of 9
Supplementary information for: Two-dimensional infrared-Raman spectroscopy as a
probe of water’s tetrahedrality
Tomislav Beguˇsi ́c
1,
*)
and Geoffrey A. Blake
1, 2,
*)
1)
Division of Chemistry and Chemical Engineering, California Institute of
Technology, Pasadena, California 91125, USA
2)
Division of Geological and Planetary Sciences, California Institute of Technology,
Pasadena, California 91125, USA
*)
Electronic mail: tbegusic@caltech.edu
*)
Electronic mail: gab@caltech.edu
1
SUPPLEMENTARY DISCUSSION 1: TWO-DIMENSIONAL MODEL
SYSTEM
Following Ito and Tanimura,
1
we simulated the spectra of two different two-dimensional
models with harmonic Hamiltonian
H
0
(
q
1
,q
2
,p
1
,p
2
) =
2
X
i
=1
p
2
i
2
+
1
2
ω
2
i
q
2
i
,
(1)
ω
1
= 0
.
5,
ω
2
= 2, and
μ
1
=
q
1
and
μ
2
=
q
2
, where
μ
1
and
μ
2
are the dipole moments
for the interaction with the first and second light pulses. For the model with mechanical
anharmonicity, we set the Hamiltonian to
H
(
q
1
,q
2
,p
1
,p
2
) =
H
0
(
q
1
,q
2
,p
1
,p
2
) +
2
X
i
=1
α
i
q
4
i
+
λq
1
q
2
2
,
(2)
where
α
i
= 2
.
5
×
10
5
ω
4
i
and
λ
= 0
.
1, and we set Π =
q
2
. The weak quartic terms are added
to ensure that the potential is bound. For the model with electrical anharmonicity, the
Hamiltonian was quadratic,
H
=
H
0
, while Π =
0
.
05
q
1
q
2
. The spectra (Supplementary
Fig. 3) were simulated with exact quantum mechanics on a grid in position representation,
similar to the benchmark simulations of Ref. 2. The response function was simulated for
maximum
t
1
and
t
2
times of 50 with a time step of 0
.
25. Exponential damping exp(
(
t
1
+
t
2
)
) with
τ
= 1 was applied to the response function before evaluating the discrete sine
transform.
SUPPLEMENTARY DISCUSSION 2: THEORY OF RII AND IIR
RESPONSE FUNCTIONS
A. Electrical anharmonic coupling
Here, we show that the nonlinear polarizability (for a RII pulse sequence) or dipole (for
IIR) responsible for the first interaction with the external electric fields does not contribute
to the response function. We must first assume that all types of anharmonicity are in the
perturbative regime, i.e., that
R
(
t
1
,t
2
)
R
mech
.
(
t
1
,t
2
) +
R
elec
.
(
t
1
,t
2
)
,
(3)
2
where
R
mech
.
(
t
1
,t
2
) is the term corresponding to the mechanical anharmonicity and considers
only linear dipole and polarizability operators, while
R
elec
.
(
t
1
,t
2
) corresponds to the electrical
anharmonicity and time evolution governed solely by the harmonic part of the Hamiltonian
H
0
. Terms that involve both types of anharmonicity simultaneously are neglected in this
perturbative picture. Now let us consider the RII signal discussed in the main text, with Π
nonlinear in coordinates and
μ
(
q
) = (
μ
)
T
·
q
a linear function of coordinates. Then,
R
RII
,
elec
.
=
1
2
[[ˆ
μ
(
t
2
+
t
1
)
,
ˆ
μ
(
t
1
)]
,
ˆ
Π(0)]
= 0
(4)
because the commutator [ˆ
q
i
(
t
1
+
t
2
)
,
ˆ
q
j
(
t
1
)] is either zero (for
i
̸
=
j
) or a time-dependent
scalar (for
i
==
j
)
2
that then commutes with the polarizability operator. Therefore, the RII
spectrum shown in Fig. 3 of the main text must be dominated by the mechanical anharmonic
coupling mechanism.
B. Comparing mechanical anharmonic coupling in IIR and RII
We now compare the RII spectrum to the IIR spectrum (which we call TIRV in the
main text because there we focused on that specific frequency region). Only mechanical
anharmonic coupling is considered, which means that the dipole and polarizability functions
are assumed to be linear functions of coordinates. For IIR we have:
R
IIR
,
mech
.
(
t
1
,t
2
) =
1
2
[[
ˆ
Π(
t
2
+
t
1
)
,
ˆ
μ
(
t
1
)]
,
ˆ
μ
(0)]
(5)
=
1
2
X
abc
Π
c
μ
b
μ
a
[[ˆ
q
c
(
t
2
+
t
1
)
,
ˆ
q
b
(
t
1
)]
,
ˆ
q
a
(0)]
,
(6)
where the sum runs over all normal modes and
μ
a
a
) is the derivative of the dipole moment
(polarizability) with respect to mode
a
. Similarly, for RII,
R
RII
,
mech
.
(
t
1
,t
2
) =
1
2
[[ˆ
μ
(
t
2
+
t
1
)
,
ˆ
μ
(
t
1
)]
,
ˆ
Π(0)]
(7)
=
1
2
X
abc
μ
c
μ
b
Π
a
[[ˆ
q
c
(
t
2
+
t
1
)
,
ˆ
q
b
(
t
1
)]
,
ˆ
q
a
(0)]
.
(8)
We see that the spectral feature due to the coupling between modes
a
and
c
is proportional
to Π
c
μ
a
in IIR and
μ
c
Π
a
in RII. Assuming that the one-dimensional IR and Raman spectra
are dominantly harmonic, we have
I
IR
(
ω
a
)
μ
2
a
and
I
Raman
(
ω
a
)
Π
2
a
, where
ω
a
denotes
3
the frequency of mode
a
. Then, the IIR and RII spectra are related by
R
IIR
,
mech
.
(
ω
1
2
) =
g
(
ω
1
2
)
R
RII
,
mech
.
(
ω
1
2
)
(9)
g
(
ω
1
2
) =
s
I
IR
(
ω
1
)
I
Raman
(
ω
2
)
I
Raman
(
ω
1
)
I
IR
(
ω
2
)
.
(10)
In the simulations presented in Fig. 3 of the main text, the RII spectrum was simulated using
only the permanent part of the dipole moment, i.e., with
μ
ind
= 0. Therefore, Supplementary
Eq. 10 was adjusted to
g
(
ω
1
2
) =
s
I
IR
(
ω
1
)
I
Raman
(
ω
2
)
I
Raman
(
ω
1
)
I
IR
,
perm
(
ω
2
)
,
(11)
where
I
IR
,
perm
(
ω
) denotes the IR spectrum simulated with the permanent dipole moment.
Also, because the IIR and RII response functions assumed
zz
-component of the polarizability
tensor (Π
zz
), the Raman spectrum in Supplementary Eq. (11) was computed as
I
Raman
(
ω
) =
ω
Im
Z
0
Π
zz
(
t
)
·
̇
Π
zz
(0)
e
t
2
/
2
σ
2
t
e
iωt
dt,
(12)
where
σ
t
is defined in the Methods section of the main text. For completeness, the IR
spectrum computed with full dipole, IR spectrum computed with permanent dipole only,
and the
zzzz
Raman spectrum are shown in Supplementary Fig. 4.
SUPPLEMENTARY DISCUSSION 3: SPECTRUM DECOMPOSITION
Assuming sufficiently small
ε
, we can expand
Π(
q
±
,t
) = Π(
q
t
)
±
ε
2
d
Π(
q
t
)
d
p
0
·
(
q
0
)
d
q
0
(13)
= Π(
q
t
)
±
ε
2
N
mol
X
i
=1
d
Π(
q
t
)
d
p
i,
0
·
(
q
0
)
d
q
i,
0
(14)
to first order in
ε
, where the sum goes over
N
mol
molecules, and
q
i
and
p
i
denote nine-
dimensional position and momentum vectors of atomic coordinates in water molecule
i
.
Then, it can be shown that
R
MD
(
t
1
,t
2
) =
N
mol
X
i
=1
R
MD
i
(
t
1
,t
2
)
,
(15)
4
where
R
MD
i
(
t
1
,t
2
) =
β
ε
[Π(
q
(
i
)
+
,t
2
)
Π(
q
(
i
)
,t
2
)] ̇
μ
(
q
t
1
)
(16)
and
q
(
i
)
±
,t
is the position vector (all 3
N
atom
coordinates) of a trajectory with initial momentum
p
(
i
)
±
,
0
=
p
0
±
ε
2
0
.
.
.
(
q
0
)
d
q
i,
0
.
.
.
0
,
(17)
i.e., a trajectory with the electric field applied only on molecule
i
. In our simulations, we
did not decompose the spectrum into individual molecules, but into two groups of molecules
exhibiting low and high tetrahedral order parameter, which was computed at time zero.
Spectra evaluated in this way were divided by the fractions of molecules of a given order
in the thermal distribution, which were
x
(
Q <
0
.
62) =
R
0
.
62
0
P
(
Q
)
dQ
= 0
.
388 and
x
(
Q >
0
.
72) =
R
1
0
.
72
P
(
Q
)
dQ
= 0
.
403 at 320 K. This temperature was chosen for the spectrum
decomposition because it contains approximately equal number of molecules in the two
groups. The tetrahedral order parameter was evaluated using the
order
code, which was
obtained from https://github.com/ipudu/order.git.
3
5
SUPPLEMENTARY FIGURES
0
500
1000
3000
3500
4000
ω
2
/
2
π
c
[cm
1
]
T
= 280K
0
500
1000
T
= 300K
0
500
1000
ω
1
/
2
π
c
[cm
1
]
T
= 320K
0
500
1000
T
= 340K
0
500
1000
T
= 360K
-0.20
-0.10
0.00
0.10
0.20
R
(
ω
1
,
ω
2
)[10
6
au]
a
0
1000
2000
0
500
1000
1500
2000
ω
2
/
2
π
c
[cm
1
]
T
= 280K
0
1000
2000
T
= 300K
0
1000
2000
ω
1
/
2
π
c
[cm
1
]
T
= 320K
0
1000
2000
T
= 340K
0
1000
2000
T
= 360K
-0.20
-0.10
0.00
0.10
0.20
R
(
ω
1
,
ω
2
)[10
6
au]
b
Supplementary Fig. 1. Statistical error. Statistical error of the IIR spectra computed with MD at
different temperatures.
a
TIRV part of the spectrum.
b
Low-frequency part of the spectrum.
0
2000
4000
0
2000
4000
E
0
= 10 V/
̊
A
0
2000
4000
E
0
= 1 V/
̊
A
0
2000
4000
Difference
-0.45
-0.30
-0.15
0.00
0.15
0.30
0.45
R
(
ω
1
,
ω
2
)[10
6
au]
ω
2
/
2
π
c
[cm
1
]
ω
1
/
2
π
c
[cm
1
]
a
b
c
Supplementary Fig. 2. Electric field parameter. Difference (
c
) between IIR spectra simulated
using MD at 280 K with
ε
= 2 (
E
0
= 10 V/
̊
A,
a
) and
ε
= 0
.
2 (
E
0
= 1 V/
̊
A,
b
). The difference is
comparable to the statistical error presented in Supplementary Fig. 1.
6
0
1
2
3
0
1
2
3
ω
2
/
2
π
c
[cm
1
]
Mechanical
0
1
2
3
Electrical
ω
1
/
2
π
c
[cm
1
]
a
b
Supplementary Fig. 3. Two-dimensional model system. IIR spectra of the two-dimensional model
systems (Supplementary Discussion 1) with mechanical (
a
) and electrical (
b
) anharmonic coupling.
0
1000
2000
3000
4000
ω/
2
π
c
[cm
1
]
0.0
0.5
1.0
0
1000
2000
3000
4000
ω/
2
π
c
[cm
1
]
0.0
0.5
1.0
×
100
Normalized intensity
a
IR Absorption spectrum
b
zzzz Raman spectrum
Full dipole
Permanent dipole
Supplementary Fig. 4. One-dimensional spectra used in Supplementary Eq. (11).
a
IR spectrum
simulated according to Eq. (12) of the main text with full dipole moment (Eq. (10) of the main
text) and permanent dipole moment (
μ
ind
= 0 in Eq. (10) of the main text).
b
zzzz
Raman
spectrum simulated according to Supplementary Eq. (12). All simulations were performed with
MD at 300 K.
7