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Effect of partial wave interference on angular distributions and sideways scattering in rearrangement collisions
Volume 79, number
CHEMICAL PHYSICS LETTERS
1
EFFECT OF PARTIAL AND SIDEWAYS
WAVE INTERFERENCE
SCATTERING
ON ANGULAR
IN REARRANGEMENT
1 Apnl 1981
DISTRIBUTIONS
COLLISIONS
S-H. SUCK and R W. EMMONS Department of Physics and Graduate Center for Cloud Phys~s Research, Rolla. Lhsoun 65401, USA Received
17 November
1980, in final form 29 December
Utuvemty
of Ahssoun-Rolla,
1980
The effect of partial wave Interference on angular dlstnbutlons IS evammed We fmd that monotomc backward or sldeways scattermg are not entirely due to contributton of small orbital angular momenta the sldeways reactwe scattermg of r + Hz at higher mcldent energes IS corwstent wth observation
changes either In The occurrence of
1 . Introduction
2. Transition
Several three-dlmenstonal DWBA (Distorted wave Born approxunation) methods [l-6] have been proposed to study the state-to-state reactive scattenng processes of atom-&atomic molecule systems. Recently Shan et al. [7] reported a DWBA study of v1bratlon-rotation product &stnbutlons for F + H2 + HF + H. Based on his earlier DWBA method [S], Suck [6] made a comparative study of angular Istnbution wth the Franck-Condon model results of Vda et al. [8], using the unperturbed molecule approxunation for the same reactive system. On the other hand, Redmon and Wyatt [9], and Wyatt [IO] recently reported a quantum resonance structure m F + H+za = 0.1~ = 0) + HF(nb = 0, all&) + H, using a close-coupling method. Kafn et al. [ 1 l] applied the plane wave Born approximation for the study of the product (HF) rotatlonai state rhstnbutlon. Lately, the molecular beam measurements [ 121 the F + H2 system appeared. The measurements of Sparks et al. [12] showed sideways scattering at a lugher collmon energy of 3.17 kcal/mole (e-14 eV) for the reactwe transitlon from the reactant molecule states of II, = 0 and iowla to the product molecule state of ti b = 2 and alllb. In our present study, we &scuss theoretical results of sideways scattering. In addition, we report the effect of mterference among partial waves on angular distnbutions.
Followmg ref. [S], the “reduced” DWBA transrtlon amphtude III -7” (transferred angular momentum) expansion 1s
t’b’yec m) m
X
amplitude
=
f’lm’ (0, m )[(2L, Lb
+ 1)(2Lb + l)] “2(LbLarnOIJrn)
~,r~(KavRa)RbRa x j- XLb(Kb,Rb)FL/;;aJ
dRbaa . (1)
Here 1111s the z component of]. K, (Kb) is Indid (final) channel wave number. J IS the Jacobran of the transfomlatlon necessary for the coordinate transformatlon 151. L, (Lb) is the uutial (final) channel orbital angular momentum. ok, (0~. b) is the phase shrft of the initial (final) distorted waves xLa Oc,,). 8, 1s the c-m. (center of mass) scattermg angle. FJbJa Lb&l is the form factor coefficient wluch arises as a result of the original form factor being transformed hke a spherical harmonic_ The DWBA [5] prer.hcts that the angular distniutlon is constructed from the mterference of both initial and final channel partial waves. 93
Volume 79, number
1
CHEMICAL
PHYSICS LETTERS
1 Apnl
1981
TIus IS unhke the case of the total cross sectlon where only the uutlai (reactant) channel partial waves yield coherence (mterference) [5,6] _ In our calculat,ons of the drfferentlal cross sectlons we report below, we used the transltlon amphtude gven m ref [ 11.
3. Partial
wave interference
and sideways
scattering
For a grven state-to-state reactive transItIon, the magrutudes of both the drfferentlal and total cross sectlons depend on the incident kmetic energy. For a gwen mcldent kmetlc energy, the transition probabllltles or cross sections vary with state-to-state reactive scattenng processes. Now, for a given reactive system, the contnbutlon of each orbital angular momentum to the cross sections will depend on the mcadent kmetrc energy For a given incident kmetlc energy, the contnbutlon of each orbltal angular momentum wffl vary Lvlth the nature of the potential energy surface and the reduced mass characterized by an atom-dKitomlc molecule system. In our present DWBA study, we focus our attention on the energy and orbital angular momentum dependence of the differential cross sectlons or angular dlstnbutlons for the specific reactive system of F + H?(rz, = O,j, = 0) + HF(rlb = 2, lb = 0) f H. Such a chorce of the rotational ground-state rransltlon enables us to converuently analyze our computed results Usmg the hiuckerman V LEPS potential energy surface [7,13], we present various angular &stnbutlons m figs. 14 Note that for an easy companson, the magnitude of each angular dlstnbutlon 1s scaled to a peak value of 1 Each angular titrlbutlon corresponds to a coherent summation to d designated orbital angular momentum, as shown m figs 14. Interzstmgly, we fiid oscfflatory structures m the angular distnbutlons at both mcldent energes of 0 1 and 0.2 eV. It occurs when the coherent summation to a certain orbital angular momentum IS not sufficient. However, dsappearance of such oscdlatory structures IS unequivocally predicted \nth further added interference with the partial waves of larger orbital angular momenta as is seen m figs 2 and 4 Thus, we have demonstrated that coherence (mterference) by all important partlclpetmg partial waves IS the cause of the generally known monotomc structures III the angular distributions of atom-&atomic molecule systems Now, the converRed (in the sense of the orbital an94
0
45
!35
90
C 11 SCATTERING
180
RNGLE
Fig I Angular dlstr:butlons obtamed from coherent summatlons to angulx momenta 1, 3 and 7 respectwely, at an mcldent kmetlc energy of 0 1 eV. Note that for an easy comparrson the magmtude of each angular dlstrlbutlon LSscaled to a peak value set as 1 The rest of the figures are drawn in the same manner
0
55 C.M.
90 SCflTTERING
135
180
RNGLE
Frg 2 Angular dlstributlons obtamed from coherent tlons to angular momenta 11, 14 and 20 respectwely, mcldent kmetlc energy of 0 1 eV.
summaat an
Volume
79, number
1
0
CHEhIICAL
45 C.M.
90 SCFITTERING
135 ANGLE
180
Fig 3. Angular &strlbutmns obtamed from coherent summauons to angular momenta 1, 3 and 7 respectwely, at an UICIdent kmettc energy of 0 2 eV
gular momentum sum) angular distributions show some marked differences between the two mcident kinetx energies; the angular distnbution at the h&er energy of 0.2 eV (4.6 kcal/mole) shows sldeways scat-
0.8
0.6
0
Y5 C.M.
90 SCQTTERING
135
180
RNGLE
Fig. 4. Angular lstributlons obtatned from coherent tions to angular momenta 11, 14 and 26 respecttvely, mctdent kmettc energy of 0 2 eV.
summaat ail
PHYSICS
LETTERS
1 Apnl 1981
tering (peak posltlon at 0,,_ P 180°, see fig. 4), wlule at the lower energy of 0.1 eV, backward scattering (peak posltion at 8, m = 1 SO”, see fig. 2) is pre&cted. In general, the osctiatory structures are seen to be more pronounced at the higher incident energy than at the lower one, when convergence m orbital angular momentum is not achieved. Another mterestmg observation is the great smulanty in the structures of the angular distributions between the two incident energies, for the coherent summation to the orbital angular momentum of approximately L = 7. Thus 1s due to a roughly identical reaction zone whxh arises at smaller orbital angular momenta. However, at increased orbltal angular momenta, the difference in the reactlon zones between the two incident kinetic energes becomes greater, yleldmg a larger probabihty for a h.~gh Incident energy proJecttie to penetrate mto the zone more prone to the reaction. For an mcldent kmetic energy of 0.2 eV, @5 partial waves were necessary for the determination of the well-converged angular distnbutlons, whde 20 partial waves were reqmred for convergence 111the case of the lower energy of 0.1 eV. However, these values are subject to change dependmg on the nature of 2 chosen potential energy surface. We fiid that the partial waves of smaller orbital angular momenta yield non-neghgble forward angle scattering as IS seen in figs. 1 and 3. Added mterference with larger orbital angular momenta may eliminate such forward angle scattering structure. Indeed, this conjecture 1s well verified from the present DWBA results as IS clearly seen m figs. 2 and 4. However, we found that although not shown here the selection of only larger orbital angular momenta leads to forward angle scattermg. Thus, m rearrangement collisions, all orbital angular momenta are important for determming the overall structures of the angular &stributIons. As mentioned earlier, Sparks et al. have observed the sldeways scattermg at the higher incident energy of 3.17 kcal/mole (a.14 eV) for the transition from the rovlbratlonal states of Na = 0 and low ja to all the rotational states of the tzb = 2 vibrational product molecule HF. From the Muckerman V LEPS potential energy surface, we find that state-to-state sideways scattermg begms to occur at 4.6 kcal/mole (x0.2 eV) for the reactive transltion from the rovlbrational ground state to the product state of q, = 2 and lb = 0. Although not presented here, our further computed 95
Volume
79
number
1
CHEMICAL
PHYSICS
results of the tronsltlons to the final product states of rotntlonai angular momenta conststently sho\%ed the sideways scattermg Backward scattering was predrcted for the transitton from il, = 0 and Ja = 0 to “,, = I and lb = 0 only below a co!frslon (inctdent) energy of *3.50 kcd/mole (=O 15 eV) On the other hand. we elaarnmed the translt1on from rt, = 0 and/, = 0 to i$, = 3 and& = 0 up I0 a colhsion energy of 13 SS kcal/ mole (0 6 eV), and our DWBA results consistently showed backward scnttermg InterestIngI;,, at a colhslon energy of 3.17 kcai/mofe (x0 13 e\l), Sparks et al [ 12 1 observed no sideways scattenng for the rextlve transitton from the low rovlbrxlonal reactant molecufe states of 13~ = 0 and Lowe, to all the rotatlonai states of erther >zb = I or I?~ = 3 product molecule HF Fuifer details wgLI be reported eisewhere.
LETTERS
1
April
1981
(4) The monotonously varying structures of angular d~strlbutions m the case of either backward or sideways scattermg are not due to contrlb~t~on solely by smaller orbit& angular momenta. We b&eve that conclusions made above WIU remam
unchanged even wth an exact theory.
Acknowledgement The authors are grateful to a referee partxularly for a comment regarding the rotatlonal states of the reactant mofecuie H, m the crossed molecular beam etrperunent of Sparks et al
References 4 Conclusion In the present DWBA study, we hJve euammed the coherence effect ofpnrtlal waves on anguIar dlstrlbutlons Important findmgs m this study are ( 1) The forward angle scattering strtxture that arises as a result of the coherent summation of an msufficient number of orbital angular mornent~ dlsappears due to added InterfererIce tvlth the partial waves of larger orbital angular momenta (see figs f-4) (2) Osclilatory structure disappears as a consequence of interference effect with added partial waves mvolvmg larger orbital angular momenta (3) Sldeways scattermg for the transltlon from low rovlbratlonal states of H, to alI rotatlonal states of the product molecule HF of the vlbrntlonal quantum number ?zb = 2 IS expected to occur, m general. as a result of the Incoherent sum of each state-to-state stdeways scattcrmg angufar dlstrlbutlon
96
I\’ H R D KT KT S H S H
Ulller, J Chem Phys 49 (1968) 2373 Lebqnc, Israel J Chem 8 (1970) 13 Tang and 51 Knrplus, Phys Rev A3 (1971) 1844 Tang and B H Char, J Chem Phys 62 (197.5) 3612 Such, Phls Rev. Al5 (1977) 1893 Such, Chem Phys Letters 77 (1981) 390 Y Shnn, B H Choi, R T Poe and K T Tang, Chem Ph! s Lerters 57 (1978) 379 C L Vda, D J Zw~ac and J Ross J Chcm Phys 70 (1979) 5362 M J Redmon 2nd R E Wyatt, Chem Phys Letters 63 (1979) 209 R C Wwtt, III Horizons rn quantum chemrstr~, eds 1; Fuhu! and B Pullman (Retdcl, Dordrecht, 1980) p 63 .-I Knfrt, Y ShImon!, R D Levme and S Alexander, Chem Phys 13 (1976) 323 R K Sparks, C C Ha> den, Ii Shobnttlhe, D hl Ne\\mmnrh and Y T Lee, ia Horizons WIquantum chemistry, eds K rukul and B Pullman (Reldel, Dordrecht, 1980) p 91 J T hluckerman, J Chem Phys 56 (1972) 2997,57 (1972) 3388
CHEMICAL PHYSICS LETTERS
1
EFFECT OF PARTIAL AND SIDEWAYS
WAVE INTERFERENCE
SCATTERING
ON ANGULAR
IN REARRANGEMENT
1 Apnl 1981
DISTRIBUTIONS
COLLISIONS
S-H. SUCK and R W. EMMONS Department of Physics and Graduate Center for Cloud Phys~s Research, Rolla. Lhsoun 65401, USA Received
17 November
1980, in final form 29 December
Utuvemty
of Ahssoun-Rolla,
1980
The effect of partial wave Interference on angular dlstnbutlons IS evammed We fmd that monotomc backward or sldeways scattermg are not entirely due to contributton of small orbital angular momenta the sldeways reactwe scattermg of r + Hz at higher mcldent energes IS corwstent wth observation
changes either In The occurrence of
1 . Introduction
2. Transition
Several three-dlmenstonal DWBA (Distorted wave Born approxunation) methods [l-6] have been proposed to study the state-to-state reactive scattenng processes of atom-&atomic molecule systems. Recently Shan et al. [7] reported a DWBA study of v1bratlon-rotation product &stnbutlons for F + H2 + HF + H. Based on his earlier DWBA method [S], Suck [6] made a comparative study of angular Istnbution wth the Franck-Condon model results of Vda et al. [8], using the unperturbed molecule approxunation for the same reactive system. On the other hand, Redmon and Wyatt [9], and Wyatt [IO] recently reported a quantum resonance structure m F + H+za = 0.1~ = 0) + HF(nb = 0, all&) + H, using a close-coupling method. Kafn et al. [ 1 l] applied the plane wave Born approximation for the study of the product (HF) rotatlonai state rhstnbutlon. Lately, the molecular beam measurements [ 121 the F + H2 system appeared. The measurements of Sparks et al. [12] showed sideways scattering at a lugher collmon energy of 3.17 kcal/mole (e-14 eV) for the reactwe transitlon from the reactant molecule states of II, = 0 and iowla to the product molecule state of ti b = 2 and alllb. In our present study, we &scuss theoretical results of sideways scattering. In addition, we report the effect of mterference among partial waves on angular distnbutions.
Followmg ref. [S], the “reduced” DWBA transrtlon amphtude III -7” (transferred angular momentum) expansion 1s
t’b’yec m) m
X
amplitude
=
f’lm’ (0, m )[(2L, Lb
+ 1)(2Lb + l)] “2(LbLarnOIJrn)
~,r~(KavRa)RbRa x j- XLb(Kb,Rb)FL/;;aJ
dRbaa . (1)
Here 1111s the z component of]. K, (Kb) is Indid (final) channel wave number. J IS the Jacobran of the transfomlatlon necessary for the coordinate transformatlon 151. L, (Lb) is the uutial (final) channel orbital angular momentum. ok, (0~. b) is the phase shrft of the initial (final) distorted waves xLa Oc,,). 8, 1s the c-m. (center of mass) scattermg angle. FJbJa Lb&l is the form factor coefficient wluch arises as a result of the original form factor being transformed hke a spherical harmonic_ The DWBA [5] prer.hcts that the angular distniutlon is constructed from the mterference of both initial and final channel partial waves. 93
Volume 79, number
1
CHEMICAL
PHYSICS LETTERS
1 Apnl
1981
TIus IS unhke the case of the total cross sectlon where only the uutlai (reactant) channel partial waves yield coherence (mterference) [5,6] _ In our calculat,ons of the drfferentlal cross sectlons we report below, we used the transltlon amphtude gven m ref [ 11.
3. Partial
wave interference
and sideways
scattering
For a grven state-to-state reactive transItIon, the magrutudes of both the drfferentlal and total cross sectlons depend on the incident kmetic energy. For a gwen mcldent kmetlc energy, the transition probabllltles or cross sections vary with state-to-state reactive scattenng processes. Now, for a given reactive system, the contnbutlon of each orbital angular momentum to the cross sections will depend on the mcadent kmetrc energy For a given incident kmetlc energy, the contnbutlon of each orbltal angular momentum wffl vary Lvlth the nature of the potential energy surface and the reduced mass characterized by an atom-dKitomlc molecule system. In our present DWBA study, we focus our attention on the energy and orbital angular momentum dependence of the differential cross sectlons or angular dlstnbutlons for the specific reactive system of F + H?(rz, = O,j, = 0) + HF(rlb = 2, lb = 0) f H. Such a chorce of the rotational ground-state rransltlon enables us to converuently analyze our computed results Usmg the hiuckerman V LEPS potential energy surface [7,13], we present various angular &stnbutlons m figs. 14 Note that for an easy companson, the magnitude of each angular dlstnbutlon 1s scaled to a peak value of 1 Each angular titrlbutlon corresponds to a coherent summation to d designated orbital angular momentum, as shown m figs 14. Interzstmgly, we fiid oscfflatory structures m the angular distnbutlons at both mcldent energes of 0 1 and 0.2 eV. It occurs when the coherent summation to a certain orbital angular momentum IS not sufficient. However, dsappearance of such oscdlatory structures IS unequivocally predicted \nth further added interference with the partial waves of larger orbital angular momenta as is seen m figs 2 and 4 Thus, we have demonstrated that coherence (mterference) by all important partlclpetmg partial waves IS the cause of the generally known monotomc structures III the angular distributions of atom-&atomic molecule systems Now, the converRed (in the sense of the orbital an94
0
45
!35
90
C 11 SCATTERING
180
RNGLE
Fig I Angular dlstr:butlons obtamed from coherent summatlons to angulx momenta 1, 3 and 7 respectwely, at an mcldent kmetlc energy of 0 1 eV. Note that for an easy comparrson the magmtude of each angular dlstrlbutlon LSscaled to a peak value set as 1 The rest of the figures are drawn in the same manner
0
55 C.M.
90 SCflTTERING
135
180
RNGLE
Frg 2 Angular dlstributlons obtamed from coherent tlons to angular momenta 11, 14 and 20 respectwely, mcldent kmetlc energy of 0 1 eV.
summaat an
Volume
79, number
1
0
CHEhIICAL
45 C.M.
90 SCFITTERING
135 ANGLE
180
Fig 3. Angular &strlbutmns obtamed from coherent summauons to angular momenta 1, 3 and 7 respectwely, at an UICIdent kmettc energy of 0 2 eV
gular momentum sum) angular distributions show some marked differences between the two mcident kinetx energies; the angular distnbution at the h&er energy of 0.2 eV (4.6 kcal/mole) shows sldeways scat-
0.8
0.6
0
Y5 C.M.
90 SCQTTERING
135
180
RNGLE
Fig. 4. Angular lstributlons obtatned from coherent tions to angular momenta 11, 14 and 26 respecttvely, mctdent kmettc energy of 0 2 eV.
summaat ail
PHYSICS
LETTERS
1 Apnl 1981
tering (peak posltlon at 0,,_ P 180°, see fig. 4), wlule at the lower energy of 0.1 eV, backward scattering (peak posltion at 8, m = 1 SO”, see fig. 2) is pre&cted. In general, the osctiatory structures are seen to be more pronounced at the higher incident energy than at the lower one, when convergence m orbital angular momentum is not achieved. Another mterestmg observation is the great smulanty in the structures of the angular distributions between the two incident energies, for the coherent summation to the orbital angular momentum of approximately L = 7. Thus 1s due to a roughly identical reaction zone whxh arises at smaller orbital angular momenta. However, at increased orbltal angular momenta, the difference in the reactlon zones between the two incident kinetic energes becomes greater, yleldmg a larger probabihty for a h.~gh Incident energy proJecttie to penetrate mto the zone more prone to the reaction. For an mcldent kmetic energy of 0.2 eV, @5 partial waves were necessary for the determination of the well-converged angular distnbutlons, whde 20 partial waves were reqmred for convergence 111the case of the lower energy of 0.1 eV. However, these values are subject to change dependmg on the nature of 2 chosen potential energy surface. We fiid that the partial waves of smaller orbital angular momenta yield non-neghgble forward angle scattering as IS seen in figs. 1 and 3. Added mterference with larger orbital angular momenta may eliminate such forward angle scattering structure. Indeed, this conjecture 1s well verified from the present DWBA results as IS clearly seen m figs. 2 and 4. However, we found that although not shown here the selection of only larger orbital angular momenta leads to forward angle scattermg. Thus, m rearrangement collisions, all orbital angular momenta are important for determming the overall structures of the angular &stributIons. As mentioned earlier, Sparks et al. have observed the sldeways scattermg at the higher incident energy of 3.17 kcal/mole (a.14 eV) for the transition from the rovlbratlonal states of Na = 0 and low ja to all the rotational states of the tzb = 2 vibrational product molecule HF. From the Muckerman V LEPS potential energy surface, we find that state-to-state sideways scattermg begms to occur at 4.6 kcal/mole (x0.2 eV) for the reactive transltion from the rovlbrational ground state to the product state of q, = 2 and lb = 0. Although not presented here, our further computed 95
Volume
79
number
1
CHEMICAL
PHYSICS
results of the tronsltlons to the final product states of rotntlonai angular momenta conststently sho\%ed the sideways scattermg Backward scattering was predrcted for the transitton from il, = 0 and Ja = 0 to “,, = I and lb = 0 only below a co!frslon (inctdent) energy of *3.50 kcd/mole (=O 15 eV) On the other hand. we elaarnmed the translt1on from rt, = 0 and/, = 0 to i$, = 3 and& = 0 up I0 a colhsion energy of 13 SS kcal/ mole (0 6 eV), and our DWBA results consistently showed backward scnttermg InterestIngI;,, at a colhslon energy of 3.17 kcai/mofe (x0 13 e\l), Sparks et al [ 12 1 observed no sideways scattenng for the rextlve transitton from the low rovlbrxlonal reactant molecufe states of 13~ = 0 and Lowe, to all the rotatlonai states of erther >zb = I or I?~ = 3 product molecule HF Fuifer details wgLI be reported eisewhere.
LETTERS
1
April
1981
(4) The monotonously varying structures of angular d~strlbutions m the case of either backward or sideways scattermg are not due to contrlb~t~on solely by smaller orbit& angular momenta. We b&eve that conclusions made above WIU remam
unchanged even wth an exact theory.
Acknowledgement The authors are grateful to a referee partxularly for a comment regarding the rotatlonal states of the reactant mofecuie H, m the crossed molecular beam etrperunent of Sparks et al
References 4 Conclusion In the present DWBA study, we hJve euammed the coherence effect ofpnrtlal waves on anguIar dlstrlbutlons Important findmgs m this study are ( 1) The forward angle scattering strtxture that arises as a result of the coherent summation of an msufficient number of orbital angular mornent~ dlsappears due to added InterfererIce tvlth the partial waves of larger orbital angular momenta (see figs f-4) (2) Osclilatory structure disappears as a consequence of interference effect with added partial waves mvolvmg larger orbital angular momenta (3) Sldeways scattermg for the transltlon from low rovlbratlonal states of H, to alI rotatlonal states of the product molecule HF of the vlbrntlonal quantum number ?zb = 2 IS expected to occur, m general. as a result of the Incoherent sum of each state-to-state stdeways scattcrmg angufar dlstrlbutlon
96
I\’ H R D KT KT S H S H
Ulller, J Chem Phys 49 (1968) 2373 Lebqnc, Israel J Chem 8 (1970) 13 Tang and 51 Knrplus, Phys Rev A3 (1971) 1844 Tang and B H Char, J Chem Phys 62 (197.5) 3612 Such, Phls Rev. Al5 (1977) 1893 Such, Chem Phys Letters 77 (1981) 390 Y Shnn, B H Choi, R T Poe and K T Tang, Chem Ph! s Lerters 57 (1978) 379 C L Vda, D J Zw~ac and J Ross J Chcm Phys 70 (1979) 5362 M J Redmon 2nd R E Wyatt, Chem Phys Letters 63 (1979) 209 R C Wwtt, III Horizons rn quantum chemrstr~, eds 1; Fuhu! and B Pullman (Retdcl, Dordrecht, 1980) p 63 .-I Knfrt, Y ShImon!, R D Levme and S Alexander, Chem Phys 13 (1976) 323 R K Sparks, C C Ha> den, Ii Shobnttlhe, D hl Ne\\mmnrh and Y T Lee, ia Horizons WIquantum chemistry, eds K rukul and B Pullman (Reldel, Dordrecht, 1980) p 91 J T hluckerman, J Chem Phys 56 (1972) 2997,57 (1972) 3388