of 57
S
1
Supporting Informatio
n
The Impact of Ligand Field Symmetry on Molecular Qubit
Coherence
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, California 91125, United States
*Corresponding Author:
rghadt@caltech.edu
S
2
Table of
Contents
1. Computational methods.
................................
................................
................................
........................
2
Table S1
................................
................................
................................
................................
...................
3
Figures S1
-
S7
................................
................................
................................
................................
...........
4
2.
T
1
model
................................
................................
................................
................................
..................
6
Figure
................................
................................
................................
................................
.......................
6
3. Choice of spin
-
phonon coupling coefficients
................................
................................
........................
8
Figures S11
-
S18
................................
................................
................................
................................
.......
8
4. Choice of vibrational energy cutoff
................................
................................
................................
....
17
5. Scaling of
T
1
model predictions
................................
................................
................................
...........
19
6. Tabulation of vibrational modes and spin
-
phonon coupling coefficients
................................
........
21
Tables S
................................
................................
................................
................................
..................
21
7. Group theory tables
................................
................................
................................
.............................
39
Tables S1
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................................
................
39
8. Example input files
................................
................................
................................
...............................
41
9. Equilibrium coordinates used for spin
-
phonon coupling calculations
................................
............
47
10. References
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................................
................................
................................
...........
56
1. Computational
m
ethods.
All DFT
spin
-
phonon coupling
(SPC)
calculations were performed in ORCA
1
3
4.2.1 using the B3LYP functional.
4
7
Vanadyl phthalocyanine (VOPc)
and
copper phtha
locyanine (CuPc)
were optimized in the gas phase prior to SPC calculations
8,9
using the def2
-
TZVP basis set for all atoms in
the first coordination sphere
,
the def2
-
SVP basis set for all remaining atoms
, and the def2/J auxiliary basis
set
.
[VO(dmit)
2
]
2
-
was
optimized in the gas phase prior to SPC calculations
8,9
using the def2
-
TZVP basis
set
and the def2/J auxiliary basis set
for all atoms
.
Frequency calculations were conducted using identical
basis sets.
Owing to non
-
negligible crystal packing distort
ions, geometries used for [V(bdt)
3
]
2
-
, [V(bds)
3
]
2
-
, [Cu(bdt)
2
]
2
-
,
and
[Cu(bds)
2
]
2
-
were taken from literature crystal structures
10
and H
-
atom optimi
zed with the
def2
-
TZVP basis set.
SPC calculations were conducted using the ZORA relativistic correction in all cases,
with the ZORA
-
def2
-
TZVP
/
ZORA
-
def2
-
SVP
basis set partitioning for VOPc and CuPc as described
above and the ZORA
-
def2
-
TZVP basis set for
all other compounds
, with an auxiliary basis set of SARC/J
in all cases
.
DFT grid 7 and TightSCF convergence criteria were used throughout, giving a convergence
tolerance of 10
-
8
Hartree.
The % Hartree
-
Fock exchange
(%HFX)
included in the B3LYP functional was
spectroscopically calibrated to experimental
values
(Table
S
1)
, in accordance with
an established
literature
procedure
.
11,12
Smaller values of %HFX are
required
for
accurate modeling of highly
covalent
complexes, including heavy
-
atom ligands and copper complexes.
We d
id
not consider a dependence of the
molecular geometry on temperature. While bond lengths in VOPc crystal structures increase by 0.02 Å
fro
m 150K to 295K,
13,14
this should have a negligible impact on
/
휕푄
that is well within the inherent
errors and approximations of the model.
No symmetry constraints were imposed in ORCA at any point
S
3
during the geometry optimizations, vibrational mode calculations, or spin
-
phonon coupling
calculations.
All irreducible representations in the text are based on manual assignment to the
most approp
riate
point
group.
Table S1: Comparison of experimental and calculated g
values used for %HFX DFT calibration.
Molecule
%HFX
!
"
#
Expt.
Calc.
Expt.
Calc.
Expt.
Calc.
CuPc
38
2.199
2.166
2.052
2.051
2.052
2.051
[Cu(bdt)
2
]
2
-
20
2.085
2.047
2.019
2.015
2.019
2.016
[Cu(bds)
2
]
2
-
20
2.082
2.089
2.018
2.053
2.018
2.031
VOPc
60
1.966
1.961
1.989
1.982
1.989
1.982
[V(bdt)
3
]
2
-
60
1.988
1.963
1.970
1.947
1.970
1.947
[V(bds)
3
]
2
-
20
1.950
1.945
1.955
1.939
1.960
1.935
[VO(dmit)
2
]
2
-
60
1.970
1.951
1.988
1.980
1.986
1.978
Table S1
Vibrational modes were defined according to the dimensionless normal coordinates
1,2,15
given below
16
,
which are directly comparable to previous studies
.
17
$
=
)
$
,
%
&
-
'$
훿푋
'
1
'
(
)
'
*
%
(S1)
These coordinates are implemented in ORCA 4.2.1 through the
%mtr
block, using the
ddnc
variable to
control the scan step size. We conducted all vibrational scans using single point jobs rather than
%mtr
scans
to afford greater control over the initial guess wavefunction. However, the vibrational
steps
used are
identical to those produced by
%mtr
.
Calculations of the
values were conducted over a dimensionless
normal coordinate range from
-
0.15 to +0.15 in 0.05 step increments, with the exception of [V(bdt)
3
]
2
-
,
where numerical noise required stepping from
-
0.75 to +0.75 in 0.25 increments.
/
휕푄
values were
calculated by centered finite difference about the origin (
for example,
/
휕푄
=
(
+
,
.
,.
/
,
.
,.
)
/
(
0
.
10
)
). Principal tensor values are used for
#
,
"
, and
!
, while the non
-
diagonal
tensor was used in
comparisons to
alternative approaches
(
vide infra
).
17
Raw data from the normal coordinate scans is shown
below in Figures S1
-
S7.
Figure S1: CuPc
value scans and fits.
S
4
Figures S1
-
S7
Figure S2:
[Cu(bdt)
2
]
2
-
value scans and fits.
Figure S3: [Cu(bds)
2
]
2
-
value scans and fits.
Figure S4: VOPc
value scans and fits.
S
5
Figure S5: [V(bdt)
3
]
2
-
value scans and
fits.
Figure S6: [V(bds)
3
]
2
-
value scans and fits.
Figure S7: [VO(dmit)
2
]
2
-
value scans and fits.
S
6
2.
T
1
model
.
H
-
atom optimized
crystal structures of
[V(bdt)
3
]
2
-
, [V(bds)
3
]
2
-
, [Cu(bdt)
2
]
2
-
, [Cu(bds)
2
]
2
-
, and
[VO(dmit)
2
]
2
-
were required to accurately describe molecular geometries
for SPC coefficient calculations,
as the square
-
planar Cu(II) complexes undergo tetrahedral distortion under gas
-
phase optimization and the
six
-
coordinate V(IV) complexes exhibit substantial ligand
field distortions from crystal packing.
Consequently, [V(bds)
3
]
2
-
, [Cu(bdt)
2
]
2
-
, and
[Cu(bds)
2
]
2
-
possess negative energy vibrational modes in the
gas phase frequency calculation. These modes are discarded to make the predictions in Figure 6. It is
import
ant to demonstrate that the predictions of the thermally
-
weighted T
1
model are insensitive to these
modes. To accomplish this, we have made equivalent T
1
prediction plots in which all negative energy modes
are assigned an arbitrary frequency of 20 cm
-
1
(Fi
gure
S8
). The results are qualitatively the same as in
Figure 6
the high
-
temperature ordering of all complexes is still preserved correctly, and [V(bdt)
3
]
2
-
and
[Cu(bds)
2
]
2
-
undergo a nearly identical tangent crossover point
under
100
K.
Figure S
8
:
Comparative effect of negative
-
frequency modes on T
1
predictions. (A)
T
1
predictions produced
by discarding negative
-
frequency modes. (B) T
1
predictions produced by setting all negative modes to an
arbitrary frequency of 20 cm
-
1
and including their SPC coefficients in the thermal weighting. (C)
Experimental data reproduced fro
m Ref
10
with permis
sion from the Royal Society of Chemistry
. Panels
(A) and (C) present identical data as Figure 6 in the main text, but are reproduced here for comparison.
All eight T
1
prediction curves in (A) and (B) are
scaled
by the same factor, chosen so that the
[Cu(bd
t)
2
]
2
-
prediction in (A) matches the experimental
[Cu(bdt)
2
]
2
-
data at 280 K.
Figure
S
7
Figure S9:
Thermally
-
weighted ligand field model for phthalocyanine qubits (equivalent to Figure
5
in the
main text, except
plotting model predictions and experimental data points on separate panels A and B,
respectively
). All T
1
predictions are
scaled
by the same factor, chosen for the VOPc all
-
modes prediction
to match the experimental data at 300 K.
Experimental data reproduced from Ref. 8
with permission from
the
American Chemical Society.
Figure S10:
Thermally
-
weighted ligand field model for dit
hiolate and diselenate qubits, plotted on the same
graph (equivalent to Figure
6
in the main text). All T
1
predictions are
scaled
by the same factor, chosen for
[Cu(bdt)
2
]
2
-
to match the experimental data at 280 K.
Experimental data reproduced from Ref. 10 with
permission from the Royal Society of Chemistry.
S
8
3. Choice of spin
-
phonon coupling coefficients
.
In this study, we have chosen to employ only the principal
tensor values for analysis of spin
-
phonon coupling. Previous works have employed all nine
tensor
values, thereby including not only orbital angular momentum modulation, but also rotation of the
tensor
principal axes.
17
Here we
compare and contrast the two approaches.
The principal
value derivatives for
CuPc and VOPc are given in Figures S
11
-
S1
2
. The full
tensor derivatives are given in Figures S1
3
-
S1
4
,
while the summed off
-
diagonal components are given in Figure S1
5
.
Figures
S11
-
S18
Figure
S
11
: CuPc principal
value
derivatives.
Figure
S1
2
: VOPc principal
value derivatives.
S
9
Figure
S1
3
: CuPc full
tensor derivatives.