relaxation in he+co2(01 0) and he+co cite as: j. chem ...coupled states calculations on vibrational...

J. Chem. Phys. 86, 802 (1987); https://doi.org/10.1063/1.452282 86, 802

© 1987 American Institute of Physics.

Coupled states calculations on vibrational

relaxation in He+CO2(0110) and He+CO

Cite as: J. Chem. Phys. 86, 802 (1987); https://doi.org/10.1063/1.452282Submitted: 01 July 1986 . Accepted: 17 July 1986 . Published Online: 31 August 1998

A. J. Banks, and D. C. Clary

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Coupled states calculations on vibrational relaxation in He + CO2(01 10) and He+CO

A. J. Banks and D. C. Clary University Chemical Laboratory, Lensfield Road, Cambridge, CB2 lEW, United Kingdom

(Received 1 July 1986; accepted 17 July 1986)

Computations of vibrational relaxation rate coefficients for He + CO2 (0110) and He + CO (v = 1) over the temperature range 100-300 K are reported. The calculations used the quantum mechanical coupled-channel method for vibrations and the coupled states approximation (CSA) for rotations. These calculations were used to test the accuracy of the vibrationally close coupled, rotationally infinite order sudden approximation (VCC-IOSA) and a semiclassical method. For He + CO2, the VCC-IOSA results compare very well with those obtained using the CSA, while the agreement is not so good for He + CO. This is because CO has a much larger rotational constant and vibrational frequency than CO2,

I. INTRODUCTION

The field of vibrational energy transfer has been exten-sively investigated by both experimental I and theoretical2

means. In recent years the theoretical studies based on quan-tum mechanical3 and semiclassical methods4 have come to the fore. The quantum mechanical solutions to the problem of vibrational relaxation have been centered mainly on the close-coupling approach,s where the wave function describ-ing the scattering process is expanded in a suitable set of basis functions and the problem is solved numerically. The main difficulty with applying the close-coupling methods to problems other than that of He + H2 (v = 1)6 is that the vibrational-rotational basis sets required become too large for the calculation to be feasible. A set of approximations under the general heading of angular momentum decoupling approximations 7 have been introduced in order to render the problems tractable. The two most widely used approxima-tions, the vibrationally close-coupled, rotationally infinite order sudden approximation (VCC-IOSA) and the more accurate, but also more computationally expensive, coupled states approximation (CSA) have been tested against exact calculations on the He + H2 system.6 This test and a recent comparison of the two approximations8 on the He + N2 (v = 1) system suggest that the VCC-IOSA cannot be used indiscriminately.

This paper presents calculations of rate coefficients over the temperature range 100-300 K for the systems,

He + CO(v = l)~He + CO(v = 0) and

He + C02(01l0)~He + CO2(OOoO). For the He + CO system, a comparison of the VCC-IOSA and the semiclassical method of Billing has already been made.9 The semiclassical (SC) method treated the vibra-tions quantum mechanically, while dealing with the rota-tional and translational motions in a classical manner, and the resulting SC rate coefficients were found to be much greater than those calculated using the VCC-IOSA. We ex-tend this comparison to include the CSA in order to assess the accuracy of the more approximate quantal and semiclas-sical methods. The CO rotational constant (Be = 1.9313

cm- I ) and vibrational frequency (we = 2170 cm-l)lO are similar to those of N2. In He + CO (v = 1) we would there-fore expect similar discrepancies to exist between the CSA and VCC-IOSA as those found in He + N2 (v = 1).8

We also report CSA calculations on He + CO2 (0110). This is the first time that the CSA has been applied to study vibrational relaxation in an atom + polyatomic system. Much emphasis has been placed on examining the vibration-al relaxation in CO/ l,12 because of the importance of the CO2 gas laser

l3 and the role CO2 (01 10) plays in maintaining the thermal balance in the upper atmosphere. 14 The VCC-IOSA has been applied to the He + CO2(01 l0) system be-fore 12 and good agreement with the photoacoustic results of Lepoutre et al. 15 was found. The potential used was based on an ab initio SCF calculation with the calculated points fitted to a simple sum of atom-atom exponential functions. The good agreement between theory and experiment could, therefore, be a fortuitous one generated by an approximate potential. Here we compare our CSA rate coefficients with those obtained in the experiment and those calculated using the VCC-IOSA. We also compare the CSA and VCC-IOSA at the level of individual rovibrational cross sections. For CO2, the lower rotational constant (Be = 0.3895 cm -I) and bending vibrational frequency (we = 667.3 cm -I) 16 imply that the VCC-IOSA should be a better approximation for this system than for the He + CO or He + N2 systems. We consider He + CO2 to be a standard and typical test case for atom + polyatomic molecular scattering. By comparing re-sults for He + CO and He + CO2 in the same paper we wish to emphasize the significant differences between vibrational relaxation in diatomic and poly atomic systems.

Finally we attempt to gain a quantitative guide to the applicability of the VCC-IOSA. We do this by varying the molecular rotational constant, vibrational frequency and collisional reduced mass in turn and comparing the CSA and VCC-IOSA results as a function of the varying system pa-rameters. This should provide a useful guide to the expected accuracy ofthe VCC-IOSA.

Section II gives the theory needed to apply the CSA to vibrational relaxation in both diatomic and triatomic mole-cules. The computational details of the calculations are pre-

802 J. Chern. Phys. 86 (2), 15 January 1987 0021-9606/87/020802-11$02.10 © 1986 American Institute of Physics

A. J. Banks and D. C. Clary: Vibrational relaxation in He+CO, 803

sented along with a brief discussion of the potential energy surfaces in Sec. III. Results and discussions, including the attempt to quantitatively assess the accuracy of the VCC-IOSA, are given in Sec. IV. The conclusions drawn from this work can be found in Sec. V.

II. THEORY

A. Atom + diatom The formalism we use for describing atom-diatom

quantal inelastic scattering is well known!7 and only a brief outline is given here.

The CSA method is applied within a body-fixed (BF) axis system defined such that the BF z axis lies along the vector R which relates the center of mass of the diatom to the atom. The Hamiltonian for the system is approximated by expressing the centrifugal term asl8

(1)

where J is the total angular momentum operator, j the mo-lecular rotation angular momentum operator, and Jz andjz denote projection operators which give the z component of the total and rotational angular momenta, respectively. Here we have neglected the coupling between the n values, where n is the projection of the total angular momentum along the BF z axis. This approximation is identical in form to the helicity decoupling approximation of Tamir and Shapiro. 19

Another way of applying the CSA is to neglect both the coupling between n values and also to approximate the diag-onal terms by L(L + 1)~.20

To employ the close-coupling method, the partial wave functions are expanded in terms of translational f ~o (R ) , vibrational hVj (r) and rotational Y b. (0,0), functions,

t/JJO = J.... L Lf~O(R) hVj (r) Y~ (0,0), (2) R v j

where r is the intramolecular coordinate and 0 the angle between the vectors Rand r. This expansion leads to cou-pled-channel equations for the translational functions, f~o(R), which are solved subject to appropriate boundary conditions. These boundary conditions define a S matrix, S~~-Vj' which is used to calculate the vibrational relaxation cross sections,

a(vj--+v') = 2 ~ ~LL (21+ 1)IS~!l_vjI2, k vj (2J + 1) J J °

(3)

where flkv} is the momentum associated with the initial vj state. A rate coefficient, k(vj--+v'), is obtained by calculating these cross sections over a range of energies and Maxwell-Boltzmann averaging. A comparison with experimental rate coefficients can be obtained by Boltzmann averaging the k(vj--+v') coefficients over the initialj states to give the rate coefficient, k(v--+v').

In the VCC-IOSA method, the Hamiltonian is obtained by making the approximation,

IJ-jI2=~J(J+1); j2=0. (4)

The coupled-channel expansion,

t/JJ = J.... Lf;(R;O) hv (r), R v

(5)

is then made where the translational term is now parametri-cally dependent on 0 and no rotational functions are re-quired. The S matrix is S ;Lv (0) and the vibrational relaxa-tion cross sections are now given by,

a(v--+v') = ~ L (21 + 1) [sin 0 dO IS:'-. (0) 12. 2k~ J °

(6)

B. Atom + linear triatomic molecule To apply the CSA to a linear triatomic molecule we

continue to work in the BF reference frame which is related to the space-fixed (SF) frame through the Euler angles (a, P, r) .21 The BF frame is defined so that the BF z axis lies along the vector R which defines the position of the atom relative to the center of mass of the molecule. The wave func-tion in the SF frame is related to that in the BF frame by

t/JJM = L D ~M (a, P,r)t/JJO, (7) °

where M is the projection of the total angular momentum on the SF z axis and D ~M (a, P,r) is a Wigner rotation func-tion.21

The BF Hamiltonian for this system can be written

H=_!!:..J....£'(R)+IJ-W+I-I +V 2p, R aR 2 2p,R 2 "aco, tnt'

(8)

where,u is the reduced mass of the system, J the total angular momentum operator for the atom + triatomic molecule sys-tem, and j the operator for the total CO2 molecular angular momentum in the BF frame. Hco, is the molecular Hamilto-nian and V;nt the intermolecular potential. We again apply the CSA by setting

IJ - JI2 = J2 +f - 2Jz jz, (9)

where Jz gives the component ofthe total angular momen-tum along the BF z axis and jz gives the component of the molecular angular momentum along this axis. To describe the molecule, we work within the rigid rotor harmonic oscil-lator model and use the isomorphic Hamiltonian for CO2, applied by Whitehead and Handy22 and based on the deriva-tion of Watson.23 Previous VCC-IOSA calculations24 per-formed on the He + CO2 (0110) system indicate that the ne-glect of anharmonicity in the intramolecular potential is a good approximation for the 01 10_00°0 vibrational relaxa-tion. The Hamiltonian is given by

1 {2 2 2 2 [IT; + IT; ] Heo, ="2 PI +P2a +P2b +P3 + [0

+ 11.1 Q i + 11.3 Q ~ + 11.2 (Q t, + Q ~b ) } , ( 10) where the P j are momentum operators corresponding to the normal coordinate Qj, and [0 is the moment of inertia of the molecule. Here the indices 1 and 3 refer to the symmetric and

J. Chern. Phys., Vol. 86, No.2, 15 January 1987

804 A. J. Banks and D. C. Clary: Vibrational relaxation in He+CO,

z

)tB x / I

/f


means we can drop the E label from the basis set. Operating on the basis set of Eq. (21), with the Hamil-

tonian of Eq. (8), generates the coupled equations for f~'i(R),

{£.+k2 _ [J(J+ 1) +j(j+ 1) -2n2]}fJO(R) dR2 v);' R2 v);'

=~ ~ V~;'i'o(R)fjf;,.,(R). 7r v71.'

(24)

Here the potential matrix elements are given by, (eo (211' (211'

v~;'i,o (R) = Jo Q2 dQ2 Jo d/) Jo d¢

X So'" sin 0 dO G vj;'O· (Q2'O,/),¢)

(25)

where

Gvj;'O(Q2'O,/),¢) = IvjAnE)

The wave vector kvj;' in Eq. (24) is given by,

k ~j;' = ~ (E - Evj;')' (27) where E is the total energy of the collision and E - Ev);' is the translational energy, The angles ¢ and /) are both in the x-y plane, see Fig. 1, and hence, it is possible to expand the inter-molecular potential in a Fourier series, II

~nt (R,Q2'O,¢,/) = I hn (R,Q2'O) cos n(¢ - /), (28) n

It is readily shown, by evaluating the integrals over this po-tential using the basis set IvjAnE), that the coupled channel matrix elements ofEq. (25) are independent of the integra-tion over ¢. It is therefore possible to set ¢ to 0 in the basis functions when evaluating these matrix elements. Thus the rotational functions D h;. (O,O,¢) can be replaced by the re-duced rotation matrices,21 d ~ (0), and the four dimension-al integral of Eq. (25) is reduced to the three dimensional integral,

V~;f,o = leo Q2 dQ2 f1l'd/)

where

X [ sin 0 dO Fvp..0·(Q2'O,/)

X ~nt F V'/;' 'o( Q2'O,/), (29)

Fvp..O(Q2'O,/) = CE [gv;. (Q2'/) d h;. (0) + Egv_;' (Q2'/)

Xd'h_;.(8)] (2i; 1 )112. (30) The coupled channel equations of Eq. (24) are solved

subject to the boundary conditions,

fjf;,.,(O) =0

and

(31)

where S ;7;. '-v);' is the S matrix. The normal phase factors have been omitted from these boundary conditions as the vibrational relaxation cross sections depend on the square of the S-matrix elements as demonstrated below,

U(vjA--I>V'A ') = 1T k ~j;' (2j + 1)

XIII (2J+ 1)IS~?;"-vj;.12. (32) l J 0

The method of applying the VCC-IOSA to the atom + linear triatomic molecule problem has been discussed

previously.11·12 Here we only present sufficient details to il-lustrate the major differences between the VCC-IOSA and theCSA.

The primitive basis set used in the VCC-IOSA calcula-tions is

g;.p Q~-Iexp (-!aQD

with

g;'p=e(/) = cos A/) g;.p=o(/) = sin A/),

(33)

where e stands for even and 0 for odd. For the 01 10 ..... 0000 transition only the p = e set are required. II This primitive basis set is used in diagonalizing the molecular vibrational Hamiltonian from which an orthogonal vibrational basis set is obtained which can be used in the scattering problem. The centrifugal term in the Hamiltonian is again replaced by an eigenvalue expression, Eq. (4). The resulting coupled equa-tions are solved to give an angle dependent S matrix S ~';' 'p-v). P (0) which leads to an expression for the vibration-al relaxation cross sections,

U(VA--I>V'A ') = ~ bv;' I I (2J + 1) 2k~ p J

where

bv;. = 1 for A = 0, bv). =! for A> O. The bv;. term allows for degeneracy of the levels with A > O.

III. NUMERICAL DETAILS AND POTENTIAL ENERGY SURFACES

A. He-CO system

The He-CO intermolecular potential was taken from Ref. 27. It is a dumbbell fit to the elastic ab initio potential of Thomas et al.28 and is expressed in the atom-atom form,

V;nt(R,r,8) =.I [Aj exp( -aj R j) - C~], (35) .=1.2 R j

where

RI = R - Elr cos 8 + ! ~ r R -I sin28, R2 = R + E2 r cos 8 + ! ~ r R -1 sin2 0

(36a)

(36b)


806 A. J. Banks and D. C. Clary: Vibrational relaxation in He+C02

and the parameters take the values A I = 358.32 eV, A2 0_ 1 0_1 = 710.92 eV, a l = 3.442 A ,a2 = 3.751 A ,CI = 3.569

0 6 06 eV A , C2 = 5.014 eV A ,EI = 0.7232, and E2 = 0.3631. The CO molecule is described by a Morse potential,

VCr) =D.{1-exp[ -au(r-r.)]P, (37)

with De = 21.69 eV, au = 1.655 A -I, and re = 1.128 A. It is unrealistic to expect a dumbbell fit to an elastic

potential to give accurate rate coefficients for vibrational re-laxation and a recent paper has demonstrated the weakness of this surface. 29 However, we are comparing methods of treating the dynamics and not attempting a comparison with experiment.

In evaluating the coupled-channel matrix elements,

(huj(r) YA (0,0)1 J';nt Ihv:t (r) Yb(O,O»,

16 primitive harmonic oscillator eigenfunctions were used to construct the molecular vibrational eigenfunctions hUj (r). The integration over r was performed using a 24 point Gauss-Hermite quadrature. The potential was expanded in terms of Legendre polynomials and analytical formulae were used in the integration over O. Eight terms in this Le-gendre expansion were sufficient to give cross sections con-verged to within 5% at a translational energy of 0.05 eV.

The convergence with respect to basis set size is illus-trated for two translational energies in Table I. The basis set description (xo, Xl' X2) denotes that Xi rotational functions are associated with the ith vibrational level. At 0.3 eV, the j = 0 cross section is converged to within better than 1 % using the (35,35,5) basis while in the worst case, conver-gence is still better than 1.5%. At the lower energy all the cross sections are converged to within 1 %.

The coupled-equations were solved using the R-matrix propagator method.30•31 The propagation space was divided into 85 equally spaced sectors for values of R between 2.5 and 16.500 , The summation over the total angular momen-tum was carried out by incrementing the J values with a

TABLE I. Convergence tests on He-CO basis sets."

Translational energy = 0.3 eV

Basisb j=O j=4 j=9 j= 18

25,25,5 2.304( - 4)C 3.989( - 4) 5.178( - 4) 3.934( - 4) 30,25,5 2.826( - 4) 6.389( - 4) 2.07l( - 4) l.OO3( - 2) 25,30,5 3.280( - 4) 9.868( - 4) 2.857( -4) 7.81l( - 3) 35,30,5 2.091( - 4) 3.296( -4) 3.380( - 4) 2.755( - 4) 40,30,5 2.094( - 4) 3.295( - 4) 3.387( - 4) 2.712( -4) 40,35,5 2. 105 ( -4) 3.327( - 4) 3.407( - 4) 2.235( - 4) 35,40,5 2. 105 ( - 4) 3.328(-4) 3.416( - 4) 2.272( -4) 35,35,5 2.105( -4) 3.328( - 4) 3.414( -4) 2.262( - 4)


Basis j=O j=5 j= 10

30,25,5 l.043( - 7) l.480( - 7) 3.473( - 8) 25,30,5 1.043( - 7) l.480( -7) 3.469( - 8)

"The numbers displayed are the summed cross sections u(v = I,j-v') (in units of a~) calculated for n = O.

b The basis set description is defined in the text. C Numbers in parentheses are powers of 10.

stepsize of four and fitting the summed cross sections to a spline function which allowed interpolation to give the inter-mediate J values. The summation over J was continued until a less than 1 % increase in the total cross section occurred.

Cross sections, 0'( v = 1, j -+ Vi = 0), at nine energies between 0.0119 and 0.3 eV were calculated with.n values up to 18 considered. It was found to be computationally more efficient to calculate the cross sections at all values of the translational energies required for a given .n and then, after repeating for all .n values, carrying out the summation and averaging over the .n contributions. To calculate one ofthe more expensive cross sections, 0'( v = 1, j = 18 -+ v = 0), at an energy of 0.3 e V required typically 30 minutes of Cray-1 S CPU time.

The initialj dependent rate constants were calculated by fitting the natural logarithm of the cross sections to a cubic spline function and interpolating to obtain sufficient points for a Simpson rule integration over the translational ener-gies. This initialj dependent rate constant was Boltzmann averaged to give the final vibrational relaxation rate con-stant. It is worth noting that the previous CSA calculation29

on vibrational relaxation in the He-CO system did not treat fully the .n contributions to the cross sections so that rate coefficients Boltzmann averaged over j could not be ob-tained. B. He-COz system

The intermolecular potential for the He-C02 system has been described previously.12 It was obtained by a least squares fit to SCF calculations performed at the STO-6G level. The points were fitted to the function

J';nt =A [exp( -BRI) +exp( -BR3)]

(38)

with

Ri = [(Xi - R sin 0 cos tP)2 + (Yi - R sin 0 sin tP)2

+ (Zi - R cos 0)2] 1/2 (39)

being the distance from the colliding atom to an atom i in the molecule and the coefficients A, B, C, and D taking the val-ues 79.28 Eh , 2.49000- I, 30.19 Eh , and 2.141 00- I, respec-tively.

To evaluate the coupled-channel matrix elements ofEq. (29), numerical integration over the variables Q2' 8, and 0 was performed. A Gauss-Legendre scheme over 0 was em-ployed using 64 quadrature points, four equally spaced points between 0-21T were required for integration over 8 and four Gauss-Laguerre points were needed for integration over the Q2 variable. The Gauss-Laguerre scheme was modified in accordance with Ref. 22. The program to inform the integrations was written in such a way that it benefited from the vector processing capabilities of the Cray-1S com-puter. This resulted in a substantial saving in CPU time.

The vibrational-rotational CSA basis set used to obtain the results in Sec. IV contained 55 basis functions with 15 rotational functions in v = 00°0 giving a maximum value ofj equal to 28 (only even values ofj exist if A = 0) and 35 and 5 functions in the 0110 and 02°0 levels, respectively. Details of the convergence tests on the basis set at two energies are given in Table II. The convergence is good to within 3% for

J. Chern. Phys., Vol. 86, No.2. 15 January 1987

A. J. Banks and D. C. Clary: Vibrational relaxation in He+CO. 807

TABLE II. Convergence tests on He-C02 basis sets. a


Basisb j=1 j=8 j= 16

15,25,3 1.388( - 2)· 1.351(-4) 6.958( - 6) 15,30,3 1.388( - 2) 1.351(-4) 6.967( - 6) 15,35,3 1.388( - 2) 1.351( - 4) 6.967( - 6) 9,30,3 1.449( - 2) 1.400( - 4) 9.138( - 6)

11,30,3 1.391( - 2) 1.354( - 4) 7.108( - 6) 17,30,3 1.388( - 2) 1.351( -4) 6.967( - 6)


Basis j=1 j=8 j= 16

15,30,3 4.052( - 1) 4.403( - 3) 5.245(-4) 15,35,3 4.031( - 1) 4.339( - 3) 5.197( -4) 15,40,3 4.031( - 1) 4.339( - 3) S.208( - 4) 13,35,3 4.010( -1) 4.347( - 3) 5.179(-4) 17,35,3 4.019( -1) 4.333( - 3) 5.182( -4)

• The numbers displayed are the summed cross sections u(OI'O,j - 0000), in units of a~ calculated for n values of 0 + 1.

b The basis set description is defined in the text.


17.-----.-----.-----.-----r---~

4-~CO CSA (j= 0, 4,S,I2 J

18 -IlJj

0 e

1111 ""e u --.:.:: 12 8' 19 8 --' = 4-,

0

20+-----.----.r---~-----r--~ ~ 100 150 200 2SO 300

Temp(KJ

FIG. 3. The initialj dependent CSA rate coefficients, k( v = l,j--+v = 0) for He + CO.

found that semiclassical method gives rate coefficients that are a factor of 4-8 larger than the VCC-IOSA coefficients over the temperature range 100-300 K. The CSA rate coeffi-cients are in good agreement with the semiclassical results.

The discrepancy between the VCC-IOSA and CSA co-efficients can be attributed to two main factors. Firstly the inclusion of rotational energy levels allows a closer energy matching between the ground and first vibrationally excited state. This is reflected in the cross section data of Table III. Secondly the initial j dependent rate constant k(v = 1, j-+I/ = 0) increases with increasingj as illustrated in Fig. 3. Boltzmann averaging over these initialj values leads to an enhanced vibrational relaxation rate coefficient.

B.He+C02 For He + CO2, CSA cross sections 00(01 10, j

= 1-00°0) are compared in Table IV with the VCC-IOSA cross sections, oo( v = OliO-v' = 00°0). The absolute values

TABLE IV. Cross sections for He + CO2,

Energy"

om 0.02 0.03 0.047 0.065 0.1 0.15 0.2

CSA(j= l)b

6.08( - 4)C 2.05( - 3) 4.78( - 3) 1.33( - 2) 2.88( - 2) 8.ll( - 2) 2.09( - I) 3.91( - 1)

"Initial translational energy in e V.

VCC-IOSA

4.28( - 4) 1.60( - 3) 3.95( - 3) 1.l2( - 2) 2.59( - 2) 7.54( - 2) 1.97( - I) 3.71(-1)

CSA

VCC-IOSA

1.42 1.28 1.21 1.19 1.11 1.08 1.06 1.05

b The cross sections are in units of a~. The CSA cross sections are described by u(OI'O,j = 1---00"0).

cNumbers in parentheses are powers of 10.

125-r---.---r----r-----,r------,

+

a.Expt. b. V((IOSA

14 ( c. (SA

14'5, '-00--'---2'0-0 ----.----,30

r-:"0--

TemplKl

FIG. 4. The rate coefficients, k(OI'o---oo"O), for He + CO2 (01'0). The crosses represent the experimental results of Lepoutre et al. (Ref. 15).

of the CSA cross sections vary by three orders of magnitude over the translational energy range 0.01-0.2 e V and they are much less sensitive to translational energy than those for He + CO. The He + CO2 cross sections are also several or-ders of magnitude larger than the He + CO cross sections, as would be expected on the basis of simple energy gap theor-ies32•33; the vibrational frequency for CO is 2170 cm -- I and that for CO2 667.3 cm -- I. The most interesting feature of the results in Table IV is the good agreement between the CSA and VCC-IOSA cross sections. The CSA cross sections are larger than the VCC-IOSA values by a factor of between 1.05-1.5. The agreement between the CSA and VCC-IOSA cross sections is far superior to that obtained in the He + CO case.

The rate coefficients obtained by Boltzmann averaging these CSA and VCC-IOSA cross sections are presented in Fig. 4 along with the experimental results of Lepoutre et al. IS The agreement between the experimental and calculated rate coefficients lies within 20%-50% and the two sets of calculated values agree to within 10%.

Table V compares the rate coefficients in Fig. 4 with those calculated using the CSA (j = 1) cross sections only. It can be seen that the VCC-IOSA rate coefficients are larger

TABLE V. Vibrational relaxation rate constants for He + CO2(01'0).

T(K) Expt. CSA" (j= 1) CSA VCC-IOSA

153 2.06( - 14)b 1.74( - 14) l.37( - 14) Ul( -14) 178 2.95( - 14) 2.66( - 14) 2.12( - 14) 2.34( - 14) 193 3.20( - 14) 3.34( - 14) 2.69( - 14) 2.97( -14) 213 5.73( - 14) 4.40( - 14) 3.59( -- 14) 3.95( - 14) 233 5.31( - 14) 5.67( - 14) 4.66( - 14) 5.1l( -14) 253 7.60( - 14) 7.13( - 14) 5.90( - 14) 6.46( - 14) 300 1.l5( - 13) 1.l4( - 13) 9.53( - 14) 1.04( - 13)

'CSA(j = 1) rate coefficient is described by, k(OI'O,j - 1---.0000) while the CSA rate coefficient is obtained by Boltzmann averaging over j. The units are cm3 s-' molecule-'.

bNumbers in parentheses are powers of 10.

J. Chem. Phys .• Vol. 86, No.2. 15 January 1987


35

if5-x

... ..hI 0 e ~

u

~-

5-

4He+(~(010) j-dependert rate ca-.stants

80 120 T(K)

160 200

FIG. 5. The initialj dependent CSA rate coefficients, k(OI'O,j---OOO) for He + CO2 ,

than those calculated using the full CSA method, although the CSA (j = 1) rate coefficients are larger than the VCC-IOSA results. In both these cases the extent of the discrepan-cies is small and of the order of 5%-10%. These results provide an interesting contrast with the He + CO case where, although the CSA (j = 0) rate coefficients were larger than those calculated by the VCC-IOSA method, the full Boltzmann averaged coefficients were larger still. To in-vestigate the matter further, the CSA calculations ofinitialj dependent rate coefficients for He + CO2 are displayed in Fig. 5. Comparing with the results of Fig. 3 shows that for He + CO2 the rate coefficient decreases with increasing j while for He + CO the coefficients increase with increasing j. A possible physical explanation for this behavior could lie with the steric hindrance caused by the rotating molecule. To deactivate the bend the most favored collision will have the incoming atom colliding with the carbon atom while at-tacking the molecule perpendicUlarly. II If the molecule ro-tates it becomes more difficult for this position to be obtained and the effectiveness of the collisions at producing vibration-al relaxation of the bend is decreased. A similar steric effect has been observed in the study of ion-molecule reactions.34

To compare VCC-IOSA and CSA calculations at a more detailed level, individual rovibrational cross sections were calculated. Two sets of such results are illustrated in Fig. 6. In Fig. 6(a) the cross sections u(OOoO,j = 2-+01 10, j') as a function ofj' are presented while Fig. 6(b) shows the rotational cross sections within the vibrationally excited state u(01 I O,j = 2-+01 I O,j'). At low values ofj' the agree-ment between the CSA and VCC-IOSA methods is good to within 20%, however the differences between the cross sec-tions increase at higher values of j'. This behavior is to be

100

~ ..q b 50

00

15

10

o

j'

4He+(O (Olo) 2

(SA

V( CIOSA

10

ETRANS=D131oV

15 20

ETRANS=00481eV

j' 15 20

FIG. 6. Individual rovibrational cross sections for He + CO2 : (a) Rota-tional excitation cross sections for He + CO2 (OOoO, j = 2)---+He + CO2 (OI'O,l). (b) Rotational transitions in the first vibrationally excit-

ed state, He + CO2 (Ol'O,j = 2)---+He + CO2 (OlIO./).

expected as the energy sudden approximation in the VCC-IOSA breaks down for higherj' values. The oscillating struc-ture in these cross sections has also been deduced from pro-pensity rules35 based on a consideration of the Wigner 3-j symbols which arise in the VCC-IOSA theory for He + CO2, The CSA results confirm this predicted struc-ture.

The validity of the VCC-IOSA, as judged by compari-son with the CSA for the contrasting cases of He + CO and He + CO2, would appear to depend on molecular param-eters such as the rotational and vibrational energy level spac-ings and also on system parameters such as the potential and reduced mass. The next section attempts to define some quantitative limits within which the VCC-IOSA method will yield accurate results.

c. Variation of system parameters The VCC-IOSA method has been applied to many

atom-polyatomic molecule vibrational relaxation prob-lemsll.12.24.32.36-39 and a quantitative estimate ofthe expect-ed accuracy of these calculations would clearly be valuable. The qualitative requirement that the rotational period of the molecule should be long compared to the collision time is well known, but there are no well defined limits detailing the validity of the VCC-IOSA. In an attempt to produce such limits the rotational constant, vibrational frequency, and

J. Chern. Phys .• Vol. 86, No.2, 15 January 1987


collision reduced mass which assist in defining the He + CO2 system were varied and CSA and VCC-IOSA cal-culations were then performed with the new parameters. The agreement between the two methods is then examined as a function of the varying parameter.

1. Rotational constant

The rotational constant Be was varied from 0.01 to 5 cm -I, and a series of CSA calculations were performed in order to obtain cross sections 0'(01 10, j = 1---+00°0), as a function of Be. The VCC-IOSA calculations are indepen-dent of Be' and given that the vibrational frequency was kept constant at 667.3 cm -I the VCC-IOSA results will be repre-

10 ETRANS=(}02eV

7·5 X B=O'3695em'

25

-2 -1 -1 log (S/em )

ETRANS=0·047eV 2·5 -

"';'0 2'0 -

15

,. 0 7"-------,,.---------,,------,--' - 2 -1

Log (B leJ1l'1 )

FIG. 7. CSA cross sections, O'(Ol'O,j = 1-+00°0) calculated for different rotational constants. The cross sections were calculated at a translational energy of (a) 0.02 eV; (b) 0.047 eV. The cross marks the value of the CO2 rotor constant.

sented by cross sections at the asymptotic limit Be-o. The results for a translational energy of 0.02 eV are plotted in Fig. 7(a). For Be values between 0.01-{).6 cm- I the curve shows little deviation from the VCC-IOSA cross section. For Be values larger than this upper limit the curve rises steeply as the VCC-IOSA breaks down, a factor of2 differ-ence resulting from a Be value of approximately 2 cm -I. The systems described in this paper conform to these predictions, the rotational constant of CO2 being 0.3895 cm -I and that of CO 1.931 cm -I . However the nature of the curve depends on the translational energy and vibrational frequency of the sys-tem considered and at higher translational energies and hence larger cross sections the applicability of the VCC-IOSA extends to higher values of Be. This is illustrated in Fig. 7(b) where the translational energy has been increased to 0.047 eV. The factor of 2 difference between the VCC-IOSA and CSA cross sections now occurs at a Be value of 4 cm- I •

2. Vibrational frequency

The force constant for the harmonic oscillator was var-ied to give a range of frequencies from 667 to 2650 cm - I. VCC-IOSA and CSA calculations on the He + CO2 system were performed over this range of frequencies and the cross sections as a function of frequency are plotted in Fig. 8 for a translational energy of 0.047 eV. As the frequency increases, the cross sections become smaller and the VCC-IOSA meth-od becomes less accurate. At a frequency of 2200 cm - 1 this model predicts the ratio of the CSA and VCC-IOSA cross sections to be about 2.5. This agrees quite well with the cal-culations on He + CO (see Table III) where a ratio of 3.7 was calculated for a translational energy of 0.047 eV, when the rovibrational coupling terms were included along the off diagonals of the close-coupling matrix, and a ratio of 2.5 when these terms are omitted.

2

3

7

8

a. (SA

b. V(CroSA

500 1000 1500 2000 1 2500 Frequency (ern )

a b

FIG. 8. CSA cross sections, 0'(01'0, j = 1---00"0) calculated for different vibrational frequencies. The cross sections were calculated at a translational energy of 0.047 eV. The VCC-IOSA cross sections are also shown.



2·0

- 3·0 "'nf b .s

I 4.0 a. (SA(j=1) b. V(CIOSA

a

5.0 +----.-----r---....-_-r-=~b::.......J o 5 10 15 20

Mass(a.m.u)

FIG. 9. CSA cross sections, u(OI IO,j = l~oO) calculated for different masses of the colliding atom. The cross sections were calculated at a transla-tional energy of 0.047 eV. The VCC-IOSA cross sections are also shown.

3. Collisional mass

The reduced mass of the system was altered by changing the mass of the colliding atom over a range of values from 4.0026 to 22.0 amu. The CSA and VCC-IOSA cross sections are plotted as a function of mass in Fig. 9 for a translational energy of 0.047 eV. As the mass increases the cross sections become smaller and the discrepancy between the VCC-IOSA arid CSA increases from a factor of 1.15 at a mass of 4.0amu to 2.6 at amass of22.0 amu. ForNe + CO2 (01

10) a comparison of the VCC-IOSA and semiclassical (SC) meth-ods has been made.40 The SC rate coefficients are approxi-mately a factor of 4 higher than the VCC-IOSA coefficients at 300 K. However our comparisons of Fig. 9 are for a single j dependent cross section at one energy and, as for He + CO (v = 1), averaging over the initialj dependent rate constants will have a significant effect. It would have been too expen-sive computationally for us to obtain converged Boltzmann averaged rate constants for Ne + CO2 using the CSA.

V. CONCLUSION

We have presented calculations of vibrational relaxa-tion rate coefficients over the temperature range 100-300 K for the systems He+CO (v= 1) and He+C02 (01

10). The coefficients have been calculated using both the CSA and VCC-IOSA methods. For He + CO a comparison was made with the results obtained using the semiclassical meth-od of Billing. 9 The SC method was found to be more accurate than the VCC-IOSA method for this particular system. The study ofthe He + CO2 (0110) system represents the first at-tempt at applying the CSA to the problem of vibrational relaxation in an atom + polyatomic molecule system. It is also interesting in that the vibrational relaxation occurs from a bending mode. The agreement ofthe VCC-IOSA calcula-tion both with the CSA method and with the experimental rate coefficients for He + CO2 is very encouraging.

Having discovered that the VCC-IOSA method is accu-rate for one system He + CO2 (0110) but rather inaccurate

for another He + CO (v = 1), we attempted to find where the VCC-IOSA breaks down in terms ofthe system param-eters. To do this we varied the rotational constant, reduced mass and vibrational frequency in tum and compared the VCC-IOSA and CSA calculations at the cross section level. We conclude that if the vibrational frequency is less than 1000 cm -1, the rotational constant is less than 4 cm -I and the colliding atom possesses a light mass, then the CSA and VCC-IOSA cross sections will be in reasonable agreement. We expect that the results of VCC-IOSA calculations on more complicated systems such as He + cyclopropane,41 where the vibrational frequencies42 and rotational constants are small16 can be treated with confidence.

ACKNOWLEDGMENTS

This work was supported by the Science and Engineer-ing Research Council and the NATO Stimulation Action Programme. The calculations were performed on the Cray-IS computer at the University of London Computer Center.

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J. Chern. Phys .• Vol. 86. No.2. 15 January 1987

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relaxation in he+co2(01 0) and he+co cite as: j. chem ...coupled states calculations on vibrational...

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