Supplemental Material to
Oscillatory reaction cross sections caused by normal mode sampling
in quasiclassical trajectory calculations
Tibor Nagy, Anna Vikár and György Lendvaya)
Institute of Materials and Environmental Chemistry, Research Centre for Natural Sciences, Hungarian
Academy of Sciences, Magyar tudósok körútja 2., H-1117 Budapest, Hungary
NORMAL MODE ANALYSIS, NORMAL MODE SAMPLING AND ASSIGNMENT OF
NORMAL MODE QUANTUM NUMBERS TO AN ARBITRARY STATE
After applying the harmonic approximation to the Hamiltonian in Eq. (1), it can be written in a
simpler form1 by introducing mass-scaled deformation coordinates g  M1 / 2 x  xe  and taking
into account Hamilton’s equation x  M 1p which gives g  M 1 / 2p :
H harm ( x, p)  12 g T g  12 g T Φe g ,
(ES1)
where Φe  M1/ 2 Fe M1/ 2  (d 2V / dg 2 )g 0 is the mass-scaled force-constant matrix at the
equilibrium geometry. Matrix M-1/2 is symmetric (diagonal) and positive definite, matrix Fe is
symmetric and positive semidefinite, thus matrix Φe is also symmetric and positive semidefinite
(positive definite for irredundant internal coordinate system), and therefore it can be
diagonalized by a similarity transformation with an O orthogonal matrix (O–1=OT):
Λ  O 1Φ eO ,
(ES2)
where =diag(1,.., 3N) diagonal matrix contains the nonnegative eigenvalues of Φe , and the
column vectors of matrix O are the corresponding orthonormal eigenvectors. Introducing normal
coordinates (deformations) as components of vector Q  OT g  (Q1 ,..., Q3 N ) and conjugate
  ( P ,..., P ) , the Hamiltonian decomposes into 1D harmonic oscillators (HO)
momenta P  Q
1
3N
Hamiltonians shown in Eq. (2). Based on the energy quantization condition for HO shown in Eq.
(3), the amplitudes of the oscillation in both normal mode deformation (Qi,max) and momentum
( Pi , max   i Qi , max ) can be calculated considering phases of vibrations when the total energy is
either fully in potential or fully in kinetic form.
1D
2 2
2
1
1
H harm
, i  2 i Qi , max  2 Pi , max .
a)
Author to whom correspondence should be addressed. Electronic mail: lendvay.gyorgy@ttk.mta.hu.
(ES3)
For generating initial conditions a phase i is selected randomly from uniform distribution for
each vibrational mode, with which the normal mode coordinate and momentum of oscillator i is:
Qi  Qi , max cos i
(ES4)
Pi  Q i  iQi , max sin i .
(ES5)
Transforming these normal mode deformations (vector Q) and momenta (vector P) using
matrices M and O gives the normal mode sampled Cartesian initial conditions x and p:
x  x e  M 1 / 2OQ ,
(ES6)
p  M1 / 2OP .
(ES7)
The eigenvectors corresponding to rotations are geometry dependent, therefore they describe
pure rotation only at the equilibrium geometry (xe). After any finite non-shape-preserving
deformation the rotational eigenvectors will not be orthogonal to the vibrational normal modes,
thus normal mode sampling in Cartesian coordinates inherently generates some angular
momentum. The standard procedure is that one removes this spurious angular momentum by
adjusting the momenta of each atom (i=1,…,N) according to the formula:
pi  p i  miri  Θ 1L ,
(ES8)
where L and  are the spurious angular momentum vector and the instantaneous moment of
inertia matrix in the center-of-mass system.
In order to assign normal mode vibrational quantum numbers to an arbitrary classical state of
a molecule ( ri and pi , where i=1,...,N), translation and rotation should be removed, which is
equivalent to transforming the molecule into the frame of a cotranslating-corotating reference
molecule, called Eckart frame. This can be achieved by appropriate adjustments of the position
(center-of-mass displacement with r), the orientation (rotation with matrix R), the momentum
(center-of-mass boost with velocity v) and the angular velocity (spin with angular velocity )
of the molecule:
ri R(rir)
pi  R(pi  mi v)  mi ω  (R(ri  r)) ,
and
(ES9)
in such a way that the final coordinates (ri) and momenta (pi) fulfill the Eckart conditions2–4 and
their time derivatives:
m r  0
i i
and
i
m r
i i ,e
p
i
 0,
i
 ri  0 and
i
r
i,e
i
2
 pi  0 .
(ES10)
(ES11)
Here ri,e denotes the Cartesian coordinates of atom i in the reference equilibrium molecule with
given orientation for which the normal mode analysis was carried out. Translation can be exactly
separated, whereas separation of rotation from vibrations (Coriolis coupling energy) is only
approximate. After Eckart transformation, the remaining displacements (r-ri,e) and momenta (pi)
of atoms are vibrational deformations and deformation velocities with respect to the reference
molecule. This vibration can be decomposed into normal modes by the inverse of the
transformations in Eq. (ES6) and (ES7) and normal mode quantum numbers can be assigned
based on energy correspondence in Eq. (2).
1
E.B. Wilson, J.J.C. Decius, and P.C. Cross, Molecular Vibrations: The Theory of Infrared and
Raman Vibrational Spectra (Dover Publications, New York, 1980).
2
C. Eckart, Phys. Rev. 47, 552 (1935).
3
A.Y. Dymarsky and K.N. Kudin, J. Chem. Phys. 122, 124103 (2005).
4
V. Szalay, J. Chem. Phys. 140, 234107 (2014).
Table S1. (a) Experimentally determined and normal mode (CBE and ZBB3 PESs) frequencies of symmetric stretch
modes (A1: either local or group mode for the bonds highlighted in red) for CH nD4-n methane isotopologs.
Frequency of ensemble average bond length oscillation (determined by fitting a sine function) in CH nD4-n methane
isotopologs on both PESs. Frequency of reaction probability oscillation (b=0, Ecoll=40 mEh) as a function of flight
time (determined by fitting a sine function) determined for isotopolog reaction CH nD4-n+H on both PESs. (b) The
period of oscillations corresponding to frequencies in (a). Values in the brackets are 2 statistical errors obtained
from the fits.
(a)
(b)
Freq. (cm-1)
CBE PES
Experi
Isotopolog mental normal bond reaction
and bond
mode length probability
2093
C–D4 2109 2118 2075
2089
HC–D3 2142 2168 2119
2126
H2C–D2 2202 2221 2154
2166
H3C–D 2200 2279 2193
2917
2994
2897
2897
H4–C
2945
3046
2930
2919
H3–CD
3008
H2–CD2 2974 3091 2967
2993
3130
3054
3074
H –CD3
CBE PES
Period (0)
Experi
Isotopolog mental normal bond reaction
and bond
mode length probability
651 665(1.3) 659(10)
C–D4 654
636 651(1.4) 660(13)
HC–D3 644
621 640(1.8) 649(14)
H2C–D2 626
627
605 629(2.4) 637(10)
H3C–D
473
461 476(0.5) 476(2.5)
H4–C
468
453 471(0.6) 472(3.6)
H3–CD
464
446 465(1.1) 458(4.8)
H2–CD2
461
441 452(1.8) 449(11)
H –CD3
3
ZBB3 PES
normal bond reaction
mode length probability
2141
2087
2091
2187
2132
2127
2235
2162
2156
2287
2194
2159
3026
2900
2916
3068
2924
2932
3104
2955
2966
3137
3049
3063
ZBB3 PES
normal bond reaction
mode length probability
644 661(0.9) 659(7.7)
631 647(1.3) 648(9.2)
617 638(1.5) 639(11)
603 628(1.8) 640(13)
456 475(0.2) 473(4.5)
449 472(0.4) 470(4.0)
444 467(0.8) 465(6.2)
440 452(1.0) 450(9.4)
FIG. S1
FIG. S1 Time dependence of the mean C–H (left panel) and C–D (right panel) bond lengths (gray lines; equilibrium
value: 2.0595 a0) on the ZBB3 PES in ground-state methane isotopologs CHnD4-n (n=0,..,4), obtained from the time
evolution of 103 normal mode sampled states. There is a scale change at 5103 0 from linear to logarithmic. The
black curve is the moving average with a one-period wide time window.
4
FIG. S2
FIG. S2 The correlation between periods of oscillation τreaction probability (of the CHnD4-n+H reaction probability
oscillation as a function of flight time at b=0 a0, Ecoll=40 mEh and τmean bond length (of the oscillation of ensemble
average bond length in CHnD4-n as a function of relaxation time) calculated on the CBE and ZBB3 PESs. Periods of
oscillation were determined by fitting sine functions to the evolution of reaction probability and of mean bond
length in the Rini range of 12-24.9 a0, and the corresponding time range of 1,000-3,000 0 (see Figs. 1, 5 and S1).
Error bars correspond to 2 statistical error.
5