- Email: [email protected]
Rotational excitation of diatomic molecules in exothermic processes
Chemical Physics 111 (1987) 17-20 North-Holland, Amsterdam
17
ROTATIONAL EXCITATION OF DIATOMIC MOLECULES IN EXOTHERMIC PROCESSES Dirk POPPE Institut ftir Physikalische Chemie, Freie Universitiit Berlin, Takustrasse 3, I000 Berlin 33, FRG Received 30 June 1986
We describe a simple model which is capable of explaining the rotational distribution of Nz after the quenching reaction Na(3p) + Nz + Na(3s) + NZ. Restrictions by energy and total angular momentum conservation together with the dynamics in the product valley of the potential surface determine the rotational energy transfer.
1. Introduction Energy transfer processes and chemical reactions between atoms and molecules are usually accompanied by the rotational excitation of the products. In this article we would like to present a simple model which explains qualitatively the rotational distribution of a diatomic molecule B’C’ produced in an exothermic reaction A+BC+A’+B’C’+AhE,
(1)
where the exothermicity is denoted by AE. The model is applicable if the following conditions are fulfilled. (1) The exothermicity is large compared to the initial collision energy. (2) The rotational excitation is mainly generated during the second half of the collision. Then the system has entered the exit channel of the reaction where the products begin to separate. The first part of the collision does not produce any significant rotational excitation. (3) The dynamics in the second half of the collision is impulsive and can be well described by the infinite-order-sudden (10s) approximation of rotationally inelastic scattering. The higher the exothermicity is the more appropriate this approximation will be.
(4) Vibrational excitation plays only a minor role during the second part of the scattering process. This condition is not essential for our model. It merely facilitates the calculation of the rotational excitation. In section 2 we shall discuss the details of the approach. For simplicity we confine ourselves to a classical version although the idea can certainly be worked out quantum mechanically. We shall discuss how it is related to Schinke’s reflection principle [1,2]. The model is applied to the quenching of Na(3p) in collisions with N,.
2. The model The energy balance for reaction (1) reads E totd=E+EBC(n, --, E’ + EB&‘,
i)+AE j’),
(2)
where n, j denotes the vibrational and the rotational quantum number, respectively. The translational energy is denoted by E. Unprimed (primed) quantities correspond to the initial (final) states. We divide now an individual collision into two parts. The first part is the motion in the entrance valley of the potential surface. Rotational excita-
0301-0104/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
18
D. Poppe / Rotational excitution
tion during this part is likely to be small since the initial collision energy is small. In any case the intermediate rotational energy is bounded by E,,, < Etotal - u,
of diatomic molecules
j’( b', a’) = -m’
(3)
where U is the minimum of the potential that has to be passed so that the reaction takes place. The second part starts when the system has ultimately reached the valley of the products. The potential is now basically repulsive. The final translational energy is much larger than the initial collision energy. If the anisotropy is large enough we expect large rotational excitation. Our model for the dynamics of rotational excitation is based upon the following idea: The motion in the product valley corresponds approximately to the second half of an inelastic non-reactive collision in the product part of the potential surface. The second half of this non-reactive collision begins after the turning point has been passed. To calculate the rotational excitation we replace in the language of classical scattering theory - an exact reactive trajectory T by the second half of a non-reactive trajectory To. This replacement is of course never exact. If one averages over an ensemble of trajectories as we do, the effect of the inaccuracy is expected to be small. The initial condition for To is a product in the rotational ground state, the final condition is given by a product in the rotational state observed. The collision energy is E” = Et_, - EBlc,(n’, 0) since vibrational excitation in the exit valley is neglected. We confine ourselves to initially nonrotating molecules. Then the total angular momentum J equals the initial orbital angular momentum 1 for T. The trajectory To has to be determined for the final orbital angular momentum 1’ = J. The rotational excitation of To is calculated in the 10s approximation. Since the 10s approach conserves the orbital angular momentum during the collision, neglects the rotational energy transfer and treats the relative orientation of the colliding species (Yas a fixed parameter we may identify I’ = 1, E’ = E”, and 0~’= a’. The excitation function j’(b’, a’) as a function of the impact parameter b’ = 1/(2~n’E’)‘/~ and (Y’ prior to the entrance into the exit valley [3] is
JROm dR
(G’/&z’)
x {2m’E’[l
- I’( R, a’)/E’
- ( b’)‘/R2]
} -1’2,
(4)
where a factor 2 has been omitted since we are dealing with only a half collision. The classical integral cross section is calculated from a(0,
n-j’,
n’) =
n da’ sin (Y’ J0 hmaxb’db’6( j’ -j’(
X
J0 xP(
b’cdln + n’),
b’, a’)) (5)
where P is an (unknown) weighting factor. It expresses the conditional probability for b’ and CY’ for a given vibrational transition. The probability P depends on the total reaction probability as well as on the geometries that contribute to the reaction. Before we apply eq. (5) to exothermic reactions it is useful to look first at pure rotational excitation in a repulsive potential. Then P has to be omitted from eq. (5) and a factor 2 must be added in eq. (4). Since we are interested primarily in the generic properties of the cross sections it is sufficient to model j’(b’, a’) using the experience from many numerical calculations [4] for homonuclear molecules j’(b’,
a’) =w(b’)
]sin ICY’],
(5’)
where o( b’) decreases monotonically with increasing impact parameter b’. The j’ dependence of (I depends an the slope of w( b’). If w( 6’) decays very rapidly with increasing b’ as observed for short-range potentials the cross section peaks at j’ = 0 and decreases with rising j’. A rather slow decrease of w(b’) is followed by a (I that has a maximum at j’ # 0. This can be easily seen from the limiting case w(b’) = C if 0 < b’ < b,, and w( b’) = 0 otherwise. The cross section can be evaluated analytically
e( j’)
0: [ (1 + 4Y2
+ (1 - qY*]
with the abbreviation cross section increases
/q,
[l - ( j’/C)2]‘/2. The with increasing j’ with a q =
D, Poppe / Rotational excitation
singularity at the largest classically accessible j’ = C. However, in all investigated realistic cases of pure rotational excitation discussed so far w(b’) decreases so fast with increasing b’ that always j’ = 0 is most probably populated. We return to eq. (5) and apply it to an exothermic reaction. The conditional probability P weights different (Y’ and b’ and thereby modifies the rotational distribution. If P depends sensitively on the dynamics then eq. (5) is of no practical advantage with respect to a full scattering calculation. In case of a process with a large cross section one might expect that P does not depend critically on (Y’ and b’. An example which fulfills all conditions of section 1 is the quenching of Na(3p) in low-energy (E = 0.16 eV) non-adiabatic collisions with nitrogen. It is an exothermic reaction with a released energy of BE = 2.1 eV which is converted into vibrational and rotational energy of nitrogen and translational energy of ground state Na. Calculations of surface hopping trajectories using realistic potential surfaces [5] for the electronic ground and excited state have shown a large cross section for quenching. The entrance into the product valley of the potential can be identified with the last hop to the ground state surface, The rotational excitation in the first part of the collision can be estimated from the rotational distribution if no quenching occurs. Usually then only states j’ < 8 are populated. The trajectory calculations show that b’ and (Y’ contribute according to their statistical weight giving P = constant. The (Y’ values around (Y’= O” and (Y’= 90” are weakly disfavoured due to a smaller non-adiabatic coupling being however of no numerical significance. The trajectory calculations yield integral and differential rotational distributions which exhibit peaks at j’ f 0 well below the energetic threshold at j’ = 100 (see fig. 1). Experimental data by Reiland et al. [6] from molecular beam experiments support this result. Eq. (5) offers a simple physical interpretation. Suppose all initial impact parameters 0 < b -c bmax contribute to the quenching process. The initial orbital angular momentum is conserved during the first part of the collision since there rotational excitation is neglected. For the final I’ we have therefore I’ = 1. The final impact parameter 6’
of diatomic molecules
19
000
400
0
0
20
40
60
I’
Fig. 1. Rotational distributions as a function of j’. Dots correspond to surface hopping trajectory calculations for Na(3p)+Nz [S]. Squares and crosses show the results of a simulation using w ( b’) a exp( - 3.5 b’/b,,) with 0 < b’ < b,,,, and 0 < b’ < 0.27b_, respectively. The distributions are normalized to the same value at the maxima.
entering eqs. (4) and (5) is given by b’(b)
= 1/(2mE’)“* =b{i
+ [AE+
-E,&‘,
E&I, 0)]/E}-1’2.
0) (6)
Since E,,( n, 0) = E,,,,( n’, 0) we have a dramatic reduction of the final b’ by a factor of = 0.27. In other words P( b’, a’ ) n --, n’) = 0 for b’ > b’( b,,,,). Compared to pure rotational excitation large impact parameters are missing which leads to a substantial reduction of scattering into low j’ states. The distribution exhibits. then a peak at medium j’ values. The largest j’ can be calculated from eq. (4) in the hard-shell limit [3] J;,
= (2mE’)“*(
B - A),
where B and A are the two half axes of the hard ellipsoid. For E’ = 1.75 eV we estimate j;, = 42 which is in reasonable agreement with the trajectory calculations (fig. 1). They show a somewhat larger j;, = 60, however, states with j’ > 42 are only weakly populated. Comparison is also made with a simple analytical expression w( b’) a exp( -3.5b’/b,,). The cross section a( j’) is determined for 0 < b’ < bmax and 0 < b’ -z 0.27b,, using eq. (5) with P = constant. The shift of the maximum of the j’
D. Poppe / Rotutional excitution of diatomic molecules
20
distribution is clearly demonstrated. The analytical model agrees well with the trajectory results. It is the conservation of the total angular momentum and energy that constitutes a “window” for the impact parameter. This effect is related to a model for photodissociation invoked by Schinke [l] and Schinke and Engel [2]. The rotational excitation of OH after photofragmentation of H,O is also basically determined by a half collision. It takes place on the potential surface of excited H,O and can also be described by eq. (5) in a classical approximation. Only (Y’ values contribute that correspond to configurations populated by the photoexcitation process. Moreover, for each final j’ only one b’ = l'/(2m'E1)'/2 h as to be taken into account because 1’ = j’ since the (total) angular momentum of H,O vanishes. The “window” for the photofragmentation process in the (Y’as well as in the b’ domain is much smaller than in our case. Schinke (21 explained successfully the rotational distribution of OH and similarly of CO after photodissociation of H,CO. We believe that our simple model provides some insight into the physical origin of rotational distributions in exothermic reactions. It may be easily extended to other reactions. Since the main arguments are based on energy and angular momentum conservation the model is also applicable to collisions where the 10s conditions are not fulfilled. Just another example where this model is applicable is the reaction
rotational distributions for various final vibrational states n’ that are highly inverted. For the vibrational ground state n’ = 0 one finds that the largest energetically allowed j’ is most probably populated while for n’ > 0 the peak is shifted to smaller j’ values. The reduction of the final impact parameter for this system is similar to that in the quenching process discussed above. The enhanced rotational excitation is at least partly due to the fact that the product OH is heteronuclear. The shift to smaller j’ with increasing n’ is caused by two factors. Firstly, the final translational en= w: n’( w; = 0.46 eV) decreases and ergy E’ = Etotal leads to less rotational excitation. Secondly, the final impact parameter is less reduced which “opens the window” for small j’ values.
Acknowledgement The author gratefully acknowledges financial support from Sonderforschungsbereic 161 “Hyperfeinwechselwirkungen” and from “Fonds der Chemischen Industrie”.
References (11 [2] [3] [4] [5]
O(‘D,)
+ HCl + OH + Cl. [6]
The exothermicity of AE = 2.38 eV is large compared with the initial collision energy E = 0.1 eV. Calculations [7] and experiments [8] show product
[7] [8]
R. S&ix&e, Chem. Phys. Letters 120 (1985) 129. R. Schinke and V. Engel, J. Chem. Phys. 83 (1985) 5068. H.J. Korsch and D. Poppe, Chem. Phys. 69 (1982) 99. R. Schinke and J.M. Bowman, in: Molecular collision dynamics, ed. J.M. Bowman (Springer, Berlin, 1983). D. Poppe, D. Papierowska-Kaminski and V. BonJiEKoutecky, J. Chem. Phys., to be published. W. Reiland, C.P. Schulz, H.U. Tittes and I.V. Hertel Chem. Phys. Letters. 91 (1982) 329. R. Schinke, J. Chem. Phys. 80 (1984) 5510. A.C. Lutz, J. Chem. Phys. 73 (1980) 5393.
17
ROTATIONAL EXCITATION OF DIATOMIC MOLECULES IN EXOTHERMIC PROCESSES Dirk POPPE Institut ftir Physikalische Chemie, Freie Universitiit Berlin, Takustrasse 3, I000 Berlin 33, FRG Received 30 June 1986
We describe a simple model which is capable of explaining the rotational distribution of Nz after the quenching reaction Na(3p) + Nz + Na(3s) + NZ. Restrictions by energy and total angular momentum conservation together with the dynamics in the product valley of the potential surface determine the rotational energy transfer.
1. Introduction Energy transfer processes and chemical reactions between atoms and molecules are usually accompanied by the rotational excitation of the products. In this article we would like to present a simple model which explains qualitatively the rotational distribution of a diatomic molecule B’C’ produced in an exothermic reaction A+BC+A’+B’C’+AhE,
(1)
where the exothermicity is denoted by AE. The model is applicable if the following conditions are fulfilled. (1) The exothermicity is large compared to the initial collision energy. (2) The rotational excitation is mainly generated during the second half of the collision. Then the system has entered the exit channel of the reaction where the products begin to separate. The first part of the collision does not produce any significant rotational excitation. (3) The dynamics in the second half of the collision is impulsive and can be well described by the infinite-order-sudden (10s) approximation of rotationally inelastic scattering. The higher the exothermicity is the more appropriate this approximation will be.
(4) Vibrational excitation plays only a minor role during the second part of the scattering process. This condition is not essential for our model. It merely facilitates the calculation of the rotational excitation. In section 2 we shall discuss the details of the approach. For simplicity we confine ourselves to a classical version although the idea can certainly be worked out quantum mechanically. We shall discuss how it is related to Schinke’s reflection principle [1,2]. The model is applied to the quenching of Na(3p) in collisions with N,.
2. The model The energy balance for reaction (1) reads E totd=E+EBC(n, --, E’ + EB&‘,
i)+AE j’),
(2)
where n, j denotes the vibrational and the rotational quantum number, respectively. The translational energy is denoted by E. Unprimed (primed) quantities correspond to the initial (final) states. We divide now an individual collision into two parts. The first part is the motion in the entrance valley of the potential surface. Rotational excita-
0301-0104/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
18
D. Poppe / Rotational excitution
tion during this part is likely to be small since the initial collision energy is small. In any case the intermediate rotational energy is bounded by E,,, < Etotal - u,
of diatomic molecules
j’( b', a’) = -m’
(3)
where U is the minimum of the potential that has to be passed so that the reaction takes place. The second part starts when the system has ultimately reached the valley of the products. The potential is now basically repulsive. The final translational energy is much larger than the initial collision energy. If the anisotropy is large enough we expect large rotational excitation. Our model for the dynamics of rotational excitation is based upon the following idea: The motion in the product valley corresponds approximately to the second half of an inelastic non-reactive collision in the product part of the potential surface. The second half of this non-reactive collision begins after the turning point has been passed. To calculate the rotational excitation we replace in the language of classical scattering theory - an exact reactive trajectory T by the second half of a non-reactive trajectory To. This replacement is of course never exact. If one averages over an ensemble of trajectories as we do, the effect of the inaccuracy is expected to be small. The initial condition for To is a product in the rotational ground state, the final condition is given by a product in the rotational state observed. The collision energy is E” = Et_, - EBlc,(n’, 0) since vibrational excitation in the exit valley is neglected. We confine ourselves to initially nonrotating molecules. Then the total angular momentum J equals the initial orbital angular momentum 1 for T. The trajectory To has to be determined for the final orbital angular momentum 1’ = J. The rotational excitation of To is calculated in the 10s approximation. Since the 10s approach conserves the orbital angular momentum during the collision, neglects the rotational energy transfer and treats the relative orientation of the colliding species (Yas a fixed parameter we may identify I’ = 1, E’ = E”, and 0~’= a’. The excitation function j’(b’, a’) as a function of the impact parameter b’ = 1/(2~n’E’)‘/~ and (Y’ prior to the entrance into the exit valley [3] is
JROm dR
(G’/&z’)
x {2m’E’[l
- I’( R, a’)/E’
- ( b’)‘/R2]
} -1’2,
(4)
where a factor 2 has been omitted since we are dealing with only a half collision. The classical integral cross section is calculated from a(0,
n-j’,
n’) =
n da’ sin (Y’ J0 hmaxb’db’6( j’ -j’(
X
J0 xP(
b’cdln + n’),
b’, a’)) (5)
where P is an (unknown) weighting factor. It expresses the conditional probability for b’ and CY’ for a given vibrational transition. The probability P depends on the total reaction probability as well as on the geometries that contribute to the reaction. Before we apply eq. (5) to exothermic reactions it is useful to look first at pure rotational excitation in a repulsive potential. Then P has to be omitted from eq. (5) and a factor 2 must be added in eq. (4). Since we are interested primarily in the generic properties of the cross sections it is sufficient to model j’(b’, a’) using the experience from many numerical calculations [4] for homonuclear molecules j’(b’,
a’) =w(b’)
]sin ICY’],
(5’)
where o( b’) decreases monotonically with increasing impact parameter b’. The j’ dependence of (I depends an the slope of w( b’). If w( 6’) decays very rapidly with increasing b’ as observed for short-range potentials the cross section peaks at j’ = 0 and decreases with rising j’. A rather slow decrease of w(b’) is followed by a (I that has a maximum at j’ # 0. This can be easily seen from the limiting case w(b’) = C if 0 < b’ < b,, and w( b’) = 0 otherwise. The cross section can be evaluated analytically
e( j’)
0: [ (1 + 4Y2
+ (1 - qY*]
with the abbreviation cross section increases
/q,
[l - ( j’/C)2]‘/2. The with increasing j’ with a q =
D, Poppe / Rotational excitation
singularity at the largest classically accessible j’ = C. However, in all investigated realistic cases of pure rotational excitation discussed so far w(b’) decreases so fast with increasing b’ that always j’ = 0 is most probably populated. We return to eq. (5) and apply it to an exothermic reaction. The conditional probability P weights different (Y’ and b’ and thereby modifies the rotational distribution. If P depends sensitively on the dynamics then eq. (5) is of no practical advantage with respect to a full scattering calculation. In case of a process with a large cross section one might expect that P does not depend critically on (Y’ and b’. An example which fulfills all conditions of section 1 is the quenching of Na(3p) in low-energy (E = 0.16 eV) non-adiabatic collisions with nitrogen. It is an exothermic reaction with a released energy of BE = 2.1 eV which is converted into vibrational and rotational energy of nitrogen and translational energy of ground state Na. Calculations of surface hopping trajectories using realistic potential surfaces [5] for the electronic ground and excited state have shown a large cross section for quenching. The entrance into the product valley of the potential can be identified with the last hop to the ground state surface, The rotational excitation in the first part of the collision can be estimated from the rotational distribution if no quenching occurs. Usually then only states j’ < 8 are populated. The trajectory calculations show that b’ and (Y’ contribute according to their statistical weight giving P = constant. The (Y’ values around (Y’= O” and (Y’= 90” are weakly disfavoured due to a smaller non-adiabatic coupling being however of no numerical significance. The trajectory calculations yield integral and differential rotational distributions which exhibit peaks at j’ f 0 well below the energetic threshold at j’ = 100 (see fig. 1). Experimental data by Reiland et al. [6] from molecular beam experiments support this result. Eq. (5) offers a simple physical interpretation. Suppose all initial impact parameters 0 < b -c bmax contribute to the quenching process. The initial orbital angular momentum is conserved during the first part of the collision since there rotational excitation is neglected. For the final I’ we have therefore I’ = 1. The final impact parameter 6’
of diatomic molecules
19
000
400
0
0
20
40
60
I’
Fig. 1. Rotational distributions as a function of j’. Dots correspond to surface hopping trajectory calculations for Na(3p)+Nz [S]. Squares and crosses show the results of a simulation using w ( b’) a exp( - 3.5 b’/b,,) with 0 < b’ < b,,,, and 0 < b’ < 0.27b_, respectively. The distributions are normalized to the same value at the maxima.
entering eqs. (4) and (5) is given by b’(b)
= 1/(2mE’)“* =b{i
+ [AE+
-E,&‘,
E&I, 0)]/E}-1’2.
0) (6)
Since E,,( n, 0) = E,,,,( n’, 0) we have a dramatic reduction of the final b’ by a factor of = 0.27. In other words P( b’, a’ ) n --, n’) = 0 for b’ > b’( b,,,,). Compared to pure rotational excitation large impact parameters are missing which leads to a substantial reduction of scattering into low j’ states. The distribution exhibits. then a peak at medium j’ values. The largest j’ can be calculated from eq. (4) in the hard-shell limit [3] J;,
= (2mE’)“*(
B - A),
where B and A are the two half axes of the hard ellipsoid. For E’ = 1.75 eV we estimate j;, = 42 which is in reasonable agreement with the trajectory calculations (fig. 1). They show a somewhat larger j;, = 60, however, states with j’ > 42 are only weakly populated. Comparison is also made with a simple analytical expression w( b’) a exp( -3.5b’/b,,). The cross section a( j’) is determined for 0 < b’ < bmax and 0 < b’ -z 0.27b,, using eq. (5) with P = constant. The shift of the maximum of the j’
D. Poppe / Rotutional excitution of diatomic molecules
20
distribution is clearly demonstrated. The analytical model agrees well with the trajectory results. It is the conservation of the total angular momentum and energy that constitutes a “window” for the impact parameter. This effect is related to a model for photodissociation invoked by Schinke [l] and Schinke and Engel [2]. The rotational excitation of OH after photofragmentation of H,O is also basically determined by a half collision. It takes place on the potential surface of excited H,O and can also be described by eq. (5) in a classical approximation. Only (Y’ values contribute that correspond to configurations populated by the photoexcitation process. Moreover, for each final j’ only one b’ = l'/(2m'E1)'/2 h as to be taken into account because 1’ = j’ since the (total) angular momentum of H,O vanishes. The “window” for the photofragmentation process in the (Y’as well as in the b’ domain is much smaller than in our case. Schinke (21 explained successfully the rotational distribution of OH and similarly of CO after photodissociation of H,CO. We believe that our simple model provides some insight into the physical origin of rotational distributions in exothermic reactions. It may be easily extended to other reactions. Since the main arguments are based on energy and angular momentum conservation the model is also applicable to collisions where the 10s conditions are not fulfilled. Just another example where this model is applicable is the reaction
rotational distributions for various final vibrational states n’ that are highly inverted. For the vibrational ground state n’ = 0 one finds that the largest energetically allowed j’ is most probably populated while for n’ > 0 the peak is shifted to smaller j’ values. The reduction of the final impact parameter for this system is similar to that in the quenching process discussed above. The enhanced rotational excitation is at least partly due to the fact that the product OH is heteronuclear. The shift to smaller j’ with increasing n’ is caused by two factors. Firstly, the final translational en= w: n’( w; = 0.46 eV) decreases and ergy E’ = Etotal leads to less rotational excitation. Secondly, the final impact parameter is less reduced which “opens the window” for small j’ values.
Acknowledgement The author gratefully acknowledges financial support from Sonderforschungsbereic 161 “Hyperfeinwechselwirkungen” and from “Fonds der Chemischen Industrie”.
References (11 [2] [3] [4] [5]
O(‘D,)
+ HCl + OH + Cl. [6]
The exothermicity of AE = 2.38 eV is large compared with the initial collision energy E = 0.1 eV. Calculations [7] and experiments [8] show product
[7] [8]
R. S&ix&e, Chem. Phys. Letters 120 (1985) 129. R. Schinke and V. Engel, J. Chem. Phys. 83 (1985) 5068. H.J. Korsch and D. Poppe, Chem. Phys. 69 (1982) 99. R. Schinke and J.M. Bowman, in: Molecular collision dynamics, ed. J.M. Bowman (Springer, Berlin, 1983). D. Poppe, D. Papierowska-Kaminski and V. BonJiEKoutecky, J. Chem. Phys., to be published. W. Reiland, C.P. Schulz, H.U. Tittes and I.V. Hertel Chem. Phys. Letters. 91 (1982) 329. R. Schinke, J. Chem. Phys. 80 (1984) 5510. A.C. Lutz, J. Chem. Phys. 73 (1980) 5393.