El Bouzaidi and El Gridani /
Chemistry International 2(2) (2016) 7079
Study of the vibronic coupling in the ground state of Methylthio radical
R. Drissi El Bouzaidi^{1,2,*} and A. El Gridani^{1}
^{1}Laboratoire
de chimie physique, Faculté des Sciences, B.P. 8106, Université Ibn Zohr, 80000, Agadir, Maroc.
^{2}Centre
Régional des Métiers de L’Education et de Formation (CRMEF), Souss Massa Daraa, Inezgane, Maroc.
*Corresponding author’s E. mail: drissi.rachid1@hotmail.fr, Tel.: 212672222253, Fax: 212524882123
For figures refer to PDF at: http://bosaljournals.com/chemint/images/pdffiles/28.pdf
ABSTRACT
Using a methodology based on the
crude adiabatic approximation, we study the complete linear and quadratic vibronic coupling in the ground state of SCH_{3}
radical. In order to build the representation of the hamiltonian, we evaluated 30 integrals intervening in
the formulation of the vibronic coupling. Diagonalization of this representation gives the vibronic levels. For the lowest vibronic
states, the implied modes are Q_{1} (symmetric CS
stretching) and Q_{4} (CH_{3} rocking). Energy
gaps A_{1}A_{2} and A_{2}ε resulting from the
splitting due to the JahnTeller
coupling E U e = A_{1} + A_{2}+ ε are evaluated to 250 and 169 cm^{1}, respectively.
Essential coupling parameters are surrounded to simplify the study of highly vibronic states.
Keywords: Vibronic coupling, Emission spectrum, Ground state, Methylthio radical, Crude adiabatic
Capsule Summary: The linear and quadratic vibronic coupling in the ground state of SCH_{3}
radical was studied using adiabatic approximation, the firstorder parameters, β_{1} is the most important and diagonalization allowed to determine
the nature of first vibronic levels.
Cite This
Article As: R.
Drissi El Bouzaidi and A.
El Gridani. 2016. Study of the vibronic coupling
in the ground state of Methylthio radical. Chemistry International 2(2) 7079
INTRODUCTION
The development of analytical methods, mainly the
laserinduced fluorescence, significantly reduced the analysis time while
providing emission spectra well resolved vibrationally.
The study of shortlived species has been considered. Thus, a number of studies
on simple organic and organometallic radicals have
been published (Kochi, 1978; Reilly et al., 2008; Murakami et al., 2007; Fu et
al., 2005; Gravel et al., 2004). The precise interpretation of the results
requires methods appropriate to the calculations of open layers species. Pankratov (2004; 2005; 2012) reported scientific bases of
the analytical characteristics prediction for azo
coupling reactions.
Our work focuses on the organic
radical CH_{3}S. This radical is an important chemical intermediate in
the environmental chemistry. It has been suggested that it may be an
intermediate in the atmospheric oxidation by OH and NO_{3 }(Mellouki et al., 1977) of organic sulfides such as CH_{3}SCH_{3},
CH_{3}SSCH_{3} and CH_{3}SH. It has been the subject of
numerous spectroscopic studies, such as emission (Ohbayashi
et al., 1977), the laser photodetachment (Janousek et al., 1980; Engelking
et al., 1978), the electron paramagnetic resonance (EPR) (Gillbro , 1974), the IR
matrix (Jacox , 1983), the laserinduced fluorescence
(Chiang et al., 1991; Hsu et al., 1989; Suzuki et al., 1984) and the microwave
(Endo et al., 1986).
In a previous paper (El Bouzaidi et al., 2000), we carried
out a structural and vibrational study of CH_{3}S
radical in the ground state and in the first excited state. It turned out that
the static JahnTeller effect in the ground state was
low, in the order of 84 cm^{1}. The relaxation of the C_{3v}
symmetry structure, in the same state, led to two C_{s} symmetry
structures, A’and A’’. A’ is slightly more stable than A’’.
The lifting of degeneracy of the two potential energy surfaces may cause a
significant vibronic coupling. In this work, taking
into account all modes of vibration, we used the approximation "crude
adiabatic" to determine the first vibronic
states and the nature of the modes involved in each state.
In the framework of this
approximation, Marinelli and Roch
(1986) have formally treated the T_{2}U (a_{1}+ e+ 2t_{2}) coupling problem with all linear and quadratic parameters
in the case of JahnTeller instability for XY_{4}
molecules with T_{d} configuration. Numerical applications have
been done for NH_{4} in a type p Rhydberg
^{2}T_{2} excited state (Cardy
et al., 1988) and for CH4^{+}^{ }in the ^{2}T_{2}_{
}state (Marinelli and Roche, 1990) formed by the
removal of 1t_{2} electron from the methane ground state. In the
first case theoretical results confirm the assignment by Herzberg of the
emission spectrum of the ammonium radical. In the second case theoretical
results led to a calculated photoelectron spectrum of CH_{4} in
agreement with the experimental one. Recently, El Bouzaidi
et al. (2015) have studied the complete linear and quadratic vibronic coupling in the first excited state of MgCH_{3}
radical
COMPUTATIONAL DETAILS
Construction of the Hamiltonian
representation
The
vibronic stationary states may be obtained by solving
the Schrödinger equation
where q and Q are
respectively the electronic and the nuclear coordinates for the vibrational motions.
Currently
H_{e}(q,Q)
is expanded to secondorder near the reference nuclear configuration Q_{0}.
with
and
The
chosen model to process the vibronic coupling
implies, therefore, two preliminary choices: The geometry of reference and
coordinates of displacement which will allow tending the space around the
reference point.

Choice of the reference structure
We
can a priori choose any structure of the potential surface. But the choice,
which seems the most logical considering the necessity to curtail the basis of
electronic functions of manner to reduce the dimension of the problem, consists
of choosing the JahnTeller point (C_{3v} structure
of the ground state (El Bouzaidi et al., 2000) where
cross the two potential surfaces A’ and A”. The excitation
allowing describing this structure is built with SCF molecular orbitals of the ground state. This reference is shown as
the one which assures the continuity of the potential energy surface near the JahnTeller point.

Choice of displacement coordinates
We
have determined the displacement coordinates as follows:
At
the C_{3v} JahnTeller instability
point (reference point) which correspond to a certain matrix G in the
Wilson’s method (Wilson et al., 1955), we associate a matrix F
respecting the C_{3v} group properties of symmetry. In this
work, this matrix F has been built by regrouping on the one hand, the
matrix of force constants linked to the
block of a_{1} symmetry
calculated for the ground state (since the instability concerns only e symmetry modes) and on the other
hand that of the e symmetry block of the C_{3v} symmetry
first excited state.
The
diagonalization of the matrix (GF) defines a
system of normal coordinates: Q_{1}, Q_{2}, Q_{3}
(a_{1} symmetry) and (Q_{4x}, Q_{4y}), (Q_{5x},
Q_{5y}), (Q_{6x}, Q_{6y}) (e
symmetry). The normal coordinates, thus described, define a reference potential
V_{0 }supposed harmonic, centered on JahnTeller
point.
In
Table 1, we have summarized the vibration frequencies (El Bouzaidi
et al., 2000) associated with different modes, together with the corresponding
constants intervening in the definition of the reference potential V_{0}.
In
the crude adiabatic approximation, the matrix elements of H can be
expressed as follows:
limiting the
electronic functions basis to degenerate functions
The
vibrational Xj(Q) are taken as the Eigen functions of an
arbitrary C_{3v} harmonic
Hamiltonian T(Q)+V_{0}.
The
evaluation of matrix elements
For
this we need the terms
with
(firstorder parameters)
(quadratic and bilinear parameters)
In
the present case there are a priori 162 integrals to be evaluated.
The
application of group theory and C_{3v} symmetry of the reference
configuration proves that there are only 30 integrals (Cardy
et al., 1988), distributed as follow:

Firstorder parameters: β_{1}, β_{2} and β_{3} (

Noncrossed secondorder parameters:
k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6},
k’_{4}, k”_{4}, k’_{5}, k”_{5}, k’_{6}
and k”_{6}
Table 1: Definition of harmonic reference potential V_{0}
Q_{i} 
Q_{1} 
Q_{2} 
Q_{3} 
Q_{4x},Q_{4y} 
Q_{5x},Q_{5y} 
Q_{6x},Q_{6y} 
ν_{i} (cm^{1}) 
738 
1149 
3030 
605 
1485 
3198 
k_{i} (a.u.) 
0,3365.10^{2} 
0,5239.10^{2} 
0,1381.10^{1} 
0,2759.10^{2} 
0,6772.10^{2} 
0,1461.10^{1} 
Table 2: Values of parameters intervening in the formulation of vibronic coupling
K_{1} 0,3484.10^{2} 
K_{2} 0,5299.10^{2} 
K_{3} 0,1384.10^{1} 
K_{4} 0,2759.10^{2} 
K_{5} 0,6772.10^{2} 
K_{6} 0,1461.10^{1} 
K’_{4} 0,3628.10^{2} 
K”_{4} 0,6704.10^{2} 
K’_{5} 0,4580.10^{2} 
K”_{5} 0,5328.10^{2} 
K’_{6} 0,1429.10^{1} 
K”_{6} 0,1421.10^{1} 
β_{1} 0,1145.10^{2} 
β_{2} 0,4128.10^{3} 
β_{3} 0,3100.10^{3} 
γ_{14} 0,3548.10^{3} 
γ_{15} 0,6156.10^{4} 
γ_{16} 0,4744.10^{3} 
γ_{24} 0,2962.10^{3} 
γ_{25} 0,4820.10^{3} 
γ_{26} 0,3423.10^{4} 
γ_{ 34} 0,2237.10^{3} 
γ_{ 35} 0,1272.10^{3} 
γ_{ 36} 0,3179.10^{4} 
γ’_{45} 0,3155.10^{3} 
γ’_{46} 0,3178.10^{5} 
γ’_{56} 0,1282.10^{3} 
γ”_{45} 0,1406.10^{3} 
γ”_{46} 0,1891.10^{3} 
γ”_{56} 0,1390.10^{3} 
Table 3: Study of the three first excited vibronics
levels convergence
NT 
3 
6 
9 
12 
15 
Vibronics A_{1} 
0.038547 0.041381 0.046143 
0.038005 0.041128 0.044269 
0.037986 0.040883 0.043162 
0.037929 0.040885 0.042682 
0.037928 0.040876 0.042662 
Vibronics A_{2} 
0.040159 0.041861 0.046107 
0.039215 0.041576 0.044540 
0.039084 0.041435 0.043870 
0.039055 0.041418 0.043161 
0.039051 0.041413 0.043111 
Vibronics ε 
0.035571 0.041205 0.041505 
0.035306 0.040329 0.041287 
0.035267 0.039916 0.041072 
0.035255 0.039749 0.041058 
0.035252 0.039710 0.041017 
(

Crossed secondorder parameters: γ_{14}, γ_{15}, γ_{16}, γ_{24}, γ_{25}, γ_{26}, γ_{34}, γ_{35}, γ_{36}, γ’_{45}, γ’_{46},
γ’_{56}, γ”_{45},
γ”_{46} and γ”_{56 }
These
integrals may be considered as adjustable parameters so that the two model
potential surfaces E_{1} and E_{2} which obey the
equation
Evaluation of the integrals
intervening in the formulation of vibronic coupling
The
used procedure can be summarized in three points:

Choice of a cup in the potential
surface (activation of the mode). This choice is
guided by the aimed parameters.

Calculation, at MP2 level, of
A’ and A’’ states in a number of points of the direction in the
potential surface defined by the activated modes. We built the excitation
allowing describing the two structures with SCF ground state M.O.

Simultaneous adjustment, by a less
square method, of
_{ }Table 4: Reduction of the vibrational basis to Q_{1} and Q_{4} modes_{}
Symmetry of the considered
state 
Energies (a.u.) 
ΔE^{(14)} (cm^{1}) 
1ε(a_{1}) (Z.P.E. state) 
0.035589 
 
2ε(a_{1}) 
0.038177 

1A_{1}(e) 
0.038917 

1A_{2}(e) 
0.039318 

3ε(e) 
0.040330 



Table 5: Put in evidence the essential coupling
parameters
Symmetry of the considered
state 
Energies (a.u.) 
ΔE^{(14)} (cm^{1}) 
ΔE^{(21)} (cm^{1}) 
1ε(a_{1}) (Z.P.E. state) 
0.035589 
 
 
2ε(a_{1}) 
0.038177 


1A_{1}(e) 
0.038917 


1A_{2}(e) 
0.039318 

 
3ε(e) 
0.040330 

 


Table 6: Study of Q_{5} coupling
Symmetry of the considered
state 
Energies (a.u.) 
ΔE (cm^{1}) 
ΔE_{exp}_{ } (cm^{1}) (Chiang and Lee, 1991) 
ε(a_{1}) (Z.P.E. state) 
0.035802 
 
 
ε(a_{1}) 
0.039128 


A_{2}(e) 
0.042576 


A_{1}(e) 
0.042749 

 
ε(e) 
0.042826 

 
Calculation of vibronic
coupling in SCH_{3} radical’s ground state ^{2}E
: Diagonalization of the hamiltonian
Computer
codes
A
first code gives matrices containing the coefficients of symmetric vibronics
A
second code constructs the matrix
Finally
the three different blocks of
RESULTS AND DISCUSSION
Table
2 shows the values of the 30 parameters. We have carried the values resulting
from an arithmetic mean when different determinations were possible and which
are kept for the calculation of vibronic coupling.
The
obtained results show that:

(i) At the
level of firstorder parameters, the parameter β_{1} is the
most important. (ii) The gap
Limitation of the problem to e
symmetry normal modes
In
a first time, we have activated only normal modes of e symmetry (Q_{4},
Q_{5} and Q_{6}), assets in JhanTeller
effect (El Bouzaidi et al., 2000), by exciting them
equally (V=V_{4}=V_{5}= V_{6}). The vibrational excitation number of a_{1}
symmetry modes are posed equal to zero (V_{1}=V_{2}=V_{3}=0).
Then, we varied the vibrational quantum number V
from 1 to 5, therefore NT from 3 to 15 where
The
obtained results are recorded in Table 3 (for each symmetry we have carried the
three lowest vibronic states).
From
the analysis of this table, it was observed that whatever the value of NT
considered, the hierarchy of vibronic levels was the
same. This result was in agreement with that of a previous work (El Bouzaidi et al., 2015).
On
Figure 1, we carried the relative position of the different vibronic
levels of symmetry A_{1}, A_{2}, ε calculated in the framework of the application NT=21.
We adopted for the vibronic levels the following
notation: n X(x)
Where,
n: the
number of state in each Irreducible representation (states are classified by
ascendant order of the energy).
X: the symmetry of the vibronic state.
x: the
symmetry of the implied vibration mode.
A
priori, particularly it was noted that the second vibronic
excited state 2ε(a_{1}) involves an a_{1} symmetry
mode, while the first 1A_{1}(e), the third 1A_{2}(e), and the fourth 3ε(e) excited vibronic
states involve as through an e symmetry mode.
To
specify the nature of these implied modes, it is necessary to analyze the vibronic function of each state.
Analysis of the vibronic function of the
first four excited states
To
simplify notations, we reduced the writing of the vibrational
functions, product of nine polynomial of Hermite, to
only the active modes with in exponent the value of the vibrational
quantum number associated with each mode and the symbol (^{*})
to be able to differentiate, thereafter, the basis of the modes for the excited
state from that of the ground state.
·
Vibronic function of the state 2ε(a_{1})
This
development carries essentially on the Q_{1} mode with a weak
contribution of Q_{4x} and Q_{4y} modes.
Therefore the vibration movement implied in this state is that of CS stretching.
·
Vibronic functions of excited states 1A_{1}(e), 1A_{2}(e)
and 3ε(e)
The
analysis of these three vibronic functions shows that
the implied vibration modes in each of corresponding states are the same namely
Q_{4x} and Q_{4y}. Consequently, the active
vibration movement is the methylrocking one.
We
can therefore conclude that these three states result from the coupling of Q_{4x}
and Q_{4y} modes (e modes) by means of electrons movement
(electronic state E). Therefore a coupling of type:
and schematized
in Figure 2.
In
the case where the vibronic coupling is neglected
(part a of Figure 2), we represented transitions, towards the lowest vibrational level a_{1} of the ground state,
on the one hand from the lowest vibrational level a_{1}(Δ_{1}) of the
first excited state and on the other hand from the vibrational
level e (Δ_{2}) implying Q_{4x}
and Q_{4y} modes of this same
state. The value of the frequency ν_{4} associated with Q_{4x}_{ }and Q_{4y}
modes, deduced experimentally by emission spectroscopy from the difference (Δ_{2}Δ_{1}), is estimated
in this work at 605 cm^{1}. The raising of degeneracy of these modes (Q_{4x}
and Q_{4y}) by vibronic coupling (part
b of Figure 2) leads respectively to three equivalents transitions of energies
In
summary, beside transitions from 1A_{1}(e),
we predicted equivalent transitions from 1A_{2}(e) and 3ε(e) levels
which will be distant of 250 cm^{1} (822572) and 172 cm^{1} (994822),
respectively.
Table 4: Study of Q_{6} coupling_{}
Symmetry of the considered
state 
Energies (a.u.) 
ΔE (cm^{1}) 
ΔE_{exp}_{ }(cm^{1}) (Chiang and Lee, 1991) 
ε(a_{1}) (Z.P.E. state) 
0.035897 
 
 
ε(a_{1}) 
0.039223 


A_{1}(e) 
0.048659 


A_{2}(e) 
0.048718 

 
ε(e) 
0.048750 

 
Put in evidence the essential
coupling parameters
In
the preceding paragraph, we analyzed the vibronic
functions for the lowest four excited states 1A_{1}(e),
1A_{2}(e), 2ε(a_{1}) and 3ε(e). This
analysis has shown that these states involve Q_{1}, Q_{4x}
and Q_{4y} modes. We can, therefore, hope to reduce the
dimension of the vibrational basis and to take into
account only these modes.
For
this, we consider the applications NT=2,4,6,8,10,12
of
The
results of the Hamiltonian diagonalization relative
to the application NT=14, for which we have obtained a convergence of 10^{6}
a.u., are summarized in Table 4.
These
results show that, by report to 1ε(a_{1})
(Z.P.E.) level, the energy gaps
We
can therefore conclude that the vibrational basis (Q_{1},
Q_{4}) is sufficient for the determination of the lowest vibronic levels.
At
this level, we can easily list parameters that appear essential for the
calculation of these states. These parameters
are as follows:

Parameters k_{1}, k_{2}, k_{3}, k_{4}, k_{5}
and k_{6} intervening in the definition of the reference
harmonic potential V_{0}.

The firstorder parameter
corresponding to Q_{4}, namely β_{1}.

The secondorder parameters k’_{4},
k”_{4}, k’_{5}, k”_{5}, k’_{6}, k”_{6}
and γ_{14} (coupling
term since Q_{1} and Q_{4} mode are active).
In
these conditions, the results of the Hamiltonian diagonalization
are regrouped in Table 5. We obviously verified that the first obtained states
as well as the corresponding energy values are rigorously the same as those of
Table 4. It was convenient therefore to note that the taken fourteen parameters
are well adapted to the calculation of the first vibronic
levels involving Q_{1} and Q_{4} modes.
Vibronic
coupling study of Q_{4} and Q_{5} modes by means of electron
movement
In
the preceding paragraph, we have shown that the vibrational
basis (Q_{1},Q_{4}) is
sufficient to the study of Q_{4} mode coupling. In the
following, we take into account only Q_{1} and Q_{5}
modes to the coupling study of Q_{5} mode and only Q_{1}
and Q_{6} modes to that of Q_{6} mode.
o
Case of the vibrational
basis (Q_{1}, Q_{5})
We
have considered applications NT=2,4,6,8,10,12,14 of
To
the extent where the vibrational functions basis is
limited to Q_{1} and Q_{5} modes, the new list of
coupling parameters, which seems to be compatible with this basis, is as
follows:

k_{1}, k_{2}, k_{3}, k_{4}, k_{5} and k_{6} parameters defining the reference
potential V_{0}.

Parameter of firstorder
corresponding to Q_{5} (β_{2}).

Parameters of secondorder k’_{4},
k”_{4}, k’_{5}, k”_{5}, k’_{6}, k”_{6}
and γ_{15}.
A
satisfying convergence of the order of 10^{6} a.u.
is obtained for the application NT=14 (
The
energy gap,
Furthermore,
the analysis of the vibronic function of the three
other obtained excited levels A_{1}(e),
A_{2}(e) and ε(e) shows the
implication of Q_{5x} and Q_{5y} modes (antisymmetric CH stretching).
These
states are therefore the result of the coupling of these modes by means of
electrons movement (coupling of type
If
 Z.P.E. state and A_{2}(e)
state.
 A_{2}(e) and
A_{1}(e).

A_{2}(e) and ε(e).
beside
transitions from A_{2}(e) level at 1567 cm^{1}
o
Case of the vibrational
basis (Q_{1}, Q_{6})
In
this case, the vibrational functions basis was
limited to Q_{1} and Q_{6} modes and the adapted
parameters are as follows:

Parameters defining the reference
potential V_{0}.

The firstorder parameter
corresponding to Q_{6} (β_{3}).

Secondorder parameters k’_{4},
k”_{4}, k’_{5}, k”_{5}, k’_{6}, k”_{6}
and γ_{16}.
The
obtained results are summarized in Table 7.
We
signal that we have obtained a convergence of the order of 10^{6} a.u. for the application NT=14 (
Considering
 The Z.P.E. level and the first excited level ε (a_{1})
implying the Q_{1} mode.
 The Z.P.E. level and the second level A_{1}
(e) involving the Q_{6} mode (antisymmetric
CH bending).

Vibronic states A_{1} (e) and A_{2}(e) (implying also the Q_{6}
mode).

Vibronic states A_{1} (e) and ε (e) (implying
the same mode Q_{6}).
where 
State 1A_{1}(e) 
State 1A_{2}(e) 
State 3ε(e)
The
last three excited levels are therefore obtained by coupling of the Q_{6}
mode by means of the electrons movement (coupling
CONCLUSION
In this paper, we have performed a vibronic
coupling study in the ground state of SCH_{3} radical using a
methodology based on the crude adiabatic approximation. Firstly, we evaluated
the electronic integrals intervening in the formulation of the vibronic coupling and which are used to build the
Hamiltonian. The obtained results showed that:

At
the level of firstorder parameters, the parameter β_{1} was the most important.

The
gap
We then diagonalized the
representation of the Hamiltonian. This diagonalization
allowed to determine the nature of first vibronic
levels, therefore the implied modes in this coupling (Q_{1}, Q_{4x}
and Q_{4y}), and to surround essential coupling parameters.
Energy gaps, A_{1}A_{2} and A_{2}
ε, resulting of splitting of the level e (Q_{4x}
and Q_{4y} modes) by means of electrons movement, have been
evaluated at 250 and 169 cm^{1}, respectively.
Concerning the high excited vibronic
states implying the Q_{5} and Q_{6} modes, the
revealed splitting due to the JahnTeller coupling
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