
-
dimensional inelastic neutron
scattering
,
21
ascertaining the spin
-
phonon coupling of those
modes remains an outstanding
experimental challenge. In lieu of experimental evidence, several studies have sought to assign the
most impactful spin
-
phonon coupling modes through computational studies.
17,22
–
24
There exists an
emerging recognition of the importance of the symmetry of the vibrational mode, with recent
stud
ies
empirically concluding that
gerade
modes exhibit heightened spin
-
phonon coupling over
ungerade
modes
for square plan
ar
compounds
.
22,24
However, no general theory yet exists for
predicting which
vibrational symmetries exert the greatest spin
-
phonon coupling
and
modeling the
imp
lications for
temperature
-
dependent T
1
.
This
hinders rational molecular design and constitutes
an
important
challenge in the field.
6
Here
we
derive
group theor
y
selection rules for determining vibrational modes that are
active for spin
-
phonon coupling.
We show that the
coupling modes are those that are group
5
theoretically allowed to undergo ligand field excited state distortions. These
vibrational modes
dynamically change the amount of ground state orbital angular momentum.
W
e
then
show
that a
simple thermal weighting of the
se
molecular vibrational mode spin
-
p
honon coupling coefficients
furnishes an excellent agreement with experimen
tal spin
-
lattice relaxation
rate
trends
,
thus
describing how different vibrations dominate
T
1
over
different temperature regimes.
2. Results and Analysis
2.
1.
Symmetry effects on spin
-
phonon coupling
.
Spin
-
phonon coupling arises when some
portion of the spin Hamiltonian is modulated by a vibrational mode.
6,25
The
푔
tensor,
품
,
describing
the Zeeman effect
has been implicated as
a
major source of spin
-
phonon coupling
in molecular
qubits
.
15,24
Therefore,
to understand the impact of symmetry on spin
-
phonon
coupling
, we first
turn to the molecular origins of the
푔
values in a transition metal complex.
A free electron has an isotropic
푔
value of
푔
!
=
2
.
0023
owing to its intrinsic spin angular
momentum; deviations from this value arise when the electron addit
ionally possesses
ground state
orbital angular momentum, as quantified by the Land
é
formula. While the presence of a ligand
field quenches orbital angular momentum in
tetragonal
transition metal complexes
, spin
-
orbit
coupling with ligand field excited states reintroduces orbital angular momentum into the ground
state. Thus, changes in the
푔
value arise from changes in spin
-
orbit coupling. In order for
the
i
th
vibrational mode to have a nonzero
first
-
order
spin
-
phonon coupling coefficient
,
휕
품
/
휕
푄
"
, the
magnitude of spin
-
orbit coupling must therefore change as a function of the vibrational mode
coordinate
푄
"
.
The expression for the
푔
value of a transition metal complex due to the spin
-
orbit
perturbation is given by
26
6
푔
"
=
푔
!
−
2
휆
.
/
훹
#
1
푳
3
풊
1
훹
!
4
/
훹
!
1
푳
3
풊
1
훹
#
4
퐸
!
−
퐸
#
!
%
#
(
1
)
where
휆
is the
many
-
electron
spin
-
orbit
coupling constant
,
훹
#
and
훹
!
are the ground and excited
states with energies
퐸
#
and
퐸
!
,
respectively,
푳
3
풊
is an orbital angular momentum operator, and
푖
=
푥
,
푦
,
푧
refer to the
푔
tensor principal axes and the molecular quantization frame,
which are
aligned
for
the
tetragonal qubit
s
considered in this work
. Equation 1 shows that the
푔
values
have a
sensitive
depend
ence
on the
energy gap between the ground and excited states involve
d in the
spin
-
orbit coupling. (The precise excited states involved can be determined from double groups
(
Figure 1
B
)
using
Tables S11
and
S12
and tables of d
-
orbital rotations.
6,27
)
If the ground and
excited state potential energy surfaces reach a minimum at the same value of the vibrational
coordinate
푄
"
, then the energy gap
퐸
!
−
퐸
#
can at most vary quadratically as a function of
푄
"
,
implying
휕
품
/
휕
푄
"
=
0
(
Figure 2
B
)
. However, if the equilibrium geometry of the excited state is
different than
that of
the ground state equilibrium geometry along
푄
"
, the energy gap
퐸
!
−
퐸
#
can
va
ry linearly as a function of
푄
"
and give rise to
휕
품
/
휕
푄
"
≠
0
(
Figure 2
B
)
.
We refer to such modes
as the distorting modes.
27
The first
-
order coupling coefficient
휕
품
/
휕
푄
"
is predicted to exert the
leading influence on spin
-
lattice relax
ation times.
15,24
Therefore, the most important vibrational
modes for spin
-
phonon coupling are
precisely the
se
distorting modes
.
22
7
Figure 2
.
The excited state origins of
ground state
spin
-
phonon coupling.
(
A
) Schematic potential
energy surfaces for
the
푏
&
'
bending mode in
CuPc
. The ground and excited state potential energy
minima coincide, implying
no excited state distortion and thus no linear spin
-
phonon coupling.
(
B
)
Schematic potential energy surfaces for
the
푎
(
#
symmetric stretch in
CuPc
.
The ground and
excited state minima are offset, implying excited state distortion and linear
ground state
spin
-
phonon coupling.
Crucially, the excited state distortion can be
expressed through a matrix element
27
involving vibrational perturbations of the ligand field Hamiltonian
(
퐻
)*
)
:
Δ
푄
"
=
−
@
휓
!+!,
!
B
C
휕
퐻
)*
휕
푄
"
D
-
B
휓
!+!,
!
E
푘
"
(
2
)
8
Here
Δ
푄
"
gives the
excited state distortion
along the vibrational mode
푄
"
,
푘
"
is the force constant,
and
휓
!+!,
!
is the excited state wave
function that spin
-
orbit couples into the
ground
state
.
T
he matrix
element is evaluated at the ground state equilibrium geometry.
The
key
utility of this expression
lies in the application of group theory symmetry selection rules to the integral. The state symmetry
of
휓
!+!,
!
and
푄
"
(
Γ
!+!,
and
Γ
.
!
, respectively)
can be assigned through textbook techniques.
28
The
ligand field
Hamiltonian always has the totally symmetric irreducible representation in the
molecular point group, so the derivative has the symmetry
Γ
.
!
. Therefore, the symmetry of the
integrand
is given
27
by
a
direct
triple product
. For the integral to be nonzero, Equation 3
a
must
contain the totally symmetric irreducible representation
:
(
Γ
!+!,
×
Γ
.
!
×
Γ
!+!,
)
=
푎
(
+
⋯
(3
a
)
Here
푎
(
in Equation 3a
signifies the totally symmetric representation in the desired point group
,
and
t
he excited state is
group theoretically
allowed to undergo distortion when the condition in
Equation 3b
is met
:
[
Γ
!+!,
×
Γ
!+!,
]
=
Γ
.
!
(3
b
)
T
he
square brackets
in Equation 3b
denote the symmetric direct product operation, appropriate for
the product of
Γ
!+!,
with itself
, and
Γ
.
!
represents all mode symmetries that are allowed to
couple
.
27,29,30
This
selection rule
enables
facile
calculation of which vibrational symmetries will
be able to exhibit linear spin
-
phonon
coupling terms
for a given coordination geometry and
9
electronic structure.
This analysis shows that the critical modes to consider are those that are group
theoretically allowed to undergo ligand field excited state distortions.
For nondegenerate states,
o
nly the totally symmetric modes will couple, while other nontotally symmetric modes can couple
for degenerate excited states.
We note that this consideration is
a
more general
basis for
understanding
forces in molecules (i.e.
,
the
Hellmann
-
Feynman force
31
), including those of
relevance for
transition metal photophysics
32,33
and
those predicted by
the Jahn
-
Teller theorem
to
give rise to
the
instability
of
orbitally degenerate states.
29
To illustrate the power of this approach
in understanding spin
-
phonon coupling
contributions to decoherence
in molecular qubits
, we turn to a comparison
between
vanadyl
phthalocyanine (
VOPc
) and copper phthalocyanine (
CuPc
) (
Figure 1
A
).
19
VOPc
belongs to the
non
-
centrosymmetric point group
C
4v
, while
CuPc
belongs to the centrosymmetric point group
D
4h
.
The electronic ground state of
VOPc
has the state symbol
2
B
2
(d
xy
), which spin
-
orbit couples
with the
2
B
1
(d
x2
-
y2
) excited state to introduce orbital angular momentum into
푔
0
. The situation is
reversed in
CuPc
owing to the hole formalism, with a
2
B
1g
(d
x2
-
y2
) ground state and a
2
B
2g
(d
xy
)
excited state. The
relevant
lowest lying
excited state
for
푔
0
is
nond
egenerate
in both cases. Because
the direct product of any nondegenerate irreducible representation
with itself gives the totally
symmetric irreducible representation, Equation 3
b
reduces to
푎
(
=
Γ
.
!
for
VOPc
in order for
휕
푔
0
/
휕
푄
"
≠
0
. An identical analysis holds for
CuPc
, where
푎
(
#
is the
totally symmetric
representation in
D
4h
. Thus, th
e group theory model predicts that the strongest spin
-
phonon
coupling for
푔
0
should arise from totally symmetric vibrational modes
. Indeed, previous
computational studies
have observed that
푎
(
#
or
푎
(
modes exhibit
large
coupling coefficients,
22,24
with
D
2d
CuCl
4
2
-
possessing more spin
-
phonon coupling than
D
4h
CuCl
4
2
-
owing to a greater
number of totally symmetric modes.
22
10
Though
totally symmetric vibrational modes
dominate
푔
0
coupling
for
both
VOPc
and
CuPc
,
the change in point group between
C
4v
and
D
4h
nonetheless
has important consequences for
spin
-
phonon coupling.
CuPc
displays a single
푎
(
#
mode below 400 cm
-
1
corresponding to the
totally
symmetric Cu
-
N stretch
(
Figure 3
A
)
. Owing to the reduced number of irreducible
representations in the C
4v
point group,
VOPc
displays five total
푎
(
vibrational modes below 400
cm
-
1
, encompassing mixtures of both the symmetric stretch and
metal out
-
of
-
plane motion
(
Figure
3B
)
.
The portion of the vibrational density of states which matters for spin
-
phonon coupling is thus
very different
:
CuPc
possesses a lone linear coupling mode at
262
cm
-
1
, while
VOPc
possesses
five
spin
-
phonon active modes
below 400
cm
-
1
(
Table 1
)
.
Calculation of the
휕
푔
0
/
휕
푄
"
coefficients
for
CuPc
and
VOPc
via calibrated density functional theory (DFT)
34
according to a previous
procedure
22
(see also Supporting
Information
, Section 1
)
shows that the totally symmetric
vibrations have the largest coefficients by orders of magnitude, confirming the group theory
analysis
(
Figure 3
C
)
.
The coefficient for
CuPc
is
an order of magnitude
larger than
those for
VOPc
owing to the larger spin
-
orbit
coupling
constant of Cu
(II) relative to V(IV)
.
19
For both
VOPc
and
CuPc
, only a very small portion of the vibrational density of states contributes to spin
-
phonon
coupling
for
푔
0
(
Figure 3A
,
B
)
11
Table 1.
Linear
푔
0
spin
-
phonon coupling
modes for
VOPc
and
CuPc
. All modes have the
totally symmetric representation.
VOPc
CuPc
E (cm
-
1
)
(
휕
푔
0
/
휕푄
)
&
E (cm
-
1
)
(
휕
푔
0
/
휕푄
)
&
4
2
5.5 × 10
-
8
26
2
2.8 × 10
-
5
17
8
1.5 × 10
-
6
26
2
6.3 × 10
-
7
317
2.9 × 10
-
6
39
5
1.9 × 10
-
6
12
Figure 3
.
Impact of symmetry on spin
-
phonon coupling
.
(
A
)
Normalized v
ibrational density of
states (lavender
, left y
-
axis
) and spin
-
phonon
coupling
active vibrations (red
, right y
-
axis
) for
CuPc
. (
B
)
Normalized v
ibrational density of states (lavender
, left y
-
axis
) and spin
-
phonon
coupling
active vibrations (red
, right y
-
axis
) for
VOPc
.
(
C
) Analysis of selected modes for
VOPc
.
Arrows indicate atomic displacements
; additional pictures
are
provided in
T
ables S2
-
S6
.
Symmetry
selection rules
are evaluated for the
2
B
1
(d
x2
-
y2
) excited state (
푔
0
spin
-
phonon coupling)
via Equation 3
b
.
1×10
-
10
constitutes the limit of
numerical
precision.
13
A similar analysis can be performed for
휕
푔
1
/
휕
푄
"
. For both
VOPc
and
CuPc
, orbital
angular momentum is introduced to
푔
1
principally via spin
-
orbit coupling with the d
xz
/d
yz
excited
state
s
,
which are orbitally doubly degenerate and have
the representa
tion
s
2
E in
C
4v
and
2
E
g
in
D
4h
.
Evaluation of Equation 3
b
for
VOPc
now yields
(
푎
(
+
푏
(
+
푏
&
)
=
Γ
.
!
, showing that
푎
(
,
푏
(
, and
푏
&
vibrational modes are able to have
휕
푔
1
/
휕
푄
"
≠
0
by symmetry.
(
푎
&
is produced by the
antisymmetric direct product and is therefore discarded.)
30
Similarly, Equation 3
b
for
CuPc
yields
O
푎
(
#
+
푏
(
#
+
푏
&
#
P
=
Γ
.
!
, showing that
multiple
nondegenerate
gerade
modes are able to couple
for
푔
1
.
Note that the
gerade
selection rule would hold true even if the electronic state symmetry
were
ungerade
, because Equation 3
b
contains the electronic symmetry twice.
While group theory
states which modes are allowed to couple by symmetry, as with any selection rule, this
does
no
t
guarantee a large nonzero coefficient.
27
Comparison
between
the coupling modes for
CuPc
and
[Cu(bdt)
2
]
2
-
(
bdt = 1,2
-
benzenedithiolate
)
illustrates the impact of descending in symmetry from
D
4h
to
D
2h
(
Figure 4
)
.
Lower than
400 cm
-
1
,
CuPc
displays a single active mode with
휕
푔
0
/
휕
푄
"
, the
푎
(
#
symmetric
stretch. Two modes for
CuPc
display nonzero
휕
푔
1
/
휕
푄
"
, including both the
푎
(
#
symmetric stretch
and the
푏
(
#
antisymmetric
stretching mode
.
The presence of
the linearly coupling
푏
(
#
mode is
enabled by the degeneracy of the
2
E
g
electronic state. However, no degenerate irreducible
representations exist in the
D
2h
point group, so the d
xz
and d
yz
orbitals are split into the
B
&
#
and
B
2
#
representations. All electronic states implic
ated in the
푔
1
and
푔
3
spin
-
phonon coupling are
nondegenerate for
[Cu(bdt)
2
]
2
-
, implying that only totally symmetric
푎
#
vibrational modes will
display linear coupling
for all three canonical orientations. Indeed, examination of the spin
-
phonon
coupling coefficients for
[Cu(bdt)
2
]
2
-
shows that the most prominent coupling modes are the same
14
for both
휕
푔
0
/
휕
푄
"
and
휕
푔
1
/
휕
푄
"
and possess
푎
#
symmetry as predicted
(
Figure 4
)
.
The coupling
푏
(
#
mode from
CuPc
correlates to a
푏
(
#
mode in
[Cu(bdt)
2
]
2
-
, implying that the linear coupling of
this
antisymmetric stretch
mode has been turned off by the descent in symmetry. Conversely, the
푏
&
#
in
-
plane scissoring mode in
CuPc
correlates to
푎
#
symmetry for
[Cu(bdt)
2
]
2
-
and is activated
for
푔
0
coupling
. Thus, descent in symmetry from
D
4h
to
D
2h
retains the total number of linear
coupling modes for
푔
1
, but changes the identity of those modes
(
Figure 4
)
.
Similar behavior is
observed for the
C
2v
qubit [VO
(
dmit
)
2
]
2
-
, with many
푎
(
modes exhibiting coupling for both
푔
1
and
푔
0
.
G
lobal molecular symmetry can impact the spin
-
phonon coupling modes even for apparently
similar coordination geometries, a surprising result elucidated by group theory.
This result
establishes control of de
generate electronic excited states as an important design consideration for
controlling activation of spin
-
phonon coupling vibrational modes.
Figure 4.
Orientation
-
dependent spin
-
phonon coupling coefficients for
CuPc
,
[Cu(bdt)
2
]
2
-
, and
[VO(dmit)
2
]
2
-
.
15
A previous study of
two
D
4h
Cu(II) complex
es
empirically
concluded that
gerade
modes
exhibited the strongest coupling
.
24
Our work differs in two important ways. First, the present
approach
provides a predictive group theory analysis not
dependent on a centrosymmetric point
group
. In addition to the
C
nv
point groups consider in this work, this will also
enabl
e
extension of
spin
-
phonon coupling symmetry analysis to qubits with trigonal coordination environments.
35,36
By analogy to
gerade
/
ungerade
,
point groups containing the prime
/
double prime representati
ons
should see coupling only from the single
-
prime vibrational modes, as the double direct product of
the electronic excited state in Equation 3
b
will yield a single
-
prime representation irrespective of
the electronic representation, and the totally symmetric representation will always have a single
-
prime value.
Furthermore, evaluation of Equation 3
b
for the
D
4h
point group reveals that the
푎
&
#
m
ode is not predicted to exhibit linear coupling despite possessing
gerade
symmetry. This
prediction is in agreement both with previous calculations
24
and our own.
Second, a
point of
variance
with the previous study
24
arises over
the role of the degenerate
푒
#
vibrations, which are found to couple in that study, but not predicted to couple by the present
group theory analysis. This is because
the present analysis has considered the spin
-
phonon
coupling coefficients corresponding to the canonical orientations of
the
푔
tensor; namely,
푔
1
,
푔
3
,
and
푔
0
. By contrast,
Santanni et al. averaged all nine
휕
품
/
휕푄
values for the non
-
diagonalized
푔
tensor.
24
Nonzero off
-
diagonal derivatives correspond to dynamic rotation of the principal axes of
the
푔
tensor. Indeed, the
푅
1
and
푅
3
rotation operators transform as
퐸
#
in
D
4h
, and
pictures
of the
푒
#
vibrational modes show that the first coordination sphe
re undergoes a rigid rotation out of the
xy
-
plane
(
Table S
4
)
.
A minimal
square
-
planar coordination environment such as
D
4h
CuCl
4
2
-
does
not posses
푒
#
normal modes
,
22
as these would correspond to
pure
rotational degrees o
f freedom.
In
CuPc
, however, counterrotation of the phthalocyanine ligand framework enables
푒
#
normal