Real-World Predictions from Ab Initio Molecular Dynamics Simulations - Multiscale Molecular Methods in Applied Chemistry - page 133

3.6 Properties: IR, NMR and EXAFS

3.6.1 IR Spectra

Due to the costs associated with the electronic structure calculations, AIMD
simulations always suffer from short simulation times; see also previous Sect. 3 .
In 2005, Iftimie and Tuckerman devised a method that allows well-converged
results for IR spectra from small AIMD systems and short trajectories [ 72 ]. The
frequency-(

n )-dependent Beer-Lambert absorptivity coefficient a ( n ) is given as

Z 1

qm d t

aðnÞ¼ pn½

1 exp ðb h 2 pnÞ

ntÞ M ð 0 Þ M ðtÞ

exp ð 2 p i (48)

3 hVcn ðnÞE 0

1

with b being 1/( k b T ), V the sample's volume, c the speed of light, n ( n ) the index of

e 0 the vacuum permittivity, and M the quantum mechanical total dipole
moment operator. Iftimie and Tuckerman applied the harmonic approximation
expressed in
Z 1

refraction,

Z 1

qm d

b h 2 pn

ntÞ M ð Þ M ðtÞ exp ð 2 p i 0 t ¼ exp ð 2 p i ntÞ

1 exp ðb h 2 pnÞ

1 1

(49)

h M ð 0 Þ M ðtÞ i cl d t:

In the last line “cl” denotes a classical ensemble average, i.e., phase space
integration. Next, the authors suggested the application of integration by parts:

Z 1

cl d

2 p c E 00 ¼ aðnÞ 1

_ M

Þ _ M n ðnÞ¼ exp ð 2 p i ntÞ ð 0 ðtÞ t: (50)

6 cV E 0 k B T

1

Applying this expression with the four-term Blackman windowed Fourier trans-
form approach led to sufficient accuracy based on a relatively short trajectory
(10 ps), i.e., the authors found excellent agreement between the experimentally
obtained spectra for liquid water and ice. Using this approach and decomposing the
total dipole moment

X
A m A

M ¼ (51)

the contribution of a molecule A was calculated via cross-correlation:

Z 1

cl d

1

_ M

E 00 ¼ exp ð 2 p i ntÞ ð 0 Þ m A ðtÞ t: (52)

12 p V E 0 k B T

1