J. Phys. Chem. 1981, 85,3187-3198
interactions and higher polymeric species such as a cyclic tetramer may attain an observable concentration. It does seem likely that, in any solvent in which self-association occurs, the cyclic dimer will be an important species.
Acknowledgment. The author thanks Dr. H. Bradford
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Thompson for helpful and stimulating discussions.
Supplementary Material Available: Tables of dielectric constants, refractive indices, and NMR chemical shifts as a function of concentration are available (3 pages). Ordering information is given on any current masthiad page.
Atom-Diatomic Molecule Collisions at Very Low Energies. 1. A Computational Study of the Adiabatic and Effective Potential Approximations V. Sethuraman and Stuart A. Rice” The Department of Chemlstty and The James Franck Institute, The Unlverslty of Chlcago, Chlcago, Illnols 60637 (Recelved: March 17, 1981; In Flnal Form: June 16, 1981)
As a preliminary step in the study of rovibrational scattering of atoms and molecules at very low energies (1-6 cm-’), we examine the usefulness of the adiabatic (LZCES) and effective potential (EP) approximations in the description of these collisions. He-, Ne-, and Ar-12 collisions, in the rigid-rotor approximation, are examined here to determine the behavior of the low-energy limit of scattering and the range, depth, and anisotropy of the potential over which the LZCES and EP approximations yield useful results. The adiabaticapproximation is found to be valid for energies greater than 1.5, 5, and -20 cm-l, respectively, for He, Ne, and Ar projectiles and is observed to break down with increasing range and anisotropy of the potential. The effective potential approximation is observed to be valid for energies greater than -3 cm-l for He and Ne, and its accuracy is noted to be much less sensitive to the collision energy, mass of the projectile, and parameters of the potential in comparison with the adiabatic approximation.
I. Introduction Recent experiments by Sulkes, Tusa, and Rice’ have shown that very low-energy collisions between a He, Ne, or Ar atom and a vibrationally excited 311iu I2molecule induce efficient vibrational relaxation. Although these data refer to collisions involving electronically excited 12,a variety of less specific experiments also lead to the inference that very low-energy collisions of atoms with molecules in their ground electronic states induce efficient vibrational relaxation. That very low-energy collisions should be particularly efficient in prompting vibrational relaxation has been suggested many times, but heretofore not directly observed. It is usually assumed that “complexes”, or van der Waals molecules, exist a t low temperature, where the average collision energy is small relative to the vibrational spacing, and that vibrational relaxation of a complex is rapid. Sulkes, Tusa, and Rice suggest, instead, that the enhancement of the cross section for vibrational relaxation in very low-energy collisions is due to scattering resonances in the energy spectrum of the system (i.e., not to boundstate formation). An experimental verification of that suggestion requires observationof the change in vibrational relaxation cross section as the collision energy crosses a putative scattering resonance region; theoretical support for the suggestion must be based on a demonstration that the behavior seen is consistent with quantum scattering theory for a potential surface that supports collision resonances. This paper is the first of several devoted to the theoretical basis for the Sulkes-Tusa-Rice proposal, The scattering of atoms by diatomic molecules has been thoroughly discussed in a series of papers by Pack2 and
Kouri3and their co-workers. The technical problem which must be solved is well-known: for massive collision partners at room temperature, so many final-state channels are open that approximation schemes which limit the size of the basis set used to represent the states of the system must be introduced if any use is to be made of the theory. The approximation that the projection of the rotational angular momentum of the diatom along the atom-diatom axis is conserved during the collision is appropriate for a diatomic molecule whose force field has only short-range anisotropy. Since this approximation involves the neglect of perturbations due to centrifugal forces, it is called the j , conserving centrifugal sudden (JZCCS) approximation. A different approximation, namely, that the diatom remains fixed in space during the collision, conserves the projection of the orbital angular momentum along the axis of the diatom. Since the rotational energy of the diatom remains unchanged in this case, it is called the 1, conserving energy sudden (LZCES) approximation; it is clearly appropriate for high-energy collisions of atoms with massive diatomic molecules which have closely spaced rotational energy levels. In practice, imposition of conservation of an average 1 in the JZCCS approximation, or an average j in the LZCES approximation, has been found to yield better results than the original forms of these approximations. When both an 1 and a j are taken to be conserved during the collision, the calculational scheme is called the “infinite order sudden” (10s)approximation. Secrest6has shown, through a transformation method, how all these approximations can be derived systematically from the close-coupling (CC) scattering equations.
(1) M. Sulkes, J. Tuaa, and S. A. Rice, J. Chem. Phys., 72,573 (1980). (2) T. P. Tsien and R. T. Pack, Chem. Phys. Lett., 8,579 (1971); T. P. Tsien, G.A. Parker, and R. T. Pack, J. Chem. Phys., 59,5373 (1973). See G. A. Parker and R. T. Pack, J. Chem. Phys., 68,1585 (1978) for a more complete list of references.
(3) P. McGuire and D. J. Kouri, J . Chem. Phys., 60, 2488 (1974); R. T. Pack, ibid. 60, 633 (1974). (4) K. Takayangi, h o g . Theor. Phys., Suppl., 25,40 (1963). (5) Don Secrest, J. Chem. Phys., 62, 710 (1975).
0022-3654/8i/2085-3 187$01.25/0 0 1981 Amerlcan Chemical Society
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The approximation that the colliding system evolves adiabatically along the scattering coordinate R, with respect to the slow internal motion of the target, was first considered by Chase; it is clearly analogous to the adiabatic approximation for the separation of electronic and nuclear motions in the bound electronic states of molecules. Application of this approximation to the collision of an atom and a diatom is equivalent to fixing the orientation in space, 2, of the diatom and solving for the scattering amplitudes that are elastic with respect to the rotational state of the diatom. These adiabatic scattering amplitudes depend parametrically on 2. Amplitudes for the scattering from the initial j,m of the diatom to the final state j’,m’ are then obtained by integrating the P-dependent adiabatic scattering amplitudes over the initial and final rotational states of the diatom. The coupled equations of the adiabatic approximation, stated in a rotating frame which has its 2 axis parallel to the diatom, conserve the projection of 1 along that 2 axis, and the rotational energy of the diatom, and are usually called the LZCES equations. Chu and Dalagarno’ were the first workers to compare the results of the LZCES approximationwith those of the JZCCS approximation and the effective potential approximation (EPA) described below. They studied the rotational excitation in H-CO collisions for collision energies of 0.01 and 0.05 eV and concluded that, for this system, the adiabatic approximation implemented as described above was inaccurate for low energies and timeconsuming for high energies, at least when compared with the results of the JZCCS and the effective potential approximations. Kharea has made a detailed examination of the expressions for the trqsition matrix elements and scattering amplitudes fymJjm(R)in the adiabatic approximation and related the latter for general initial Gm) and final G’m? rotational states to the transition matrix elements obtained by solving the LZCES equations in a rotating frame with its 2 axis parallel to the axis of the diatom. This work, on which we draw extensively, greatly simplifies the calculation of cross sections and opacity functions in the LZCES approximation. The effective potential approximation introduced by Rabitzg aims at avoiding the description of the spatial effects associated with the magnetic quantum numbers of the target and the projectile. The introduction of this approximation is motivated by the fact that most experiments do not prepare the colliding system in specific magnetic substates, nor do they analyze the magnetic substates of the particles in the exit channels. Furthermore, the cross sections aGm j m ? are not strongly dependent on m and m’.Though the prescription given by Rabitz for construction of an effective collision operator by averaging over the variable associated with the magnetic quantum numbers does not appear to be unique, he has succeeded in defining a matrix for the effective collision operator in terms of the matrix elements of the complete collision operator. Solution of the scattering wave equations appropriate to the corresponding effective Hamiltonian naturally involves coupled equations of greatly reduced dimensionality. Zarur and Rabitzgcompared the cross sections calculated with the EP approximation with those of the close-coupling method for the He-H2 system, for collision energies greater than 320 cm-‘. They con-
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(6)D.M.Chase, Phys. Rev., 104, 838 (1956). (7)S.I. Chu and A. Dalgarno, R o c . R. SOC.London, Ser. A, 342,191 (1975). (8)V. Khare, J. Chem. Phys., 68, 4631 (1978). (9)H.Rabitz, J. Chem. Phys., 67,1718(1972);G. Zarur and H.Rabitz, ibid., 60,2057 (1974).
Sethuraman and Rice
jectured that the EP approximationmight break down for very low collision energy because of coherence among the limited number of magnetic substates of the low-lying rotational states of the target. However, the validity of the EP approximation does not seem to depend on the collision energy in any obvious way, and it is not clear how the properties of quasibound states of the target-projectile system will be affected by the averaging over the angles appropriate to the projection quantum numbers. Neither of these approximations appears to have been examined for ita validity at the low energies of interest (1-5 cm-l) in the Sulkes-Tusa-Rice experiments. It is interesting to note that, to study the validity of the adiabatic approximation for rovibrational scattering involving vibrational deexcitation, it is necessary to examine the limits of its usefulness for rigid-rotor scattering in the incident vibrational channel. This is also true for the EP approximation, provided we can establish that the EP approximation is indifferent, or becomes better, as the collision energy increases. This paper is concerned with the accuracy of the EP and adiabatic approximations for very low collision energy; it is organized as follows. Section I1 and Appendix A review the adiabatic approximation calculation of the collision cross section, including vibration. Section I11 and Appendix C rederive the effective potential approximation to the collision cross section for the case of a general potential of interaction. In Section N, after a brief summary of the computational aspects of the problem, we compare and analyze the results of computations of rigid-rotor scattering in the close-coupling, adiabatic, and EP approximations. We shall present in another paper calculations of rovibrational scattering for very low collisional energies. 11. Adiabatic Approximation
A. Atom-Vibrotor Collision. The scattering of an atom by a diatomic vibrotor is described most directly in a laboratory system of coordinates X’,Y’,Z’ by the wave equation [E - TF- V(R’,r’) - %Jr? - 7fr(2?]Q(R’,r’) = 0 (1) where 2’is taken to coincide with the direction of the incident wave vector, R’is the vector joining the center of mass of the diatom and the atom (translated to make its origin coincide with the center of mass of the entire system), r’ is the vector denoting the bond of the diatom, and Svand 3f, are the vibrational and rotational Hamiltonians of the diatom. Primes and hats over vectors signify laboratory frame and unit vectors, respectively. The separation of the variable ?’from r (length of r’) and R’, based on the adiabatic approximation of Chase? is discussed in Appendix A. This approximation, which is an analogue of the Born-Oppenheimer approximation in the theory of molecular bound states, enables us to obtain solutions for the scattering states neglecting the effect of the angular velocity of the diatom on the dynamics of the atom. By analogy with the approximate separation of bound-state molecular eigenfunctions as the product of a function of the nuclear coordinates and a function of the electronic coordinates for fixed nuclear positions, one separates a diatomic rotational wave function (taken to be independent of the collision coordinate R’)from \k(R’,r’), leaving an adiabatically evolving part @(R’,r’),which is taken to be parametrically dependent on 2’ Furthermore, just as in the case of the Born-Oppenheimer approximation, this assumption is understood to imply that the derivatives of @ with respect to the angular coordinates of the diatom are negligible. We have, then
The Journal of Physical Chemlstty, Vol. 85, No. 21, 1981 3189
Atom-Diatomic Molecule Collisions
WR’,r’) = Yjm(YMjm(R’,r’)
(2)
Expanding @jm in terms of partial waves as ajm(R’,r’)= alR-l~,(R,i?ll,~(r)Yl,x~(ti? q’ = ( ~ ’ 1 % ’ ) 19,’
the close-couplingequations of the adiabatic approximation may be obtained from eq 1 as [d2/dR2+ K’2 - 111’ + 1)/R2]f:,= C ~ , , , @ , P ? f 9 u (3) dl
Here Kt2 = 2p(E - E,) and u = 2pV. As shown in Appendix A, integration of eq 3 over i’after pre- and postmultiplication by ?+,,and Y.,,,, respectively, yields a set of coupled equations that difier from the exact equations only in the repacement of 6,the eigenvalues of Ffr, by zero. The S matrix of the adiabatic approximation is related to the Vdependent S matrix obtained by solution of eq 3 O,(R,i’) = K’-1/2[6q,te-iBKq - S4,9(i’)eiBdq‘] (4)
TWQ= I d ? ’ Y;m,Tqrq(3?Yjm =
C (li]lil)1/2[Jl-1(j’m‘Z’m - mlJm) x JR
(j’ol QIJQ)(jmlOlJm)(jOlQlJQ) Z”$$u~6m-ml,x~ &’= (uy’m’l\’} (7) where, in performing the integral, we have used the result in eq 6 and the fact that i’ has 0 = @ and 3. = a while 2 = [a,P,O]. Applying the result in eq B-6 for the specific case j = m = 0 of eq 7, we obtain T$q:dl= Cliq1/2[ l]-l(j’O1’ QIlQ) T$$)J,ul (8) R
Following Khare, we now relate the scattering amplitude to fili,t,ualand thus to futjm,* From the asymptotic form WR’,r’)RzYjm( i; 9[?,( r)eiKZ+ Cf,& ,’? 9R-leiK’Rqd]
futj’mt,vjm
U’
= R-’Yjm C alK’-1/2~dYl,x.(~? [6,t,e-ieK~- S,,,e”d/]
6 ~ ’=~ K’R ’ - y&‘T
l,,’
we find
as
fu,u(ti:i? = c
Sdj’mtlth,;ujml E SqQ = I d ? ’ q:,S,,(i? Yjm
Il’h‘R
[i”’+l(T[l] /KK’) 1/21 YpA!DfJ,fiy@5;I
= [.. I Ypy(-)”[1]1/2( l O l ’ A ’ ~ o A ’ ) D ~ o ~ ~ ~ , , , o(9) 1 It is convenient to reformulate this scattering problem in 11 %’j0 a rotated frame_X,Y,Z,in which the axis of the diatom 2‘ In obtaining the second line of eq 9, we have used the coincides with 2. The wave equation in the rotated frame relation inverse of eq 8 and the additive property of the differs from that in the laboratory frame only through the rotation matrixes, while [.. I signifies the contents of the potential V(R’,r’) going over to V(R,r),where the unsquare bracket in the first line. The adiabatic approxiprimed vectors are in the rotated frame. The wave funcmation to the amplitude for scattering from ujm to u,j’m’ tion takes the form may be obtained from futu in eq 9 as 9(R,r) = E 6mnR-1g~?~(R,i)Y~~n(B)DtR(n)s, (5) f u t ~ m t , u j m ( f i ?= S d i ‘ q,m,(ti’,P?fd~(R‘,P?Yjm 11X’O
In eq 5 R is the set of Euler angles that rotate the laboratory frame to the molecule fixed frame, and al and bln are determined from the initial condition a1 = i ’ + ’ ( ~ [ l ] / K ) ~ bln / ~ = alDbn(n)* [ l ] = 21 1
+
It is easy to verify, by using the results in Appendix B, that the T-matrix elements in the laboratory and the rotated frames are related by the transformation Tuvht,ul
o ( i ? = CD!,’qiTL,uPb;, n
(6)
In this approximation, in the molecule fixed frame, Q , the projection of the orbital angular momentum 1 along the diatom axis, is a constant of the motion. The main advantage in the use of the molecule fixed frame for calculation is the ease of evaluation of the angular integrals in the matrix element Vdtgl; this advantage is somewhat offset by the need to transform the asymptotic form of the numerical solution before matching it to a Bessel function to extract the elements of the T matrix. B. T-Matrix Elements and Cross Sections. In this section we rederive a result of Khare8 that expresses the as T-matrix element for an arbitrary transition, fi$&ull, a linear combination of which are, in turn, iinear transforms of the T-matrix elements of the adiabatic approximation, T$$ur. These relationships ultimately translate into simple results for total and differential cross sections ady,,,j, etc., in terms of the T$),,,l)s. The latter matrix elements are obtained from solution of the adiabatic approximation equations and are therefore easily studied for their convergence with respect to 1,l’ as a function of such parameters as the incident energy, anisotropy, and depth of the potential. The transition matrix elements in the adiabatic approximation are given by
qJdf,ual(s
=
C [...]Ylf,m-m~([1]Ij]/li~)1/2(101’m - m’/jom - m’) x
wo
(jmjom’- mli ’m’) (j ojoov’0 ) (-) r’+m-m’T(9. UJd’,dl (lo)
From eq 10 it is clear that fu~jp+n,u&? =E ~1*,m-m~([~1/lio1)~’~ x 11‘
[**.I
(lOl‘m-m‘[jom- m’)(-)l’+m-m’T( ;$&:dl (11) while from eq 10 and 11 we obtain (Khare, eq 36)
9
f u ~ ~ m ~ , u j m (= R
Clo (Go1lil / li1)1/2(imiom’- mli ‘m’) (jOjoOliQ )fu*jmt-m,m (12) All of these results have been obtained previously by Khare with differences due to exclusion of vibration of the rotor (which we have included). Differential and integral cross sections u’j’- u j and u’ u transitions may be obtained in terms of ~ l ~ o l , , u aand l 7$?{J,ul within the adiabatic approximation as usual. We find da(u’j‘,uj)/dQ = (K’/K)u]-l E, I f u t j ’ m , u j m 12 =
-
mm
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The Journal of Physical Chemistry, Vol. 85, NO. 21, 1981
In eq 14 wG)is the probability of occupation of the jth rotational level. For the purpose of comparison, we also set down the exact results for these cross sections: fdpm,ujm(fi’) = ( ~ / K j K y ’ ) lC / ~[1]1/2i””+1(jm10JJm) X Jll‘
( j ’m2 ’m-m ‘1J m ) Yllm-m,(R ’) !l$$’T:vil a(u)j’,uj) = JdQ ( K y ’ / K j ) v ] - l C
mm ‘
W,u)=
Ifutymf,ujm(fi’)12
C wG)u(uIj’,uj)
(15)
(16)
jj‘
111. Effective Potential Approximation The effective potential approximation was first formulated by Rabitz9for a general diatom-diatom collision and was applied to the case of He-H2 collisions in the energy range 0.04-0.8 eV by Zarur and R a b i t ~ .Chu ~ and Dalgarno‘ compared the EP, adiabatic, and JZCCS approximations for H-CO collisions at 0.01 and 0.05 eV and concluded that for that system the JZCCS and the EP approximations have useful accuracy, while the adiabatic approximation was found to be inaccurate at low energies and time-consuming at high energies. The EP approximation recognizes the fact that most experiments do not distinguish the magnetic substates of the initial and final scattering states, and therefore replaces the true potential with an effective potential obtained by “summing over” the magnetic substates of the system. Appendix C contains details of the specialization of Rabitz’s derivation of the effective Hamiltonian to the case of an atom-diatom collision with a potential of a general form. A prescriptipn for the matrix elements of an “effective operator” 8 between states that do not carry any information on the magnetic substates, in terms of the matrix elements of the corresponding operator d between the states pm), has been proposed by Rabitz. This prescription was obtained by studying the relationship between the rn-independent reduced matrix element (jll and the total matrix element (jmlw’m’) for the case of a potential that can be resolved into a sum of products of radial and angular parts; that relationship was then generalized and used to define an effective zero-order Hamiltonian that leaves the eigenvalues of the target (and/or projectile) unaltered. Constrainingthe effective potential operator to be local and its matrix to be real and symmetric, one may obtain a consistent (though not unique) definition of its matrix elements between the “states” The reduction in the dimensionality of the scattering problem arising from use of the effective Hamiltonian is due to the absence of any m dependence of the rotational states of the target, and the diagonal nature of V in 1 and X and its independence of A. Thus, for a rigid-rotor scattering problem, a close-coupling calculation involves the solution of (Jm=- Jmin+ 1) equations, each of dimensionality C,[2 X min ( J j ) + ll, with the sum over j running from jmin to,,j whereas the EP approximation involves solution of only ( I , - lmin + 1)equations, each - jmin + 1). of dimensionality (jmm In Section IV-D we present a comparison of total cross sections for rotational transitions for rigid-rotor scattering of He-12 and Ne-12 obtained by the close-coupling the effective potential methods. From eq 16 and 17 of Appendix C, partial wave cross sections q(EP) and (~J(CC) may be defined as the individual terms of the 1 and J sums, respectively. Their dependences on the constants of the motion l,J in the two approximations are shown in Figures
v)
vu’)
v).
Sethuraman and Rice
8-14 for the case of parameters appropriate to He-I2 and Ne-I2 collisions.
IV. Computations A. Procedure. The coupled equations of the LZCES approximation, eq 3, have to be solved subject to the scattering boundary conditions of eq 4 and the condition that the amplitude vanishes at the origin. In our first computations the algorithm of Gordon9 was employed; it was later replaced by the R-matrix algorithm of Light and co-workers,1° since the latter has been found to be - 5 times faster than the Gordon integrator. Both the Rmatrix and Gordon methods divide the region of integration into intervals across which the potential is approximated by a Taylor series expansion relative to the midpoint. Whereas Gordon’s method propagates the wave function @, and its derivative W,across such intervals by the method of variation of parameters, the R-matrix method similarly propagates the ratio matrix @/@’. To be successful, Gordon’s method requires stabilization of \k,W in the region where closed channels are present; normally such stabilization is not required in the R-matrix method at every interval. For the error-control parameters recommended for the two methods, for the computations reported below, the T matrixes obtained agreed to within 3%. Both methods failed to yield T-matrix elements convergent with respect to inclusion of closed channels for very low energies (e.g.,
