Supplemental Online Material
Energy levels and spectra of tetrahedral molecules
To Journal of Chemical Physics paper:
Rotation of methane and silane molecules in He droplets
Hiromichi Hoshina,a) Dmitry Skvortsov,b) Boris G. Sartakovc) and Andrey F. Vilesov
Department of Chemistry, University of Southern California, Los Angeles, CA 90089
USA.
a) Present address: RIKEN, 2-1 Hirosawa, Wako 351-0198, Japan.
b) Present address: DCG Systems, Inc., 45900 Northport Loop East, Fremont, CA 94538
c) On leave from: Russian Academy of Sciences, A.M. Prokhorov General Physics
Institute, RAS, Vavilov str. 38, 119991 Moscow, Russia.
The rotational structure of vibrational bands of molecules embedded in 4He droplets
was found to be similar to that in free molecules. However, the values of spectroscopic
constants of embedded molecule differ from those in the gas phase.
At present, the
Hamiltonian for molecules surrounded by liquid helium cannot be derived from the first
principles. Therefore, here we have used an effective rotational-vibrational Hamiltonian
of free molecules for empirical fits of the infrared spectra of the embedded molecules.
There are two main assumptions on the effective Hamiltonian which are usually used to
account for the measured spectra of embedded molecules. First, is that the interaction
between a molecule and superfluid helium environment does not change the symmetry of
the effective Hamiltonian of the embedded molecule.
Second, is that the effective
Hamiltonian of a molecule in helium can be expanded into power series of vibration and
rotation operators using the ordering scheme, which is similar to the Born-Oppenheimer
scheme used for the Hamiltonian of free molecules.
Here we consider the fundamental  3 mode of a methane-like molecule XY4 which
has a tetrahedral symmetry.
The vibration-rotation interaction between the  3 and other
vibrational states will be neglected as in helium droplets only low rotational states of light
molecules (J<3) are populated. In the framework of the Born-Oppenheimer scheme the
effective Hamiltonian of a molecule can be expanded into power series of the angular
momentum operators and the vibrational operators as:1
H ( 0 )  H 2, 0  H 0, 2 ,
H (1)  H 2,1 ,
H ( 2 )  H 2 , 2  H 0, 4 ,
2
H ( 3)  H 2 , 3 ,
where H(i) is the i-th order part of the Hamiltonian and H n ,m terms collect all the power
terms in which the total power of the vibrational operators is n and the total power of the
rotational operators is m. This scheme was originally proposed and verified for the free CH4
molecules having rotational quantum numbers J10. In free molecules the order of
magnitude of the terms can be estimated to be:
H (0) ~ 1000 cm1 ,
H (1) ~ 1 cm1 ,
H ( 2) ~ 102
cm1 ,
H (3) ~ 105
cm1 .
The effective Hamiltonian of molecule in helium usually shows slower convergence.
Nevertheless, third order terms in the effective Hamiltonian are usually of about
H (3) ~ 102  103
cm1 , which is sufficient for the fitting of the experimental spectra.
The leading H n ,m terms are:





H 2,0   3 p32  q32 / 2   3 3  1/ 2 is the harmonic oscillator Hamiltonian,

H 0,2  B  J 2 is the rigid rotor Hamiltonian,

H 2,1  2 B   3  J  l 3 is the Coriolis interaction term, where l is the angular
3
momentum operator of the 3 vibrational mode.
3
Hereafter we shall assume that the effective vibrational parameters, i.e. the frequency of the
 3 mode and the Coriolis parameter  3 are not affected significantly by the interaction
between the embedded molecule and the helium environment as the molecule-helium
interaction is weak and does not change appreciably the force field of the molecule itself.
Zero order and first order terms of the effective Hamiltonian are of spherical symmetry.
The rotational spectrum of a molecule in the ground vibrational state in the first order of
approximation is:
E 0  B J ( J  1)
where J is the angular momentum quantum number. The first order energy spectrum of the
three fold degenerate vibration 3 state is complicated by the Coriolis interaction of the
total angular momentum and the vibrational angular momentum of the 3 state (l3=1) and
results in the splitting of each of the J rotational levels into three sublevels. These sublevels
are commonly designated with the quantum number R which is the eigenvalue of the pure
rotational angular momentum operator R  J  l 3 . The value of R runs from |J-l3| to J+l3
. Thus, each rotational state having J1 has three R sublevels J-1, J and J+1, which are also
designated as F(-), F(0) and F(+) sublevels, respectively.2
In the first order and omitting zero point energy, energy levels in the ground and in
the 3 states are given by the following equations:
E (  0, J , R  J )  B  J ( J  1)
E ( 3 , J , R)   3  B  J ( J  1)  B   3  R( R  1)  J ( J  1)  2 .
Eigenfunctions of the first order Hamiltonian are obtained as a linear combination3
4
3 , l3 , J , M , R, K R   C JR,,KK,l ,m  J ,M ,K  * ,l ,m .
R
3
3
3 3
(1)
K ,m
Here J , M , K are symmetric top wavefunctions, where M and K are the quantum numbers
of the projections of the total angular momentum on the space fixed axis and on the
molecule fixed axis, respectively.
degenerated harmonic oscillator.

3 ,l 3 , m3
are the wavefunctions of the triply
In our case the vibrational quantum number 3 =1, and
the vibrational angular momentum quantum number l3 =1; m3 is the quantum number of
the projection of the vibrational angular momentum on the molecule fixed axis. Finally,
CJR,,KK,Rl3 , m are the Clebsch-Gordan coefficients. In the ground vibrational state R=J and
K R  K . In the first order of approximation the eigenvalues of the ground state do not
depend on K R and each R-sublevel is 2R+1 times degenerate.
The power expansion of the dipole momentum operator over the vibrational coordinates
in the molecule fixed system of coordinates reads as:

 

  0  
 qi  ... ,
qi
i

where 0 is the permanent dipole moment which equals zero in tetrahedral molecules.
The term linear in qi yields nonzero matrix elements for the infrared active vibrational
transitions  i   i  1 . The higher order nonlinear terms are successively diminishing in
the magnitude in accordance with the Born-Oppenheimer ordering scheme. For the 3
mode the formula for the dipole moment operator can be written as:


  2  d01  q3 ,
5
where d 01 is the dipole moment matrix element of the fundamental transition of the

vibrational mode and q3 is the dimensionless coordinate vector of the mode. The
transformation from the molecule fixed system of coordinates to the space fixed system of
coordinates yields the following expression for the space fixed components of the dipole
moment
   D1,* ( ,  ,  )  
(2)

where D1,* ( ,  ,  ) is Wigner function, , ,  are Euler angles of the molecule
orientation,  are the spherical components of the dipole moment operator with respect
to the space fixed system of coordinates,  are the spherical components of the dipole
moment operator with respect to the molecule fixed system of coordinates.
According to
Eq. (2)  is a scalar product of the pure rotational function D1,* ( ,  ,  ) and the
vibrational vector  . Therefore, in the first order of approximation, the dipole moment
matrix elements of   must obey the selection rule of R=0 and the R-, Q- and Pbranches of the 3 band include only the rotational-vibrational transitions to/from F(-), F(0)
and F(+) Coriolis sublevels, respectively, as shown in Fig. 2. The selection rule R=0 is
sensitive to the perturbations which break the spherical symmetry of the first order
Hamiltonian. Therefore, the absence of the rotational-vibrational transitions other than
having R=0 can be considered as an evidence of the weakness of the high order terms in
the effective Hamiltonian.
The second and third order terms of the Hamiltonian of tetrahedral molecules have been
analyzed in Refs.3-5 The centrifugal distortion part of the Hamiltonian reads as:3
6
H 0, 4   Ds  J 4  Dt  Oˆ JJJJ ,
where the first term is a scalar and does not break the spherical symmetry of the first order
Hamiltonian. The second term is a tensor allowed by the tetrahedral symmetry of the
molecule. The tensor term lifts the K R degeneracy and leads to the splitting of the
R-manifold of levels into the tetrahedral sublevels  . In the case of Td group the
symmetry species  span A1 , A2 , E, F1 , F2 symmetry species. The H 2 , 2 term of the
effective Hamiltonian in the case of the 3 mode is expressed as: 3
H 2, 2  Y3  3  J 2  Z 3s  Oˆ JJ 33 ( scalar )  Z 3t  Oˆ JJ 33 (tensor ) ,
where only the last term is tensoral. The explicit formulas for operators Ô are given in
Ref.3
Finally, the third order terms read as:
H (3)  F3s  J 2  (J  l 3 )  F3t  Oˆ JJJ 3 .
The tensoral Ô operators, which lead to the breaking of spherical symmetry in the 3
state, appear in the second and higher order terms of the Hamiltonian.
In the case of
fundamental 0   3 transition, these terms can be considered as a small perturbation with
respect to the Coriolis splitting.
Therefore, the manifold of the tetrahedral sublevels can
be characterized by the good quantum numbers R.
The matrix elements of the tensor operators of the third order Hamiltonian can be
expressed using symmetry adopted Clebsch-Gordan coefficients F 4 (see Ref.4). The
tetrahedral sublevels are labeled as k=p where  is the symmetry of sublevel and p is the
running index used to enumerate the eigenstates of the same symmetry  within the
manifold of levels J, R.
In the studied molecules, the tensor terms are too weak to
7
significantly perturb the Coriolis sublevels. Therefore, the approximate selection rule
k  0 determines the strong transitions in the 3 band. 4
Ten parameters  3 , B ,  3 , Y3 , Z 3s , Z 3t , F3s , F3t , Ds and Dt appear in the
third order Hamiltonian of tetrahedral molecules
constants
of the
potential energy surface.3
and they can be expressed via the force
The frequencies of the allowed
R  0 vibrational-rotational transitions 0, J   3 , J ' (R = J) can be parameterized by six
spectroscopic constants as: 6
 PR ( J , k )  m  nX  pX 2  qX 3  g 
C( J )
 ( 1) J FA(14kk, J' , J ) ,
A( X )
where X=J+1 and X=-J for the R- and P-branches, respectively.
 Q ( J , k )  m   J ( J  1)  2 g 
(3)
For the Q branch lines:
C( J )
 ( 1) J FA(14kk, J' , J ) ,
B( J )
(4)
where
A(X)  2X(2X  1) ,
B(X)  2X(2X  2) ,
C(X)  [ (2X - 3)(2X  5)]1/2 .
F4 coefficients have been tabulated elsewhere.4,5
Here we have neglected constant h,6
which makes negligible contribution to νP,R at J < 3.
Hougen’s
notation7,8
reads
as:
k  pA1  k '  pA2
The selection rule k  0
in
k  pF1  k '  pF2
or
or
k  pE  k '  pE , see Fig. 2.
Six spectroscopic constants m, n, p, q, , g can be expressed in terms of ten molecular
constants of the effective Hamiltonian as follows:
m   3  2 Beff   3  Z 3s / 2 ,
8
n  2 Beff  (1   3 )  Y3  (5 / 6)  Z 3 s  F3 s ,
p  Y3  Z 3s / 3  2F3s ,
  Y3  (2 / 3)  Z 3s  F3s ,
g   3 / 7  Z3t  10  Dt  4  F3t  ,
q  4 Ds  F3s .
All ten molecular constants of the Hamiltonian can not be unambiguously obtained from
the spectra of strong transitions due to the limitations implied by the selection rules
k  0 and R=0 and usually require observation of additional weak R  0 and k  0
transitions.
The
dipole
moment
i  3  0, J , M , R  J , k
matrix
element
state and the final
of
transition
between
the
initial
f  3  1, J ' , M ' , R' , k ' state is
J 'M '
f d i  d fi  d01   k 'k   R'R  CJM
,1
The spectral dependence of the absorption cross section of the transition f  i is
 fi ( )  4 2  d 2fi  fi / c g fi (  fi ) ,
where  fi is the frequency of the line and g (   fi ) is the normalized line form-factor.
The acceptable fit of the observed spectra was reached with the Lorentzian form-factor
g (  fi ) 
 Lor
2
 [(  fi ) 2   Lor
]
where  Lor is an empirical broadening parameter.
Finally the absorption spectra are simulated using the following expression:
9


2
S ( )  4 2  d 01
 3 / 3c  
 Lor
2 J '1

 NJ k
2
2 J  1  [(   fi ) 2   Lor
]
(5)
where N J k is the total population of initial k=p symmetry sublevel of the J rotational
level in the ground vibrational state. Upon derivation of the Eq. (5) we used the sum rule
 C

J 'M ' 2
JM ,1
 (2 J '1) / 3 .
M ,M '
The integral intensity of a spectral line is determined by the factor:
2 J '1
 N Jk .
2J 1
The value of N Jk is determined by the Boltzmann factor of rotational level J and by the
nuclear spin weight of symmetry sublevel . At room temperatures the molecules populate
high J states where the partition numbers   of sublevels of symmetry  are in ratio:
 A : A : E : F : F  1 : 1 : 2 : 3 : 3
1
2
1
2
Accordingly, the nuclear spin weights w ratio for XH4 is:
wA1 : wA2 : wE : wF1 : wF2  5 : 5 : 2 : 3 : 3 ,
and the nuclear spin weights w ratio for XD4 is:
wA1 : wA2 : wE : wF1 : wF2  15 : 15 : 12 : 18 : 18
Upon cooling to the ambient temperature in the helium droplet of 0.37 K the spin species
remain unrelaxed during the flight from the pick-up chamber to the mass-spectrometer. On
the other hand, the relaxation between rotational levels is fast enough to yield the
equilibrium rotational-vibrational distributions of population for the A , E and F spin
species.
Therefore, because the energy spacing between rotational levels of XH4 and
XD4 molecules is much larger than droplet’s temperature of 0.37 K, all the A1 and A2
spin species are in the J=0, = A1 state; E
spin species are in the J=2, = E states and
10
all the F1 and F2 spin species are in the J=1, = F1 states. Thus the relative population
of the rotational levels of XH4 in the droplets is:
N J  0, A : N J 1, F : N J  2, E  5 : 9 : 2
The relative population or rotational levels of XD4 in the droplets is:
N J  0, A : N J 1, F : N J  2, E  5 : 18 : 4
Therefore, because the symmetry of the tetrahedral molecule is preserved in helium
droplets the ratio of the integral intensities of spectral lines for XH4 in helium droplets is
expected to be:
I P 2 : I P1 : I Q 2 : I Q1 : I R 0 : I R1 : I R 2  12 : 30 : 20 : 90 : 150 : 150 : 28
and for XD4 to be:
I P 2 : I P1 : I Q 2 : I Q1 : I R 0 : I R1 : I R 2  12 : 30 : 20 : 90 : 75 : 150 : 28 .
Literature
1.
2.
M. R. Aliev and J. K. G. Watson, J. Mol. Spectrosc. 61, 29-52 (1976).
G. Herzberg, Molecular spectra and molecular structure, II Infrared and Raman
spectra of polyatomic molecules. (Van Nostrand, Princeton, New Jersey, London,
1968).
3.
4.
5.
6.
7.
K. T. Hecht, J. Mol. Spectrosc. 5, 355-389 (1961).
J. Moret-Bailly, L. Gautier, and J. Montagutelli, J. Mol. Spectrosc. 15, 355-377
(1965).
B. J. Krohn, Los-Alamos Scientific Report LA-6554 MS (1976).
B. Bobin and K. Fox, Journal De Physique 34, 571-582 (1973).
J. T. Hougen, MTP International Review if Science, Physicsl Chemistry Series Two,
Vol. 3 (Spectroscopy), Edited by D.A. Ramsay (Butterworths). (1976).
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8.
J. T. Hougen, J. Chem. Phys. 39, 358-365 (1963).
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