- Email: [email protected]
Raman Q and S line broadening and shifting coefficients: some commonly used assumptions revisited
5 February 1999
Chemical Physics Letters 300 Ž1999. 275–280
Raman Q and S line broadening and shifting coefficients: some commonly used assumptions revisited M.-L. Dubernet
a,)
, P.A. Tuckey
b
a
b
Laboratoire de Physique Moleculaire, UMR CNRS 6624, UniÕersite´ de Franche-Comte, ´ ´ F-25030 Besanc¸on Cedex, France Laboratoire d’Astrophysique, UPRES-A 6091 CNRS, ObserÕatoire de Besanc¸on, UniÕersite´ de Franche-Comte, ´ 41 bis aÕenue de l’ObserÕatoire, BP 1615, 25010 Besanc¸on Cedex, France Received 17 September 1998; in final form 23 November 1998
Abstract We use quantum calculations to study two features in the behaviour of Raman SŽ j . and QŽ j . broadening and shifting coefficients of the H 2 –He system: the influence of the centrifugal distortion of the potential, and the j and E dependences of the dephasing contribution to the QŽ j . broadening coefficients. We stress that certain approximations used in modelling experimental data must be reconsidered, and we examine how to properly account for these effects within a semiclassical model. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction In the present Letter, we study two questions: the effect of the centrifugal distortion of the potential energy surface ŽPES. on broadening and shifting coefficients of Raman QŽ j . and SŽ j . lines, and the j and E dependences of the dephasing contributions to the broadening coefficients of QŽ j . lines. The centrifugal distortion of the PES is often ignored in quantum and semiclassical calculations, for the sake of simplicity and because few PES give the full dependence on the internal stretching coordinates of the collision partners. However, the PES of the H 2 –He system w1,2x does give information on the stretching coordinate of the H 2 molecule, allowing a thorough investigation of the influence of the cen-
)
Corresponding author. E-mail: [email protected]
trifugal distortion to be carried out on this system. This is a widely studied system because of its simplicity and because it is a first step towards the understanding of more complex systems, such as pure H 2 or H 2 –H 2 O. The spectra of the H 2 molecule perturbed by H 2 , O 2 and H 2 O are used in combustion applications for the determination of local temperatures and concentrations. The analysis of such spectra requires an accurate knowledge of the broadening and shifting coefficients at high temperature. Predicted values are usually obtained by quantum or semiclassical calculations or by extrapolation of experimental data. Quantum calculations are often more limited than semiclassical models in predicting values at high temperature. The H 2 –He system is an exception, owing to its simplicity and to the large rotational constant of H 2 , and quantum close-coupling calculations can be carried out up to temperatures of about 3000 K. This makes it possible to test
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M.-L. Dubernet, P.A. Tuckey r Chemical Physics Letters 300 (1999) 275–280
the effects of the PES centrifugal distortion on broadening and shifting coefficients of the Raman QŽ j . and SŽ j . lines. We find that for SŽ j . lines the centrifugal distortion has only a small effect on the broadening but has a strong effect on the shift. We show how to add this effect to the Robert–Bonamy ŽRB. semiclassical model w3x and confirm that this leads to the correct behaviour in the calculated cross-sections. Quantum calculations of the dephasing contributions to the broadening of QŽ j . lines of H 2 –He were performed in Ref. w4x and showed a strong dependence on the rotational quantum number j. The possibility of such a j-dependence is usually ignored in the modelling of experimental QŽ j . broadening cross-sections w4,5x. The main reason for this is that the simplicity of the models used allows the estimation of various parameters from the experimental data via a fit; in particular the total inelastic crosssections are extracted in this manner. The approximation of j-independence is also due to a misinterpretation w5x of the definition of the dephasing and inelastic rotational contributions to QŽ j . broadening, based on the RB semiclassical model w3x. It is interesting to investigate further the behaviour of the dephasing contribution in order to assess the reliability of the parameter estimations from experimental data. We find both j and E dependencies which are inconsistent with the experimental models, meaning that comparison of the inelastic cross-sections thus obtained with directly measured values would be inappropriate. We compare our quantum results with the DePristo–Rabitz ŽDeP–R. semiclassical model w6x, and also examine how a correct description of the dephasing contribution may be obtained within the RB model w3x. We performed close-coupling and coupled states calculations with the scattering code MOLSCAT w7x using the hydrid propagator of Alexander and Manolopoulos w8x. These calculations were performed using the same conditions as in Refs. w9,4x. Semiclassical calculations followed the prescription of Robert and Bonamy w3,9x except when stated otherwise. Section 2 is devoted to the effect of centrifugal distortion of the potential. The j and E dependences of the dephasing contribution are studied in Section 3.
2. Centrifugal distortion of the potential The potential energy surface of H 2 –He is available as an expansion in terms of Legendre polynomials: V Ž R ,r , u . s
Ý
Vl Ž R ,r . Pl Ž cos u . ,
Ž 1.
ls0,2,4
where R is the distance joining the atom to the center of mass of the diatom and u is the angle between the intermolecular vector R and the monomer bond vector r of length r. The Vl Ž R,r . are tabulated w2x for 48 values of R and 5 values of r, and the long-range parts are given by van der Waals coefficients C6 , C8 , C10 . At each R, the five potential data points are well fitted w9x by a second-order polynomial in r. The rovibrational wavefunctions of the diatomic molecule H 2 are calculated with the ‘level 6.0’ program w10x and the H 2 PES w11x of Schwartz and Le Roy. We calculated broadening and shifting coefficients with the full PES Ž l s 0, 2,4. as well as with the isotropic PES Ž l s 0. only. When the centrifugal distortion of the potential was to be neglected, the Vl Ž R,r . terms were averaged for a given vibrational level Õ over the rovibrational wavefunction Õ, j s 0. 2.1. Raman S(j) lines A Raman SŽ j .-line corresponds to an optical transition between the rotational levels j and j q 2 for a given vibrational state of the optical molecule. The contribution of the isotropic PES to the SŽ j .-line broadening and shifting coefficients is zero when the centrifugal distortion of the potential is ignored. When the centrifugal distortion is taken into account, the contributions from the isotropic PES to the broadening are non-zero and strongly j-dependent, but remain small ŽFig. 1.. Overall, the centrifugal distortion has a small effect on the broadening cross-sections. As expected, the effect is stronger for higher values of j, ranging from 1% for the SŽ1. line to 5% for the SŽ3. line at 5000 cmy1 . In contrast, the centrifugal distortion has a strong effect on the shifting cross-sections, they are increased by a factor of 2–3 at 5000 cmy1 ŽFig. 2.. Further, the major contributions come from the isotropic term Ž l s 0. in the PES; they represent 60%
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277
part of the PES is given by the first-order term S1,w0 jq2,0 j x which is the difference between the contributions of elastic collisions when the optical molecule is in its initial and final states: S1,w0 jq2,0 j x s S1,0 jq2 y S1,0 j .
˚2 Fig. 1. SŽ j .-line broadening close coupling cross-sections in A versus kinetic energy E: effect of the centrifugal distortion of the PES. Dotted lines labelled ‘SŽ j . – V0 ’ show the broadening crosssections calculated with the isotropic potential only and centrifugal distortion Žnote the right-hand scale applies to these curves.. The cross-sections calculated with a full PES including the centrifugal distortion of the PES are given in solid lines labelled ‘SŽ j .’, while long-dashed lines labelled ‘SŽ j . –no CD’ display the cross-sections calculated with a full PES neglecting the centrifugal distortion.
of the total cross-section for the SŽ1.-line and 80% for the SŽ3.-line. This means that one should be careful with the approximations often used in semiclassical calculations such as the RB model w3x. The RB expression of the SŽ j . shifting cross-section for a given kinetic energy E is
Ž 4.
Eq. 8 of Ref. w3x approximates S1, Õ j as S1,Õ , involving the average of the isotropic potential over initial or final vibrational states of the optical molecule and thus neglecting the centrifugal distortion of the potential. This leads to a zero value of the first-order term S1,w0 jq2,0 j x. Instead, S1,Õ j should be written as: q`
S1,Õ j s "y1
Hy` d t ² Õj V
iso
r Ž t . Õj : ,
Ž 5.
where < Õj : is the rovibrational state. With this expression, the first-order term S1,w0 jq2,0 j x is no longer zero for SŽ j . lines, thus accounting for the effects of the centrifugal distortion of the isotropic potential, which, as we have seen, is an important contribution. Fig. 2 shows the RB shifting cross-sections for the SŽ1. line calculated with and without the first-order term, demonstrating the improvement given by this term. The effect of the centrifugal distortion of the
sdRB Ž Õ s 0 j q 2,Õs0 j; E . s²eyR ew S 2 ,w0 jq 2 ,0 j xŽ b , E .x sin h Ž b, E . :b
Ž 2.
with
h Ž b, E . s S1, w 0 jq2,0 j x Ž b, E . q Im S2, w 0 jq2,0 j x Ž b, E .
4, Ž 3.
where b is the impact parameter, the terms Rew S2,w0 jq2,0 j x x and Imw S2,w0 jq2,0 j x x involve the anisotropic part of the potential; Rew S2,w0 jq2,0 j x x is a sum of contributions due to elastic and inelastic collisions on the initial j and final j q 2 optical states, and Imw S2,w0 jq2,0 j x x expresses the difference between the inelastic contributions of the initial and final optical states. The contribution of the isotropic
˚2 Fig. 2. SŽ j .-line shifting close coupling cross-sections in A versus kinetic energy E: effect of the centrifugal distortion of the PES. Comparison between cross-sections calculated with the isotropic potential only and centrifugal distortion Ždotted lines labelled ‘SŽ j . – V0 ’., with the full PES and centrifugal distortion Žsolid lines labelled ‘SŽ j .’. and with the full PES neglecting centrifugal distortion Žlong-dashed lines labelled ‘SŽ j . –no CD’.. The semiclassical SŽ1.-line shifting coefficients w5x with and without centrifugal distortion of the PES are given by the open circles Žlabelled ‘RB–CD’ and ‘RB–no CD’, respectively..
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278
potential will obviously be stronger when j increases. 2.2. Raman Q(j) line A Raman QŽ j . line corresponds to an optical transition between two vibrational states Õ i and Õf when the optical molecule is in a given rotational state j. The isotropic contributions to the broadening and shifting cross-sections are identical when the centrifugal distortion of the potential is ignored. When the centrifugal distortion is introduced, it has a small effect of about 1–2% on all QŽ j . cross-sections and can thus be safely ignored in a first approximation.
3. j and E dependence of the dephasing contribution in Raman Q(j) lines For Raman QŽ j . lines, the close coupling broadening cross-section can be exactly written w12,4x as the sum of the total average inelastic cross-section s i Ž E . and of a dephasing cross-section sd Ž E .:
P sd Ž E . s
2k
Ž 2 J q 1. 2
Ý Ž 2 j q 1. Jll X
S ÕJi jlX ;Õ i jl y S ÕJf jlX ;Õ f jl
2
.
Ž 6. Close-coupling calculations showed w4x that the dephasing contribution sd Ž E . had a strong j-dependence. The experimental broadening coefficients of Ref. w4x were modelled as the sum of an inelastic broadening coefficient and a j-independent dephas-
Table 1 Vibrational elastic dephasing contributions w10y3 cmy1 amagaty1 x for H 2 –He broadening coefficients. Comparison between gÕ ŽEXP-FIT. extracted from experimental data ŽTable 8, Ref. w1x., gÕ ŽCC-FIT. extracted from close-coupling calculations using the same modeling as for experimental data ŽTable 9, Ref. w1x. and gÕCC ŽT . calculated with close-coupling methods and with the isotropic part of the potential T
296 K
600 K
795 K
995 K
gÕ ŽEXP-FIT. gÕ ŽCC-FIT. gÕCC ŽT .
1.58 1.45 1.45
3.32 2.89 2.81
4.49 3.62 3.58
5.38 4.28 4.31
Fig. 3. QŽ j .-line: comparison between the total dephasing contribution to the broadening cross-section Žsolid lines labelled ‘QŽ j . – dep’. and the vibrational dephasing contribution Ždotted lines labelled ‘QŽ j . – V0 ’.. The effect of neglecting the centrifugal distortion of the PES on the dephasing contributions is shown in big open symbols and labelled ‘QŽ j . –dep–no CD’. Centrifugal distortion lifts the degeneracy of the vibrational dephasing contributions.
ing contribution gÕ ŽEXP-FIT. thought of as a vibrational broadening. The gÕ ŽEXP-FIT. term was viewed as arising solely from elastic collisions on the isotropic potential, whereas the inelastic coefficient was supposed to give the state-to-state rotational energy transfer rate. Table 1 shows that the quantum theoretical isotropic dephasing gÕCC ŽT . calculated with the isotropic potential is very close both to the gÕ ŽEXP-FIT. w4x extracted from the experimental data and to the gÕ ŽCC-FIT. w4x extracted from close coupling calculations by use of the same model. However, as shown in Fig. 3, the total dephasing term sd Ž E ., calculated with either of the close-coupling or coupled states methods, is very different from the vibrational dephasing gÕCC Ž E .. Firstly, it has a strong j-dependence, increasing as j increases over the whole range of energy. Secondly, the total dephasing term increases with energy up to the opening of the first inelastic channel and then decreases, unlike the isotropic contribution which rises monotonically with E. This E-dependence is consistent with the prediction of DePristo and Rabitz w6x, that the total dephasing term decreases when the inelastic contributions increase. These authors used the coupled states expression of the relaxation cross-section and a semiclassical approximation of
M.-L. Dubernet, P.A. Tuckey r Chemical Physics Letters 300 (1999) 275–280
279
the total inelastic probability in order to express the real part of the relaxation cross-section in the form:
s Ž Õi j;Õf j; E . , 2 P bj2 q
2P k2
Ý Ž 2 l q 1. l
= 1 y exp y Ž l q 12 . rkbj
½
=Ý V
ž
j V
j yV
0 0
5
/
=sin2 Ž hjÕVi l y hjÕVf l . .
Ž 7.
The first term on the right-hand side is the total inelastic cross-section, where bj has the interpretation of an effective impact parameter for inelastic collisions of the molecule in state j, and the second term is the dephasing cross-section where hjÕVl is the phase of the S-matrix element S jÕVl , j V . As a semi-independent demonstration of our results we have estimated the DeP–R dephasing term from our coupled states calculations: the effective impact parameter bj was extracted from the inelastic cross-sections and the term D s hjÕVi l y hjÕVf l was calculated from the S-matrix elements in the Õ i s 0 and Õf s 1 states. Table 2 shows the good agreement between the CS, CC and estimated DeP–R dephasing contributions for the QŽ0. line. We conclude that the modelling of experimental data carried out in Refs. w4,5x overestimates more and more the dephasing contribution to the QŽ j . broadening coefficients as the temperature increases and for the lower values of j. This implies that a comparison between the extracted state-to-state
Fig. 4. QŽ0.-line: various contributions from Eq. Ž10. to the semiclassical dephasing cross-section sdRB . The long dashed line shows the close coupling vibrational dephasing cross-section g VCC .
rotational energy transfer rates and directly measured rates would be inappropriate. It is interesting to ask if it is possible to obtain a correct description of the dephasing contribution within the semiclassical RB model w3x, where the broadening cross-section expression is w9x
sgRB Ž 1 j,0 j; E . s²1 y eyR ew S 2 ,w1 j,0 j xŽ b , E .x cos h Ž b, E . :b Ž 8. with h Ž b, E . given by Eq. Ž3.. If the terms h1 s S1,w1 j,0 j x , h 2 s Imw S2,w1 j,0 j x x and S2 s Rew S2,w1 j,0 j x x are small, the above expression may be expanded as sgRB Ž 1 j,0 j; E .
¦
, S2 Ž inelastic. q S2 Ž elastic.
Table 2 ˚ 2 . to the broadening crossQŽ0.-line: dephasing contributions ŽA sections as a function of the relative kinetic energy E. Comparison between our estimation of the DePristo and Rabitz dephasing cross-section ŽDeP–R. w12x and our calculated values obtained with close-coupling ŽCC. and coupled-states ŽCS. calculations E Žcmy1 .
CC
CS
DeP–R
20 60 100 250 500 1000 2000 3000 5000
0.075 0.148 0.212 0.377 0.479 0.539 0.527 0.477 0.375
0.075 0.148 0.212 0.376 0.478 0.532 0.510 0.453 0.347
0.075 0.148 0.212 0.384 0.478 0.526 0.513 0.471 0.380
h12
h 22
;
q h1h 2 . Ž 9. 2 2 b We further neglect the centrifugal distortion of the potential since quantum calculations showed that it has a small effect, so the term S2 Želastic. is zero. The term S2 Žinelastic. corresponds to the inelastic contributions to the broadening cross-section. Therefore, following the quantum separation, the dephasing contribution sdRB is identified with the remaining terms: h12 h 22 sdRB s q q h1h 2 . Ž 10 . 2 2 b The Boltzmann average over energy of ²h12r2: b is the vibrational elastic dephasing term and is equivalent to the experimental gÕ ŽEXP-FIT. term and to q
¦
q
;
280
M.-L. Dubernet, P.A. Tuckey r Chemical Physics Letters 300 (1999) 275–280
Fig. 5. QŽ j .-line: dephasing contributions to the broadening cross-sections obtained with close coupling calculations, sdCC , Žlong dashed lines. and with the semiclassical Eq. Ž10., sdRB , Žsolid lines..
the quantum gÕCC ŽT . term calculated with the isotropic potential. ²wh 22r2 q h1h 2 x: b is the contribution to dephasing arising from inelastic collisions. In the case of H 2 –He studied here, the addition of the term ²h1h 2 : b gives the required behaviour for the dephasing contribution. h 2 is negative and ²h1h 2 : b damps the term ²h12r2: b as soon as the first inelastic channel is opened ŽFig. 4.. Fig. 5 shows the good qualitative agreement between dephasing cross-sections calculated with close-coupling methods, sdCC , and those obtained with Eq. Ž10..
4. Conclusion We showed, for the system H 2 –He, that the common approximation of neglecting the centrifugal distortion of the potential leads to a wrong determination of the shifting coefficients for Raman SŽ j . lines by a factor of 2–3. We showed how to include the important contribution from the centrifugal distortion of the isotropic potential in the semiclassical RB model. We then showed, again for H 2 –He, that the simple j-independent dephasing contribution assumed in
modelling of experimental QŽ j . broadening coefficients is not correct. Further, the E-dependence of the dephasing contribution is qualitatively different from that given by the modelling of experimental data. We compared our results with the DeP–R semiclassical model, finding good agreement. We considered how to obtain the dephasing contribution correctly within the RB model, obtaining good agreement for the system studied. More experimental and theoretical studies are currently being performed on Raman QŽ j . and SŽ j . lines of H 2 perturbed by more complex colliders such as H 2 , N2 and H 2 O. It is therefore important to be aware of the defaults of the above approximations both in theoretical and experimental approaches, before going to more complicated calculations and to further approximations.
References w1x W. Meyer, P.C. Hariharan, W. Kutzelnigg, J. Chem. Phys. 73 Ž1980. 1880. w2x J. Schaefer, W.E. Kohler, Physica A 129 Ž1985. 469. ¨ w3x D. Robert, J. Bonamy, J. Phys. ŽParis. 40 Ž1979. 923. w4x X. Michaut, R. Saint-Loup, H. Berger, M.L. Dubernet, P. Joubert, J. Bonamy, J. Chem. Phys. 109 Ž1998. 951. w5x K.C. Smyth, G.J. Rosasco, W.S. Hurst, J. Chem. Phys. 87 Ž1987. 4413. w6x A.E. DePristo, H. Rabitz, J. Quant. Spectrosc. Radiat. Transfer 22 Ž1979. 65. w7x J.M. Hutson, S. Green, MOLSCAT computer code,version 12 ŽCollaborative Computational Project No. 6 of the Science and Engineering Research Council, UK, 1993.. w8x M.H. Alexander, D.E. Manolopoulos, J. Chem. Phys. 86 Ž1987. 2044. w9x P. Joubert, M.-L. Dubernet, J. Bonamy, D. Robert, J. Chem. Phys. 107 Ž1997. 3845. w10x R.J. Le Roy, LEVEL 6.0 A Computer Program Solving the Radial Schrodinger Equation for Bound and Quasibound ¨ Levels, and Calculating Various Expectation Values and Matrix Elements ŽChemical Physics Research Report CP-555, University of Waterloo, 1995.. w11x C. Schwartz, R.J. Le Roy, J. Mol. Spectrosc. 121 Ž1987. 420. w12x R. Blackmore, S. Green, L. Monchick, J. Chem. Phys. 88 Ž1988. 4113.
Chemical Physics Letters 300 Ž1999. 275–280
Raman Q and S line broadening and shifting coefficients: some commonly used assumptions revisited M.-L. Dubernet
a,)
, P.A. Tuckey
b
a
b
Laboratoire de Physique Moleculaire, UMR CNRS 6624, UniÕersite´ de Franche-Comte, ´ ´ F-25030 Besanc¸on Cedex, France Laboratoire d’Astrophysique, UPRES-A 6091 CNRS, ObserÕatoire de Besanc¸on, UniÕersite´ de Franche-Comte, ´ 41 bis aÕenue de l’ObserÕatoire, BP 1615, 25010 Besanc¸on Cedex, France Received 17 September 1998; in final form 23 November 1998
Abstract We use quantum calculations to study two features in the behaviour of Raman SŽ j . and QŽ j . broadening and shifting coefficients of the H 2 –He system: the influence of the centrifugal distortion of the potential, and the j and E dependences of the dephasing contribution to the QŽ j . broadening coefficients. We stress that certain approximations used in modelling experimental data must be reconsidered, and we examine how to properly account for these effects within a semiclassical model. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction In the present Letter, we study two questions: the effect of the centrifugal distortion of the potential energy surface ŽPES. on broadening and shifting coefficients of Raman QŽ j . and SŽ j . lines, and the j and E dependences of the dephasing contributions to the broadening coefficients of QŽ j . lines. The centrifugal distortion of the PES is often ignored in quantum and semiclassical calculations, for the sake of simplicity and because few PES give the full dependence on the internal stretching coordinates of the collision partners. However, the PES of the H 2 –He system w1,2x does give information on the stretching coordinate of the H 2 molecule, allowing a thorough investigation of the influence of the cen-
)
Corresponding author. E-mail: [email protected]
trifugal distortion to be carried out on this system. This is a widely studied system because of its simplicity and because it is a first step towards the understanding of more complex systems, such as pure H 2 or H 2 –H 2 O. The spectra of the H 2 molecule perturbed by H 2 , O 2 and H 2 O are used in combustion applications for the determination of local temperatures and concentrations. The analysis of such spectra requires an accurate knowledge of the broadening and shifting coefficients at high temperature. Predicted values are usually obtained by quantum or semiclassical calculations or by extrapolation of experimental data. Quantum calculations are often more limited than semiclassical models in predicting values at high temperature. The H 2 –He system is an exception, owing to its simplicity and to the large rotational constant of H 2 , and quantum close-coupling calculations can be carried out up to temperatures of about 3000 K. This makes it possible to test
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 1 3 3 4 - 7
276
M.-L. Dubernet, P.A. Tuckey r Chemical Physics Letters 300 (1999) 275–280
the effects of the PES centrifugal distortion on broadening and shifting coefficients of the Raman QŽ j . and SŽ j . lines. We find that for SŽ j . lines the centrifugal distortion has only a small effect on the broadening but has a strong effect on the shift. We show how to add this effect to the Robert–Bonamy ŽRB. semiclassical model w3x and confirm that this leads to the correct behaviour in the calculated cross-sections. Quantum calculations of the dephasing contributions to the broadening of QŽ j . lines of H 2 –He were performed in Ref. w4x and showed a strong dependence on the rotational quantum number j. The possibility of such a j-dependence is usually ignored in the modelling of experimental QŽ j . broadening cross-sections w4,5x. The main reason for this is that the simplicity of the models used allows the estimation of various parameters from the experimental data via a fit; in particular the total inelastic crosssections are extracted in this manner. The approximation of j-independence is also due to a misinterpretation w5x of the definition of the dephasing and inelastic rotational contributions to QŽ j . broadening, based on the RB semiclassical model w3x. It is interesting to investigate further the behaviour of the dephasing contribution in order to assess the reliability of the parameter estimations from experimental data. We find both j and E dependencies which are inconsistent with the experimental models, meaning that comparison of the inelastic cross-sections thus obtained with directly measured values would be inappropriate. We compare our quantum results with the DePristo–Rabitz ŽDeP–R. semiclassical model w6x, and also examine how a correct description of the dephasing contribution may be obtained within the RB model w3x. We performed close-coupling and coupled states calculations with the scattering code MOLSCAT w7x using the hydrid propagator of Alexander and Manolopoulos w8x. These calculations were performed using the same conditions as in Refs. w9,4x. Semiclassical calculations followed the prescription of Robert and Bonamy w3,9x except when stated otherwise. Section 2 is devoted to the effect of centrifugal distortion of the potential. The j and E dependences of the dephasing contribution are studied in Section 3.
2. Centrifugal distortion of the potential The potential energy surface of H 2 –He is available as an expansion in terms of Legendre polynomials: V Ž R ,r , u . s
Ý
Vl Ž R ,r . Pl Ž cos u . ,
Ž 1.
ls0,2,4
where R is the distance joining the atom to the center of mass of the diatom and u is the angle between the intermolecular vector R and the monomer bond vector r of length r. The Vl Ž R,r . are tabulated w2x for 48 values of R and 5 values of r, and the long-range parts are given by van der Waals coefficients C6 , C8 , C10 . At each R, the five potential data points are well fitted w9x by a second-order polynomial in r. The rovibrational wavefunctions of the diatomic molecule H 2 are calculated with the ‘level 6.0’ program w10x and the H 2 PES w11x of Schwartz and Le Roy. We calculated broadening and shifting coefficients with the full PES Ž l s 0, 2,4. as well as with the isotropic PES Ž l s 0. only. When the centrifugal distortion of the potential was to be neglected, the Vl Ž R,r . terms were averaged for a given vibrational level Õ over the rovibrational wavefunction Õ, j s 0. 2.1. Raman S(j) lines A Raman SŽ j .-line corresponds to an optical transition between the rotational levels j and j q 2 for a given vibrational state of the optical molecule. The contribution of the isotropic PES to the SŽ j .-line broadening and shifting coefficients is zero when the centrifugal distortion of the potential is ignored. When the centrifugal distortion is taken into account, the contributions from the isotropic PES to the broadening are non-zero and strongly j-dependent, but remain small ŽFig. 1.. Overall, the centrifugal distortion has a small effect on the broadening cross-sections. As expected, the effect is stronger for higher values of j, ranging from 1% for the SŽ1. line to 5% for the SŽ3. line at 5000 cmy1 . In contrast, the centrifugal distortion has a strong effect on the shifting cross-sections, they are increased by a factor of 2–3 at 5000 cmy1 ŽFig. 2.. Further, the major contributions come from the isotropic term Ž l s 0. in the PES; they represent 60%
M.-L. Dubernet, P.A. Tuckey r Chemical Physics Letters 300 (1999) 275–280
277
part of the PES is given by the first-order term S1,w0 jq2,0 j x which is the difference between the contributions of elastic collisions when the optical molecule is in its initial and final states: S1,w0 jq2,0 j x s S1,0 jq2 y S1,0 j .
˚2 Fig. 1. SŽ j .-line broadening close coupling cross-sections in A versus kinetic energy E: effect of the centrifugal distortion of the PES. Dotted lines labelled ‘SŽ j . – V0 ’ show the broadening crosssections calculated with the isotropic potential only and centrifugal distortion Žnote the right-hand scale applies to these curves.. The cross-sections calculated with a full PES including the centrifugal distortion of the PES are given in solid lines labelled ‘SŽ j .’, while long-dashed lines labelled ‘SŽ j . –no CD’ display the cross-sections calculated with a full PES neglecting the centrifugal distortion.
of the total cross-section for the SŽ1.-line and 80% for the SŽ3.-line. This means that one should be careful with the approximations often used in semiclassical calculations such as the RB model w3x. The RB expression of the SŽ j . shifting cross-section for a given kinetic energy E is
Ž 4.
Eq. 8 of Ref. w3x approximates S1, Õ j as S1,Õ , involving the average of the isotropic potential over initial or final vibrational states of the optical molecule and thus neglecting the centrifugal distortion of the potential. This leads to a zero value of the first-order term S1,w0 jq2,0 j x. Instead, S1,Õ j should be written as: q`
S1,Õ j s "y1
Hy` d t ² Õj V
iso
r Ž t . Õj : ,
Ž 5.
where < Õj : is the rovibrational state. With this expression, the first-order term S1,w0 jq2,0 j x is no longer zero for SŽ j . lines, thus accounting for the effects of the centrifugal distortion of the isotropic potential, which, as we have seen, is an important contribution. Fig. 2 shows the RB shifting cross-sections for the SŽ1. line calculated with and without the first-order term, demonstrating the improvement given by this term. The effect of the centrifugal distortion of the
sdRB Ž Õ s 0 j q 2,Õs0 j; E . s²eyR ew S 2 ,w0 jq 2 ,0 j xŽ b , E .x sin h Ž b, E . :b
Ž 2.
with
h Ž b, E . s S1, w 0 jq2,0 j x Ž b, E . q Im S2, w 0 jq2,0 j x Ž b, E .
4, Ž 3.
where b is the impact parameter, the terms Rew S2,w0 jq2,0 j x x and Imw S2,w0 jq2,0 j x x involve the anisotropic part of the potential; Rew S2,w0 jq2,0 j x x is a sum of contributions due to elastic and inelastic collisions on the initial j and final j q 2 optical states, and Imw S2,w0 jq2,0 j x x expresses the difference between the inelastic contributions of the initial and final optical states. The contribution of the isotropic
˚2 Fig. 2. SŽ j .-line shifting close coupling cross-sections in A versus kinetic energy E: effect of the centrifugal distortion of the PES. Comparison between cross-sections calculated with the isotropic potential only and centrifugal distortion Ždotted lines labelled ‘SŽ j . – V0 ’., with the full PES and centrifugal distortion Žsolid lines labelled ‘SŽ j .’. and with the full PES neglecting centrifugal distortion Žlong-dashed lines labelled ‘SŽ j . –no CD’.. The semiclassical SŽ1.-line shifting coefficients w5x with and without centrifugal distortion of the PES are given by the open circles Žlabelled ‘RB–CD’ and ‘RB–no CD’, respectively..
M.-L. Dubernet, P.A. Tuckey r Chemical Physics Letters 300 (1999) 275–280
278
potential will obviously be stronger when j increases. 2.2. Raman Q(j) line A Raman QŽ j . line corresponds to an optical transition between two vibrational states Õ i and Õf when the optical molecule is in a given rotational state j. The isotropic contributions to the broadening and shifting cross-sections are identical when the centrifugal distortion of the potential is ignored. When the centrifugal distortion is introduced, it has a small effect of about 1–2% on all QŽ j . cross-sections and can thus be safely ignored in a first approximation.
3. j and E dependence of the dephasing contribution in Raman Q(j) lines For Raman QŽ j . lines, the close coupling broadening cross-section can be exactly written w12,4x as the sum of the total average inelastic cross-section s i Ž E . and of a dephasing cross-section sd Ž E .:
P sd Ž E . s
2k
Ž 2 J q 1. 2
Ý Ž 2 j q 1. Jll X
S ÕJi jlX ;Õ i jl y S ÕJf jlX ;Õ f jl
2
.
Ž 6. Close-coupling calculations showed w4x that the dephasing contribution sd Ž E . had a strong j-dependence. The experimental broadening coefficients of Ref. w4x were modelled as the sum of an inelastic broadening coefficient and a j-independent dephas-
Table 1 Vibrational elastic dephasing contributions w10y3 cmy1 amagaty1 x for H 2 –He broadening coefficients. Comparison between gÕ ŽEXP-FIT. extracted from experimental data ŽTable 8, Ref. w1x., gÕ ŽCC-FIT. extracted from close-coupling calculations using the same modeling as for experimental data ŽTable 9, Ref. w1x. and gÕCC ŽT . calculated with close-coupling methods and with the isotropic part of the potential T
296 K
600 K
795 K
995 K
gÕ ŽEXP-FIT. gÕ ŽCC-FIT. gÕCC ŽT .
1.58 1.45 1.45
3.32 2.89 2.81
4.49 3.62 3.58
5.38 4.28 4.31
Fig. 3. QŽ j .-line: comparison between the total dephasing contribution to the broadening cross-section Žsolid lines labelled ‘QŽ j . – dep’. and the vibrational dephasing contribution Ždotted lines labelled ‘QŽ j . – V0 ’.. The effect of neglecting the centrifugal distortion of the PES on the dephasing contributions is shown in big open symbols and labelled ‘QŽ j . –dep–no CD’. Centrifugal distortion lifts the degeneracy of the vibrational dephasing contributions.
ing contribution gÕ ŽEXP-FIT. thought of as a vibrational broadening. The gÕ ŽEXP-FIT. term was viewed as arising solely from elastic collisions on the isotropic potential, whereas the inelastic coefficient was supposed to give the state-to-state rotational energy transfer rate. Table 1 shows that the quantum theoretical isotropic dephasing gÕCC ŽT . calculated with the isotropic potential is very close both to the gÕ ŽEXP-FIT. w4x extracted from the experimental data and to the gÕ ŽCC-FIT. w4x extracted from close coupling calculations by use of the same model. However, as shown in Fig. 3, the total dephasing term sd Ž E ., calculated with either of the close-coupling or coupled states methods, is very different from the vibrational dephasing gÕCC Ž E .. Firstly, it has a strong j-dependence, increasing as j increases over the whole range of energy. Secondly, the total dephasing term increases with energy up to the opening of the first inelastic channel and then decreases, unlike the isotropic contribution which rises monotonically with E. This E-dependence is consistent with the prediction of DePristo and Rabitz w6x, that the total dephasing term decreases when the inelastic contributions increase. These authors used the coupled states expression of the relaxation cross-section and a semiclassical approximation of
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the total inelastic probability in order to express the real part of the relaxation cross-section in the form:
s Ž Õi j;Õf j; E . , 2 P bj2 q
2P k2
Ý Ž 2 l q 1. l
= 1 y exp y Ž l q 12 . rkbj
½
=Ý V
ž
j V
j yV
0 0
5
/
=sin2 Ž hjÕVi l y hjÕVf l . .
Ž 7.
The first term on the right-hand side is the total inelastic cross-section, where bj has the interpretation of an effective impact parameter for inelastic collisions of the molecule in state j, and the second term is the dephasing cross-section where hjÕVl is the phase of the S-matrix element S jÕVl , j V . As a semi-independent demonstration of our results we have estimated the DeP–R dephasing term from our coupled states calculations: the effective impact parameter bj was extracted from the inelastic cross-sections and the term D s hjÕVi l y hjÕVf l was calculated from the S-matrix elements in the Õ i s 0 and Õf s 1 states. Table 2 shows the good agreement between the CS, CC and estimated DeP–R dephasing contributions for the QŽ0. line. We conclude that the modelling of experimental data carried out in Refs. w4,5x overestimates more and more the dephasing contribution to the QŽ j . broadening coefficients as the temperature increases and for the lower values of j. This implies that a comparison between the extracted state-to-state
Fig. 4. QŽ0.-line: various contributions from Eq. Ž10. to the semiclassical dephasing cross-section sdRB . The long dashed line shows the close coupling vibrational dephasing cross-section g VCC .
rotational energy transfer rates and directly measured rates would be inappropriate. It is interesting to ask if it is possible to obtain a correct description of the dephasing contribution within the semiclassical RB model w3x, where the broadening cross-section expression is w9x
sgRB Ž 1 j,0 j; E . s²1 y eyR ew S 2 ,w1 j,0 j xŽ b , E .x cos h Ž b, E . :b Ž 8. with h Ž b, E . given by Eq. Ž3.. If the terms h1 s S1,w1 j,0 j x , h 2 s Imw S2,w1 j,0 j x x and S2 s Rew S2,w1 j,0 j x x are small, the above expression may be expanded as sgRB Ž 1 j,0 j; E .
¦
, S2 Ž inelastic. q S2 Ž elastic.
Table 2 ˚ 2 . to the broadening crossQŽ0.-line: dephasing contributions ŽA sections as a function of the relative kinetic energy E. Comparison between our estimation of the DePristo and Rabitz dephasing cross-section ŽDeP–R. w12x and our calculated values obtained with close-coupling ŽCC. and coupled-states ŽCS. calculations E Žcmy1 .
CC
CS
DeP–R
20 60 100 250 500 1000 2000 3000 5000
0.075 0.148 0.212 0.377 0.479 0.539 0.527 0.477 0.375
0.075 0.148 0.212 0.376 0.478 0.532 0.510 0.453 0.347
0.075 0.148 0.212 0.384 0.478 0.526 0.513 0.471 0.380
h12
h 22
;
q h1h 2 . Ž 9. 2 2 b We further neglect the centrifugal distortion of the potential since quantum calculations showed that it has a small effect, so the term S2 Želastic. is zero. The term S2 Žinelastic. corresponds to the inelastic contributions to the broadening cross-section. Therefore, following the quantum separation, the dephasing contribution sdRB is identified with the remaining terms: h12 h 22 sdRB s q q h1h 2 . Ž 10 . 2 2 b The Boltzmann average over energy of ²h12r2: b is the vibrational elastic dephasing term and is equivalent to the experimental gÕ ŽEXP-FIT. term and to q
¦
q
;
280
M.-L. Dubernet, P.A. Tuckey r Chemical Physics Letters 300 (1999) 275–280
Fig. 5. QŽ j .-line: dephasing contributions to the broadening cross-sections obtained with close coupling calculations, sdCC , Žlong dashed lines. and with the semiclassical Eq. Ž10., sdRB , Žsolid lines..
the quantum gÕCC ŽT . term calculated with the isotropic potential. ²wh 22r2 q h1h 2 x: b is the contribution to dephasing arising from inelastic collisions. In the case of H 2 –He studied here, the addition of the term ²h1h 2 : b gives the required behaviour for the dephasing contribution. h 2 is negative and ²h1h 2 : b damps the term ²h12r2: b as soon as the first inelastic channel is opened ŽFig. 4.. Fig. 5 shows the good qualitative agreement between dephasing cross-sections calculated with close-coupling methods, sdCC , and those obtained with Eq. Ž10..
4. Conclusion We showed, for the system H 2 –He, that the common approximation of neglecting the centrifugal distortion of the potential leads to a wrong determination of the shifting coefficients for Raman SŽ j . lines by a factor of 2–3. We showed how to include the important contribution from the centrifugal distortion of the isotropic potential in the semiclassical RB model. We then showed, again for H 2 –He, that the simple j-independent dephasing contribution assumed in
modelling of experimental QŽ j . broadening coefficients is not correct. Further, the E-dependence of the dephasing contribution is qualitatively different from that given by the modelling of experimental data. We compared our results with the DeP–R semiclassical model, finding good agreement. We considered how to obtain the dephasing contribution correctly within the RB model, obtaining good agreement for the system studied. More experimental and theoretical studies are currently being performed on Raman QŽ j . and SŽ j . lines of H 2 perturbed by more complex colliders such as H 2 , N2 and H 2 O. It is therefore important to be aware of the defaults of the above approximations both in theoretical and experimental approaches, before going to more complicated calculations and to further approximations.
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