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Semiclassical model for energy transfer in polyatomic molecules. XIV. He- and Ar-glyoxal collisions
Chemical Physics 173 (1993) 167-175 North-Holland
Semiclassical model for energy transfer in polyatomic molecules. XIV. He- and Ar-glyoxal collkions G.D. Billing Department of Chemistry, H.C. 0rsted Institute, University of Copenhagen, 2100 Copenhagen 0, Denmark Received 9 February 1993
Cross sections and rate constants for vibrational excitation of the v, mode in glyoxal by collisions with He and Ar atoms are calculated by a quantum-classical model. Comparison with approximate quantum calculations and experimental data is made.
1. Introdwction We have in a series of papers (I-XIII) developed and applied a semiclassical model for energy transfer in polyatomic molecules in collision with atoms or molecules. The model is based upon the following strategy: The translational motion and the rotational motion of the two colliding molecules are treated by classical mechanics. The vibrational motion of both molecules are treated quantally. However, in some cases also the soft vibrations can be included in the classical part of the problem [ I]. The quantum mechanical treatment is based upon a zeroth order solution: that of a set of linearly and quadratically forced harmonic oscillators (including also the so-called dynamical Coriolis coupling). The anharmonic inter- and intramolecular terms and the static Coriolis coupling are included using state expansion and perturbation technique [ 2 1. However, in some cases a much simpler overlap model can be used to include the effect of anharmonicity [ 31. Often it suffices to consider just the harmonic model for which the semiclassical approach becomes very simple to use. This is the approach taken in the present paper where energy transfer to all 12 vibrational and the rotational modes of glyoxal is included in the calculation. However, only state to state cross sections and rate constants for excitation of the v7 mode are calculated. These cross sections and rates have previously been calculated by Kroes and Rettschnik [ 41 using the AVCC-10s (azimuthal and vibrational close-coupled, infinite order sudden) method. In this approach the vibrations and part of the rotational motion are treated within a close-coupling method, whereas the remaining rotational motion is decoupled by invoking the 10s approximation. As far as the vibrational modes are concerned only the vg, v7 and us modes were included in these calculations. The present approach includes all the vibrational modes quantally but the translational and rotational motion within a classical mechanical description. In ref. [4] poor agreement between the experimental data for Ar-glyoxal was found, whereas the agreement for He-glyoxal was good. It was then speculated whether the poor agreement for Ar-glyoxal could be due to the 10s approximation, which previously was found to be less accurate for heavy mass systems [ 5 1. This conjecture made in ref. [ 41 was one of the reasons for studying the same system with another method using the same potential energy surface as used in the previous calculations.
2. Theory Although the theory has been given in several previous publications we shall outline the main features in the 0301-0104/93/S 06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
G.D. Billing/ Chemical Physics173 (I 993) 16 7-175
168
present paper. For a collision between an atom and a polyatomic molecule we consider the following approximate Hamiltonian: HZ fro + uanh+ Hc, + Vi”, + H&t + Tkin9
(1)
where & is the zeroth order Hamiltonian for the vibrational motion of the polyatomic molecule, vti anharmanic intramolecular coupling terms, Hco the Coriolis coupling term, Hfot the zeroth order rotational Hamiltonian for the polyatomic molecule and Vintthe intermolecular potential which in the present case will be expanded as
(2) The last term in eq. ( 1) is the kinetic energy of the relative motion. The Coriolis coupling term depends upon time through the classical rotational motion, i.e. (3) where (Y=x, y, z and pa the vibrational angular momentum Pa=
FgF
lV’lj~~(QkPk’-QklPk)
(4)
-
When solving the equations for the quantum amplitudes as a function of time it is possible to include the socalled dynamical Coriolis coupling term [ 31. The dynamical Coriolis coupling will due to the rotational motion of the polyatomic molecule (eventually induced by the collision) couple the various vibrational modes. Furthermore we include the following creation/annihilation operators a:, ak and al& (k, k’= 1, .... 3N-6), where N is the number of atoms. Then the time-dependent Schriidinger equation is solvable and we obtain the following expression for the state to state transition amplitudes: <{$]u]{m)>=
2
<{n}lUVVl{k})({k}lUVTI{m})
(5)
9
for the VV matrix ekInentS where {m} denote a set of quantum numbers ml, .... m&v_&The general eXpreSSiOn are given in ref. [ 61. But for large polyatomic molecules the VV intramolecular processes induced by the intermolecular potential are often small. Thus within a first order treatment of these processes the following approximate expression is obtained:
+
C JmQplexp
IZP
c5 #lAP
Q,nl+(n;+l)Q~+(n6-1)Q~~
x~nb-1IV:P:In,~~n~+ll~~I~~~,~~,~~;I~~I~,~ A
1 3
(6)
where M= 3N- 6, (7) and
G.D. Billing/Chemical Physics I73 (1993) 167-l 75
(n;]U#
169
(-llPPR (8) p~~p!(n;-n*+p)!(nr-p)! ’
Ink)=exp(igk-_~k)(n;!nk!)1’2(i~k+)”~--n*
where (9) to
The classical equations of motion, i.e. the relative translational and the rotational motion are coupled to the vibrational excitation processes through an effective Hamiltonian: &r=(VHIU
(10)
*
In this manner the vibrational excitation process is coupled to the translational and rotational motion such that Z&r is conserved. Thus we have [ 7 ]
+
c
[F,$(n~Qk,+n~Q~+c$cr~
+...)+c.c.]
(11)
,
krl
where the dots indicate additional smaller terms [ 81 and C.C.the complex conjugate of the first term. In eq. ( 11) we have F;
=bk--eqeXp(
aV
+imkt)
(12)
and
cu;=-I~
I
dtF,f,
where bk = d-.
(13)
We furthermore have
exp[+i(c&-o,)t] and the functions QU are obtained as solutions to an MxMmatrix
(14) equation:
ifih=FR,
(15)
where Q= R - I and R ( to) = I. The classical equations of motion are obtained from eq. ( 11)) i.e.
(16) (17) plus corresponding equations for the Y and Z components of the relative motion. In eq. ( 16 ) p is the reduced mass. For the rotational motion we introduce the Euler angles and the corresponding momenta [ 81, i.e.
170
G.D. Billing / Chemical Physics I 73 (I 993) I6 7-175
sin ry -COS
V/csc
cos v/
e
cos w cot e
sin wcsc e -sin v/cot e
(18)
and aH,, ae - (P+ cos e-pw) [ (PJZL)
Ik3=-
cos y-
(PJZ;,)
sin w] sin -28,
aH,,
.
(20)
a@ 9
Pqb=-
aH,, aw
Pw=-
-A
VJzL,)
(19)
(p@ csc
e-p,
cot
cos w- (P,lz~,)
e) [ (P,/Zk) sin
sin w+ (P,/z;,)
cos ~1
wl .
(21)
The 12 classical equations of motion are integrated together with eqs. ( 15). From the solution to these equations we obtain energy transfer to the rotational motion of the molecule and to the various normal modes, mk
= fiwkpk
(22)
,
where Pk=I&i2.
(23)
Also the individual state to state probabilities are through eqs. (5)-(9) above equations.
given in terms of the solution to the
3. Potential energy surface As mentioned above the potential energy surface used for the semiclassical calculations was the same as the one used in the AVCC-10s calculations. Thus the He-glyoxal and Ar-glyoxal interaction was approximated with a sum of pair potentials, I’(Ri)=Ai exp( -BiR,)-C,D(R,)/Rf
,
.(24)
where Ri denote the distance from atom “i” in the molecule to the atom He or Ar. D(Ri) is a switching function, D(Ri) = 1.0,
=exp[ - (1.28R,,/R,
R, a 1.28R,, , - 1.0)2],
R, < 1.28R,,
.
(25)
The potential parameters are given in ref. [ 4 1. The semiclassical method used here also needs the normal mode frequencies and the transformation matrix between Cartesian and normal mode coordinates. This information is obtained by diagonalizing the force-constant (second derivative) matrix obtained from the intramolecular interaction potential [ 91. We have in table 1 given the principal moments of inertia and the normal mode frequencies obtained by diagonalizing the force constant matrix for deuterated glyoxal ( glyoxal-d2 ) .
G.D. Billing /Chemical Physics I73 (1993) 167-175 Table I Normal mode frequencies (cm-’ ) and moments of inertia (amu “spectroscopic” designation is given for the modes 6,7 and 8
171
A*)for glyoxal-& The frequencies are given in increasing order - the Moment of inertia
1 2 3 4 5 6 7 8 9 10 11 12
u,
237 414 536 684 739 947 999 1006 1656 1776 2368 2380
19.90 82.61 135.9
v6
V0
4. Results
In order to compare with the calculations of ref. [ 41 the calculations were carried out on deuterated glyoxal (glyoxal-d,). Thus cross sections for the vibrational excitation of the v7 mode from the ground state to the first sol and second excited state ao2 as well as rate constants for deactivation of the first excited state klo( 7’) are calculated. Also average energy transfer to the vibrational and rotational modes of glyoxal is reported. The maximum impact parameter was set to 5 A and the average energy transfer is therefore for trajectories chosen with impact randomly in the range 0 to 5 A. Cross sections are obtained as 0n*ni= s’Tdl(2/Cl) n 0
j.dB4dmj^dvsinBjdP,PrdP~~dP=~~ 0
0
0
0
0
luRk,,;12,
(26)
0
where 8, @and VIare the Euler angles specifying the initial random orientation of the polyatomic molecule, N the number of trajectories, I the orbital angular momentum, and k:=
$
(E-E,,-E,,,)
,
(27)
where E,, denotes the initial vibrational energy, E,, the rotational energy and ,u the reduced mass. In eq. (26) we have also used that dp, dp, dp, = sin 8 dP,dP, dP, .
(28)
The momenta Pa (a=~, y, z) are chosen randomly such that [ 61 E
_I z+p:+p: KID’ - 2 ( I:, Zh
ZZZ >*
(29)
In order to obtain vibrationally resolved rate constants we average over a Boltzmann distribution for the relative translational and the rotational motion. This is most conveniently done by introducing the so-called average cross section [ 10 1,
G.D. Billing / Chemical Physics 173 (1993) I6 7- I 75
172
where U is the sum of the kinetic and rotational energy, i.e.
where E is the total energy and E,, the vibrational energy of the initial state. Q,, is the rotational partition function evaluated at the reference temperature T,, and unk,ni,the amplitude for the vibrational transition from state nk to state n; in mode k. The rate constants are then obtained as (32)
We notice that To is just an arbitrary reference temperature included in the definition of the average cross section in order to obtain the correct unit (A*) for this quantity. It disappears from the final expression when the classical limit of the rotational partition function at T= T,,:
where &, = fi2/2kZga is introduced. In order to obtain a converged cross section (to within about 10%) a total of 150 to 200 trajectories at each energy are run. The initial Euler angles, impact parameter and momenta P, are selected randomly. The results are reported in tables 2 and 3. The semiclassical values for the cross section cxol are in figs. 1 and 2 compared with the quantum AVCC-10s calculations of ref. [4]. We notice that the agreement is good - but that the semiclassical values lie slightly above the quantum for He-glyoxal and even more for Ar-glyoxal in the low energy range but below the quantum values for He-glyoxal at higher energies. One reason for the discrepancy could be that our trajectories are allowed to lose energy to all the vibrational modes and tables 2 and 3 show that the energy loss to the low frequency mode (our mode 2) is substantial ( 10% at least). At the turning point for the radial motion these other low frequency modes will be even more “excited”, hence the impact velocity will be effectively smaller than if these vibrational degrees of freedom had been omitted. Especially the less efficient Ar-glyoxal collisions would be affected by these dynamical features. At Table 2 Energy transfer to the vibrational modes and the rotational mode of glyoxal in collisions with He. The normal modes are listed in increasing order of the vibrational frequency, w1=237 cm-‘, @=414 cm-’ and w,=536 cm-‘. Cross sections in A’ for excitation of the lowest frequency mode from the ground state to the first and second excited state are also given. Energies are in units of 100 kJ/mol. The trajectories with impact parameters between 0 and 5 A and the energy Eti, refer to the O-l cross sections (see text)
&ii7
%I
%2
Ml
0.0367 0.0459 0.0652 0.114 0.214
0.125 0.261 0.681 1.52 2.79
0.002 1 0.011 0.040 0.20 0.61
1.13x 2.25x 5.06x 1.54x 5.10x
lo-4 10-4 1O-4 10-3 10-S
a
&
6.17~10-~ 1.41x10-5 l.l3xlo-4 6.82x lO-4 3.06x lo-’
5.32x 1.98x 1.38x 1.24x 6.27x
alb 1O-8 IO-6 10-3 lO-4 lO-4
1.19x 2.42x 6.36x 2.40x 9.57x
fwcl, lo-4 lO-4 lO-4 lo-’ 10-X
3.17x 10-s 3.31 x 10-r 7.99x 10-x 1.43x 10-2 2.21 x10-2
Table 3 Same as table 1 but for Ar-glyoxal collisions
&¶I
%I
002
m
0.0367 0.0449 0.0652 0.114 0.214
0.0785 0.156 0.341 0.969 4.14
3.4x IO-’ 1.38x lo-’ 0.71 x10-2 4.73 x 10-r 6.51 x lo-’
4.57x 8.88x 2.63x 8.95x 3.47 x
a
10-S 10-S 1O-4 1O-4 lo-’
3.45 x 6.46x 2.22x 9.92x 9.24x
IO-6 lO-6 10-S lO-5 IO-’
fi3
A&b
2.98x lo-’ 9.13x10-’ 3.98x lO-6 2.87x lO-5 2.25 x lo-’
4.95x 9.66x 2.90x 1.05x 4.68x
hE,
10-5 lO-5 lO-4 10-3 lO-3
6.90x lo-’ 9.50x lo-” 1.21x10-2 1.93x 10-2 3.16x10-*
173
G.D. Billing / Chemical Physics I 73 (I 993) 16 7-175
I
I
I
10 L
I
I
I I
Ar-Glyoxol
He-qlyoxol
0
0.010.0
I
I
0.2
0.1
I
0.3
&in/c Fig, 1. Cross sections uO,in A2excitation of the Y, mode in giyoxal by He collisions as a function of kinetic energy. I Z= 100 kJ/mol. The solid line indicates quantum AVCC-10s calculations from ref. [4].
0.01 0.0
I 0.1
I 0.2
I 0.3
&in/f Fig. 2. Same as fig 1 but for Ar-glyoxal collisions.
higher energies the effect should show up also in the large cross sections obtained for the He-glyoxal system, where we (see table 2) have that the energy transfer to mode 2 is almost the same as to mode 1. Indeed, fig. 1 shows a smaller cross section for He at higher energies. Based upon our previous comparisons between the semiclassical and the VCC-10s quantum method [ 5 ] we would expect that the VCC-10s method underestimate the cross sections - and mainly for the heavy mass systems, i.e. Ar-glyoxal in the present case. The reason that good agreement is obtained at higher energies is we think due to the energy transfer effect mentioned above, i.e. energy transfer to modes not included in the quantum calculations. The calculations are carried out at a given initial “semiclassical” kinetic energy U and the “quantum” kinetic energy bound by introducing the usual symmetrization principle [ 6 1. Therefore the kinetic energy given in tables 1 and 2 is for a specific transition (the O-l transition). To obtain the kinetic energy for the O-2 transition one has to add 0.0 1429 to the kinetic energies given in tables 2 and 3. The cross sections given in tables 2 and 3 were obtained with the initial rotational state equal to zero for the glyoxal molecule. Thus in order to compute the rate constant for the vibrational deactivation process in mode v7 (our model 1) we used expression (30) which allows for a Boltzmann distribution of the rotational energy of the glyoxal molecule. These so-called average cross sections are given in table 4 and used to calculate the rates shown in fig. 3. The above-mentioned symmetrization is now introduced when calculating the rate constants. We notice that our rate constants are about 30% above the AVCC-10s rate for Ar-glyoxal at 300 K and 25% below for the He-glyoxal system. The reason for these discrepancies is we think due to the fact that the 10s approximation tends to underestimate the rates for heavy mass systems and that we in the case of He-glyoxal have considerable energy transfer to the low frequency modes not included in the AVCC-10s calculations. The experimental data at 300 K show a very small difference between the Ar and He deactivation rate. In this respect it should be noted that the cross section (see fig. 2) for Ar-glyoxal increases more sharply with energy than for He-glyoxal. Thus at some temperature the two rates will be comparable. But with the present potential energy
G.D. Billing / Chemical Physics173 (1993) I6 7-175
174
I
PExpenment3
Table 4 Average cross sections ( uc, ) (A*)for He- and Ar-glyoxal collisions as a function of energy iJ, where U is the sum of kinetic energy and rotational energy of the glyoxal molecule. Energies in units of 100 kJ/mol and between 100 and 200 trajectories were used at each energy V (100 kJ/mol)
:s
(He)
(Ar) (A*)
0.0250 0.0300 0.0400 0.0500 0.100 0.150 0.200 0.300 0.500
1.02x IO-2 3.45x10-2 2.06x 10-l 3.27x 10-l 2.15 1.00x 10’ 2.22 x 10’ 8.35x 10’ 4.95x 102
1.20x 10-2 2.38x lo-* 6.98x lo-* 1.30x 10-l 2.31 1.12x10’ 5.00x 10’ 1.97x lo* 1.18x lo3
Fig. 3. Bate constants for vibrational deactivation of the Y, mode as a function of temperature. The solid lines indicate the values obtained semi-classically for He- and Ar-glyoxal. The numbers are compared with quantum AVCC-IOS calculations [ 41 and experimental data from ref. [ 111. The upper experimental number is for He-glyoxal; the lower for Ar-glyoxal at room temperature.
surface it happens well above room temperature. Thus even at 500 K the ratio is 1.6, i.e. still higher than the one found experimentally at room temperature. Whether it is possible to construct a surface which does give this small difference in the rates is of course still an open question. The present calculations indicate that it is necessary to include all or at least all the low frequency vibrational modes (including w5 in table 1) in the numerical calculations. The present calculation allows the molecule to rotate according to the dynamics of the system thus there are no dynamical constraints as far as this degree of freedom is concerned. This is especially important if the potential. is strongly anisotropic and if the vibrational excitation of the molecule depends strongly upon the approach angle. Previous comparisons between this semiclassical method and the VCC-10s method [ 51 for He- and Ne-C02 collisions showed that the VCC-10s method underestimated the rates from an order of magnitude at 100 K to a factor of two for Ne-CO2 but much less for He-C02. That we in the present case get much better agreement between the AVCC-10s method [ 121 and the semi-classical values both for Ar and He is due to the fact that the cross sections and hence the rates are much larger (several orders of magnitude at low energies/temperatures) in the present case. Large cross sections or probabilities for vibrational excitation are of course not as, sensitive to such dynamical details as those mentioned above. However, this is not true if the potential is very anisotropic [ 131, but this is not the case for the system investigated here.
Acknowluigement
This research was supported by the Danish Natural Science Research Council. Dr. G.-J. Kroes is acknowledged for supplying me with the force constant matrix and cross sections used for figs. 1 and 2.
References [ 11 G.D. Billing, Chem. Phys. Letters 89 ( 1982) 337. [2]G.D.Billing,Chem.Phys.61 (1981)415(paperVI).
G.D. Billing/ChemicalPhysics173(1993) 167-175 [ 31 G.D. Billing, J. Chem. Sot. Faraday 86 (1990) 1663. [ 4 ] G.-J. Kroes and R.P.H. Rettschnick, Chem. Phys. 156 ( 1991) 293. [ 51G.D. Billing and D.C. Clary, Chem. Phys. Letters 90 ( 1982) 27. [6] G.D. Billing, Computer Phys. Rept. 1 (1984) 237. [7] G.D. Billing, Chem. Phys. 46 (1980) 123 (paper II). [ 81 H. Goldstein, Classical mechanics (Addison-Wesley, Reading, MA, 1950 ) . [ 91 G.-J. Rroes, private communication. [lo] G.D. Billing, Chem. Phys. 76 (1983) 315 (paper VIII). [ 111 G. De Leeuw, Ph.D. Thesis, Universiteit van Amsterdam, Amsterdam, The Netherlands ( 198 1), [ 121 D.C. Clary, J. Chem. Phys. 81 (1984) 4466. [ 131 EA. Gianturco, S. Sema, A. Palma, G.D. Billing and V. Zenevich, submitted for publication.
175
Semiclassical model for energy transfer in polyatomic molecules. XIV. He- and Ar-glyoxal collkions G.D. Billing Department of Chemistry, H.C. 0rsted Institute, University of Copenhagen, 2100 Copenhagen 0, Denmark Received 9 February 1993
Cross sections and rate constants for vibrational excitation of the v, mode in glyoxal by collisions with He and Ar atoms are calculated by a quantum-classical model. Comparison with approximate quantum calculations and experimental data is made.
1. Introdwction We have in a series of papers (I-XIII) developed and applied a semiclassical model for energy transfer in polyatomic molecules in collision with atoms or molecules. The model is based upon the following strategy: The translational motion and the rotational motion of the two colliding molecules are treated by classical mechanics. The vibrational motion of both molecules are treated quantally. However, in some cases also the soft vibrations can be included in the classical part of the problem [ I]. The quantum mechanical treatment is based upon a zeroth order solution: that of a set of linearly and quadratically forced harmonic oscillators (including also the so-called dynamical Coriolis coupling). The anharmonic inter- and intramolecular terms and the static Coriolis coupling are included using state expansion and perturbation technique [ 2 1. However, in some cases a much simpler overlap model can be used to include the effect of anharmonicity [ 31. Often it suffices to consider just the harmonic model for which the semiclassical approach becomes very simple to use. This is the approach taken in the present paper where energy transfer to all 12 vibrational and the rotational modes of glyoxal is included in the calculation. However, only state to state cross sections and rate constants for excitation of the v7 mode are calculated. These cross sections and rates have previously been calculated by Kroes and Rettschnik [ 41 using the AVCC-10s (azimuthal and vibrational close-coupled, infinite order sudden) method. In this approach the vibrations and part of the rotational motion are treated within a close-coupling method, whereas the remaining rotational motion is decoupled by invoking the 10s approximation. As far as the vibrational modes are concerned only the vg, v7 and us modes were included in these calculations. The present approach includes all the vibrational modes quantally but the translational and rotational motion within a classical mechanical description. In ref. [4] poor agreement between the experimental data for Ar-glyoxal was found, whereas the agreement for He-glyoxal was good. It was then speculated whether the poor agreement for Ar-glyoxal could be due to the 10s approximation, which previously was found to be less accurate for heavy mass systems [ 5 1. This conjecture made in ref. [ 41 was one of the reasons for studying the same system with another method using the same potential energy surface as used in the previous calculations.
2. Theory Although the theory has been given in several previous publications we shall outline the main features in the 0301-0104/93/S 06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
G.D. Billing/ Chemical Physics173 (I 993) 16 7-175
168
present paper. For a collision between an atom and a polyatomic molecule we consider the following approximate Hamiltonian: HZ fro + uanh+ Hc, + Vi”, + H&t + Tkin9
(1)
where & is the zeroth order Hamiltonian for the vibrational motion of the polyatomic molecule, vti anharmanic intramolecular coupling terms, Hco the Coriolis coupling term, Hfot the zeroth order rotational Hamiltonian for the polyatomic molecule and Vintthe intermolecular potential which in the present case will be expanded as
(2) The last term in eq. ( 1) is the kinetic energy of the relative motion. The Coriolis coupling term depends upon time through the classical rotational motion, i.e. (3) where (Y=x, y, z and pa the vibrational angular momentum Pa=
FgF
lV’lj~~(QkPk’-QklPk)
(4)
-
When solving the equations for the quantum amplitudes as a function of time it is possible to include the socalled dynamical Coriolis coupling term [ 31. The dynamical Coriolis coupling will due to the rotational motion of the polyatomic molecule (eventually induced by the collision) couple the various vibrational modes. Furthermore we include the following creation/annihilation operators a:, ak and al& (k, k’= 1, .... 3N-6), where N is the number of atoms. Then the time-dependent Schriidinger equation is solvable and we obtain the following expression for the state to state transition amplitudes: <{$]u]{m)>=
2
<{n}lUVVl{k})({k}lUVTI{m})
(5)
9
for the VV matrix ekInentS where {m} denote a set of quantum numbers ml, .... m&v_&The general eXpreSSiOn are given in ref. [ 61. But for large polyatomic molecules the VV intramolecular processes induced by the intermolecular potential are often small. Thus within a first order treatment of these processes the following approximate expression is obtained:
+
C JmQplexp
IZP
c5 #lAP
Q,nl+(n;+l)Q~+(n6-1)Q~~
x~nb-1IV:P:In,~~n~+ll~~I~~~,~~,~~;I~~I~,~ A
1 3
(6)
where M= 3N- 6, (7) and
G.D. Billing/Chemical Physics I73 (1993) 167-l 75
(n;]U#
169
(-llPPR (8) p~~p!(n;-n*+p)!(nr-p)! ’
Ink)=exp(igk-_~k)(n;!nk!)1’2(i~k+)”~--n*
where (9) to
The classical equations of motion, i.e. the relative translational and the rotational motion are coupled to the vibrational excitation processes through an effective Hamiltonian: &r=(VHIU
(10)
*
In this manner the vibrational excitation process is coupled to the translational and rotational motion such that Z&r is conserved. Thus we have [ 7 ]
+
c
[F,$(n~Qk,+n~Q~+c$cr~
+...)+c.c.]
(11)
,
krl
where the dots indicate additional smaller terms [ 81 and C.C.the complex conjugate of the first term. In eq. ( 11) we have F;
=bk--eqeXp(
aV
+imkt)
(12)
and
cu;=-I~
I
dtF,f,
where bk = d-.
(13)
We furthermore have
exp[+i(c&-o,)t] and the functions QU are obtained as solutions to an MxMmatrix
(14) equation:
ifih=FR,
(15)
where Q= R - I and R ( to) = I. The classical equations of motion are obtained from eq. ( 11)) i.e.
(16) (17) plus corresponding equations for the Y and Z components of the relative motion. In eq. ( 16 ) p is the reduced mass. For the rotational motion we introduce the Euler angles and the corresponding momenta [ 81, i.e.
170
G.D. Billing / Chemical Physics I 73 (I 993) I6 7-175
sin ry -COS
V/csc
cos v/
e
cos w cot e
sin wcsc e -sin v/cot e
(18)
and aH,, ae - (P+ cos e-pw) [ (PJZL)
Ik3=-
cos y-
(PJZ;,)
sin w] sin -28,
aH,,
.
(20)
a@ 9
Pqb=-
aH,, aw
Pw=-
-A
VJzL,)
(19)
(p@ csc
e-p,
cot
cos w- (P,lz~,)
e) [ (P,/Zk) sin
sin w+ (P,/z;,)
cos ~1
wl .
(21)
The 12 classical equations of motion are integrated together with eqs. ( 15). From the solution to these equations we obtain energy transfer to the rotational motion of the molecule and to the various normal modes, mk
= fiwkpk
(22)
,
where Pk=I&i2.
(23)
Also the individual state to state probabilities are through eqs. (5)-(9) above equations.
given in terms of the solution to the
3. Potential energy surface As mentioned above the potential energy surface used for the semiclassical calculations was the same as the one used in the AVCC-10s calculations. Thus the He-glyoxal and Ar-glyoxal interaction was approximated with a sum of pair potentials, I’(Ri)=Ai exp( -BiR,)-C,D(R,)/Rf
,
.(24)
where Ri denote the distance from atom “i” in the molecule to the atom He or Ar. D(Ri) is a switching function, D(Ri) = 1.0,
=exp[ - (1.28R,,/R,
R, a 1.28R,, , - 1.0)2],
R, < 1.28R,,
.
(25)
The potential parameters are given in ref. [ 4 1. The semiclassical method used here also needs the normal mode frequencies and the transformation matrix between Cartesian and normal mode coordinates. This information is obtained by diagonalizing the force-constant (second derivative) matrix obtained from the intramolecular interaction potential [ 91. We have in table 1 given the principal moments of inertia and the normal mode frequencies obtained by diagonalizing the force constant matrix for deuterated glyoxal ( glyoxal-d2 ) .
G.D. Billing /Chemical Physics I73 (1993) 167-175 Table I Normal mode frequencies (cm-’ ) and moments of inertia (amu “spectroscopic” designation is given for the modes 6,7 and 8
171
A*)for glyoxal-& The frequencies are given in increasing order - the Moment of inertia
1 2 3 4 5 6 7 8 9 10 11 12
u,
237 414 536 684 739 947 999 1006 1656 1776 2368 2380
19.90 82.61 135.9
v6
V0
4. Results
In order to compare with the calculations of ref. [ 41 the calculations were carried out on deuterated glyoxal (glyoxal-d,). Thus cross sections for the vibrational excitation of the v7 mode from the ground state to the first sol and second excited state ao2 as well as rate constants for deactivation of the first excited state klo( 7’) are calculated. Also average energy transfer to the vibrational and rotational modes of glyoxal is reported. The maximum impact parameter was set to 5 A and the average energy transfer is therefore for trajectories chosen with impact randomly in the range 0 to 5 A. Cross sections are obtained as 0n*ni= s’Tdl(2/Cl) n 0
j.dB4dmj^dvsinBjdP,PrdP~~dP=~~ 0
0
0
0
0
luRk,,;12,
(26)
0
where 8, @and VIare the Euler angles specifying the initial random orientation of the polyatomic molecule, N the number of trajectories, I the orbital angular momentum, and k:=
$
(E-E,,-E,,,)
,
(27)
where E,, denotes the initial vibrational energy, E,, the rotational energy and ,u the reduced mass. In eq. (26) we have also used that dp, dp, dp, = sin 8 dP,dP, dP, .
(28)
The momenta Pa (a=~, y, z) are chosen randomly such that [ 61 E
_I z+p:+p: KID’ - 2 ( I:, Zh
ZZZ >*
(29)
In order to obtain vibrationally resolved rate constants we average over a Boltzmann distribution for the relative translational and the rotational motion. This is most conveniently done by introducing the so-called average cross section [ 10 1,
G.D. Billing / Chemical Physics 173 (1993) I6 7- I 75
172
where U is the sum of the kinetic and rotational energy, i.e.
where E is the total energy and E,, the vibrational energy of the initial state. Q,, is the rotational partition function evaluated at the reference temperature T,, and unk,ni,the amplitude for the vibrational transition from state nk to state n; in mode k. The rate constants are then obtained as (32)
We notice that To is just an arbitrary reference temperature included in the definition of the average cross section in order to obtain the correct unit (A*) for this quantity. It disappears from the final expression when the classical limit of the rotational partition function at T= T,,:
where &, = fi2/2kZga is introduced. In order to obtain a converged cross section (to within about 10%) a total of 150 to 200 trajectories at each energy are run. The initial Euler angles, impact parameter and momenta P, are selected randomly. The results are reported in tables 2 and 3. The semiclassical values for the cross section cxol are in figs. 1 and 2 compared with the quantum AVCC-10s calculations of ref. [4]. We notice that the agreement is good - but that the semiclassical values lie slightly above the quantum for He-glyoxal and even more for Ar-glyoxal in the low energy range but below the quantum values for He-glyoxal at higher energies. One reason for the discrepancy could be that our trajectories are allowed to lose energy to all the vibrational modes and tables 2 and 3 show that the energy loss to the low frequency mode (our mode 2) is substantial ( 10% at least). At the turning point for the radial motion these other low frequency modes will be even more “excited”, hence the impact velocity will be effectively smaller than if these vibrational degrees of freedom had been omitted. Especially the less efficient Ar-glyoxal collisions would be affected by these dynamical features. At Table 2 Energy transfer to the vibrational modes and the rotational mode of glyoxal in collisions with He. The normal modes are listed in increasing order of the vibrational frequency, w1=237 cm-‘, @=414 cm-’ and w,=536 cm-‘. Cross sections in A’ for excitation of the lowest frequency mode from the ground state to the first and second excited state are also given. Energies are in units of 100 kJ/mol. The trajectories with impact parameters between 0 and 5 A and the energy Eti, refer to the O-l cross sections (see text)
&ii7
%I
%2
Ml
0.0367 0.0459 0.0652 0.114 0.214
0.125 0.261 0.681 1.52 2.79
0.002 1 0.011 0.040 0.20 0.61
1.13x 2.25x 5.06x 1.54x 5.10x
lo-4 10-4 1O-4 10-3 10-S
a
&
6.17~10-~ 1.41x10-5 l.l3xlo-4 6.82x lO-4 3.06x lo-’
5.32x 1.98x 1.38x 1.24x 6.27x
alb 1O-8 IO-6 10-3 lO-4 lO-4
1.19x 2.42x 6.36x 2.40x 9.57x
fwcl, lo-4 lO-4 lO-4 lo-’ 10-X
3.17x 10-s 3.31 x 10-r 7.99x 10-x 1.43x 10-2 2.21 x10-2
Table 3 Same as table 1 but for Ar-glyoxal collisions
&¶I
%I
002
m
0.0367 0.0449 0.0652 0.114 0.214
0.0785 0.156 0.341 0.969 4.14
3.4x IO-’ 1.38x lo-’ 0.71 x10-2 4.73 x 10-r 6.51 x lo-’
4.57x 8.88x 2.63x 8.95x 3.47 x
a
10-S 10-S 1O-4 1O-4 lo-’
3.45 x 6.46x 2.22x 9.92x 9.24x
IO-6 lO-6 10-S lO-5 IO-’
fi3
A&b
2.98x lo-’ 9.13x10-’ 3.98x lO-6 2.87x lO-5 2.25 x lo-’
4.95x 9.66x 2.90x 1.05x 4.68x
hE,
10-5 lO-5 lO-4 10-3 lO-3
6.90x lo-’ 9.50x lo-” 1.21x10-2 1.93x 10-2 3.16x10-*
173
G.D. Billing / Chemical Physics I 73 (I 993) 16 7-175
I
I
I
10 L
I
I
I I
Ar-Glyoxol
He-qlyoxol
0
0.010.0
I
I
0.2
0.1
I
0.3
&in/c Fig, 1. Cross sections uO,in A2excitation of the Y, mode in giyoxal by He collisions as a function of kinetic energy. I Z= 100 kJ/mol. The solid line indicates quantum AVCC-10s calculations from ref. [4].
0.01 0.0
I 0.1
I 0.2
I 0.3
&in/f Fig. 2. Same as fig 1 but for Ar-glyoxal collisions.
higher energies the effect should show up also in the large cross sections obtained for the He-glyoxal system, where we (see table 2) have that the energy transfer to mode 2 is almost the same as to mode 1. Indeed, fig. 1 shows a smaller cross section for He at higher energies. Based upon our previous comparisons between the semiclassical and the VCC-10s quantum method [ 5 ] we would expect that the VCC-10s method underestimate the cross sections - and mainly for the heavy mass systems, i.e. Ar-glyoxal in the present case. The reason that good agreement is obtained at higher energies is we think due to the energy transfer effect mentioned above, i.e. energy transfer to modes not included in the quantum calculations. The calculations are carried out at a given initial “semiclassical” kinetic energy U and the “quantum” kinetic energy bound by introducing the usual symmetrization principle [ 6 1. Therefore the kinetic energy given in tables 1 and 2 is for a specific transition (the O-l transition). To obtain the kinetic energy for the O-2 transition one has to add 0.0 1429 to the kinetic energies given in tables 2 and 3. The cross sections given in tables 2 and 3 were obtained with the initial rotational state equal to zero for the glyoxal molecule. Thus in order to compute the rate constant for the vibrational deactivation process in mode v7 (our model 1) we used expression (30) which allows for a Boltzmann distribution of the rotational energy of the glyoxal molecule. These so-called average cross sections are given in table 4 and used to calculate the rates shown in fig. 3. The above-mentioned symmetrization is now introduced when calculating the rate constants. We notice that our rate constants are about 30% above the AVCC-10s rate for Ar-glyoxal at 300 K and 25% below for the He-glyoxal system. The reason for these discrepancies is we think due to the fact that the 10s approximation tends to underestimate the rates for heavy mass systems and that we in the case of He-glyoxal have considerable energy transfer to the low frequency modes not included in the AVCC-10s calculations. The experimental data at 300 K show a very small difference between the Ar and He deactivation rate. In this respect it should be noted that the cross section (see fig. 2) for Ar-glyoxal increases more sharply with energy than for He-glyoxal. Thus at some temperature the two rates will be comparable. But with the present potential energy
G.D. Billing / Chemical Physics173 (1993) I6 7-175
174
I
PExpenment3
Table 4 Average cross sections ( uc, ) (A*)for He- and Ar-glyoxal collisions as a function of energy iJ, where U is the sum of kinetic energy and rotational energy of the glyoxal molecule. Energies in units of 100 kJ/mol and between 100 and 200 trajectories were used at each energy V (100 kJ/mol)
:s
(He)
0.0250 0.0300 0.0400 0.0500 0.100 0.150 0.200 0.300 0.500
1.02x IO-2 3.45x10-2 2.06x 10-l 3.27x 10-l 2.15 1.00x 10’ 2.22 x 10’ 8.35x 10’ 4.95x 102
1.20x 10-2 2.38x lo-* 6.98x lo-* 1.30x 10-l 2.31 1.12x10’ 5.00x 10’ 1.97x lo* 1.18x lo3
Fig. 3. Bate constants for vibrational deactivation of the Y, mode as a function of temperature. The solid lines indicate the values obtained semi-classically for He- and Ar-glyoxal. The numbers are compared with quantum AVCC-IOS calculations [ 41 and experimental data from ref. [ 111. The upper experimental number is for He-glyoxal; the lower for Ar-glyoxal at room temperature.
surface it happens well above room temperature. Thus even at 500 K the ratio is 1.6, i.e. still higher than the one found experimentally at room temperature. Whether it is possible to construct a surface which does give this small difference in the rates is of course still an open question. The present calculations indicate that it is necessary to include all or at least all the low frequency vibrational modes (including w5 in table 1) in the numerical calculations. The present calculation allows the molecule to rotate according to the dynamics of the system thus there are no dynamical constraints as far as this degree of freedom is concerned. This is especially important if the potential. is strongly anisotropic and if the vibrational excitation of the molecule depends strongly upon the approach angle. Previous comparisons between this semiclassical method and the VCC-10s method [ 51 for He- and Ne-C02 collisions showed that the VCC-10s method underestimated the rates from an order of magnitude at 100 K to a factor of two for Ne-CO2 but much less for He-C02. That we in the present case get much better agreement between the AVCC-10s method [ 121 and the semi-classical values both for Ar and He is due to the fact that the cross sections and hence the rates are much larger (several orders of magnitude at low energies/temperatures) in the present case. Large cross sections or probabilities for vibrational excitation are of course not as, sensitive to such dynamical details as those mentioned above. However, this is not true if the potential is very anisotropic [ 131, but this is not the case for the system investigated here.
Acknowluigement
This research was supported by the Danish Natural Science Research Council. Dr. G.-J. Kroes is acknowledged for supplying me with the force constant matrix and cross sections used for figs. 1 and 2.
References [ 11 G.D. Billing, Chem. Phys. Letters 89 ( 1982) 337. [2]G.D.Billing,Chem.Phys.61 (1981)415(paperVI).
G.D. Billing/ChemicalPhysics173(1993) 167-175 [ 31 G.D. Billing, J. Chem. Sot. Faraday 86 (1990) 1663. [ 4 ] G.-J. Kroes and R.P.H. Rettschnick, Chem. Phys. 156 ( 1991) 293. [ 51G.D. Billing and D.C. Clary, Chem. Phys. Letters 90 ( 1982) 27. [6] G.D. Billing, Computer Phys. Rept. 1 (1984) 237. [7] G.D. Billing, Chem. Phys. 46 (1980) 123 (paper II). [ 81 H. Goldstein, Classical mechanics (Addison-Wesley, Reading, MA, 1950 ) . [ 91 G.-J. Rroes, private communication. [lo] G.D. Billing, Chem. Phys. 76 (1983) 315 (paper VIII). [ 111 G. De Leeuw, Ph.D. Thesis, Universiteit van Amsterdam, Amsterdam, The Netherlands ( 198 1), [ 121 D.C. Clary, J. Chem. Phys. 81 (1984) 4466. [ 131 EA. Gianturco, S. Sema, A. Palma, G.D. Billing and V. Zenevich, submitted for publication.
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