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An experimental estimate of Rydberg-valence mixing in conjugated dienes
Volume 188, number 5,6
CHEMICAL PHYSICS LETTERS
17 January 1992
An experimental estimate of Rydberg-valence mixing in conjugated dienes Ruth McDiarmid Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, MD 20892, USA
Received 16 September 1991 ; in final form 15 October 1991
The displacements of the B2 3p Rydberg state of cyclopentadiene and the assumed Bu 3p Rydberg state of butadiene from their "unperturbed" values have been experimentally determined. These experimental energies and energy differences have been used to estimate the extent of B~ (B2) Rydberg state-valence state mixing in butadiene and cyclopentadiene. Far more extensive mixing was found in butadiene than in cyclopentadiene. This result supports conclusions drawn from ab initio calculations of these states. The extensive Rydberg state-valence state mixing found in butadiene has bearing on previous interpretations of the electron energy loss spectrum of this molecule.
N u m e r o u s ab initio calculations have been carried out on the electronic excited states o f b u t a d i e n e [ 19]. Since the first o f these calculations, the lowest Bu state of s-trans- 1,3-butadiene, henceforth butadiene, has always been found to be R y d b e r g or semiRydberg in character. This result appears to conflict with the experimentally characterized valence nature o f the lowest energy Bu state o f b u t a d i e n e [ 10 ]. The fewer calculations carried out on the analogous B2 state o f s - c i s - l , 3 - b u t a d i e n e have f o u n d this state to be more compact, thus more valence-like, than in the s-trans conformer o f b u t a d i e n e [ 7,11 ]. Here we report an e x p e r i m e n t a l d e t e r m i n a t i o n o f the extent o f R y d b e r g - v a l e n c e mixing in b u t a d i e n e and cyclopentadiene, a physically stable s-cis diene. All three 3p Rydberg states o f cyclopentadiene have recently been located a n d experimentally assigned [ 12,13 ]. Previously, when only the two B species 3p Rydberg states o f cyclopentadiene had been located, the energies o f these states were c o m p a r e d with those o f the two observed 3p Rydberg states o f butadiene. The large displacement o f the upper 3p Rydberg state o f b u t a d i e n e from the lower a n d from those o f cyclopentadiene was explained by assigning the u p p e r 3p Rydberg state o f b u t a d i e n e as the Bu 3p R y d b e r g state and assuming it to be displaced by interaction with the nearby Bu valence excited state [ 12 ]. How-
ever, it was not possible to estimate the magnitude o f this interaction. N o w that the energies o f all three 3p R y d b e r g states o f cyclopentadiene are known, the energy manifolds o f this molecule and b u t a d i e n e can be c o m p a r e d with that o f benzene, a similar molecule to cyclopentadiene. Benzene does not have adj a c e n t valence states o f the same symmetries as the 3p Rydberg states so should not have the Rydberg s t a t e - v a l e n c e state mixing that is assumed to be present in the dienes. A c o m p a r i s o n o f their 3p energy manifolds will, therefore, p r o v i d e a means o f experimentally estimating the R y d b e r g s t a t e - v a lence state i n t e r a c t i o n - i n d u c e d d i s p l a c e m e n t s o f the Rydberg states o f the dienes. The transition energies from the ground to the 3p Rydberg states o f benzene, cyclopentadiene, and butadiene are presented in fig. 1 as term values, I P - t r a n s i t i o n energies. Term values are used in preference to transition energies because they are ind e p e n d e n t o f the ground state energy levels, which differ widely between molecules. In benzene the two excited states in which in-plane 3p-Rydberg orbitals are occupied belong to the E2u s y m m e t r y species, hence are degenerate. They lie slightly higher in energy (at lower term value) than the third, the Azu state. Because there are no nearby states o f A2u symmetry, the latter is assumed to be relatively unper-
0009-2614/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
423
Volume 188, number 5,6
CHEMICALPHYSICS LETTERS Bu
2.00 -
17 January 1992
3p Rydberg and valence states, R and v. Thus, ~/JR ~---CROR "~CvR~v ,
~uv=CRv~R +C,,Ov,
>~ &.l 23
.~ 2 . 2 0 -
E _
E2o
A2u'
12/ A2
~B~
2.40 Benzene
Cyclopentadiene
Au Butadiene
Fig. 1. Comparisonof the term values (IP- transition energy) of the 3p Rydberg~,Xtransitions of benzene, cyclopentadiene,and butadiene. (Benzene [14]; cyclopentadiene [11,12,15]; butadiene [16].) turbed. In cyclopentadiene the excited states in which the in-plane 3p Rydberg orbitals are occupied are almost degenerate. In addition, they have essentially the same term values as does the Azu 3p Rydberg state of benzene. From these observations we infer that the in-plane 3p Rydberg states of cyclopentadiene interact only slightly with the other states of the molecule and deduce that the 2.32 eV term value observed for these states and the A2u state of benzene is a good estimate of the unperturbed term value of a 3p Rydberg state in a planar, conjugated molecule. The third 3p Rydberg state in cyclopentadiene, the state in which an out-of-plane 3p-Rydberg orbital is occupied, is displaced slightly to lower term value (higher relative energy) than the "unperturbed" term value. This displacement probably reflects an interaction between the valence B 2 state and the B2 3p Rydberg state in cyclopentadiene, as between the analogous states of butadiene. As before, the upper 3p Rydberg state of butadiene is strongly displaced from its "unperturbed" position. The lower is only slightly displaced. The transition energies from the ground to the valence and 3p Rydberg states of butadiene and cyclopentadiene and their displacements from their unperturbed values can be used to experimentally estimate the magnitudes of the Rydberg-valence state interactions in these molecules. For simplicity we assume a two configuration description of the B2 (Bu) 424
where v and R are the valence and Rydberg states, respectively. This, the assumption that the upper 3p Rydberg state of butadiene is the Bu state, and the validity of our conclusion that 2.32 eV is the appropriate term value for an unperturbed 3p Rydberg state in a planar conjugated molecule are the only assumptions made in this analysis. Equations for the energies of the Rydberg and valence states are obtained by applying the Schrfdinget equation to the above wavefunctions. This generates two 2 × 2 secular equations, one for each experimental energy Ev or ER. The determinants of these equations are E ° -E HvR
HvR E ° -E
"
E are the state energies, one for each state, Ev° and E ° are the unperturbed energies of the valence and Rydberg state, respectively, and HvR is the R, v coupling constant. Experimental values exist for all terms in these determinants except HvR for butadiene and cyclopentadiene. Ev and ER are the experimental transition energies. E ° is obtained by subtracting 2.32 eV, the experimental term value of an unperturbed 3p Rydberg state, from the first ionization potential of each molecule of interest. E °, in this two state model, is obtained by adding E R - E ° to the observed vertical valence state energy, Ev. (The coupling constant is dependent on nuclear configuration. The vertical transition energy is the one energy in each state at which the molecule is known to have the same nuclear configuration.) These experimental values are presented in table 1. The values of the coupling constants obtained by diagonalizing the secular determinants and inserting these energies are presented in the 6th line of table 1. Finally, the percent "foreign" character in each wavefunction R or v, calculated from appropriate secular equation using the experimentally derived coupling constants and the experimental energies, are also given in table 1. We see, table 1, that the experimentally deduced Rydberg-valence coupling constant is approxi-
Volume 188, number 5,6
CHEMICAL PHYSICS LETTERS
Table 1 Experimental values of the energies and displacements of the B2 3p Rydberg and valence states of cyclopentadiene and the Bu 3p Rydberg and valence states ofbutadiene a~ Parameter
Butadiene Cyclopentadiene
3p Rydberg state displacement observed Rydberg energy b) EOR observed valence energy b) E o c~ HRv % second configuration
0.33 7.07 6.74 5.92 6.25 0.52 29%
0.06 6,31 6.25 5.27 5.33 0.24 6%
a) Energies in eV. References for data given in the caption to fig. 1. b) Vertical energies. c~ Eo___Ev+(E,_Eo).
mately twice as large in b u t a d i e n e as in cyclopentadiene. The u n p e r t u r b e d energy difference is approximately twice as great in cyclopentadiene as in butadiene. Thus the resultant R y d b e r g c o m p o n e n t of the valence state or valence c o m p o n e n t o f the Rydberg state in this two state m o d e l is approximately five times as large in b u t a d i e n e as in cyclopentadiene. These results can be c o m p a r e d with those obtained in an extensive ab initio calculation on several low-lying states o f cis- and t r a n s - b u t a d i e n e [ 7 ]. The conclusions o f this calculation were that the 1B2 state o f cis-butadiene "... is o f the "classical" n-on* type, with the exception that the n* orbital is somewhat more diffuse ..." whereas the 1Bo state o f transb u t a d i e n e is found "... to be o f m i x e d v a l e n c e Rydberg character ...". In a different calculation on trans-butadiene, the 3p c o n t r i b u t i o n to the valence state was found to be 17% [ 17 ]. Although the twostate a p p r o x i m a t i o n used here is far less c o m p l e t e than the multireference configuration interaction description used in the calculations, the conclusions o f the calculations are fully s u p p o r t e d by the experimental estimates o f R y d b e r g - v a l e n c e mixing in cyclopentadiene and b u t a d i e n e r e p o r t e d here. The much greater R y d b e r g - v a l e n c e mixing calculated for the trans isomer than for the cis is not an artifact o f the calculation but a correct representation o f the physical situation. The conclusion that the Bu 3p R y d b e r g state o f b u tadiene contains a significant valence c o m p o n e n t has
17 January 1992
implications for our previous interpretation o f the electron impact spectrum o f b u t a d i e n e [ 18 ]. There we interpreted the experimental p a r a m e t e r dependence o f the electron scattering spectrum o f the 7 eV region o f the spectrum to indicate the presence o f a valence transition and suggested that it m a y be the transition from the ground to the 2Ag state o f butadiene. The results presented in the current investigation suggest that the " v a l e n c e " character previously d e d u c e d in the 7 eV electron energy loss b a n d is the valence c o m p o n e n t of the Bu 3p Rydberg,--X transition and does not constitute evidence for the 2 A g , - l A g transition in butadiene. In s u m m a r y , based on the observed transition energies to the 3p Rydberg states o f benzene, cyclopentadiene, and b u t a d i e n e and an assumed two-state m o d e l we have been able to experimentally estimate the extent o f R y d b e r g - v a l e n c e mixing in the lowest two B2 states o f cyclopentadiene and Bu states of butadiene. Our results, that there is extensive mixing in the trans conformer but little in the cis conformer, fully support the more extensive ab initio descriptions o f these states. The a u t h o r thanks Dr. W. Siebrand for helpful c o m m e n t s and suggestions.
References [ 1] R.J. Buenker and J.L. Whitten, J. Chem. Phys, 49 ( 1968 ) 5381. [2] R.P. Hosteny, T.H. Dunning Jr., R.R. Gilman, A. Pipano and I. Shavitt, J. Chem. Phys. 62 (1975) 4764. [3] S. Shih, R.J. Buenker and S.D. Peyerimhoff, Chem. Phys. Letters 16 (1972) 244. [4] R.J. Buenker, S. Shih and S.D. Peyerimhoff, Chem. Phys. Letters 44 (1976) 385. [5] M.A.C. Nascimento and W.A. Goddard III, Chem. Phys. 36 (1979) 147. [6] M.A.C. Nascimento and W.A. Goddard III, Chem. Phys. 53 (1980) 251. [7] R.J. Cave and E.R. Davidson, J. Phys. Chem. 91 (1987) 4481. [8] O. Kitao and H. Nakatsuji, Chem. Phys. Letters 143 (1988) 528. [9] P.G. Szalay, A. Karpfen and H. Lischka, Chem. Phys. 130 (1989) 219. [ 10] E.P. Carr, L.W. Pickett and H. Stiicklen, Rev. Mod. Phys. 14 (1942) 260; R.S. Mulliken, Rev. Mod. Phys. 14 (1942) 265. 425
Volume 188, number 5,6
CHEMICAL PHYSICS LETTERS
[ 11 ] P.G. Szalay, A. Karpfen and H. Lischka, Chem. Phys. 141 (1990) 355. [12] R. McDiarmid and A. Sabljic, J. Phys. Chem. 95 (1991) 6455. [ 13] A. Sablji6, R. McDiarmid and A. Gedanken, in preparation. [ 14] P.M. Johnson and G.M. Korenowski, Chem. Phys. Letters 97 (1983) 53.
426
17 January 1992
[15]A. Sablji6 and R. McDiarmid, J. Chem. Phys. 93 (1990) 3850. [16]R. McDiarmid, J. Chem. Phys. 64 (1976) 514, and references therein. [ 17 ] R.L. Graham and K.F. Freed, J. Chem, Phys., in press. [ 18 ] J.P. Doering and R. McDiarmid, J. Chem. Phys. 73 ( 1980 ) 3617;75 (1981) 2477.
CHEMICAL PHYSICS LETTERS
17 January 1992
An experimental estimate of Rydberg-valence mixing in conjugated dienes Ruth McDiarmid Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, MD 20892, USA
Received 16 September 1991 ; in final form 15 October 1991
The displacements of the B2 3p Rydberg state of cyclopentadiene and the assumed Bu 3p Rydberg state of butadiene from their "unperturbed" values have been experimentally determined. These experimental energies and energy differences have been used to estimate the extent of B~ (B2) Rydberg state-valence state mixing in butadiene and cyclopentadiene. Far more extensive mixing was found in butadiene than in cyclopentadiene. This result supports conclusions drawn from ab initio calculations of these states. The extensive Rydberg state-valence state mixing found in butadiene has bearing on previous interpretations of the electron energy loss spectrum of this molecule.
N u m e r o u s ab initio calculations have been carried out on the electronic excited states o f b u t a d i e n e [ 19]. Since the first o f these calculations, the lowest Bu state of s-trans- 1,3-butadiene, henceforth butadiene, has always been found to be R y d b e r g or semiRydberg in character. This result appears to conflict with the experimentally characterized valence nature o f the lowest energy Bu state o f b u t a d i e n e [ 10 ]. The fewer calculations carried out on the analogous B2 state o f s - c i s - l , 3 - b u t a d i e n e have f o u n d this state to be more compact, thus more valence-like, than in the s-trans conformer o f b u t a d i e n e [ 7,11 ]. Here we report an e x p e r i m e n t a l d e t e r m i n a t i o n o f the extent o f R y d b e r g - v a l e n c e mixing in b u t a d i e n e and cyclopentadiene, a physically stable s-cis diene. All three 3p Rydberg states o f cyclopentadiene have recently been located a n d experimentally assigned [ 12,13 ]. Previously, when only the two B species 3p Rydberg states o f cyclopentadiene had been located, the energies o f these states were c o m p a r e d with those o f the two observed 3p Rydberg states o f butadiene. The large displacement o f the upper 3p Rydberg state o f b u t a d i e n e from the lower a n d from those o f cyclopentadiene was explained by assigning the u p p e r 3p Rydberg state o f b u t a d i e n e as the Bu 3p R y d b e r g state and assuming it to be displaced by interaction with the nearby Bu valence excited state [ 12 ]. How-
ever, it was not possible to estimate the magnitude o f this interaction. N o w that the energies o f all three 3p R y d b e r g states o f cyclopentadiene are known, the energy manifolds o f this molecule and b u t a d i e n e can be c o m p a r e d with that o f benzene, a similar molecule to cyclopentadiene. Benzene does not have adj a c e n t valence states o f the same symmetries as the 3p Rydberg states so should not have the Rydberg s t a t e - v a l e n c e state mixing that is assumed to be present in the dienes. A c o m p a r i s o n o f their 3p energy manifolds will, therefore, p r o v i d e a means o f experimentally estimating the R y d b e r g s t a t e - v a lence state i n t e r a c t i o n - i n d u c e d d i s p l a c e m e n t s o f the Rydberg states o f the dienes. The transition energies from the ground to the 3p Rydberg states o f benzene, cyclopentadiene, and butadiene are presented in fig. 1 as term values, I P - t r a n s i t i o n energies. Term values are used in preference to transition energies because they are ind e p e n d e n t o f the ground state energy levels, which differ widely between molecules. In benzene the two excited states in which in-plane 3p-Rydberg orbitals are occupied belong to the E2u s y m m e t r y species, hence are degenerate. They lie slightly higher in energy (at lower term value) than the third, the Azu state. Because there are no nearby states o f A2u symmetry, the latter is assumed to be relatively unper-
0009-2614/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
423
Volume 188, number 5,6
CHEMICALPHYSICS LETTERS Bu
2.00 -
17 January 1992
3p Rydberg and valence states, R and v. Thus, ~/JR ~---CROR "~CvR~v ,
~uv=CRv~R +C,,Ov,
>~ &.l 23
.~ 2 . 2 0 -
E _
E2o
A2u'
12/ A2
~B~
2.40 Benzene
Cyclopentadiene
Au Butadiene
Fig. 1. Comparisonof the term values (IP- transition energy) of the 3p Rydberg~,Xtransitions of benzene, cyclopentadiene,and butadiene. (Benzene [14]; cyclopentadiene [11,12,15]; butadiene [16].) turbed. In cyclopentadiene the excited states in which the in-plane 3p Rydberg orbitals are occupied are almost degenerate. In addition, they have essentially the same term values as does the Azu 3p Rydberg state of benzene. From these observations we infer that the in-plane 3p Rydberg states of cyclopentadiene interact only slightly with the other states of the molecule and deduce that the 2.32 eV term value observed for these states and the A2u state of benzene is a good estimate of the unperturbed term value of a 3p Rydberg state in a planar, conjugated molecule. The third 3p Rydberg state in cyclopentadiene, the state in which an out-of-plane 3p-Rydberg orbital is occupied, is displaced slightly to lower term value (higher relative energy) than the "unperturbed" term value. This displacement probably reflects an interaction between the valence B 2 state and the B2 3p Rydberg state in cyclopentadiene, as between the analogous states of butadiene. As before, the upper 3p Rydberg state of butadiene is strongly displaced from its "unperturbed" position. The lower is only slightly displaced. The transition energies from the ground to the valence and 3p Rydberg states of butadiene and cyclopentadiene and their displacements from their unperturbed values can be used to experimentally estimate the magnitudes of the Rydberg-valence state interactions in these molecules. For simplicity we assume a two configuration description of the B2 (Bu) 424
where v and R are the valence and Rydberg states, respectively. This, the assumption that the upper 3p Rydberg state of butadiene is the Bu state, and the validity of our conclusion that 2.32 eV is the appropriate term value for an unperturbed 3p Rydberg state in a planar conjugated molecule are the only assumptions made in this analysis. Equations for the energies of the Rydberg and valence states are obtained by applying the Schrfdinget equation to the above wavefunctions. This generates two 2 × 2 secular equations, one for each experimental energy Ev or ER. The determinants of these equations are E ° -E HvR
HvR E ° -E
"
E are the state energies, one for each state, Ev° and E ° are the unperturbed energies of the valence and Rydberg state, respectively, and HvR is the R, v coupling constant. Experimental values exist for all terms in these determinants except HvR for butadiene and cyclopentadiene. Ev and ER are the experimental transition energies. E ° is obtained by subtracting 2.32 eV, the experimental term value of an unperturbed 3p Rydberg state, from the first ionization potential of each molecule of interest. E °, in this two state model, is obtained by adding E R - E ° to the observed vertical valence state energy, Ev. (The coupling constant is dependent on nuclear configuration. The vertical transition energy is the one energy in each state at which the molecule is known to have the same nuclear configuration.) These experimental values are presented in table 1. The values of the coupling constants obtained by diagonalizing the secular determinants and inserting these energies are presented in the 6th line of table 1. Finally, the percent "foreign" character in each wavefunction R or v, calculated from appropriate secular equation using the experimentally derived coupling constants and the experimental energies, are also given in table 1. We see, table 1, that the experimentally deduced Rydberg-valence coupling constant is approxi-
Volume 188, number 5,6
CHEMICAL PHYSICS LETTERS
Table 1 Experimental values of the energies and displacements of the B2 3p Rydberg and valence states of cyclopentadiene and the Bu 3p Rydberg and valence states ofbutadiene a~ Parameter
Butadiene Cyclopentadiene
3p Rydberg state displacement observed Rydberg energy b) EOR observed valence energy b) E o c~ HRv % second configuration
0.33 7.07 6.74 5.92 6.25 0.52 29%
0.06 6,31 6.25 5.27 5.33 0.24 6%
a) Energies in eV. References for data given in the caption to fig. 1. b) Vertical energies. c~ Eo___Ev+(E,_Eo).
mately twice as large in b u t a d i e n e as in cyclopentadiene. The u n p e r t u r b e d energy difference is approximately twice as great in cyclopentadiene as in butadiene. Thus the resultant R y d b e r g c o m p o n e n t of the valence state or valence c o m p o n e n t o f the Rydberg state in this two state m o d e l is approximately five times as large in b u t a d i e n e as in cyclopentadiene. These results can be c o m p a r e d with those obtained in an extensive ab initio calculation on several low-lying states o f cis- and t r a n s - b u t a d i e n e [ 7 ]. The conclusions o f this calculation were that the 1B2 state o f cis-butadiene "... is o f the "classical" n-on* type, with the exception that the n* orbital is somewhat more diffuse ..." whereas the 1Bo state o f transb u t a d i e n e is found "... to be o f m i x e d v a l e n c e Rydberg character ...". In a different calculation on trans-butadiene, the 3p c o n t r i b u t i o n to the valence state was found to be 17% [ 17 ]. Although the twostate a p p r o x i m a t i o n used here is far less c o m p l e t e than the multireference configuration interaction description used in the calculations, the conclusions o f the calculations are fully s u p p o r t e d by the experimental estimates o f R y d b e r g - v a l e n c e mixing in cyclopentadiene and b u t a d i e n e r e p o r t e d here. The much greater R y d b e r g - v a l e n c e mixing calculated for the trans isomer than for the cis is not an artifact o f the calculation but a correct representation o f the physical situation. The conclusion that the Bu 3p R y d b e r g state o f b u tadiene contains a significant valence c o m p o n e n t has
17 January 1992
implications for our previous interpretation o f the electron impact spectrum o f b u t a d i e n e [ 18 ]. There we interpreted the experimental p a r a m e t e r dependence o f the electron scattering spectrum o f the 7 eV region o f the spectrum to indicate the presence o f a valence transition and suggested that it m a y be the transition from the ground to the 2Ag state o f butadiene. The results presented in the current investigation suggest that the " v a l e n c e " character previously d e d u c e d in the 7 eV electron energy loss b a n d is the valence c o m p o n e n t of the Bu 3p Rydberg,--X transition and does not constitute evidence for the 2 A g , - l A g transition in butadiene. In s u m m a r y , based on the observed transition energies to the 3p Rydberg states o f benzene, cyclopentadiene, and b u t a d i e n e and an assumed two-state m o d e l we have been able to experimentally estimate the extent o f R y d b e r g - v a l e n c e mixing in the lowest two B2 states o f cyclopentadiene and Bu states of butadiene. Our results, that there is extensive mixing in the trans conformer but little in the cis conformer, fully support the more extensive ab initio descriptions o f these states. The a u t h o r thanks Dr. W. Siebrand for helpful c o m m e n t s and suggestions.
References [ 1] R.J. Buenker and J.L. Whitten, J. Chem. Phys, 49 ( 1968 ) 5381. [2] R.P. Hosteny, T.H. Dunning Jr., R.R. Gilman, A. Pipano and I. Shavitt, J. Chem. Phys. 62 (1975) 4764. [3] S. Shih, R.J. Buenker and S.D. Peyerimhoff, Chem. Phys. Letters 16 (1972) 244. [4] R.J. Buenker, S. Shih and S.D. Peyerimhoff, Chem. Phys. Letters 44 (1976) 385. [5] M.A.C. Nascimento and W.A. Goddard III, Chem. Phys. 36 (1979) 147. [6] M.A.C. Nascimento and W.A. Goddard III, Chem. Phys. 53 (1980) 251. [7] R.J. Cave and E.R. Davidson, J. Phys. Chem. 91 (1987) 4481. [8] O. Kitao and H. Nakatsuji, Chem. Phys. Letters 143 (1988) 528. [9] P.G. Szalay, A. Karpfen and H. Lischka, Chem. Phys. 130 (1989) 219. [ 10] E.P. Carr, L.W. Pickett and H. Stiicklen, Rev. Mod. Phys. 14 (1942) 260; R.S. Mulliken, Rev. Mod. Phys. 14 (1942) 265. 425
Volume 188, number 5,6
CHEMICAL PHYSICS LETTERS
[ 11 ] P.G. Szalay, A. Karpfen and H. Lischka, Chem. Phys. 141 (1990) 355. [12] R. McDiarmid and A. Sabljic, J. Phys. Chem. 95 (1991) 6455. [ 13] A. Sablji6, R. McDiarmid and A. Gedanken, in preparation. [ 14] P.M. Johnson and G.M. Korenowski, Chem. Phys. Letters 97 (1983) 53.
426
17 January 1992
[15]A. Sablji6 and R. McDiarmid, J. Chem. Phys. 93 (1990) 3850. [16]R. McDiarmid, J. Chem. Phys. 64 (1976) 514, and references therein. [ 17 ] R.L. Graham and K.F. Freed, J. Chem, Phys., in press. [ 18 ] J.P. Doering and R. McDiarmid, J. Chem. Phys. 73 ( 1980 ) 3617;75 (1981) 2477.