the energy level of its lowest lying degenerate vibration is only e = 171 cm , consequently according to the boltzman distribution law it has a population of 58% ( 29%2 for vibrational degeneracy ) of the ground state at dry ice temperature ( 200 k ) . therefore the rotational spectrum in the v10 = 1 state is also strong and it is not complicated by interactions with nearby vibrational states . whereas it is a prolate top , its rotational constants a  5.7 and b  2.9 ghz are not large . so , it is a possible candidate for the observation of the a1-a2 splitting in the ground state . the 
 qt+ is rather large and a  b  a is small and it is possible to observe direct -type resonance transitions in the microwave range . these situation have been previously observed only for pf3 , v4 = 1 , for cf3h and cf3d , v6 = 1 , cf3cl , v6 = 1   and ch3cf3 , v12 = 1 . the rotational spectra of cf3ccd , the isotopomer of this molecule in the ground , v10 = 1 and v10 = 2 states have been measured and studied by several authors [ 711 ] . three identical fluorine atoms ( i = 1/2 ) in cf3cch have important consequences in the determination of relative intensities of signals . the symmetry operations like exchange of nuclei , have symmetric effect on the total wavefunction for nuclei with integral or zero spin ( bose particles ) . but for nuclei with half integral spin ( fermi particles ) the total wavefunction is antisymmetric . the rotation about the c3 axis through 2/3 or 4/3 rotation is equal to two exchanges of nuclei . therefore it is symmetric ( a ) for fermi particles as well as bose particles . a rotation of 2/3 about top axis will not change rot states with k = 3n ( n = integer , including n = 0 ) since rot  e. for the symmetric state , the species are a and in antisymmetric state ( k  3n ) the species are e. there are eight different possibilities for combinations of three fluorine atoms with spin 1/2 . two of them can occur when three spins are the same 1/2 or 1/2 , these are totally symmetric because they remain unchanged by any permutation of nuclei which are equivalent to a rotation . but this situation can not take place for the remaining 6 arrangements , which are 2a1 + 2e . the statistical weight depends on the ratio between the spin function of symmetry a and the spin function of symmetry e. consequently : for symmetric top molecules like cf3cch , the ground vibrational state rotational energies are given up to sextic terms by equation ( 1 ) and by application of selection rules j = + 1 and k = 0 , the frequencies are given by equation ( 2 ) . the spectra in this state are simple , so the different k values ( k = 0,1,2,3,4 ... ) for each j transition are assigned easily . this is illustrated in figure 1 , which shows part of the observed transition j = 24  25 , and compares it with the calculated spectrum . the centrifugal distortion produces a band head to high frequency at k = 0 , with a spread to lower frequency with higher k figure 2 . due to negative value of djk for ch3cl , this situation is different figure 3 . if |k| values are plotted against frequency for this state , the fortrat diagrams are produced which are shown in figures 2,3 . in these diagrams the splitting increase as k increases and this is due to the djk parameter . djk is 6.27 and 2.517 khz for cf3cch and chcl3 respectively [ 2 , 12 ] . the sextic centrifugal distortion constants were accurately determined for the first time for the ground state , and result shows that hjk > |hkj| and hkj < 0 . for this molecule have found amount of this value , which is 
 h3=2.310 - 21b4a - b . the predicted value of h3 for cf3cch , 2.61011 mhz , gives a splitting about 12 mhz for the highest - j transition ( j = 112 ) . because v10 state of this molecule has a high population , the rotational spectra in this state are also strong . it is isolated in frequency from other vibrational states , so is not complicated by interactions with nearby states . rotational frequencies for transitions j  j + 1 in the excited degenerate vibrational state vt = 1 , t =  1 of molecules with axial symmetry were calculated by nielsen . although the nielsen theory was fairly satisfactory for the rotation spectrum of these type of molecules , some of the calculated frequencies by this method were different from the observed frequencies . this formula was extended to the case of higher j value by gordy et al . . they also interpreted the transitions j = 4  5 and j = 8  9 of cf3cch in the excited state v10 = 1 . then the frequencies of lines for transition j  j + 1 were investigated and calculated for vt = 1 and vt = 2 states [ 15 , 16 ] . the frequency of transition j  j + 1 for molecules belonging to the point group c3v in singly excited vibrational state vt = 1 , is given by the approximate perturbation expression , equation ( 3 ) . where  has the value  qt ( j + 1 ) if ( k  1 ) = 0 (  - type doubling ) or where is the -type doubling constant , b and a are rotational constants , and  is the z - coriolis constant for the vibrational state vt . the degree of -resonance thus largely depends on the ratio of 
 qt+ to the resonance denominator ( ba + a ) . with low to moderate values of this denominator , the spectrum usually found consists of two extreme outer lines for ( k-1 ) = 0 ( the - doublets ) and a center group of lines for ( k-1 )  0 whose structure depends very strongly on other parameters such as the centrifugal distortion constants . as the strength of the resonance increase , the lines of the center group spread out until the low |k-1| transitions approach the positions of the -doublets . so for large 
 qt+ and small ( a  b  a = 458 mhz ) value it is possible to observe direct -resonance transitions in the microwave range . this is a typical pattern for c3v doubly degenerate states and is predominantly due to the combinations of the splitting of the positive and negative ( k-1 ) series by - type resonance [ which is inversely proportional to ( k-1 ) ] and the shift to low frequency due to djk [ which is proportional to ( k-1 ) ] . thus one series goes from high frequency at low k to low frequency at high k , while the other is at low frequency both for high and low k , with a head at some intermediate value . in order to obtain more accuracy than can be obtained from perturbation theory in rotational energies for singly excited vibrational states it is necessary to set up the rotational hamiltonian as a matrix ( h ) in equation h = e and diagonalise to obtain the energy ( 2,10,11,17,18,21,22 ) . this is particularly necessary when[b  a + ( a ) ] is small in the above equation ; the diagonalization avoids the errors implicit in perturbation theory or the need to go to higher orders of perturbation . the hamiltonian is set up for a symmetric top molecule like cf3cch [ 2 , 11 ] . this rotationvibrational hamiltonian has two different blocks belong to the different  = + 1 and  = 1 series . k and  are no longer good quantum numbers but ( k -  ) or ( k - 1 ) may be used to distinguish between the symmetry species ; those levels with ( k - 1 ) = 3n , where n is an integer , are of species a1 or a2 . if ( k- 1 )  3n the species are e. the qt produces a first order splitting of the ( k - 1 ) = 0 , a1a2 pair which are the familiar  - doublets , as shown by grenier - besson and amat and others [ 2 , 10 , 11 , 15 , 19 ] . the main difference from the ground state spectra is the splitting of the |k - | = 0 into two widely separate  - doublets and the splitting , due to  - resonance . where  is the vibration angular momentum quantum number and  is the coriolis coupling coefficient . hence k and  are not good quantum numbers , though k - 1 may be used as a satisfactory label . the diagonal elements of the hamiltonian used were : in addition there are off - diagonal elements given by : in equation 5 , the dj , djk , dk are quartic , hj , hjk , hkj are sextic centrifugal distortion terms respectively . is the coriolis coupling coefficient between the two components of the degenerate state and j represents the centrifugal distortion of ( a ) . the general effect of the additional parameters to complicate the spectrum relative to that of a non - degenerate state . if |k 1| values are plotted against frequency for this state , the fortrat - diagram is produced which is shown for ch3cch in figure 4 . this diagram shows the '  ' doublet and two different series =  1 . so the spectrum is more complex compared with ground state . in the present example the rotational hamiltonian was diagonalised to give exact -resonance shifts . the spectrum in the millimetre - wave should consist of two widely spaced lines , the -doublets , whose separation is given by equation ( 4 ) , and a central group of transitions with |k 1| > 0 , whose frequencies are determined by equation ( 3 ) . in the latter set the transitions with low values of |k 1| will tend to lie towards the outsides of the group since their -resonance shifts as predicted by equation ( 3 ) will be larges . there are two different series in spectrum , making it more difficult to assign than the ground state figure 4 . the vibrational levels closest to v10 = 2 are v5 = 1 and v9 = 1 at 536 and 452 cm , respectively [ 2 , 9 , 2022 ] . as the analysis will show , the level can be considered as being isolated and an effective hamiltonian for a vt = 2 level is suitable for the analysis of the data . the general features of the direct -resonance transitions in the vt = 2 level have been discussed by harder et al . . there is a good agreement for the obtained rotational constants b and the  -dependence  = ( b2  b0)/4 . because the main information is obtained from the data of carpenter et al . excellent agreement is obtained for the values of 
 g=ev2-ev0/4 , because the parameter a constrained in its value . the b and other spectroscopic constants determined accurately up to v10 = 4 and show a regular dependence on the vibrational quantum number figure 5[2 , 20 , 22 , 23 ] . these spectra show no obvious regularity from the point of view of frequencies as well as intensities , because there are three series for each j since the vibrational angular momentum quantum number  can have values 0 and  2 . therefore it is difficult to identify the quantum numbers k and  associated with each absorption in these spectra . however from the point of view of experimental work , correspondence between the lines of different j value should be observed with respect to their relative position as well as their intensities . in order to obtain more accuracy in rotational energies for excited vibrational state vt = 2 it is necessary to set up the rotational hamiltonian as a matrix ( h ) in eigenvalue equation , h = e , and diagonalise to obtain the energy [ 2022 ] . this hamiltonian can set up for a symmetric top molecule such as cf3cch , cf3ccd , cf3cn , and opf3[10 , 2030 ] . it has three different vibrational blocks related to  = 2 , 0 and 2 figure 6 . the vibrational anharmonicity x separates the  =  2 and  = 0 blocks and they are coupled by the matrix elements qt and rt . in order to explain the form of the spectra we need to consider the energy levels in vt = 2 as given by the hamiltonian matrix . the diagonal elements are : where 0 = pure vibrational frequency which is independent of j , k ,  and x is the anharmonicity factor in . the main off - diagonal elements are the -doubling elements equation ( 6 ) . plotting of 
 2(j+1 ) against ( j + 1 ) can help us to analysis of signals in excited states . which is described below ? from the theoretical point the frequency expression for transition j  j + 1 may be put in following effective form : where a(j , k ,  ) and b(j , k ,  ) are complicated functions of j , k and . for analysis and investigation of signals , equation ( 7 ) can be changed after some abbreviation and alteration to the following form : if both sides of eq ( 13 ) are divided by 2(j + 1 ) , then : hence plotting 
 2(j+1 ) as a function of ( j + 1 ) will produce a series of lines ; two of them relate to k - 1 = 0 with slope - 2dj and intercept 
 b - djkk212qt+ . the k - 1  0 states will produce a series of lines with slope and intercept b - djkk . some of them are nearly parallel to each other and some of them show curvature due to strong  resonance . equation ( 10 ) can be further simplified by taking the term in dj to the right hand side . this is then an effective b value for the given transition : hence termed bplot . plotting of beff as a function of ( j + 1 ) will produce a series of lines , each of which belongs to a different ( k - 1 ) value . correlation of observed lines in different j can then be used to help with the assignment . in vt = 2 state the hamiltonian blocks off into factors which for a c3v molecule are of symmetry species a1 , a2 and e. k and  are no longer good quantum numbers but ( k -  )  s remains a good symmetry label . in this label s is sign of  , s = + 1 for   0 and s = 1 for  < 0 . the spectra are labled by ( k- ) such as 0 , -1 and 0 in the spectrum of j = 19 figure 7 . accidental resonances can occur when the  = 0 series are coupled via the qt term with  = 2 series . this resonance will affect the frequencies of the |j , k ,  > | j + 1 , k ,  > transitions . the maximum resonance will occur when the energy of state  = 2 , k + 2 equals that of  = 0 , k i.e. after omitting terms which are the same from both sides of equation ( 8) this gives : if the rotational constants of cf3cch from table 1 are put in equation ( 18 ) then k is found to be 9.7 . experimentally the resonance is indeed observed between k = 9 and k = 10 for this molecule and is independent of j , as expected from equation ( 18 ) . the calculated spectrum of cf3cch in j = 20  21 show this resonance figure 8 . like this resonance take places in different k values for different molecules which depends on their parameters table 2 . figure 9 shows the energy levels as  = 2 , 0 and 2 stacks . the lines between states in adjacent stacks represent the -type interaction ; for example k = 4 in  =  2 is linked to k = 6 ,  = 0 and this also interacts with k = ( 6 + 2 ) = 8 ,  = + 2 . the interaction results in the energy levels being pushed apart ; for example k = 4 is shifted up and k = 6 is shifted down . the extent of this shift depends on how nearly equal are the energy levels . when they are close , a large shift is obtained . the maximum shift will occur when the energies are almost equal , see equation ( 18 ) and is found in the region of k  9 . the hamiltonian for v10 = 3 has four vibrational blocks correspond to the  = + 3 , + 1 , 1 , 3 figure 10 . the  =  3 and  =  1 blocks are separated in vibrational energy by the anharmonic term . there are , however , couplings between + 3 and + 1 , + 1 , 1 and 1 and 3 , due to the off - diagonal matrix elements qt and rt ( abbreviated as q and r , respectively , in figures . 6 , 10 ) . the labelling of the eigenvalues of this hamiltonian by ( k- ) s where s is 1 for  = + 3 and + 1 , and 1 for  = 3 and  1 ; the symmetry of a given transition can then be easily deduced from the fact that the extension to the v10 = 4 is straightforward . the possible values are + 4,+ 2 , 0 ,  2 , and  4 , so that hamiltonian consists of five blocks and five series are apparent in the spectra . it is worth noting that , to the order of magnitude that was included in the hamiltonian . for v10 = 2 , there are no new parameters in either of the above extension of this hamiltonian to the cases of v10 = 3 and v10 = 4 . for the cf3cch some of the lines belonging to the  = 0 ,  = 2 and  =  2 are shown separately in different plots , ( figures 11 , 12 , 13 ) . in the =  2 series , the lines are nearly parallel to each other , and the distances between these lines become less as |k - |*s becomes smaller . the two lines 0 and 0 , which are expected to have the same frequency but observed to be are separated , the line 0 is parallel with ( j + 1 ) axis whereas 0 line is not figure 14 . the  = 0 series are located between line 9 to line 10 but most of them are on the lower frequency side of the spectrum figure 11 . the lines 1 to 9 are above the 0 line whereas the other observed lines related to 10 and greater are hidden under the 0 lines figures . line 10 is above line 0 and lines 11,12,13,14 cross line 0 but other observed lines belong to the  = 0 are under the 0 line figures . 11,13 . comparing of fortrat diagrams of v10 = 3 and v10 = 4 with corresponding for v10 = 1 figure . the   series of v10 = 3 are qualitatively similar to those of v10 = 1 , with -doublets and strong -resonance . in the same way the  = 0 and + 2 series of v10 = 4 show a similar resonance to the corresponding series in v10 = 2 while the  =  2 series is relatively unperturbed . in the case of v10 = 2 , as k increase , the  = 0 energies come into accidental resonance with those of the positive ( k -  ) s series . such a resonance is also present for v10 = 3 , but in this case , because the vibrational separation due to the x term is so much larger , this resonance only starts to become obvious at high k . the maximum resonance can occur when omitting all centrifugal distortion and higher terms and terms which are the same for both states this gives : equation ( 20 ) . if appropriate values input in equation 20 then solving this equation yields a value of k between 19 and 20 . there is a slight kink in this diagram which is caused by the accidental degeneracy of the |j , k ,  = 1 and |j , k + 4 ,  = + 3 states which are not directly connected by a matrix element but are coupled in second order by the interaction of both of them with |j , k = + 2 ,  = + 1. applying the equation 21 yields a value for the k = 8.4 which is indeed where the perturbation is observed . the  = 0 and  =  2 series have separation in v10 = 4 which are close found in the v10 = 2 vibrational state . consequently , this resonance arises in the same position found for v10 = 2 , namely between k = 9 and 10 . the  =  4 series are further away in the energy due to the x  term . in the v10 = 4 spectrum figure . 15 , the transition with ( k -  ) s = 3 for the 2 series is split by about 3 mhz in the j = 20 spectrum . this splitting is not reproduced by the parameters obtained by extrapolation of the v10 = 2 and v10 = 3 values . in order to account for this , 4. this provides a first - order coupling between the otherwise degenerate energies of the k = + 1 ,  = 2 , and k = 1 ,  = + 2 states which comprise the 3 pair of levels and leads to a splitting of this transition .
this paper deals with review of exploration of resonance in symmetric top molecules in different vibrational excited states , vt = n ( n = 1 , 2 , 3 , 4 ) . 
 calculations for cf3cch shows that resonance take place at 
 k = x+(a - b)-2aa-(a - b ) and 
 k = x+(a - b)-2aa-(a - b ) for v10 = 2 and v10 = 3 respectively . in order to account for splitting about 3 mhz for the  2 series in v10 
 = 4 is necessary to introduce the element 
 j , k , |f24| j , k + 2 ,   4 in fitting program .