Faraday Discuss. Chem. SOC., 1982, 73, 339-355 Predissociation of Weak-anisotropy Van der Waals Molecules Theory, Approximations and Practical Predictions BY ROBERT J. LE ROY, GREGORY C. COREY AND JEREMY M. HUTSON * Guelph-Waterloo Centre for Graduate Work in Chemistry, University of Waterloo, Waterloo, Ontario, Canada N2L 3Gl Received 25th January, 1982 Widths and energies for predissociating levels of Hz-inert-gas Van der Waals molecules are calculated by solving the close-coupling equations for accurate potential-energy surfaces. These exact results are used to test several approximate schemes previously employed for calculating level energies and widths. None of these is found to be entirely satisfactory, and a new method is proposed which gives more accurate results. In this new method (SEPTOC) the secular equations are solved in matrix form for the closed-channel manifold, and the open channels are treated by perturbation theory.The limited form of SEPTOC tested here gives very promising results. The level widths are found to be very sensitive to the potential used, particularly in the repulsive region. For Hz-Ar, two different potentials which both reproduce the level energies are found to give level widths which differ by ca. 30%. This is the first conclusive demonstration that measurements of predissociation widths can give potential information different from that yielded by the level energies. 1. INTRODUCTION Within the electronic Born-Oppenheimer approximation a polyatomic Van der Waals molecule can predissociate through one of two mechanisms.The simplest of these is the tunnelling-type " rotational predissociation " of a quasibound level which lies above the dissociation limit but behind a potential-energy barrier on the effective radial potential for the system. The second mechanism, the one of interest here, is associated with conversion of internal vibration-rotation energy of one of the mole- cules forming the complex into relative translational energy of the separating frag- ments. This process is induced by the same anisotropy and/or internal stretching coordinate dependence of the intermolecular potential which causes rotational and vibrational energy transfer in molecular collisions, and it has been suggested that it would provide a good way of determining those parts of the intermolecular potential which drive these inelastic processes.While detailed theoretical studies of this phenomenon were first reported some 15 years the first unambiguous observations of it did not come until a decade later.5-7 These experiments then sparked a rapid growth of theoretical work in this area *-I1 in which particular attention was paid to approximate schemes for predicting the energies and widths of predissociating levels and their use in qualitatively inter- preting and extrapolating beyond the growing body of experimental data. [Ref. (1 1) presents a detailed review of much of this work.] However, while use of these methods has provided substantial insight into the nature of the predissociation process, their reliability and the sensitivity of their predictions to the nature of the assumed potential-energy surface has not been fully explored.Moreover, since very little is known about the details of the potential-energy surfaces characterizing most systems * NATO Postdoctoral Research Fellow.340 PREDISSOCIATION OF V A N DER WAALS MOLECULES of interest, such studies can at most yield qualitative information about trends such as the dependence of predissociation lifetime on the “ energy gap ” (i.e, the amount of internal energy released) or on the vibrational quantum number. In spite of the interest in this predissociation phenomenon, only very limited predic- tions have as yet been reported 12*13 for the family of atom-diatom complexes whose potential-energy surfaces are most accurately the molecular-hydrogen- inert-gas (Ar, Kr and Xe) bimers.Moreover, most of those predictions l2 were not based on the best available potential-energy surfaces for these systems. These species are of particular interest because of the unique conjunction of three conditions: (i) accurate three-dimensional potential-energy surfaces have been determined for them from experimental data ; I4-l6 (ii) line broadening due to internal-rotational pre- dissociation has been observed; l7 and (iii) they are among the easiest systems to treat theoretically. This last condition facilitates the present use of close-coupling cal- culations to obtain accurate energies and widths of predissociating levels. These then allow comparisons with experiment to provide a quantitative test of the potential- energy surface used, and provide accurate results for testing various approximation schemes. The present paper reports calculations of the energies and widths of a variety of metastable levels of molecular-hydrogen-inert-gas complexes, with particular attention being paid to H,(v = 1, j = 2)-Ar.The spectroscopically observed l8 excited states of these species all correspond to metastable complexes formed from vibrationally excited hydrogen, and the dominant predissociation mechanism is BC(U = 1,j)-M + BC(V = 1,j’) + M (1) where j ’ = j - 2 for BC = H, or D, and j -.j’ = I and 2 for BC = HD. This process of ‘‘ internal-rotational predissociation ” is merely a special type of vi brational predissociation, the internal rotation of the BC diatom being in essence a very wide angle bending motion.Note that the process of internal-rotational predissociation defined by eqn (1) should not be confused with ordinary rotational predissociation, which has historically been defined l9 as the tunnelling type of predissociation men- tioned at the beginning of this section. The following discussion begins with a description of the potential-energy surfaces used in the present study. This is followed by a detailed comparison of a variety of methods for predicting the energies and widths of vi brationally and/or internal- rotationally predissociating levels of Van der Wads complexes. The final section then examines the potential dependence of predictions of this type, summarizes the information obtained about the molecular-hydrogen-inert-gas systems, and attempts to delineate the utility of experimental measurements of predissociation level widths, 2.POTENTIAL-ENERGY SURFACES FOR THE HZ-INERT-GAS SYSTEMS The calculations reported in the present work were all performed on potential- energy surfaces described by the BC3(6,8) function of Carley and Le R o ~ , ~ ~ , ~ ~ which is expanded as 3 where R is the distance between the atom and the diatom centre of mass, the diatomR. J . L E ROY, G . C . COREY AND J. M. HUTSON 34 1 bond length r is represented by the stretching coordinate 0.766 6348 A, and 6 is the angIe between R and the axis of the diatom. strength functions V&) all have the form : = ( r - ro)/r0 with ro = The radial D(R) = exp[-4(R,/R - for R < Ro = Rto = 1 for R 3 Ro (4) and the constants AAk and Cik are defined by the values of the depth Rtk of the minimum of T//k(R).definition of D, appearing there should read: and position {See eqn (74) and (75) of ref. (16); note that the D, F [dlfi(D(R))jdR]R=,~~.) The parameters Rtk, . . . etc.) defining the H,-Ar surface of Carley and Le Roy are listed in table V of ref. (16), and those for H2-Kr and H,-Xe are found in ref. (15). However, in practical calculations it is the vibrationally averaged (over c) forms of this surface (t.,jl V(R,<,O)[v',j'> which are relevant. Since the exponent parameters j ? ~ are independent of k, the resulting diagonal vi brationally averaged potentials are V(t.,jlR,O) = (t.,jl Y(R,<,O)[C,j) = FG(t.,jlR) + V2(Ll,jIR)P,(COS 0) ( 5 ) where the vibrationally averaged anisotropy strength functions V>.(u?jlR) have the same form as eqn (3) except that their linear scaling parameters are: for Q = A , C8 and C, where (~7,jlS~I~',j:: i s the expectation value of ck for the isolated diatom in vibration-rotation state ( u j ) .For molecular hydrogen these expectation values are readiIy calculated from the accurately known diatom potential-energy curves.2o The parameters AA, ci and ci defining the ensuing vibrationally averaged potentials for various types of H,(v,j)-Ar complexes are summarized in table I. The analogous parameters of the rigid-diatom potential defined by the k = 0 contributions to eqn (2) are also listed there. TABLE 1 .-PARAMETERS CHARACTERIZING THE VJBRATIONALLY AVERAGED CARLEY-LE ROY Ro = 3.5727 A BC3(6,8) POTENTIALS FOR Hz(v,j)-Ar; FOR ALL CASES Po = pz = 3.610 A-' AND 25 206 865 1 240 048 142 714 52.31 3.5985 9 177 581 539 933 15 797 7.59 3.843 25 065 503 1 234 302 142 185 52.19 3.5974 8 966 285 527 465 15 660 7.49 3.840 22 929 424 1 149 172 135 051 50.81 3.5772 5 452 225 13 706 320 085 5.42 3.768 22 802 769 1 143 965 134 534 50.68 3.5762 5 273 896 309 564 1 3 574 5.84 3.763 22 633 534 1138 176 134 500 50.87 3.5727 4 814 552 282 440 1 3 500 5.72 3.743342 PREDISSOCIATION OF VAN DER WAALS MOLECULES To facilitate comparisons, the parameters and Rt characterizing the depths and positions of the minima of these effective anisotropy strength functions are also listed in table 1.The changes in these quantities from one case to another, and in particular the dependence of E~ and R: on the vibrational quantum number u, provide a clear warning that this vibrational averaging must be done properly if predictions based on these surfaces are to have physical significance (see section 4A below).3 . CALCULATING THE ENERGIES AND WIDTHS OF METASTABLE LEVELS OF WEAK-ANISOTROPY ATOM-DIATOM COMPLEXES A. GENERAL CONSIDERATIONS In an exact treatment of predissociation, the energy and width of a resonance state must be deduced from the energy dependence of the total wavefunction (see below). However, the basis of most approximate treatments of predissociation phenomena is a partitioning of the full Hamiltonian for the system into two parts. The first of these, H,, provides a zeroth-order description of the dynamics of the system, and its discrete eigenvalues provide estimates of the energies of all bound and metastable states of the system.The remainder of the Hamiltonian, denoted H' = H - H,, couples the discrete eigenfunctions of Ho associated with metastable levels to the isoenergetic continuum wavefunctions associated with the open channels, leading to predissociation. Within an isolated-resonance approximation, the full width at half maximum of a predissociating level is given by the golden-rule expres- sion 21922 where tyb is the unit normalized bound-state eigenfunction of H, associated with the metastable state and tyi is a continuum eigenfunction of Ho associated with open channel i, normalized to a delta function of energy. When there is more than one open channel the total width is a sum of partial widths r i and the branching ratios are determined by the ratios of the contributing partial widths.In general there are three types of errors associated with this partitioning. The first of these arises when Ho does not provide a good description of the dynamics of the system; this is the dominant source of error in the SFD, BFD and BL approxim- ation schemes considered in sections 3C and 3D. The second is caused by neglect of the effect of the open channel(s) on the level energy; a way of estimating these level shifts is described in section 3F. The third type of error is the neglect of the effects of higher-order coupling via the open channels. A procedure for treating the second- order coupling among the open channels is described in ref.(1 1); however, it is not relevant to the present calculations of internal-rotational predissociation since the present model problem has only one open channel, B. EXACT (CLOSE-COUPLED) CALCULATIONS FOR A MODEL PROBLEM In the close-coupling formulation predissociating states are treated as scattering resonances and are located by solving the close-coupled scattering equations on a grid of energy values. Near a resonance, the phases of S-matrix elements change rapidly with energy. In the particular case where there is only one open channel, the asymp- totic phase shift increases by z over the width of the resonance, while in the many- channel case similar behaviour is shown by the S-matrix eigenphase sum.23 Only the single open channel case is considered here.R .J . LE ROY, G . C . COREY AND J . M. HUTSON 343 After separating out the motion of the centre of mass, the full Hamiltonian for an atom-diatom system may be expressed as : H(R,r) -(h2/2,L!)R-'( a2/ L'R2)R + 12/Z,UR2 + V(R,<,O) + f f d ( r ) (8) where R is the axis of the complex, Y that of the diatom, p = m,md/(m, + md) is the effective reduced mass associated with the interaction of an atom of mass ma with a diatom of mass md, l2 is, the angular-momentum operator associated with the rota- tion of the unit vector R and Hd(r) is the vibration-rotation Hamiltonian for the isolated diatom. The exact eigenfunctions of this Hamiltonian may be expanded as 1 " U" where J and M are the quantum numbers for the total angular momentum and its space-fixed projection, and O! is an index identifying a particular eigenstate of the system.? The functions {quJ(r)} are the wavefunctions for the st!etching motion of the diatom, (x;f,"(R)} are the radial channel wavefunctions, and (@i(R,E)} are a complete set of angular basis functions with quantum numbers collectively identifiedlas a.In the Arthurs and Dalgarno space-fixed r e p r e ~ ~ n t a r i o n , ~ ~ a = {j,l} and @i(R,t) is the total angular-momentum eigenfunction g;Y(R,E) which is defined as the appropria{e linear combination of the products of spherical harmonic functions Y i m j ( P ) Yrml(R) for mj + m, = M . Alternatively, in the body-fixed coordinate system, a = {j,Q}, the operator 1' is replaced by ( J - j ) 2 , and the angular basks functions are earity- adapted linear combinations of the products Y,,(E)DJ,M(R), where D ~ M ( R ) are the usual symmetric-top wavefunctions and Qh = (Klh is the magnitude of the pro- jection of the total angular momentum onto the axis of the complFx.In either case, substituting eqn (9) into eqn (8), premultiplying by [ ~ - ~ 4 3 , , ~ , ( y ) ~ ) ; i . ( R , f ) ] * and integrat- ing over r and the angular variables yields the usual set of coupled equations for an at om-d ia t om system, I- (h2/2p)d2/dR2 + U(d,a' ; v',a' ; J IR) -- EL}x!&(R) where Ei is the total energy of the system and U(u',a',"',a'' ; JIR) are the matrix elements of the angular and diatom-stretching wavefunctions with the operator U, defined as U(R,r) = Hd(r) + l2/2PR2 + v(R,<,o). (1 1) Note that in either angular basis the matrix elements of &(r) are diagonal and equal to the vibration-rotation energies of the free diatom (Ed(u,j)).Similarly, in the space-fixed representation the matrix elements of 12/2pR2 are diagonal in v',.j' and I' and equal to the simple centrifugal term "(1' + l)h2/2,uR2. The calculations reported in the present section describe the internal-rotational predissociation of the lowest (n = 0) vibrational levels of H2(1,2)-Ar and were per- formed using a realistic model potential subjected to three restrictions: (i) The matrix elements of U are assumed to be zero for u" f 0 ' ; perturbation-theory arguments show that this assumption does not significantly affect the predicted level energies and widths. (ii) The dependence of the vibrationally averaged potential onj' is neglected; the small differences between the ( u , .j = 0) and (21, j = 2) potentials seen in table 1 t In general, u is defined by the total energy of the system and its ( j i- 1 1- .I) parity; however, for weak-anisotropy molecules it may be associated with the set of " zero-coupling-limit " quantum numbers: v , j , n, 1 and J.344 PREDISSOCIATION OF VAN DER WAALS MOLECULES indicate that this too should introduce little error, (iii) The coupling to closed channels corresponding t o j >j,,, = 2 is assumed to be negligible; while this is not quite true (see section 4), it is perfectly legitimate to define a model problem in this way.As a result of these three restrictions, the potential contribution to the matrix element of Uis where 02(at,atf ; J ) is the matrix element of P2(cos 0) between the angular basis functions a;, and a;,.. For the potential-energy surface and Hamiltonian described above, even ( j + I + J ) parity metastable states associated with ( V J ) = (1,2) can dissociate into only one open channel (u = 1, j = 0, J = I ) . In this case the scattering S matrix consists of the single term S = exp[2iq(E)] where q(E) is the phase shift associated with the asympto- tic behaviour of the radial channel wavefunction for the open channel. In the present work, the coupled equations (10) were solved in space-fixed coordinates using the method of Sams and K o ~ r i , ~ ~ and the resonance energies and widths determined by fitting the resulting phase shifts to the Breit-Wigner line-shape plus a linear back- ground phase: The separations between the thresholds for the open and closed channels were taken as Ed(u,2) - E,(u,O) = 336.7 and 354.24 cm-I for u = 1 and 0, respectively, while the q(E) = b + cE + arctan [r/2(Er - E)].(13) TABLE 2.-cOMPARISON OF EXACT (cc) AND APPROXIMATE I2 = 0 RESONANCE ENERGIES OBTAINED F R O M ~ ~ ~ ~ = 2 CALCULATIONS ON THE HZ(1,2)-Ar SURFACE OF TABLE 1. THE ZERO OF ENERGY IS DEFINED AS T H E j = 2 ROTATIONAL ENERGY THRESHOLD. 1 J EYC/cm-’ y =SFD BFD BL SE SEPTOC 0 2 1 1 3 2 0 2 4 3 1 3 5 4 2 4 6 5 3 5 7 6 4 6 8 -23.238 -22.653 - 22.237 - 20.727 - 19.31 1 - 20.078 - 17.243 - 15.943 - 16.823 - 12.828 - 11.580 - 12.523 - 7.490 - 6.265 - 7.249 - 1.306 -0.113 -1.109 0.209 -0.018 0.111 - 0.074 -0.160 0.072 - 0.050 - 0.068 0.053 - 0.052 - 0.037 0.041 - 0.048 - 0.023 0.035 - 0.042 -0.017 0.028 3.453 2.412 5.780 1.265 7.990 1.157 10.074 1.076 - 0.074 - 2.461 - 4.694 a - 6.791 0.996 a -- 8.7 16 a a - 0.483 - 0.055 -0.358 - 0.005 0.41 7 0.027 0.292 -0.209 0.057 0.210 -0.158 0.053 0.150 -0.1 17 0.043 0.100 -0.083 -0.273 0.004 0.004 0.003 -0.009 - - 0.001 0.002 - 0.003 0.000 0.003 - 0.003 - 0.001 0.002 - 0.002 -0.001 0.001 - 0.002 0.000 0.002 0.029 0.002 0.023 - 0.002 0.035 0.01 9 0.001 0.032 0.01 7 0.005 0.029 0.01 5 0.006 0.028 0.01 1 0.006 0,022 0.008 mean deviation - 0.002 0.82 - 0.022 - 0.000 0.01 6 (10.082) (15.28) (f0.225) (*0.003) (&0.012) This level is incorrectly predicted to lie above the dissociation threshold.R .J . LE ROY, G , C . COREY AND J .M. HUTSON 345 values assumed for the physical constants are all incorporated in the factor h2/2p = 16,857 630/p cm-' A2, where p = 1.918 865 04 arnu for H2-Ar. The encrgics E:c and widths Ycc of the n : 0 metastable levels of H2(I,2)-Ar determined from the close-coupling (CC) calculations are tabulated in columns 1 of tables 2 and 3, respectively. Subject to the three approximations listed above, they TABLE 3.-cOMPARISON OI- EXACT (cc) WITH APPROXIMATE WIDTI-IS (IN C M - ') TOR THE H2( 1,2)-Ar ItESONANCFS CONSIDERED JN TABLE 2 1 J cc SFD BFD BL SE SEPTOC 0 2 1 1 3 2 0 2 4 3 1 3 5 4 2 4 6 5 3 5 7 6 4 6 8 0.0394 0.0530 0.0414 0.1 103 0.0235 0.0420 0.055 1 0.0237 0.041 1 0.0408 0.0219 0.0388 0.03 19 0.0191 0.035 1 0.0244 0.01 56 0.029 8 0.032 0.052 0.039 0.1 12 0.044 0.042 0.064 0.039 0.042 0.049 0.035 0,040 0.039 0.029 0.036 0.030 0.023 0.03 1 0.0 0.0 0.0 0.112 0.0 0.0 0.109 0.0 0.0 0.104 0.0 0.097 0.0 0.087 U U U U 0.001 8 0.036 0.0089 0.109 0.055 0.017 0.077 0.052 0.024 0.058 0.045 0.029 0.044 0.037 0.030 0.033 0.027 0.028 0.026 0.037 0.03 1 0.123 0.005 1 0.032 0.027 0.0072 0.034 0,022 0.01 1 0.033 0.01 3 0,0093 0.033 0.0087 0,0073 0.025 0.05 1 0.058 0,050 0.112 0.03 1 0.049 0.058 0.033 0.047 0.044 0.03 1 0.044 0.035 0.028 0.040 0.027 0.022 0.033 ~ ~~ a This lcvcl is incorrectly predicted to lie abovc the dissociation threshold.are exact results for these levels of the Carley--Le Roy BC,(6,8) potential-energy surface. The additional entries in these tables correspond to the resonance energies and widths predicted by the various approximate methods described below.Note that states of odd ( j -I-- 1 -1. J ) parity cannot predissociatc by internal rotation since there are no open ( v j ) = (1,O) channels of this parity. c. S E P A R A B L E WAVEFUNCTION APPROXIMATIONS The simplest approximate methods arc those in which Ho includes only the diagonal matrix elements of U, so that V(R,&U) is replaced by a set of channel potentials V(v,a; v,a; J[R) dcpending only on R. Jn either the spacc-fixcd or body-fixed rep- resentations, this approximation involves neglect of the coupling terms on the right- hand side of eqn (10) and hence causes the expansion of eqn (9) to collapse to a single term. If this is the only approximation made, the result is the " distortion " approx- imation of Levine et al.z-4 The resulting coupling operator H' then comprises the off- diagonal matrix dements of U appearing on the right-hand side of eqn (10).As might be expected, results obtained using these methods are quite sensitive to the choice of angular basis. This point is dramatically illustrated in tables 2 and 3, where the space-fixed distortion (SFI)) predictions are seen to be greatly superior to those obtained using the body-fixed distortion (BFD) approximation. In both cases, the dominant source of error is the neglect of the coupling among the closed channels346 PREDISSOCIATION OF VAN DER WAALS MOLECULES due to the off-diagonal matrix eIements of the operator U, and the reIative quality of the predictions may be understood from the following examination of the diagonal and off-diagonal matrix elements of this operator.In both space-fixed and body-fixed representations, the channel potentials appear- ing in Ho are the diagonal matrix elements of U, and it is the ratio of the spacing between these curves to the magnitude of the off-diagonal coupling functions which determines the relative accuracy of the various approximations. In the space-fixed description of the present model problem (with u = u’ = v” = I), the matrix elements of U are denoted U#’(JIR). For the J = 2 levels of H2(1,2)- Ar these matrix elements are plotted in fig. 1 . The upper portion of this figure shows 20 0 -20 3 I E !? 2 -LO G 10 0 -10 ’-1 7 / Ui’ 1 3.0 L.0 5.0 6.0 RIA FIG. 1 ..-Effective potentials and coupling functions for J - 2 states of H2(f ,2)-Ar in the space-fixed representation.The upper part of the figure shows the potential curves, and the lower part the coupling functions due to the potentia1 anisotropy. the functions U{{(J = ZIR), which are the effective radial potentials defining both the SFD level energies (the three upper curves) and the wavefunctions used in eqn (7); the curves in the lower portion of this figure represent the off-diagonal coupling among the closed channeIs due to the potential anisotropy. Similarly, the upper and lower portions of fig. 2 show the analogous diagonal and off-diagonal functions Uh&(JIR) associated with the body-fixed representation; in this case the off-diagonal coupling functions are solely due to the ( J - j ) 2 term in the total Hamiltonian.The results in fig. 1 show that in the SFD approximation the effective potentials are well separated relative to the magnitude of the relevant coupling functions. In contrast, fig. 2 shows that the BFD coupling functions are relatively stronger on theR. J. L E ROY, G . C . COREY AND J . M . HUTSON 347 20 0 - 20 3 I -... 2 -LO 2 C - --1 7.- 7 -- 1- -1 ---- - -. 1 ---T FIG. ?.-Effective potentials and coupling functions for J = 2 states of H2(1,2)-Ar in the body-fixed representation. The upper part of the figure shows the potential curves, and the lower part the coupling functions due to the Coriolis term in the rotational Harniltonian. interval between the classical turning points of the metastable levels, and the closud- channel potentials are much closer together; as a result the zeroth-order BFD eigen- functions provide a fairly poor description of the predissociating states of this system.Thus, the zeroth-order picture associated with the SFD approximation, that the diatom rotates freely within the complex, is much more realistic for weak-anisotropy Van der Waals complexes than the BFD assumption that the complex is a rigid symmetric top. Two other separable wavefunction approximations which have been applied to this problem are the '' central approximation " of Micha lh4 and the '' intermediate approximation " of Beswick and Requena.12 The former is effectively a crude version of the SFD approximation in which the potential appearing in Ho consists only of the vibrationally averaged isotropic term ro(v,jlR).For the H,( 1,2)-Ar levels of tables 2 and 3 it yields resonance widths roughly the same as the SFD ones, but its resonance energies are typically ca. 0.5 cm-l in error, as they neglect even the first- order splitting of the different J-states associated with a given value of 1. In general it has no advantages over the SFD method and its use should be discouraged. In the '' intermediate approximation " of Beswick and Requena,I2 the angular basis functions characterizing the closed channels for each j were expanded as linear combinations of the corresponding body-fixed basis functions, and defined by the requirement that they diagonalize the matrix Uhn((J1R) at some particular distance K = R"'. However, while this method yielded better results than the BFD approach,I2348 PREDISSOCIATION OF V A N DER WAALS MOLECULES its success depends critically on a propitious choice of the distance lP, and hence may not always be trusted.D, T H E BEST-LOCAL (BL) OR S L M P L E BORN-OPPENHEIMER APPROXlMATlON The most sophisticated of the decoupling schemes which have been applied to the present problem is the " best-local (BL) approximation, originally developed in a space-fixed formulation by Levine et ~ 1 . ~ -4 and rederived using body-fixed coordinates by Holmgren et a1.26 This approach differs from the separable approximations described above in thatfithe angular basis functions appearing in eqn (9) are parametric functions of R, @i(RIR,V"). functions are expanded in terms of a complete set of angular basis functions and are defined by the requirement that they diagonalize the operator U of eqn ( I 1).The resulting channel potentials, here denoted D#(JlR), are then the loci of the eigenvalues of U, By definition, U has no off-diagonal matrix elements in the best-local basis {@i(Rlk,P)}, and the coupling among the various channels arises from matrix elements involving these angular basis functions and the radial kinetic-energy operator ( -h2/2p)8/ aR2. The coupling functions have the form At each value of R these d dR f{j'(JIR) + gf:;'(JI.R) - and may be evaluated as described in ref. (27). In this approach, the level energies are obtained as the single-channel radial eigen- values of the closed-channel potential adiabats u#'(JJR) while the widths are obtained from cqn (7) using radial eigenfunctions associated with those effective potentials and with H' defined by eqn (14).Note that the diagonal potentials and coupling functions used in this approach are the same whether space-fixed or body-fixed basis functions are used to diagonalize U. Moreover, while the designation a = { j J } does not provide an exact description of the BL angular channels, it does suffice to identify them uniquely. For the present model problem, the energies and widths calculated in the BL approximation are listed in column 4 of tables 2 and 3. While they are distinctly better than the BFD predictions, it is perhaps surprising to see that this '' best " local approximation is inferior to the simple SFD method. However, this is readily explained by consideration of the effective potentials and coupling functions plotted in fig.3, which describe the same J = 2 case considered in fig. 1 and 2. The fact that the BL method effectively removes all coupling due to the interaction potential ensures that the adiabats o,jl'(JIR) shown in the upper segment of fig, 3 are well separated. However, the coupling functions ,f{{(JlR) [and also g{[(JJR), not shown] have large-amplitude oscillations near the point where the anisotropy strength function v2(v,jlR] changes sign. This reflects the rapid change in character of the functions (@i(RJR,P)} in the region where the preferred configuration of the complex changes from collinear to T-shaped. I n spite of the magnitude of these oscillations, the BL approximation yields reasonable level energies and widths for this system, because much of this extreme behaviour occurs at distances smaller than the classical inner turning points of the metastable levels.This argument suggests that the BL method should perform even better for the analogous states of H,-Kr and H,-Xe, since the change in sign of their V2(u,jlR) functions occurs at relatively smaller distan- ces than for H2--Ar,14.15 but it augurs ill for H,-Ne, where the sign change is expected to occur relatively farther out. Thus, the accuracy of the BL results shown in tables 2R . J . LE ROY, G . C . COREY AND J . M . HUTSON 349 3.0 4.0 5.0 6.0 N A FIG, 3,-Effective potentials and coupling functions for J : 2 states of H2(I,2)-Ar in the best-local representation. The upper part of the figure shows the potential curves, and the lower part the non-adiabat ic coup1 ing functions. and 3 is partly fortuitous, since it depends on a detail of the potential.Note too that this argument also partially explains the accuracy of the BL level energies obtained in ref. (3) and (4),* as in most of the model potentials used there V,(R) did not change sign. E. T H E SECULAR-EQUATION (SE) M E T H O D The secular-equation method of solving the coupled equations I h is in principle exact in that it chooses H, := Hand proceeds to solve the full set of coupled equations (10). However, the secular-equation method (SE) of Grabenstetter and Le Roy 28 for determining resonance energics and widths is lcss than exact because it represents the radial channel wavefunctions for both closed atdopen channels by a finite number of square integrable basis functions, These basis functions are defined as the exact eigenfunctions of some basis generating potential L’,,(R).When V,(R) is chosen appropriately a small number of these functions can provide an extremely good representation of both open- and closed-channel wavefunctions on the interval where the latter have non-negligible amplitude. 111 thc present work V,,(R) was defined in the same manner as in ref, (28), by modifying the effective isotropic potential by placing a n infinite wall at some distancc R, well outside the classically allowed region for the metastable level in question. Note that the coupling function used there for calculating the level widths is valid for the two-channel case only, and that its use in four-channel calculations [see eqn (1 I ) of r d .(4)] is incorrect. * I n ref. (3) and (4) the present BL method is called the adiabatic approximation.350 PREDlSSOCIATION OF V A N DER WAALS MOLECULES Since the ensuing representation of the open-channel wavefunction(s) cannot satisfy the appropriate scattering boundary conditions, the resonance energy and width cannot be determined in the usual manner from the asymptotic properties of the open- channel wavefunction(s). The SE method circumvents this apparent difficulty by using a Fano-type 22 decomposition of the eigenfunctions of the Hamiltonian matrix to define the metastable-level energy and width In this approach the wall position R, i s used as the variational parameter and the resonance position defined by the energy at which the amplitude or the closed-channel portion of the secular-equation eigenfunction has a local maximum, while the magnitude of this maximum is used to obtain an estimate of the resonance width.In principle the accuracy attained i n SE calculations depends on the number of radial basis functions used in the expansions for the {,y;lt(R)). In the present calcu- lations a basis set comprising the 20 lowest eigenfunctions of V,(R) was used for each case; this was more than enough to ensure convergence. However, in practice a single basis function associated with each channel usually sufficed to yield level. energies accurate to better than ca. 0.02 cm-’. The additional functions were inchded in the expansion in order to provide a realistic representation of the open-channel function, and hence of the width.The results in column 5 of table 2 show that the SE method is by far the most accurate of the approximate procedures for determining level energies considered here. However, the corresponding SE widths shown in table 3 are worse than those yielded by the SFD and BL methods, the latter both being on average some 20‘x too high while the SE widths are on average 40”/, low. The reason for this weakness is not yet clear. On the other hand, the SE method is the only one of those considered so far which mimics the pronounced /-dependence of the exact (CC) widths of levels associ- ated with a given value of J . Specifically, it predicts that the level I = J is usually much narrower than those for I : J 1).This probably occurs because the SE method provides a near-optimum representation of the closed-channeI contributions to the total wavefunction, a fact independently demonstrated by the accuracy of the level energies it yields. Thc procedure presented in the following section therefore attempts to combine this excellent representation of the closed-channel wavefunction with a better treatment of the coupling to the con- tinuum. 2 (the only exception occurs for J F. PERTURBATION-THEORY TREATMENT 0 1 7 T H E OPEN CHANNELS In the Feshbach formulation for compound state resonancqZ1 the presence of dissociative channels causes both broadening and shifting of a metastable level. Ignoring high-order coupling uia the open channels, the resulting width r and shift A are each a sum of A = 2 Re(Zi) where the index i runs over the open channels, r m raY x,(R) is the radial wavefunction associated with dosed channel c in the absence of dissociative channels, and Hire is a matrix element coupling open and closed channels. The outgoing Green’s function Gf(R,R’) is given explicitly by 29R .J . LE ROY, G . C . COREY AND J . M . HUTSON 351 = -n[x$')(R)x$')(R') + ix!')(R)x!,')(R')] , R > R' (18) where xio) and xi') are the regular and irregular solutions of the single-channel Schrodinger equation for (open) channel i, normalized to a delta function of energy. When eqn (18) is substituted into eqn (l7), eqn (1 5) collapses to the simple golden-rule level width expression of eqn (7) with Wb = 2 ~u'j'(r>Q~(R,?)xc(R> c and tpi a simple separable wavefunction for open channel i.Unfortunately, the expression for the level shift, eqn (16), does not simplify in a similar way since the real part of the Green's function is different for R > R' and R < R'. However, eqn (16) may be readily evaluated by numerical integration once the functions xio) and xj') have been determined. The present work used the uniform Airy approximation3' to represent these functions, but they could also be readily generated by direct numerical integration of the single-channel Schrodinger equation. Note too that since the present model problem has only one open channel, the sums over i in eqn (7) and (16) each collapse to a single term. Of the methods considered to this point, only the CC and SE procedures take account of the level shift due to the open channel(s).However, with the exception of the J = 0 case, for which there exists only one closed channel, use of eqn (16) to calculate corrections to the SFD, BFD or BL energies is not worth while, since their main source of error is the still-neglected coupling among the closed channels. On the other hand, as suggested at the end of the preceding section, a very promising approach would be to use an SE calculation on the closed-channel manifold to deter- mine an initial estimate of the level energy and the metastable state wavefunction tpb, and then calculate the level width and the shift due to coupling to the open channel(s) using eqn (7) and (16). This use of a secular-equation (SE) treatment of the closed channels plus perturbation-theory treatment for the open channels (PTOC) is the method identified here by the acronym SEPTOC. In the present SEPTOC calculations for H2(I,2)-Ar, tpb was expanded in terms of the SFD eigenfunctions, with the radial basis set restricted to the n = 0 level for each channel.Similarly, the regular and irregular open-channel wavefunctions xi') and 2%') were generated from the appropriate open-channel SFD potential. As is seen in the last columns of tables 2 and 3, this yields level energies which are significantly better than those obtained in the SFD, BFD or BLmethods, while the corresponding widths are distinctly better than those obtained in any of the other approximate procedures. A significant feature of the residual discrepancies in the SEPTOC energies is that they are virtually all positive.This occurs because the dominant source of error is due to the use of only one basis function in the expansion for each of the closed-channel radial wavefunctions. The improved radial behaviour obtainable in expanding these bases should also lead to improvements in the SEPTOC widths. However, this remains to be tested. G. CONCLUSIONS Several different approximate schemes have been used to calculate energies and widths for predissociating levels of H2( 1,2)-Ar in the preceding sections. The space- fixed distortion (SFD) approximation was the most successful of the simple decoupling approximations, while the SEPTOC (secdar equation with perturbation theory for open channels) method attempted to account for the neglect of coupling between352 PREDISSOCIATION OF V A N DER WAALS MOLECULES closed channels inherent in the separable approximations.The most promising method for future calculations appears to be an extension of the present SEPTOC procedure using a larger radial basis set to represent the Van der Waals stretching functions (xi:(R)). For H,-Ar, it was found that the SFD approximation was greatly superior to its body-fixed analogue (BFD), and this was rationalised in terms of the effective poten- tials and coupling functions shown in fig. 1 and 2. However, this situation will be reversed in strongly coupled Van der Waals complexes, where the body-fixed projection of the total angular momentum Q is very nearly a good quantum number. In Ar-HCI, for example, ignoring coupling for ACl + 0 gives good results for the energies of truly bound Ie~els.~’ The Born-Oppenheimer or “ best-local ” (BL) approximation gives energy levels that can be up to 0.5 cm-’ in error for H2(l,2)-Ar, since the non-adiabatic coupling functions are very sharply peaked functions of R.This reflects the rapid change in the preferred geometry of the H,-Ar complex from linear to T-shaped as the inter- molecular distance is decreased. i t should be stressed that the BL approximation is not always an improvement over the SFD and BFD approximations. The BL approximation is, however, much more accurate for systems where the preferred angular geometry is independent of R, and has also been successfully applied to strongly anisotropic systems.” Finally, it should be mentioned that all the methods tested here gibe good results for the J = 0 states of H2(v,2)-inert-gas systems; even the small errors in the SFD, BFD and BL energies for the J = 0 leveis of table 2 are almost completeIy removed if the level shifts are taken into account using eqn (16). For the type of model problem considered here, the J = 0 state has only two channels (one open, one closed), and the coupling between these channels is fairly weak.The J = 0 levels of this type of model problem have often been used in the literature for testing new theoretical methods; 3 , 1 2 * 1 3 9 2 8 , 3 2 however, as has been seen, obtaining accurate results for J = 0 states is very easy, and new methods should certainly not be tested on such states alone. 4.PRACTICAL PREDICTIONS FOR HY DROGEN-INERT-GAS VAN DER WAALS MOLECULES A. EFFECT O F DIATOM STRETCHING DEPENDENCE The least well characterized feature of most atom-diatom potential-energy surfaces is their dependence on the length of the diatom bond. Indeed, the hydrogen-inert- gas systems are the only ones for which accurate information of this type has been obtained. For H,-Ar the dependence of the potential on this degree of freedom is illustrated by the differences among the various potentiah listed in table 1 . As the diatom stretches (with increasing vibration-rotation excitation), both the isotropic and anisotropic parts of the potential become deeper and their minima are displaced to larger distances. The changes in the isotropic part of the potential affect mainly the level energies, while the much more pronounced changes in the anisotropy strength functions v,(u,jlR) affect mainly the level widths.Table 4 shows CC level energies and widths for H2(r7j = 2)-Ar obtained using potentials averaged over the diatom stretching motion in various ways. Since the widths of all IeveIs scale as matrix elements of v2(~i7jlR), the ratios of the widths for internal-rotational predissociation of divers levels should be approximately the same for any realistic model of the Hz(t!,-j = 2)-Ar potential. Relafire changes in level width due to a small change in the potential should be similar for all levels. Tn table 4 the properties of the (rz,/,J) = (0,2,0) level of H2(v7j = 2)-Ar are used toR.J . LE ROY, G . C . COREY AND J . M. HUTSON 353 TABLE 4.-CC RESULTS FOR THE (n,l,j,J) = (0’2’2’0) LEVEL OF DIFFERENT VIBRATIONALLY AVERAGED FORMS OF THE CARLEY-LE ROY l6 BC3(6,8) POTENTIAL FOR H2-Ar species E,/cm-’ T/cm - potential a Hz( 1’2)-Ar - 20.727 0.1 10 (0 = (132) Hz(0,2)-Ar - 19.567 0.043 (U’i) = (0’2) H2(0,2)-Ar - 19.474 0.040 ( V , i ) = (0,O) H2(0,2)-Ar - 19.590 0.034 k = O ~~ a All potentials taken from table 1 . illustrate the effect of diatom stretching dependence on the level widths for predissoci- ation by internal rotation. The first two rows of table 4 show that the lifetimes for internal-rotational pre- dissociation of complexes formed by Ar with H,(v = 0) will be almost three times longer than those for complexes formed from H,(v = 1).This suggests also that rotationally inelastic cross-sections for vibrationally excited H2 will be much larger than those for the ground-state species. Moreover, the difference between the second and third entries in table 4 shows that even the pure centrifugal stretching of the diatom has a significant effect on the predissociation widths. The potential used to obtain the fourth entry in this table corresponds to a fictitious rigid-rotor model with the diatom bond length fixed at its average value for ground- state H, (r-ro). While this potential does not correspond to any observable species, it is included here because these k = 0 functions have sometimes been used in cal- culations for this The substantial difference between the level widths for the third and fourth entries in table 4 shows that failure to perform proper vibrational averaging can introduce significant errors.It is clear therefore that predicted pre- dissociation level widths can have little physical significance unless the diatom stretch- ing dependence of the potential-energy surface is correctly accounted for and the vibrational averaging is performed properly. B. PHYSICAL PREDICTIONS AND 1MPLICATIONS REGARDING POTENTIALS The CC results for H2( 1,2)-Ar listed in tables 2 and 3 are essentially exact calcu- lations for the BC3(6,8) potential-energy surface of Carley and Le Roy,16 except that they neglect coupling to closed channels f o r j > 2. However, subsequent calculations showed that inclusion o f j = 4 channels shifts the levels downward by only ca.0.01 cm-’ and increases their widths by < 1 ”/. Inclusion o f j = 6 channels had no further significant effects. Thus, the constraint that j,,, = 2 does not significantly affect the accuracy of predicted H2(v,j = 2)-Ar resonance energies and widths. The main sources of inaccuracy in the physical predictions provided by the CC results of tables 2 and 3 are therefore deficiencies of the BC3(6,8) potential-energy surface of Carley and Le Roy.I6 The accuracy with which this surface reproduces the spectroscopic data and explains total differential and integral scattering cross- sections l6 implies that its (diatom stretching dependent) isotropic part is quite reliable. The accuracy of the anisotropy strength function at distances near and beyond the isotropic potential minimum is confirmed by its agreement with that determined by Zandee and Reuss 34 from their integral cross-section anisotropy measurements, but its inability to reproduce Buck’s inelastic differential cross-sections 35 suggests that its short-range behaviour is in error.This is a matter of concern here, since the short- range part of the anisotropy also determines the predicted level widths.354 PREDISSOCIATION OF VAN D E R WAALS MOLECULES In the work of Carley and Le Roy I5*l6 the exponent parameter of eqn (3) was assumed to have the same value for ;1 = 0 and 2 because the data could not determine p2 independently. More recently, Buck and Le Roy36 determined an improved estimate of the H,-Ar potential anisotropy which is in good accord with both the spectroscopic data l8 and inelastic differential cross-~ections.~~ This was obtained from fits to the spectroscopic data which held all of the isotropic (3.= 0) parameters constant at the values given in ref. (16) and fixed p2 = 3.90 A, a value suggested by the scattering data. The ratio of the widths associated with the first two entries in table 5 indicates that the predissociation level widths of this improved surface will be ca. 1/3 larger than those of the BC,(6,8) potential of Carley and Le Roy.16 Thus, the best available predictions of the predissociation level widths of H2( 1,2)-Ar are obtained by multiplying the CC results in table 3 by an approximate correction factor of 4/3. The only previously reported predictions of internal-rotational level widths for this system are values for the J = 0, 1 and 4 levels of H2(0,2)-Ar reported by Beswick and Requena.I2 Their results were obtained using the BFD and “ intermediate ” distortion approximations from a potential-energy surface based on those reported by Dunker and Gordon,37 and their predicted widths are roughly a factor of 3 too small.This discrepancy is believed to be largely due to the fact that the potential-energy surface used in ref. (12) is less realistic than the one used here. One of the most prominent features of the CC level widths shown in table 3 is the fact that the J = 0, 1 = 2 level is predicted to be more than a factor of two broader than any of the others. This prediction and most other features of the level width pattern are also expected to hold true for complexes formed from other inert-gas partners.The last two entries in table 5 show the energy and width of this J = 0 level calculated from Carley’s l5 BC3(6,8) potentials for H2(1,2)-Kr and H2( 1,2)-Xe. An interesting feature of these results is the prediction that level widths for internal- rotational predissociation decrease from H,-Ar to H2-Kr to H2-Xe. This implies that rotationalIy inelastic cross-sections will vary in this same order and hence that the system with the weakest potential anisotropy, as characterized by the anisotropy strength function depth E ~ , will have the largest inelastic transition probabilities ! This reflects the fact that level widths and rotational inelasticity are both largely determined by the strength of the anisotropy at distances shorter than the zero of the isotropic potential, and that relative to the position of these zeros the short-range repulsive waIls of V,(uJIR) are located at reIatively srnaIler distances for the heavier inert-gas partners.Thus, the anisotropy in the repulsive region is actually smaller for H,-Xe than H2-Ar, giving rise to the predicted trend in predissociation Ievel widths. The one experimental IeveI width obtained to date for the €3,-inert-gas complexes TABLE 5.-cc RESULTS FOR THE (n?/,j,J) = (0,2,2,0) LEVEL OF VARIOUS H2-INERT-GAS POTENTIAL-ENERGY SURFACES species EJcm - rlcm-1 potential H2( 1,2)-Ar - 20.71 8 0.147 Buck-modified f12 = 3.9OA-’ BC3(5? 8) H,( I ,2)-Ar - 20.727 0.110 (r7,i) = (L2) a H,(1,2tKr - 28.680 0.073 Carley’s I5 BC3(6,8) H2( 1,2)-Xe - 34.727 0.039 Carley’s * 5 BC3(6,8) ‘ From table 1 .R.J . LE ROY, G . C . COREY AND J . 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