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Collinear light-atom exchange reactions evaluated by 5-matrix propagation along delves' radial coordinate
\‘olume 76. number 1
CHEMICAL
PHYSICS
LETTERS
I December 1980
COLLINEAR LIGHT-ATOM EXCHANGE REACTIONS EVALUATED BY S-MATRIX PROPAGATION ALONG DELVES’ RADIAL COORDINATE J. MANZ Lehrsrrrlrl
fiir 77worerisclre
Chenrie.
Techische
Utzicersikir
D-SO-16 Gnrclrirrg,
Aftinchetr.
N’esr Gemmu!
and J. R6MELT Lelrrsrdrl fiir 77reoraische Chemie.
Receked
Our
28 July
novel
hght-atom
1980:
method,
exchange
Unirrrsirfir Bonrr, D-5300
in final form 27 August
S-matrix reactions.
propagation The
along
resulting
H + Hz (u s 2) - Hz (u’s 7-I+ H are similar,
1. N’esr Germnw>
1980
Delves’ radial
isotopic except
reactlon
coordinate.
allows
tunneling
simple
for H+hluH
probabilities
for different
1. Introduction Light-atom (L) exchange
Bonn
and resonance
routine
evaluations
(0<2)+HXlu
of collinear
(u’sZj+H
and
patterns.
reaction H+Hz(u~2)+Hz(~‘=z9)+H,
reactions, such as hy-
drogen transfer between two heavy particles (I-l, U’), play a very important role in chemistry, but exact quantum evaluations are likewise very difficult. Even the simplest model, i.e. the collinear triatomic reaction H+LH’(u)-,HL(u’)+H’,
(1)
where u and u’ denote the vibrational reactant and product levels, respectively, poses extreme problems for conventional methods which employ natural collision-type coordinates (4. v)_ In this paper, we solve the challenging problem (1)
simply by routine use of a nc-lel technique that we
(3
i.e., the tests presented in refs. [l, 21 provide a firm basis for the present application to systems (1) for which no comparative results are available. In contrast, the only previou; approaches to this problem either lack the necessary tests, or associated convergence problems indicate numerical limits [:6,17]. Nevertheless, we wish to bring the papers of Shipsey [16] and Baer [17] into prominence as the pioneering works in the field. As a first representative for (1) we present in section 3 definitely converged reaction probabilities _ for the muon&n isotopic variant of (II),
H + MuH(u s
2) +
Hhlu(u’ 5 2) + H ,
(3)
have recently introduced [I, 21: S-matrix propagation [3] along Delves’ radial coordinate r. Thus we use Delves’ polar coordinates (r, cp) [4,5] instead of the conventional ones, (6,~). The two sets of coordinates are compared in section 2, and all details of the technique have already been presented and are adapted directly from,refs. [l, 23. Suffice it here to say that it yields perfect agreement with both analyt-
on the Porter-Karplus surface [18]. This system is selected here because (a) the mass ratio is !q&/mH = l/9, (b) there is considerable current interest in isotopic variants of reaction (2) [9, 191, (c) there is also increasing interest in Mu chemistry (see refs. [20-261 and the reviews [27, 28]), and (d) ihe interest in
ical [6] and the standard [7-M] results for the
system for alternative
reaction
(2)
documented
in the
approaches
to light-atom 337
Volome 76, number 2
CHEMICAL
exchange reactions. In fact, all that we had to do numeric&y in going from (2) to (3) was simply to change the masses appropriately*. Section 3 also includes a preliminary comparison of reactions (2) and (3). Conclusions are in section 4.
2. Comparison of conventional and Delves’ coordins tes
Light-atom exchange reactions (1) and (3) are much more easily evaluated in terms of Delves’ coordinates (r, (o) [4,5] than conventional natural collision type coordinates (6, 7) (see ref. [30]). The two sets (r, cp)and (67~) are illustrated and defined in fig. 1 which shows the Porter-Karplus surface [18] scaled in mass-weighted soordinates (x, y) 1311. Fig. 1 indicates several variants (6 q) thar have been successfuDy used for interpretation and calculation of many collinear reactions. However, each variant has intrinsic complications which make its use very dificuit in the case of light-atom exchange reactions ( 1). * AU convergence parameters were directly adapted from ref. [2] except that here we propagate 011 (up to 7) lev& below the dissociation threshold from r = 2.5-25 A.
PHYSICS
LETTERS
1 December I980
Kzriant (5,~): In the ideal case, the reaction coordinate 5 is orthogonal to a unique set of curvilinear 7 describing internal molecu!ar motions [30, 321; this requirement induces (among others) the problem of an arbitrarily dense net 7 (cf. fig. 1). V’rianr IS, +I: The straight lines + perpendicular to 8 give rise to the well-known triple-valued problem [333. Vadmf (6, $1: The straight lines $ drawn from a turning point cut p non-orthogonally. Therefore, the basis sets representing internal motions vary dramatically. This implies completeness problems. As is evident from fig. 1, these problems increase with decreasing skewing angle ~~~~ of the potential energy surface. For light atom exchange reactions (l), 6omax= ardan [mt(m~ + mu + tn~~)ltn~m~]~‘* ,
decreases approximately as rn:” and approaches zero for extreme mass ratios. IJItimateIy then, none of the variants (5, ~1 lends itself for computations of reaction (1). In contrast, the abovemsntioned difficuIties are absent for Delves’ polar coordinates (r, cp) [I, 4,5]. They never become overdense, triple-valued or donorthogonal. Therefore, they are perfect candidates for the description of arbitrary collisions, including (1).
Fi8. 1. IIw Porter-Karplus surface [II31 plotted in mass-weighted coordinates (x, y! [31] for the reaction H +MuH(u) + HMu(u’) + H. Contours IabeHed o, 0’ = 0, 1. ,.., 6 arc drawn lor the vibrational energies of the reactant and product molecules at -4. IS. -3.fi7, -2.15, -1.40. -0.81, -0.38, and -0.11 eV, respectively. The dashed contour is for the saddle-point energy -4-35 eV. The skewing angle is I,D~,, = 0.452 = 25.9”. The insert allows the comparison of conventional natural collision type coordinates (variants labelled (6 q), (& 61, fif, $1, wi?h Delves’ coordinates (r, cp). Note that the dense-coordinate region of (4, q) and the triple-valued region of & 5) lie inside the clnssically accessible range of moderately excited reactants.
338
(4)
Volume 76. number 2
CHEMICAL
PHYSICS LE7TERS
1 December 1930
3. Results The resulting reaction probabilities PER for teactions (2) and (3) are presented and compared in figs. 2a and 2b. Note that the collision energies for reactions (2) and (3) have been’scaled-such that the thresholds for levels u = 1 and 2 coincide. With this scaling procedure, the overa energy dependences of the PEP, for reactions (2) and (3) are remarkahly similar. For example, close to the reaction threshold, there is the well-known 12, 7-151 region of nonreactivity followed by a “plateau” region of full reactivity, and finally the smooth decline towards iow reactivity at the highest energies investigated in refs. [2, 7-153. Nevertheless, we aIso note some important discrepancies of the reaction probabilities. The initial increase from zero to full reactivity starts earlier for reaction (3) than for reaction (2). indicating that tunneling on the effective potential 191 is easier for Mu than for H. (Tunneling on the static potential 191 is nsgfigible for both reactions.) Most prominent are the different resonance structures superimposed on the overall smooth energy dependence5 of the PC%. The muonium reaction (3) has a remarkable resonance peak of practically full reactivity ciose to the reaction threshold. The widtb of this peak is only -0.0004 eV, comparable to the minimum resonance width that has been observed so far in collinear reactions (0.0002 eV for the F+ Ha reaction [34]). At higher collision energies however, the resonances associated with the thresholds of higher levels decrease by orders of magnitudes. For example, for P$ the expected resonance close to the v = 2 threshold is so small that it could nat be detected even with a 0.00025 eV energy grid. The corresponding resonances are much more pronounced for reaction (2) than for (3). The translational energy dependences are similar far ground-state and vibrationally excited reactants.
4. Canclastans The evaluation of light-atom exchange reactions (1) - known as problematic cases for conventional methods - is easily carried out with our novel technique [l, 21, i.e. S-matrix propagation along Delves’
IE-0)
leV
tE--D)/eV
IE-m/ev IO
06
!.a P
PR
t 2
,
0.0-l a5
15
t 1
t II
183
1.5
t 20
1.5
30
(E - 01/ev
Fig. 2. Comparison of the reaction probabifities fur the H + Hz(u =I 2) --LH&B’< 21-e H reaction (upper panel) and its isotopic muonium variant (lower panel]. The energies E-D, D = -4,?466 eV have been scaled such that the thresholds of the vibrationa levels v = 1 and 2 agree far both reactions. Verti~l armws indicate the thresholds OFreactants in vibrational levels u = 0, 1 and 2+ The dots l give the present numerical results; far the H + Ha reaction the results are adapted from ref. 121. The results for I$! and PT?. PFF are shown in figs 2a and 2b, respective&, for clear prcsentation d the overlapping resonance patterns.
Volume
76. number
2
CHEAIICAL
radial coordinate. The results presented reaction (3) are well converged without
here for any numerical
difficulty.
Similar
results
of the most extreme reactions
for the I+HI
heavy-light-heavy
- will be presented
reaction-one
atom
elsewhere_
Perfect
numerical
results have also been obtained for the other extreme case for which the transferred atom is very heavy (and consequently, prnax = n/2) [l]. Therefore, this work confirms our previous conjecture [l] that S-matrix propagation along Delves’ radial coordinate may be used to evaluate collinear reactions with arbitrary mass ratios. We conclude that the method [l, 21 is the most flexible exact one presently available. The reaction probabilities for the muonium isotope variant (3) of the standard hydrogen reaction (2) have a similar overall energy
dependence, however, with remarkable differences in the superimposed resonances. A detailed analysis beyond the qualitative comparison presented here will be given elsewhere.
Acknowledgement
Discussions with Drs. M. Baer and G. Hauke, the support of Professors SD. Peyerimhoff and R.J. Buenker, and the financial support through a NATO project and by the Fonds der chemischen Industrie is gratefully acknowledged. The computations have been carried out at the RHRZ at the University of Bonn.
References iI] G. Hauke. J. hianz and J. RLimelt, J. Chem. [2] [3] [-I] [S]
340
be published. J. Remelt. Chem. Phys. Letters 74 (1980) J. hlanz. hIol. Phys. 28 (1974) 399. L.hi. Delves, Nucl. Phys. 9 (1959) 391. L.hi. Delves, Nucl. Phys. 20 (1960) 275.
Phys.. to
263.
PHYSICS
LE-l-I’ERS
1 December
1980
[6] J.O. Hirschfelder and K.T. Tang. J. Chem. Phys. 64 (1976) 760. [‘I] DJ. Diestler. J. Chem. Phys. 54 (1971) 4547. [83 B.R. Johnson, Chem_ Phys. Letters 13 (1972) 172. [9] S.-F_ Wu. B.R. Johnson and R.D. Levine. Mol. Phys. 25 (1973)609. [lo] J.W. Duff and D.G. Truhlar, Chem. Phys. Letters 23 (1973) 327. [Ill J.T. Adams. R.L. Smith and E.F. Hayes. J. Chem. Phys. 61 (1974) 2193. [12] A. Rosenthal and R.G. Gordon, J. Chem. Phys. 64 (1976) 1641. [13] J.C. Light and R.B. Walker, J. Chem. Phys. 65 (1976) 1272. [14] B.C. Garrett and W.H. hliller, J. Chem. Phys. 68 (1978)4051.
[15]
A. Askar, AS. Cakmak and H.A. Rabitz, Chem. Phys. 33 (1978) 267. [16] E.J. Shipsey. J. Chem. Phys. 58 (1973) 232. [17] hI. Baer, 1. Chem. Phys. 62 (1975) 305. cl83 R.N. Porter and hl. Karplus. J. Chem. Phys. 40 (1964) 1105. i19] B.C. Garrett and D.G. Truhlar. J. Chem. Phys. 71 (1980) 3460. [ZO] D-hi. Gamer. D.G. Fleming and J.H. Brewer. Chem. Phys. Letters 55 (1978) 163. 1211 D.hI. Garner, Ph.D. Thesis, University of British Columbia (1079).
1221 E. Roduner, Ph.D. Thesis, University of Zurich (1979). [23]
J.N.L. Connor, W. Jakuben and J. Maw, Chem. Phyx. Letters 45 (1977) 265. [24] J.N.L. Connor, W. Jakubetz and J. hianz, Chem. Phys. 28 (1978) 219. [ZS] J.N.L. Connor, W. Jakubetz. J. Manz and J.C. Whitehead. J. Chem. Phys. 72 (1980) 6209. [26] W. Jakubetz. J. Am. Chem. Sot. 101 (1979) 298. [27] W. Jakubetz. Hypedine Interactions 6 (1979) 387. [28] J.N.L. Connor. Computer Phys. Commun. 17 (1979) 117. [29] D.G. Truhtu and R.E. Wyatt. Ann. Rev. Phys. Chem. 27 (1976) 1. [30] G.L. Hofacker, Z. Naturfonch. 18a (1963) 607. [31] F.T. Smith, J. Chem. Phys. 31 (1959) 1352. [32] N. Agmon and R.D. Levine, J. Chem. Phys. 71 (1979) 3034. [33] J-C. Light. hlethods Comput. Phys. 10 (1971) 111. [34] J.N.L. Connor. W. Jakubetz and J. hfanz. hIol. Phys. 39 (1980) 799.
CHEMICAL
PHYSICS
LETTERS
I December 1980
COLLINEAR LIGHT-ATOM EXCHANGE REACTIONS EVALUATED BY S-MATRIX PROPAGATION ALONG DELVES’ RADIAL COORDINATE J. MANZ Lehrsrrrlrl
fiir 77worerisclre
Chenrie.
Techische
Utzicersikir
D-SO-16 Gnrclrirrg,
Aftinchetr.
N’esr Gemmu!
and J. R6MELT Lelrrsrdrl fiir 77reoraische Chemie.
Receked
Our
28 July
novel
hght-atom
1980:
method,
exchange
Unirrrsirfir Bonrr, D-5300
in final form 27 August
S-matrix reactions.
propagation The
along
resulting
H + Hz (u s 2) - Hz (u’s 7-I+ H are similar,
1. N’esr Germnw>
1980
Delves’ radial
isotopic except
reactlon
coordinate.
allows
tunneling
simple
for H+hluH
probabilities
for different
1. Introduction Light-atom (L) exchange
Bonn
and resonance
routine
evaluations
(0<2)+HXlu
of collinear
(u’sZj+H
and
patterns.
reaction H+Hz(u~2)+Hz(~‘=z9)+H,
reactions, such as hy-
drogen transfer between two heavy particles (I-l, U’), play a very important role in chemistry, but exact quantum evaluations are likewise very difficult. Even the simplest model, i.e. the collinear triatomic reaction H+LH’(u)-,HL(u’)+H’,
(1)
where u and u’ denote the vibrational reactant and product levels, respectively, poses extreme problems for conventional methods which employ natural collision-type coordinates (4. v)_ In this paper, we solve the challenging problem (1)
simply by routine use of a nc-lel technique that we
(3
i.e., the tests presented in refs. [l, 21 provide a firm basis for the present application to systems (1) for which no comparative results are available. In contrast, the only previou; approaches to this problem either lack the necessary tests, or associated convergence problems indicate numerical limits [:6,17]. Nevertheless, we wish to bring the papers of Shipsey [16] and Baer [17] into prominence as the pioneering works in the field. As a first representative for (1) we present in section 3 definitely converged reaction probabilities _ for the muon&n isotopic variant of (II),
H + MuH(u s
2) +
Hhlu(u’ 5 2) + H ,
(3)
have recently introduced [I, 21: S-matrix propagation [3] along Delves’ radial coordinate r. Thus we use Delves’ polar coordinates (r, cp) [4,5] instead of the conventional ones, (6,~). The two sets of coordinates are compared in section 2, and all details of the technique have already been presented and are adapted directly from,refs. [l, 23. Suffice it here to say that it yields perfect agreement with both analyt-
on the Porter-Karplus surface [18]. This system is selected here because (a) the mass ratio is !q&/mH = l/9, (b) there is considerable current interest in isotopic variants of reaction (2) [9, 191, (c) there is also increasing interest in Mu chemistry (see refs. [20-261 and the reviews [27, 28]), and (d) ihe interest in
ical [6] and the standard [7-M] results for the
system for alternative
reaction
(2)
documented
in the
approaches
to light-atom 337
Volome 76, number 2
CHEMICAL
exchange reactions. In fact, all that we had to do numeric&y in going from (2) to (3) was simply to change the masses appropriately*. Section 3 also includes a preliminary comparison of reactions (2) and (3). Conclusions are in section 4.
2. Comparison of conventional and Delves’ coordins tes
Light-atom exchange reactions (1) and (3) are much more easily evaluated in terms of Delves’ coordinates (r, (o) [4,5] than conventional natural collision type coordinates (6, 7) (see ref. [30]). The two sets (r, cp)and (67~) are illustrated and defined in fig. 1 which shows the Porter-Karplus surface [18] scaled in mass-weighted soordinates (x, y) 1311. Fig. 1 indicates several variants (6 q) thar have been successfuDy used for interpretation and calculation of many collinear reactions. However, each variant has intrinsic complications which make its use very dificuit in the case of light-atom exchange reactions ( 1). * AU convergence parameters were directly adapted from ref. [2] except that here we propagate 011 (up to 7) lev& below the dissociation threshold from r = 2.5-25 A.
PHYSICS
LETTERS
1 December I980
Kzriant (5,~): In the ideal case, the reaction coordinate 5 is orthogonal to a unique set of curvilinear 7 describing internal molecu!ar motions [30, 321; this requirement induces (among others) the problem of an arbitrarily dense net 7 (cf. fig. 1). V’rianr IS, +I: The straight lines + perpendicular to 8 give rise to the well-known triple-valued problem [333. Vadmf (6, $1: The straight lines $ drawn from a turning point cut p non-orthogonally. Therefore, the basis sets representing internal motions vary dramatically. This implies completeness problems. As is evident from fig. 1, these problems increase with decreasing skewing angle ~~~~ of the potential energy surface. For light atom exchange reactions (l), 6omax= ardan [mt(m~ + mu + tn~~)ltn~m~]~‘* ,
decreases approximately as rn:” and approaches zero for extreme mass ratios. IJItimateIy then, none of the variants (5, ~1 lends itself for computations of reaction (1). In contrast, the abovemsntioned difficuIties are absent for Delves’ polar coordinates (r, cp) [I, 4,5]. They never become overdense, triple-valued or donorthogonal. Therefore, they are perfect candidates for the description of arbitrary collisions, including (1).
Fi8. 1. IIw Porter-Karplus surface [II31 plotted in mass-weighted coordinates (x, y! [31] for the reaction H +MuH(u) + HMu(u’) + H. Contours IabeHed o, 0’ = 0, 1. ,.., 6 arc drawn lor the vibrational energies of the reactant and product molecules at -4. IS. -3.fi7, -2.15, -1.40. -0.81, -0.38, and -0.11 eV, respectively. The dashed contour is for the saddle-point energy -4-35 eV. The skewing angle is I,D~,, = 0.452 = 25.9”. The insert allows the comparison of conventional natural collision type coordinates (variants labelled (6 q), (& 61, fif, $1, wi?h Delves’ coordinates (r, cp). Note that the dense-coordinate region of (4, q) and the triple-valued region of & 5) lie inside the clnssically accessible range of moderately excited reactants.
338
(4)
Volume 76. number 2
CHEMICAL
PHYSICS LE7TERS
1 December 1930
3. Results The resulting reaction probabilities PER for teactions (2) and (3) are presented and compared in figs. 2a and 2b. Note that the collision energies for reactions (2) and (3) have been’scaled-such that the thresholds for levels u = 1 and 2 coincide. With this scaling procedure, the overa energy dependences of the PEP, for reactions (2) and (3) are remarkahly similar. For example, close to the reaction threshold, there is the well-known 12, 7-151 region of nonreactivity followed by a “plateau” region of full reactivity, and finally the smooth decline towards iow reactivity at the highest energies investigated in refs. [2, 7-153. Nevertheless, we aIso note some important discrepancies of the reaction probabilities. The initial increase from zero to full reactivity starts earlier for reaction (3) than for reaction (2). indicating that tunneling on the effective potential 191 is easier for Mu than for H. (Tunneling on the static potential 191 is nsgfigible for both reactions.) Most prominent are the different resonance structures superimposed on the overall smooth energy dependence5 of the PC%. The muonium reaction (3) has a remarkable resonance peak of practically full reactivity ciose to the reaction threshold. The widtb of this peak is only -0.0004 eV, comparable to the minimum resonance width that has been observed so far in collinear reactions (0.0002 eV for the F+ Ha reaction [34]). At higher collision energies however, the resonances associated with the thresholds of higher levels decrease by orders of magnitudes. For example, for P$ the expected resonance close to the v = 2 threshold is so small that it could nat be detected even with a 0.00025 eV energy grid. The corresponding resonances are much more pronounced for reaction (2) than for (3). The translational energy dependences are similar far ground-state and vibrationally excited reactants.
4. Canclastans The evaluation of light-atom exchange reactions (1) - known as problematic cases for conventional methods - is easily carried out with our novel technique [l, 21, i.e. S-matrix propagation along Delves’
IE-0)
leV
tE--D)/eV
IE-m/ev IO
06
!.a P
PR
t 2
,
0.0-l a5
15
t 1
t II
183
1.5
t 20
1.5
30
(E - 01/ev
Fig. 2. Comparison of the reaction probabifities fur the H + Hz(u =I 2) --LH&B’< 21-e H reaction (upper panel) and its isotopic muonium variant (lower panel]. The energies E-D, D = -4,?466 eV have been scaled such that the thresholds of the vibrationa levels v = 1 and 2 agree far both reactions. Verti~l armws indicate the thresholds OFreactants in vibrational levels u = 0, 1 and 2+ The dots l give the present numerical results; far the H + Ha reaction the results are adapted from ref. 121. The results for I$! and PT?. PFF are shown in figs 2a and 2b, respective&, for clear prcsentation d the overlapping resonance patterns.
Volume
76. number
2
CHEAIICAL
radial coordinate. The results presented reaction (3) are well converged without
here for any numerical
difficulty.
Similar
results
of the most extreme reactions
for the I+HI
heavy-light-heavy
- will be presented
reaction-one
atom
elsewhere_
Perfect
numerical
results have also been obtained for the other extreme case for which the transferred atom is very heavy (and consequently, prnax = n/2) [l]. Therefore, this work confirms our previous conjecture [l] that S-matrix propagation along Delves’ radial coordinate may be used to evaluate collinear reactions with arbitrary mass ratios. We conclude that the method [l, 21 is the most flexible exact one presently available. The reaction probabilities for the muonium isotope variant (3) of the standard hydrogen reaction (2) have a similar overall energy
dependence, however, with remarkable differences in the superimposed resonances. A detailed analysis beyond the qualitative comparison presented here will be given elsewhere.
Acknowledgement
Discussions with Drs. M. Baer and G. Hauke, the support of Professors SD. Peyerimhoff and R.J. Buenker, and the financial support through a NATO project and by the Fonds der chemischen Industrie is gratefully acknowledged. The computations have been carried out at the RHRZ at the University of Bonn.
References iI] G. Hauke. J. hianz and J. RLimelt, J. Chem. [2] [3] [-I] [S]
340
be published. J. Remelt. Chem. Phys. Letters 74 (1980) J. hlanz. hIol. Phys. 28 (1974) 399. L.hi. Delves, Nucl. Phys. 9 (1959) 391. L.hi. Delves, Nucl. Phys. 20 (1960) 275.
Phys.. to
263.
PHYSICS
LE-l-I’ERS
1 December
1980
[6] J.O. Hirschfelder and K.T. Tang. J. Chem. Phys. 64 (1976) 760. [‘I] DJ. Diestler. J. Chem. Phys. 54 (1971) 4547. [83 B.R. Johnson, Chem_ Phys. Letters 13 (1972) 172. [9] S.-F_ Wu. B.R. Johnson and R.D. Levine. Mol. Phys. 25 (1973)609. [lo] J.W. Duff and D.G. Truhlar, Chem. Phys. Letters 23 (1973) 327. [Ill J.T. Adams. R.L. Smith and E.F. Hayes. J. Chem. Phys. 61 (1974) 2193. [12] A. Rosenthal and R.G. Gordon, J. Chem. Phys. 64 (1976) 1641. [13] J.C. Light and R.B. Walker, J. Chem. Phys. 65 (1976) 1272. [14] B.C. Garrett and W.H. hliller, J. Chem. Phys. 68 (1978)4051.
[15]
A. Askar, AS. Cakmak and H.A. Rabitz, Chem. Phys. 33 (1978) 267. [16] E.J. Shipsey. J. Chem. Phys. 58 (1973) 232. [17] hI. Baer, 1. Chem. Phys. 62 (1975) 305. cl83 R.N. Porter and hl. Karplus. J. Chem. Phys. 40 (1964) 1105. i19] B.C. Garrett and D.G. Truhlar. J. Chem. Phys. 71 (1980) 3460. [ZO] D-hi. Gamer. D.G. Fleming and J.H. Brewer. Chem. Phys. Letters 55 (1978) 163. 1211 D.hI. Garner, Ph.D. Thesis, University of British Columbia (1079).
1221 E. Roduner, Ph.D. Thesis, University of Zurich (1979). [23]
J.N.L. Connor, W. Jakuben and J. Maw, Chem. Phyx. Letters 45 (1977) 265. [24] J.N.L. Connor, W. Jakubetz and J. hianz, Chem. Phys. 28 (1978) 219. [ZS] J.N.L. Connor, W. Jakubetz. J. Manz and J.C. Whitehead. J. Chem. Phys. 72 (1980) 6209. [26] W. Jakubetz. J. Am. Chem. Sot. 101 (1979) 298. [27] W. Jakubetz. Hypedine Interactions 6 (1979) 387. [28] J.N.L. Connor. Computer Phys. Commun. 17 (1979) 117. [29] D.G. Truhtu and R.E. Wyatt. Ann. Rev. Phys. Chem. 27 (1976) 1. [30] G.L. Hofacker, Z. Naturfonch. 18a (1963) 607. [31] F.T. Smith, J. Chem. Phys. 31 (1959) 1352. [32] N. Agmon and R.D. Levine, J. Chem. Phys. 71 (1979) 3034. [33] J-C. Light. hlethods Comput. Phys. 10 (1971) 111. [34] J.N.L. Connor. W. Jakubetz and J. hfanz. hIol. Phys. 39 (1980) 799.