El Bouzaidi and  El Gridani / Chemistry International 2(2) (2016) 70-79


Study of the vibronic coupling in the ground state of Methylthio radical


R. Drissi  El Bouzaidi1,2,* and  A. El Gridani1

1Laboratoire de chimie physique, Faculté des Sciences, B.P. 8106, Université Ibn Zohr, 80000, Agadir, Maroc.

2Centre 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


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Using a methodology based on the crude adiabatic approximation, we study the complete linear and quadratic vibronic coupling in the ground state of SCH3 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 Q1 (symmetric C-S stretching) and Q4 (CH3 rocking). Energy gaps A1-A2 and A2-ε resulting  from the splitting  due to the Jahn-Teller coupling E U e = A1 + A2+ 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 SCH3 radical was studied using adiabatic approximation, the first-order parameters, β1 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) 70-79




The development of analytical methods, mainly the laser-induced fluorescence, significantly reduced the analysis time while providing emission spectra well resolved vibrationally. The study of short-lived 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 CH3S. 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 NO3 (Mellouki et al., 1977) of organic sulfides such as CH3SCH3, CH3SSCH3 and CH3SH. 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 laser-induced 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 CH3S radical in the ground state and in the first excited state. It turned out that the static Jahn-Teller effect in the ground state was low, in the order of 84 cm-1. The relaxation of the C3v symmetry structure, in the same state, led to two Cs 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 T2U (a1+ e+ 2t2) coupling problem with all linear and quadratic parameters in the case of Jahn-Teller instability for XY4 molecules with Td configuration. Numerical applications have been done for NH4 in a type p Rhydberg 2T2 excited state (Cardy et al., 1988) and for CH4+ in the 2T2 state (Marinelli and Roche, 1990) formed by the removal of 1t2 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 CH4 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 MgCH3 radical




Construction of the Hamiltonian representation


The vibronic stationary states may be obtained by solving the Schrödinger equation        with          
        where q and Q are respectively the electronic and the nuclear coordinates for the vibrational motions.

 is the kinetic energy operator for the nucler and  is the electronic Hamiltonian, which includes the Kinetic energy operator for the electrons and all the columbic interactions.  The vibronic wavefunctions  may, in principle, be expanded in any complete vibronic basis set. In practice, this basis is severely truncated by keeping only a few electronic wavefunctions corresponding to the states that are degenerate (Jahn-Teller case) or quasi-degenerate (pseudo-Jahn-Teller case) at some Q=Q0.

Currently He(q,Q) is expanded to second-order near the reference nuclear configuration Q0.



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 Jahn-Teller point (C3v 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 Jahn-Teller point.

-          Choice of displacement coordinates

We have determined the displacement coordinates as follows:

At the C3v Jahn-Teller 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 C3v 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 a1  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 C3v symmetry first excited state.

The diagonalization of the matrix (GF) defines a system of normal coordinates: Q1, Q2, Q3 (a1 symmetry) and (Q4x, Q4y), (Q5x, Q5y), (Q6x, Q6y) (e symmetry). The normal coordinates, thus described, define a reference potential V0 supposed harmonic, centered on Jahn-Teller 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 V0.

In the crude adiabatic approximation, the matrix elements of H can be expressed as follows:


limiting the electronic functions basis to degenerate functions  at the reference point.

The vibrational Xj(Q) are taken as the Eigen functions of an arbitrary C3v harmonic   Hamiltonian  T(Q)+V0.

The evaluation of matrix elements  requires, therefore, only the elementary integrals


For this we need the terms  which we rewrite:



(first-order parameters)


(quadratic and bilinear parameters)

In the present case there are a priori 162 integrals to be evaluated.  

The application of group theory and C3v symmetry of the reference configuration proves that there are only 30 integrals (Cardy et al., 1988), distributed as follow:

-          First-order parameters: β1, β2 and β3 (   ,  i=1,2,3    j=4,5,6)

-          Non-crossed second-order parameters: k1, k2, k3, k4, k5, k6, k’4, k”4, k’5, k”5, k’6 and k”6


Table 1: Definition of harmonic reference potential V0








νi (cm-1)







ki (a.u.)









Table 2: Values of parameters intervening in the formulation of vibronic coupling











































γ 34


γ 35


γ 36















Table 3: Study of the three first excited vibronics levels convergence








        Vibronics A1

















        Vibronics A2

















        Vibronics ε

















( =    = ).

-          Crossed second-order parameters: γ14, γ15, γ16, γ24, γ25, γ26, γ34, γ35, γ36, γ45, γ46, γ56, γ45, γ46 and γ56   

These integrals may be considered as adjustable parameters so that the two model potential surfaces E1 and E2 which obey the equation fit to the same potential surfaces derived from some quantum mechanical electronic energy calculations.


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  representation, on the two calculated surfaces.

-                      Table 4: Reduction of the vibrational basis to Q1 and Q4 modes

Symmetry of the considered state





1ε(a1) (Z.P.E. state)












-          Table 5: Put in evidence the essential coupling parameters

Symmetry of the considered state







1ε(a1) (Z.P.E. state)















-          Table 6: Study of Q5 coupling

Symmetry of the considered state






(cm-1) (Chiang and Lee, 1991)

ε(a1) (Z.P.E. state)















Calculation of vibronic coupling in SCH3 radical’s ground state 2E : Diagonalization of the hamiltonian 

Computer codes

A first code gives matrices containing the coefficients of symmetric vibronics for an arbitrary choice of maximal vibrational quantum members Vi (i=1,2,….,9) and for each irreducible representation of C3v.

A second code constructs the matrix  and a third code gives the representation of the Hamiltonian in the adapted symmetry basis set for each irreducible representation.

Finally the three different blocks of  are diagonalized by the Davidson algorithm (Davidson, 1975).




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 first-order parameters, the parameter β1 is the most important. (ii) The gap  is greater than gaps  and  at the level of non-crossed second-order terms. (iii) Probably, the crossed second-order parameters,γij, will not have any effect on the vibronic coupling.


Limitation of the problem to e symmetry normal modes


In a first time, we have activated only normal modes of e symmetry (Q4, Q5 and Q6), assets in Jhan-Teller effect (El Bouzaidi et al., 2000), by exciting them equally (V=V4=V5= V6). The vibrational excitation number of a1 symmetry modes are posed equal to zero (V1=V2=V3=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 A1, A2, ε calculated in the framework of the application NT=21. We adopted for the vibronic levels the following notation: n X(x)


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(a1)  involves an a1 symmetry mode, while the first 1A1(e), the third 1A2(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(a1)

This development carries essentially on the Q1 mode with a weak contribution of Q4x and Q4y modes. Therefore the vibration movement implied in this state is that of C-S stretching.

·         Vibronic functions of excited states 1A1(e), 1A2(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 Q4x and Q4y. Consequently, the active vibration movement is the methyl-rocking one.

We can therefore conclude that these three states result from the coupling of Q4x and Q4y modes (e modes) by means of electrons movement (electronic state E). Therefore a coupling of type:

U e = A1+A2+ε

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 a1 of the ground state, on the one hand from the lowest vibrational level a1(Δ1) of the first excited state and on the other hand from the vibrational level e (Δ2)   implying Q4x and Q4y  modes of this same state. The value of the frequency ν4 associated with Q4x and Q4y modes, deduced experimentally by emission spectroscopy from the difference (Δ2-Δ1), is estimated in this work at 605 cm-1. The raising of degeneracy of these modes (Q4x and Q4y) by vibronic coupling (part b of Figure 2) leads respectively to three equivalents transitions of energies , from 1A1(e), 1A2(e)  and 3ε(e) levels implying these modes. Therefore, we predicted that there are three bands around 572, 822 and 994 cm-1 in the emission spectrum.

In summary, beside transitions from 1A1(e), we predicted equivalent transitions from 1A2(e) and 3ε(e) levels which will be distant of 250 cm-1 (822-572) and 172 cm-1 (994-822), respectively.

Table 4: Study of Q6 coupling

Symmetry of the considered state





ΔEexp (cm-1)

(Chiang and Lee, 1991)

ε(a1) (Z.P.E. state)
















Put in evidence the essential coupling parameters


In the preceding paragraph, we analyzed the vibronic functions for the lowest four excited states 1A1(e), 1A2(e), 2ε(a1)   and 3ε(e). This analysis has shown that these states involve Q1, Q4x and Q4y 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  type in which the Q1 and Q4 mode are activated of the same manner.

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(a1) (Z.P.E.) level, the energy gaps  of the lowest four excited states 1A1(e), 1A2(e), 2ε(a1)   and 3ε(e) are equal to those obtained previously for the application NT=21.

We can therefore conclude that the vibrational basis (Q1, Q4) 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 k1, k2, k3, k4, k5 and k6 intervening in the definition of the reference harmonic potential V0.

-          The first-order parameter corresponding to Q4, namely β1.

-          The second-order parameters k’4, k”4, k’5, k”5, k’6, k”6 and γ14 (coupling term since Q1 and Q4 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 Q1 and Q4 modes.


Vibronic coupling study of Q4 and Q5 modes by means of electron movement


In the preceding paragraph, we have shown that the vibrational basis (Q1,Q4) is sufficient to the study of Q4 mode coupling. In the following, we take into account only Q1 and Q5 modes to the coupling study of Q5 mode and only Q1 and Q6 modes to that of Q6 mode.

o    Case of the vibrational basis (Q1, Q5)

We have considered applications NT=2,4,6,8,10,12,14 of  type in which this time, only Q1 and Q5 modes are excited with quantum number V (varying from 1 to 7).

To the extent where the vibrational functions basis is limited to Q1 and Q5 modes, the new list of coupling parameters, which seems to be compatible with this basis, is as follows:

-          k1, k2, k3, k4, k5 and k6 parameters defining the reference potential V0.

-          Parameter of first-order corresponding to Q5 (β2).

-          Parameters of second-order k’4, k”4, k’5, k”5, k’6, k”6 and γ15.

A satisfying convergence of the order of 10-6 a.u. is obtained for the application NT=14 ( , the results of which are presented in Table 6.          

The energy gap,  (730 cm-1) of the first excited level involving the Q1 mode, by report to Z.P.E. level, was in good agreement with the experimental frequency of C-S s-stretching mode (727cm1) (Chiang and Lee, 1991). Similarly the gap  between Z.P.E. level and 1A1(e) state is very close to the measured frequency of C-H a-bending (1496 cm-1)

Furthermore, the analysis of the vibronic function of the three other obtained excited levels A1(e), A2(e) and ε(e)  shows the implication of Q5x and Q5y modes (antisymmetric C-H stretching).

These states are therefore the result of the coupling of these modes by means of electrons movement (coupling of type U e = A1+A2+ε )

If   designate energy gaps, respectively, between:

-  Z.P.E. state and A2(e) state.

-  A2(e) and A1(e).

- A2(e) and ε(e).

beside transitions from A2(e) level at 1567 cm-1 , we expected an equivalent transitions from A1(e) and ε(e) levels and which will be distant of 22 cm-1  and 34 cm-1 , respectively.


o    Case of the vibrational basis (Q1, Q6) 

In this case, the vibrational functions basis was limited to Q1 and Q6 modes and the adapted parameters are as follows:

-          Parameters defining the reference potential V0.

-          The first-order parameter corresponding to Q6 (β3).

-          Second-order parameters k’4, k”4, k’5, k”5, k’6, k”6 and γ16.

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 energy gaps respectively between:

-  The Z.P.E. level and the first excited level ε (a1) implying the Q1 mode.

-  The Z.P.E. level and the second level A1 (e) involving the Q6 mode (antisymmetric C-H bending).

- Vibronic states A1 (e) and A2(e) (implying also the Q6 mode).

- Vibronic states A1 (e) and ε (e) (implying the same mode Q6).

    (Eq. 1)

where     )

                                             (Eq. 2)

-  State 1A1(e)


-          State 1A2(e)


-          State 3ε(e)

                           (Eq. 3)
















The gaps ( ) and   are close to observed frequencies of C-S s-stretching (727 cm-1) and C-H a-bending (2706 cm-1) respectively.

The last three excited levels are therefore obtained by coupling of the Q6 mode by means of the electrons movement (coupling U e = A1+A2+ε). Therefore, we expected that for each transition from A1 (e) level at 2801 cm-1, to analogous transition from the two other levels A2(e) and ε(e) and which will be distant of 13 and 7 cm-1, respectively.




In this paper, we have performed a vibronic coupling study in the ground state of SCH3 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 first-order parameters, the parameter β1 was the most important.

-          The gap  was greater than gaps  and  at the level of non-crossed second-order terms.

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 (Q1, Q4x and Q4y), and to surround essential coupling parameters.

Energy gaps, A1-A2 and A2- ε, resulting of splitting of the level e (Q4x and Q4y modes) by means of electrons movement, have been evaluated at 250 and 169 cm-1, respectively. 

Concerning the high excited vibronic states implying the Q5 and Q6 modes, the revealed splitting due to the Jahn-Teller coupling U e = A1+A2+ε , are in this case weaker. The corresponding energy gaps are evaluated to 38 cm-1 (A2-A1), and 17 cm-1 (A2-ε) for Q5 mode, 13 cm-1 (A2-A1) and 7 cm-1 (A2-ε) for Q6 mode.




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