1,3-Cyclopentanediyl diradicals: Substituent and temperature dependence of triplet - Singlet intersystem crossing

Article abstract of DOI:10.1021/ja991362d

The lifetimes of 33 1,3-diaryl-1,3-cyclopentanediyl triplet diradicals were determined by laser flash photolysis of the corresponding azoalkane precursors. The first-order decay rate constants range from 0.8 to 16.7 × 10⁵ s^-1 in degassed solution at 294 K and exhibit a systematic, but nonlinear dependence on Brown's σ⁺ substituent constants. Analysis of spin-orbit coupling in terms of the two-electrons-in-two-orbitals model indicates that the electronic effect on the rates of intersystem crossing operates by influencing the weight of ionic contributions in the lowest singlet state wave functions of the diradicals. Counter to intuition, but in agreement with the prediction by the model, push-pull substitution does not enhance the ionic contribution. Arrhenius parameters, E_a = 2-6 kcal mol^-1 and A = 10⁷-10¹⁰ s^-1, were determined from the temperature dependences of the decay rate constants. A sensitive statistical test is used to establish that the Arrhenius parameters exhibit enthalpy-entropy compensation, that is, an approximate isokinetic relationship.

Full text of DOI:10.1021/ja991362d

J. Am. Chem. Soc. 1999, 121, 9265-9275
9265
1,3-Cyclopentanediyl Diradicals: Substituent and Temperature
Dependence of Triplet-Singlet Intersystem Crossing
Fumio Kita,† Waldemar Adam,*,† Paul Jordan,§ Werner M. Nau,‡ and Jakob Wirz*,‡
Contribution from the Institut fu¨r Organische Chemie der UniVersita¨t Wu¨rzburg, Am Hubland,
D-97074 Wu¨rzburg, Germany, and the Institut fu¨r Physikalische Chemie and the
UniVersita¨tsrechenzentrum, UniVersita¨t Basel, Klingelbergstrasse 80, CH-4056 Basel, Switzerland
ReceiVed April 26, 1999. ReVised Manuscript ReceiVed August 5, 1999
Abstract: The lifetimes of 33 1,3-diaryl-1,3-cyclopentanediyl triplet diradicals were determined by laser flash
photolysis of the corresponding azoalkane precursors. The first-order decay rate constants range from 0.8 to
16.7 × 105 s-1 in degassed solution at 294 K and exhibit a systematic, but nonlinear dependence on Brown’s
σ+ substituent constants. Analysis of spin-orbit coupling in terms of the two-electrons-in-two-orbitals model
indicates that the electronic effect on the rates of intersystem crossing operates by influencing the weight of
ionic contributions in the lowest singlet state wave functions of the diradicals. Counter to intuition, but in
agreement with the prediction by the model, push-pull substitution does not enhance the ionic contribution.
Arrhenius parameters, Ea ) 2-6 kcal mol-1 and A ) 107-1010 s-1, were determined from the temperature
dependences of the decay rate constants. A sensitive statistical test is used to establish that the Arrhenius
parameters exhibit enthalpy-entropy compensation, that is, an approximate isokinetic relationship.
Introduction
diradicals connected by extended chains (n J 7).9 The present
study deals with substituent effects on the lifetimes of 1,3-
diradicals, for which SOC is the dominant mechanism inducing
ISC.
Diradicals are ephemeral because their bifunctionality gener-
ally entails intramolecular paths for stabilization. For triplet
diradicals these reactions are, however, impeded by a “spin
barrier”, as intersystem crossing (ISC) to the singlet state is
required. This step usually determines the lifetime of triplet
diradicals, kISC ) τ-ob1sd. Spin-orbit coupling (SOC)1-7 and
hyperfine coupling8 are primarily responsible for ISC in
diradicals. Hyperfine coupling becomes important for 1,n-
Lifetimes of numerous triplet diradicals have been deter-
mined.10 Nevertheless, a comparison with theoretical predictions
for substituent effects is difficult, because calculated ISC rates
depend both on electronic and on geometrical factors such as
the relative orientation of the radical centers and of the tether.2
For flexible systems, SOC matrix elements need to be evaluated
for a distribution of molecular conformations and averaged over
wave functions of internal rotation.11 With the goal to determine
electronic substituent effects in model systems with restricted
conformational freedom, we investigated the diaryl-substituted
1,3-cyclopentanediyls 1 and 2 (Scheme 1), in which the radical
centers are embedded in a five-membered ring.12 In this series
the substituent effects on ISC rate constants follow a remarkably
simple pattern, which confirms predictions derived from a model
of two electrons in two orbitals.3 Moreover, the variation of
the ISC rate constants with temperature reveals the interplay of
substituent effects on SOC and on the thermal barrier for ISC.
† Universita¨t Wu¨rzburg.
‡ Institut fu¨r Physikalische Chemie, Basel.
§ Universita¨tsrechenzentrum, Basel.
(1) (a) McGlynn, S. P.; Azumi, T.; Kinoshita, M. In Molecular
Spectroscopy of the Triplet State; Prentice Hall: Englewood Cliffs, 1969;
Chapter 5. (b) Langhoff, S. R. In Modern Theoretical Chemistry; Schaefer,
H. F., III, Ed.; Plenum: New York, 1977; Vol. 4. (c) Levine, I. N. In
Quantum Chemistry; Allyn and Bacon: Newton, MA, 1983; p 281.
(2) Salem, L.; Rowland, C. Angew. Chem., Int. Ed. Engl. 1972, 11, 92.
(3) (a) Bonacˇic´-Koutecky´, V.; Koutecky´, J.; Michl, J. Angew. Chem.,
Int. Ed. Engl. 1987, 26, 170. (b) Michl, J.; Bonac´ic´-Koutecky´, V. In
Electronic Aspects of Organic Photochemistry; John Wiley & Sons: New
York, 1990; p 169. (c) Michl, J. In Theoretical and Computational Models
for Organic Chemistry; Formosinho, S. J., Czismadia, I. G., Arnaut, L. G.,
Eds.; Kluwer: Dordrecht, The Netherlands, 1991; p 207. (d) Michl, J. J.
Mol. Struct. (THEOCHEM) 1992, 260, 299. (e) Michl, J. J. Am. Chem.
Soc. 1996, 118, 3568.
(4) (a) Zimmermann, H. E.; Kutateladze, A. G.; Maekawa, Y.; Mangette,
J. E. J. Am. Chem. Soc. 1994, 116, 9795. (b) Zimmermann, H. E.;
Kutateladze, A. G. J. Org. Chem. 1995, 60, 6008. (c) Zimmermann, H. E.;
Kutateladze, A. G. J. Am. Chem. Soc. 1996, 118, 249.
(5) (a) Minaev, B. F.; Jonsson, D.; Norman, P.; A° gren, H. Chem. Phys.
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Quantum Chem. 1995, 55, 23.
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35, 2502. (b) Bo¨ckmann, M.; Klessinger, M.; Zerner, M. C. J. Phys. Chem.
1996, 100, 10570. (c) Klessinger, M. In Theoretical and Computational
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Carlacci, L.; Doubleday, C., Jr.; Furlani, T. R.; King, H. F.; McIver, J. W.,
Jr. J. Am. Chem. Soc. 1987, 109, 5323. (c) Caldwell, R. A.; Carlacci, L.;
Doubleday, C. E., Jr.; Furlani, T. R.; King, H. F.; McIver, J., Jr. W. J. Am.
Chem. Soc. 1988, 110, 6901. (d) Havlas, Z.; Downing, J. D.; Michl, J. J.
Phys. Chem. 1998, 102, 5681.
Results
Diradicals 1 and 2 were generated by irradiation of the
corresponding azoalkanes 3 and 4, which were synthesized by
(8) (a) De Kanter, F. F. J.; Den, Hollander, J. A.; Huizerm, A. H.;
Kaptein, R. Mol. Phys. 1977, 34, 857. (b) De Kanter, F. J. J.; Kaptein, R.
J. Am. Chem. Soc. 1982, 104, 4759. (c) Turro, N. J.; Kra¨utler, B. Acc. Chem.
Res. 1980, 13, 369.
(9) Doubleday, C., Jr.; Turro, N. J.; Wang, J.-F. Acc. Chem. Res. 1989,
22, 199.
(10) (a) Caldwell, R. A. Pure Appl. Chem. 1984, 56, 1167. (b) Johnston,
L. J.; Scaiano, J. C. Chem. ReV. 1989, 89, 521. (c) Adam, W.; Grabowski,
S.; Wilson, R. M. Acc. Chem. Res. 1990, 23, 165. (d) Caldwell, R. A. In
Kinetics and Spectroscopy of Carbenes and Biradicals; Platz, M. S., Ed.;
Plenum: New York, 1990; p 117. (e) Ichinose, N.; Mizuno, K.; Otsuji, Y.;
Caldwell, R. A.; Helms, A. M. J. Org. Chem. 1998, 63, 3176.
(11) Morita, A.; Kato, S. J. Phys. Chem. 1993, 97, 3298.
(12) Preliminary publications: (a) Adam, W.; Fro¨hlich, L.; Nau, W. M.;
Wirz, J. J. Am. Chem. Soc. 1993, 115, 9824. (b) Kita, F.; Nau, W. M.;
Adam, W.; Wirz, J. J. Am. Chem. Soc. 1995, 117, 8670.
10.1021/ja991362d CCC: $18.00 © 1999 American Chemical Society
Published on Web 09/25/1999

9266 J. Am. Chem. Soc., Vol. 121, No. 40, 1999
Kita et al.
Scheme 1
The triplet diradicals were identified by their ESR spectra,13d,18
which persist for hours in rigid glassy media at 77 K. (iv) The
diradical lifetimes are not affected by the addition of the triplet
quencher trans-piperylene in up to 0.1 M concentration. (v)
Scavenging of the triplet diradicals by molecular oxygen,
presumably to afford the corresponding peroxide trapping
products,12a occurs with rate constants in the range 0.4-2.7 ×
109 M-1 s-1, depending on the substituents X and Y. (vi) The
reaction rates are independent of azoalkane concentration. (vii)
The Arrhenius pre-exponential factors are small, log(A/s-1) )
7-10 (vide infra), which is characteristic for spin-forbidden
processes.
For derivatives with a nitro substituent, the azo n,π* bands
of 3 and 4 overlapped with end absorption by the nitrophenyl
chromophore. Solutions of these compounds with adequate
absorbance at the excitation wavelength of 351 nm (A = 0.4)
were opaque at 320 nm, and the diradicals could not be
monitored at this wavelength. However, transient absorption
spectra of these diradicals revealed strong absorption bands at
∼355 and 580 nm, which are attributed to the p-nitrobenzyl
chromophore. Similar spectra were observed upon irradiation
of azoalkane 3 (X ) Y ) p-NO2) in an EPA glass at 77 K
(Figure 2). Although optical spectra of p-nitrobenzyl radicals
have been previously recorded,19b,e,f the spectral range was
limited to the UV region, such that the strong absorption at 580
nm has apparently escaped detection prior to the present work.
To ascertain this inference, we generated p-nitrobenzyl radicals
independently by 308-nm flash photolysis of the so-called
Barton-PTOC (pyridine-2-thione-N-oxycarbonyl) precursor20 in
acetonitrile and indeed found them to exhibit about equally
strong absoption at 330 and 560 nm, in agreement with the
diradical spectra.
The lifetimes of diradicals 1 and 2 range from 0.6 to 12 µs
in degassed solutions (Table 1). Solvent effects were found to
be of minor importance; for example, the lifetimes of the triplet
diradical 2 (X ) Y ) p-NO2) were essentially the same in
benzene (11.4 ( 0.6 µs) and acetonitrile (12.0 ( 0.5 µs).
Lifetime data for hydroxyl-substituted azoalkanes were deter-
mined in acetonitrile for solubility reasons. The ISC rates of
the symmetrically disubstituted diradicals 1 with a C-C double
bond in the fused five-membered ring were generally higher
by a factor of 1.8 ( 0.1 than those of the C-C saturated
analogues 2 (Table 1).
the general route shown in Scheme 1. Unless previously
documented,13 synthetic details and compound identification are
given as Supporting Information. The azoalkanes exhibit a weak
n,π* transition (ꢀ ∼150 M-1 s-1) around 360 nm, and irradiation
affords the endo housanes 5 and 6 in quantitative yields, except
for the amino- and nitro-substituted azoalkanes, which form
secondary photoproducts upon prolonged irradiation.
Degassed benzene solutions of azoalkanes 3 and 4 were
pumped with a 351-nm excimer laser pulse (25 ns, XeF), and
in most cases the transient absorbance was monitored at 320
nm. Azoalkanes 3 and 4 undergo efficient ISC to the triplet
manifold.14 Therefore, triplet sensitization is not required and
the triplet diradicals 1 and 2 are readily generated by direct flash
photolysis of the azo precursors in solution. Two sequential
processes, the denitrogenation of the triplet azoalkanes and the
decay of the triplet diradicals, are manifested by dual exponential
decay traces in kinetic flash photolysis (Figure 1). Alternatively,
the triplet diradicals 1 and 2 were generated by triplet sensitiza-
tion of the azoalkanes with benzophenone (λexc ) 308 nm, XeCl
excimer laser). The lifetimes of the triplet diradicals determined
in this way agreed (( 5%) with those measured by direct flash
photolysis.
The temperature dependences of the ISC rate constants in
degassed benzene solution were determined in the range 10-
50 °C (Table S1, Supporting Information). The rate constants
(15) (a) Jordan, J. E.; Pratt, D. W.; Wood, D. E. J. Am. Chem. Soc. 1974,
96, 5589. (b) Neta, P.; Behar, D. J. Am. Chem. Soc. 1980, 102, 4798. (c)
Claridge, R. F. C.; Fischer, H. J. Phys. Chem. 1983, 87, 1960. (d) Tokumura,
K.; Nosaka, H.; Ozaki, T. Chem. Phys. Lett. 1990, 169, 321. (e) Tokumura,
K.; Ozaki, T.; Nosaka, H.; Saigusa, Y.; Itoh, M. J. Am. Chem. Soc. 1991,
113, 4974. (f) Mayouf, A. M.; Lemmetyinen, H.; Koskikallio, J. J.
Photochem. Photobiol., A 1993, 70, 215-221.
(16) (a) Brand, U.; Hippler, H.; Lindemann, L.; Troe, J. J. Phys. Chem.
1990, 94, 6305. (b) Boury, V. W.; Ingold, K. U. J. Am. Chem. Soc. 1992,
114, 4992. (c) Faria, J. L.; Steenken, S. J. Phys. Chem. 1992, 96, 10869.
(17) The bands of the diradicals are broad and the band maxima lie at
somewhat shorter wavelengths than the 0-0 transitions of the corresponding
monoradicals.
Evidence for the assignment of the first transient intermediate
to the excited triplet state of the azo compounds 3 or 4 has
been given previously.14b The assignment of the slower decays
to the triplet diradicals 1 or 2 is based on the following
observations: (i) The transient UV absorption spectra are similar
to those of the corresponding benzyl (λmax ) 316 nm, ꢀ ) 8800
cm-1 M-1 15 and cumyl (λmax ) 316 nm)16 radicals.17 (ii)
)
Product studies have shown that the housanes are formed
quantitatively by both direct and sensitized irradiation.13b,d,e (iii)
(18) Adam, W.; Fro¨hlich, L.; Nau, W. M.; Korth, H.-G.; Sustmann, R.
Angew. Chem., Int. Ed. Engl. 1993, 32, 1339.
(13) (a) Beck, K.; Hu¨nig, S. Chem. Ber. 1987, 120, 477. (b) Adam, W.;
Harrer, H. M.; Nau, W. M.; Peters, K. J. Org. Chem. 1994, 59, 3786. (c)
Nau, W. M.; Harrer, H. M.; Adam, W. J. Am. Chem. Soc. 1994, 116, 10972.
(d) Adam, W.; Kita, F.; Harrer, H. M.; Nau, W. M.; Zipf, R. J. Org. Chem.
1996, 61, 7056. (e) Adam. W.; Harrer, H. M.; Kita, F.; Korth, H.-G.; Nau,
W. M. J. Org. Chem. 1997, 62, 1419.
(14) (a) Adam, W.; Nau, W. M.; Sendelbach, J. S. J. Am. Chem. Soc.
1994, 116, 7049. (b) Adam, W.; Fragale, G.; Klapstein, D.; Nau, W. M.;
Wirz, J. J. Am. Chem. Soc. 1995, 117, 12578.
(19) (a) Jordan, J. E.; Pratt, D. W.; Wood, D. E. J. Am. Chem. Soc. 1974,
96, 5589. (b) Neta, P.; Behar, D. J. Am. Chem. Soc. 1980, 102, 4798. (c)
Claridge, R. F. C.; Fischer, H. J. Phys. Chem. 1983, 87, 1960. (d) Tokumura,
K.; Nosaka, H.; Ozaki, T. Chem. Phys. Lett. 1990, 169, 321. (e) Tokumura,
K.; Ozaki, T.; Nosaka, H.; Saigusa, Y.; Itoh, M. J. Am. Chem. Soc. 1991,
113, 4974. (f) Mayouf, A. M.; Lemmetyinen, H.; Koskikallio, J. J.
Photochem. Photobiol., A 1993, 70, 215-221.
(20) Barton, D. H. R.; Ferreira, J. A. Tetrahedron 1996, 52, 9347.

1,3-Cyclopentanediyl Diradicals
J. Am. Chem. Soc., Vol. 121, No. 40, 1999 9267
Figure 1. Transient absorbance changes observed at 320 nm by flash photolysis of a solution of azoalkane 4 (X ) Y ) H) in degassed benzene
at 20.5 °C with a XeF excimer laser (351 nm).
of the Arrhenius equation.
kISC ) A exp(-Ea/RT)
(1)
Discussion
Triplet diradicals 1 and 2 react quantitatively by cyclization
to the housanes 5 and 6 (Scheme 1). This spin-forbidden reaction
requires thermal activation energy as revealed by the temperature
dependence of the rate constants kISC, in particular by their
persistence for hours at 77 K. Accordingly, two factors are
considered to influence the observed triplet lifetimes: first, a
“transmission factor” accounting for the requirement of spin
flip along the reaction coordinate and, second, a small but
significant thermal barrier toward ring closure.21 Following these
considerations, we factorize the Arrhenius equation into two
contributions, a transmission factor ASOC accounting for the spin
barrier and the expression A∆ exp(-Ea/RT) reflecting the thermal
barrier (eq 2). We will first give a simple empirical treatment
Figure 2. Absorption spectra of triplet diradical 1 (X ) Y ) p-NO2);
(a) spectrum in EPA matrix at 77 K (the dotted line shows the
absorption of the azoalkane precursor 3) and (b) transient spectrum
observed with a time delay of 1 µs after nanosecond laser flash
photolysis of 3 (X ) Y ) p-NO2) in degassed benzene solution at
ambient temperature.
kISC ) ASOCA∆ exp(-Ea/RT)
(2)
of the lifetime data by means of substituent parameters. In a
second step, we will demonstrate that the substituent effects on
the lifetimes at room temperature can be adequately described
considering only SOC.22 Finally, we will consider the temper-
ature dependences of the triplet lifetimes that are described by
the remaining, “conventional” parameters of the Arrhenius eq
2, A∆ and Ea.
Empirical Analysis. Electron acceptors (NO2 or CN) reduce
the rate of ISC relative to the parent system (lengthen the triplet
lifetime), while electron donors (MeO or NH2) accelerate ISC
in the triplet diradicals 1 and 2 (shorten their lifetimes). These
electronic substituent effects are general; they hold for substitu-
tion at the para and meta positions, both for the symmetrical
(X ) Y) and nonsymmetrical (X * Y) aryl substitution patterns,
and are largely independent of solvent polarity.
Figure 3. Intersystem-crossing rate constants (kISC) of the triplet
diradicals 1 (b) and 2 (O) versus the sum of the substituent parameters
σ+ of the substituents X and Y.
(21) Adam, W.; Platsch, H.; Wirz, J. J. Am. Chem. Soc. 1989, 111, 6896.
(22) The mechanism responsible for the spin flip in the 1,3-diradicals 1
and 2 is SOC. The lifetimes of several triplet diradicals 1 and 2, which
include the longest- and shortest-lived derivatives and those with heavy
atoms, have been measured at various magnetic field strengths up to 5000
G, but no effect was observed within an error of (5%. These results preclude
any significant contributions from hyperfine coupling, the alternative
mechanism for inducing ISC (ref 8).
obeyed the Arrhenius law (eq 1). On the basis of the experience
that relatiVe errors in the experimental rate constants are
constant, the activation parameters A and Ea given in Table 1
were determined by linear regression of the logarithmic form

9268 J. Am. Chem. Soc., Vol. 121, No. 40, 1999
Kita et al.
Table 1. Triplet Lifetimes (3τ) of the Diradicals 1 and 2 and Calculated Covalent (γAB) and Polar (δAB) Perturbations in the Planar Model
Diradicals 7
substituentsa
e
f
Eac
γAB
δAB
c
diradical
X
Y
3τb (ns)
(kcal mol-1
)
log (A/s-1
)
σ+(X)d
(10-2 eV)
(eV)
1
p-I
p-I
570 ( 10
640 ( 40g
0.14
-0.91
-0.78
0.15
-0.31
-0.07
-0.19
0.35
-0.07
0.12
0.00
0.11
0.12
0.39
0.37
0.43
0.56
0.61
0.66
0.79
-3.43
-6.05
-6.13
-3.74
-4.78
-4.91
-4.91
-3.59
-4.27
-4.07
-4.23
-4.23
-4.22
-3.59
-3.36
-3.07
-3.44
-2.62
-3.31
-2.11
-3.78
-5.77
-5.73
-4.88
-4.66
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
p-OH
p-OMe
p-Br
p-Me
p-F
p-OAc
m-I
m-Me
m-OH
H
p-OH
p-OMe
p-Br
p-Me
p-F
p-OAc
m-I
m-Me
m-OH
H
p-Cl
m-OMe
m-OAc
m-Cl
m-CF3
m-CN
p-CF3
p-CN
p-NO2
p-NO2
p-NH2
p-OMe
H
3.8 ( 0.3
1.7 ( 0.1
3.6 ( 0.1
1.8 ( 0.1
1.9 ( 0.2
3.1 ( 0.2
9.0 ( 0.2
7.5 ( 0.5
8.7 ( 0.1
7.3 ( 0.1
7.4 ( 0.2
8.2 ( 0.2
670 ( 10h
960 ( 10h
1150 ( 30h
1220 ( 50h
1430 ( 60
1440 ( 30
1610 ( 160
1680 ( 220g
1700 ( 40h
1700 ( 20h
1740 ( 200
2460 ( 180
2600 ( 150
2750 ( 430
3000 ( 120
3600 ( 140
5620 ( 150
6150 ( 300h
1840 ( 40i
1320 ( 20
1420 ( 20
1620 ( 50
1940 ( 30
2000 ( 90g
2540 ( 60
2710 ( 90
3020 ( 50
4970 ( 50
5330 ( 50
5280 ( 110g
6030 ( 500
9550 ( 140
11400 ( 600
12000 ( 500g
2.0 ( 0.5
3.0 ( 0.6
1.8 ( 0.2
2.5 ( 0.1
7.3 ( 0.4
8.0 ( 0.5
7.1 ( 0.1
7.6 ( 0.1
p-Cl
m-OMe
m-OAc
m-Cl
3.8 ( 0.4
3.2 ( 0.3
2.9 ( 0.8
3.4 ( 0.2
2.7 ( 0.2
3.5 ( 0.2
5.1 ( 0.1
8.5 ( 0.3
8.0 ( 0.2
7.7 ( 0.6
8.1 ( 0.2
7.5 ( 0.2
7.9 ( 0.1
9.0 ( 0.1
m-CF3
m-CN
p-CF3
p-CN
p-NO2
p-OMe
p-NH2
p-OMe
p-NH2
p-OMe
p-OMe
p-Me
H
0
0
0
0
0.579
0
0
-0.169
-0.096
2
5.2 ( 0.2
4.4 ( 0.2
3.6 ( 0.2
2.8 ( 0.1
9.8 ( 0.1
9.2 ( 0.1
8.5 ( 0.1
7.8 ( 0.1
-1.30
-0.78
-1.30
-0.78
H
H
H
H
H
H
H
H
2.6 ( 0.1
3.6 ( 0.2
3.2 ( 0.1
3.6 ( 0.1
3.6 ( 0.2
7.5 ( 0.1
8.2 ( 0.1
7.9 ( 0.1
8.0 ( 0.1
8.0 ( 0.1
-0.31
0.00
0.11
0.49
0.66
-4.05
-3.83
-3.75
-3.03
-3.25
-0.031
0
0.103
0.260
0.262
p-Cl
p-CO2Me
p-CN
p-CN
p-NO2
p-CN
p-NO2
p-NO2
H
0.79
0.66
0.79
-2.55
-2.91
-1.71
0.476
0
0
p-CN
p-NO2
p-NO2
4.6 ( 0.1
6.1 ( 0.2
8.5 ( 0.1
9.5 ( 0.1
a The triplet diradicals were generated from the corresponding azoalkanes 3 or 4 by flash photolysis with a XeF excimer laser (351 nm). b Mean
value and standard error of samples of about twenty individual measurements in degassed benzene solutions at 20.0 °C. c Mean values and standard
errors of the Arrhenius parameters. Temperature range 10-50 °C, 5-10 measurements at each temperature. d Brown’s substituent parameters from
ref 23. e Covalent perturbation in diradicals 7 calculated by AM1; for meta-disubstituted compounds, the conformation with both substituents on
the side of the methylene bridge was chosen. The values were corrected for the underrated through-space interaction by AM1, cf. footnote 32.
f Polar perturbation in diradicals 7 (AM1); in the monosubstituted diradicals 2 orbital A is localized on the unsubstituted benzyl radical. g Values
measured at 20 °C in degassed acetonitrile. h See also ref 12a. i A ∼1:1 mixture of two regioisomers; see also ref 12b.
These trends are nicely illustrated by a Hammett plot of the
ISC rate constants versus the sum of Brown’s substituent
constants23 σ+ for the two substituents X and Y of the two sets
of diradicals 1 and 2 (Figure 3). The two series clearly follow
a systematic, albeit nonlinear trend.24 Diradicals carrying
bromine or iodine substituents, which are known to enhance
SOC through the so-called internal heavy-atom effect,25 clearly
fall out of line and were omitted from this plot.
The Two-Electrons-In-Two-Orbitals Model for SOC. The
first detailed theoretical analysis of the factors that influence
SOC in homosymmetric diradicals was carried out by Salem
and Rowland.2 These authors found a dependence on (i) the
relative orientation of the orbitals at the radical sites, (ii) the
distance between the radical centers, (iii) the atom types, and
(iv) the electronic structure of the lowest singlet state, namely,
its zwitterionic character. A more elaborate description of the
electronic structure of diradicaloids in terms of the two-
electrons-in-two-orbitals active space was formulated by Michl
and co-workers.3 Their treatment, which is employed here,
covers homosymmetric and nonsymmetric diradicals within the
same framework.
(23) Hantsch, C.; Leo, A.; Taft, R. W. Chem. ReV. 1991, 91, 165.
(24) A correlation between ISC rates and aryl substituent parameters has
been previously described for unsymmetrical Norrish-Type-II diradicals in
nonpolar solvents, generated photochemically from γ-phenylbutyrophenone
derivatives (ref 10a,d). In the polar solvent methanol, the substituents had
essentially no influence on the triplet diradical lifetimes.
(25) (a) Kavarnos, G.; Cole, T.; Scribe, P.; Dalton, J. C.; Turro, N. J. J.
Am. Chem. Soc. 1971, 93, 1033. (b) Cowan, D. O. M.; Drisko, R. L. in
Elements of Organic Photochemistry; Plenum: New York, 1975, 250. (c)
Miller, J. C.; Meek, J. S.; Strickler, S. J. J. Am. Chem. Soc. 1977, 99, 8175.
(d) Mizuno, K.; Ichinose, N.; Otsuji, Y.; Caldwell, R. A. J. Am. Chem.
Soc. 1985, 107, 5797. (e) Zimmt, M. B.; Doubleday, C., Jr.; Turro, N. J.
Chem. Phys. Lett. 1987, 134, 549. (f) Fisher, J. J.; Michl, J. J. Am. Chem.
Soc. 1987, 109, 583. (g) Ermler, W. C.; Ross, R. B.; Christiansen, P. A.
AdV. Quantum Chem. 1988, 19, 139. (h) Caldwell, R. A.; Jacobs, L. D.;
Furlani, T. R.; Nalley, E. A.; Laboy, J. J. Am. Chem. Soc. 1992, 114, 1623.
(i) Khudyakov, I. V.; Serebrennikov, Y. A.; Turro, N. J. Chem. ReV. 1993,
93, 537.
As a basis set, fully localized nonbonding molecular orbitals
A and B are chosen for minimal interorbital and maximal
intraorbital electronic repulsion. If only the through-space
interaction is taken into account, the fully localized orbitals
correspond to the separate radical (benzyl) orbitals. When the
through-bond interaction is also considered, these orbitals spread
over the adjacent σ bonds. The extent of this delocalization is
small, but it has a dominant effect on SOC.3e

1,3-Cyclopentanediyl Diradicals
J. Am. Chem. Soc., Vol. 121, No. 40, 1999 9269
According to the model of two-electrons-in-two-orbitals, the
energies of the three singlet states S0, S1, and S2 are determined
by four independent parameters of the electronic interaction
matrix (eq 3), namely, the exchange integral (KAB), the coulomb
integral function (KA′ B), and the covalent (γAB) and the polar-
izing (δAB) perturbations. The parameter δAB represents the
difference between the basis energies of orbitals A and B (δAB
( δR in Hu¨ckel theory), and the former corresponds to twice
the resonance integral used in semiempirical models (γAB ( 2â,
cf. ref 3 for details).
1
|A2 - B2
2K′ δAB
0
γAB
2KAB
AB
1
|A2 + B2
|AB
AB
δAB
2(K′ + KAB)
(3)
(
)
1
γAB
0
where
and
γAB ) 2 hAB + (AA|AB)* + (BB|BA)
δAB ) hA - hB + (JAA - JBB)/2
K′ ) [(JAA + JBB)/2 - JAB]/2
AB
Diagonalization of the matrix (3) yields the three singlet state
wave functions as linear combinations of three basis wave
Figure 4. Dependence of (a) the coefficient C2+ and (b) the energy of
the three singlet states on the electronic parameters γAB and δAB with
fixed values of KAB ) 0.007 eV and K′AB ) 1.58 eV.
functions |AB , |A2 + B2 and |A2 - B2 , the eigenfunctions
of the “perfect” diradical (δAB ) γAB ) 0). The first represents
the coValent (dot-dot) state, and the latter two describe the
zwitterionic (hole-pair) states. The one-electron SOC Hamilto-
nian mixes only the symmetrized zwitterionic 1|A2 + B2 wave
function with the triplet state 3|AB , and the interaction term is
directly proportional to C+, the coefficient of the wave function
1
1
1
of the interaction matrix, eq 3: the polarizing perturbation δAB
mixes only the two zwitterionic wave functions 1|A2 + B2 and
1
|A2 - B2 and the lowest singlet state S0 is not affected. At
δAB ) δcrit the zwitterionic singlet crosses the covalent singlet
state (lower diagram in Figure 4), and the electronic properties
of the lowest singlet state S0 change abruptly. Note that δcrit
amounts to 3.17 eV and is far beyond the range of δAB covered
by the aryl substituents, which are used in this work (Table 1).
1
|A2 + B2 in the ground state S0 of the singlet diradical: |S0
1
1
1
) C0 |AB + C+ |A2 + B2 + C- |A2 - B2 .3 According to
Fermi’s Golden Rule,3b,6c the transmission factor ASOC of eq 2
should then be proportional to the square of this interaction term,
eq 4.
Direct mixing of |A2 + B2 and |AB is induced only by the
perturbation γAB, and the nonzero slope along the axis γAB
derives from this coupling of the zwitterionic and covalent wave
functions.
1
1
ASOC C2+
(4)
We assume that the two parameters KAB and K′ are
AB
δcrit
)
K′ (K′ - K )
(5)
x
AB AB
AB
constant for the series of diradicals 1 and 2 due to their similar
topology. This is supported by semiempirical calculations (AM1,
see next section), which gave values within KAB ) 0.0074 (
The model discussed above provides a simple, qualitative
rationale for the empirical relation to the substituent constants
σ+ that is displayed in the Hammett plot (Figure 3). The
substituents’ electronegativity, which is represented by the
parameters σ+, influences the nonbonding orbital energies
(ionization energies)26 for the corresponding benzyl-type radical
fragments A and B. The fact that homosymmetric and nonsym-
metric diradicals follow the same trend demonstrates that polar
perturbations δAB, that is, differences in the basis energies of
donor- and acceptor-substituted radical centers, hardly influence
the ISC rate constants, as predicted by the model. The observed
0.0012 eV and K′ ) 1.58 ( 0.03 eV for all geometry-
AB
optimized (planar) model diradicals. The effects of the substit-
uents X and Y are, thus, represented by the covalent (γAB) and
polarizing (δAB) perturbations. The coefficient C+ is obtained
by diagonalization of the interaction matrix, eq 3, for each value
of the parameters γAB and δAB, and the dependence of C2 on
+
these two parameters is presented in Figure 4 along with the
three singlet-state energies, which represent the energy eigen-
values. All energies are defined relative to that of the triplet
state (ET ≡ 0 eV), such that the energy of the lowest singlet
state of the perfect diradical is 2KAB > 0.
trend must, therefore, arise from the covalent perturbation γAB
,
which appears to be related to the sum of the substituent
Consider the slopes of the C2+-surface starting from the
perfect diradical (δAB ) γAB ) 0) in the lower left edge of the
upper diagram of Figure 4. The slope along the δAB axis remains
0 until δAB reaches the value δcrit (eq 5), at which the weight of
the ionic-state contribution rises abruptly to its maximal value,
constants, σA + σB+. This relation is explained by the
+
qualitative orbital interaction diagram shown in Figure 5.
Electron-accepting substituents, for example, NO2, lower the
basis energy of the nonbonding molecular orbitals (NBMO’s)
(26) (a) Sim, B. A.; Milne, P. H.; Griller, D.; Wayner, D. D. M. J. Am.
Chem. Soc. 1990, 112, 6635. (b) Wayner, D. D. M.; Sim, B. A.; Dannenberg,
J. J. J. Org. Chem. 1991, 56, 4853.
2
+
C2+ ) /2; in contrast, the C -surface rises continuously along
1
the γAB axis. This behavior becomes self-evident by inspection

9270 J. Am. Chem. Soc., Vol. 121, No. 40, 1999
Kita et al.
Figure 5. Orbital interaction diagram for the NBMO’s in homosymmetric and nonsymmetric diradicals. The symbols a (antisymmetric) and s
(symmetric) refer to the symmetry properties of the molecular orbitals with respect to the plane of symmetry that is orthogonal to the plane of the
ring in the homosymmetric diradicals; A and B refer to the localized nonbonding molecular orbitals.
and thereby enhance the through-bond interaction of the
symmetric NBMO with the lower-lying pseudo π orbital of the
methylene bridge27,28 because the energy gap between the
interacting orbitals is reduced (second-order perturbation theory).
Conversely, electron-donating substituents, for example, MeO,
tend to decrease the through-bond interaction with the pseudo-π
orbital. Because of the counteracting effect of the through-space
vanishes in planar geometries, the zwitterionic state contribution
C+ is not 0. The wave functions and energies of the three lowest
singlet states at this geometry were then determined by 3 × 3
configuration interaction (CI). The energy gap between the UHF
NBMO’s, ∆ꢀ, corresponds to the covalent perturbation γAB in
homosymmetric diradicals. The eigenvalues resulting from
diagonalization of the lower right 2 × 2 block in the interaction
matrix, eq 3, are well approximated by second-order perturbation
and through-bond interactions, the covalent perturbation γAB
,
which corresponds to the resulting energy gap between the
NBMO’s, increases with decreasing through-bond interaction.
theory, because γAB , 2K′ . Hence, the coefficient C+ in the
AB
perturbed electronic ground state S0 is predicted to be propor-
tional to γAB, eq 6. In nonsymmetric diradicals the NBMO
1
As a result, the weight of the zwitterionic wave function |A2
+ B2 increases with the donating ability of the substituents
and the rate of ISC increases accordingly (eqs 2 and 4).
Molecular Dynamic Aspects of ISC and Calculation of
SOC. The fact that the decay of the triplet diradicals 1 and 2 is
associated with activation energies suggests that ISC requires
some distortion from the equilibrium geometry. Apart from the
weight of the ionic contribution C+ in the lowest singlet state
wave function of the diradicals, the size of the SOC matrix
element depends on the relative orientation of the two radical
centers, and the one-electron coupling terms vanish for the
planar conformation of 1,3-cyclopentanediyl.2 A natural choice
for a reaction coordinate that will facilitate ISC by effective
coupling of the singlet and triplet states is the soft ring puckering
distortion represented by the dihedral angle Θ shown in the
right-hand structure 7 below.
γAB
C+ ≈
(6)
2K′
AB
energy gap ∆ꢀ is no longer equal to γAB (cf. the interaction
diagram, Figure 5), but the coefficient C+ can be determined
by orthogonal transformation of the S0 wave function to the
basis set of the localized NBMO’s A and B, and eq 6 is still
valid.
The values of γAB that were calculated by the procedure
described above are small as a result of the near cancellation
of the through-space and through-bond interactions between the
radical sites (Figure 5). In such a situation, the relative ordering
of the two NBMO energies predicted by a particular model is
largely fortuitous.30 AM1 predicts the symmetric NBMO to lie
(28) The through-bond interaction is due mainly to conjugation through
the pseudo-π orbital of the methylene bridge, which affects only the
symmetric orbital (Figure 5). The ethylene bridge can be ignored in the
orbital-interaction diagram because it influences the symmetric and asym-
metric NBMOs in a similar manner.
(29) Semiempirical: (a) Dewar, M. J. S.; Zoebisch, E. G.; Healy, E. F.;
Stewart, J. J. P. J. Am. Chem. Soc. 1985, 107, 3902. (b) Rauhut, G.;
Chandrasekhar, J.; Alex, A.; Steinke, T.; Clark, T. Vamp, Version 5.0.
University of Erlangen-Nu¨rnberg, 1993. Ab initio: (c) Single-point calcula-
tions were performed to obtain the triplet (ROHF method) and singlet
[CASSCF(2,2)] energies. (d) Frisch, M. J.; Trucks, G. W.; Head-Gordon,
M.; Gill, P. M. W.; Wong, M. W.; Foresman, J. B.; Johnson, B. G.; Schlegel,
H. B.; Robb, M. A.; Replogle, E. S.; Gomperts, R.; Andres, J. L.;
Raghavachari, K.; Binkley, J. S.; Gonzalez, C.; Martin, R. L.; Fox, D. J.;
Defrees, D. J.; Baker, J.; Stewart, J. J. P.; Pople, J. A. Gaussian 92, Revision
F.4. Gaussian, Inc., Pittsburgh, PA, 1992.
Quantitative evaluation of γAB was done for the model
structures 7. Geometries were optimized by UHF calculations
for the triplet state using the semiempirical AM1 method.29a,b
The optimized structures were planar (C2V-symmetry for X )
Y and Cs-symmetry for X * Y, Θ ) 0). Although SOC strength
(27) (a) Hoffmann, R. Acc. Chem. Res. 1971, 4, 1. (b) Paddon-Row:
M. N. Acc. Chem. Res. 1982, 15, 245. (c) Goldberg, A. H.; Dougherty, D.
A. J. Am. Chem. Soc. 1983, 105, 284.
(30) Bieri, G.; Heilbronner, E.; Kloster-Jensen, E.; Schmelzer, A.; Wirz,
J. HelV. Chim. Acta 1974, 57, 1265.

1,3-Cyclopentanediyl Diradicals
J. Am. Chem. Soc., Vol. 121, No. 40, 1999 9271
Figure 7. Parameters γAB, δAB, and C+ calculated for homosymmetric
(O) and nonsymmetric (b) diradicals 7 with the same substitution
pattern as diradicals 2.
function of δAB and γAB, which is actually spanned by the
diradicals studied in this work. The parameters are those
calculated by AM1 for the model systems 7 (Table 1). In the
homosymmetric case (empty circles), the polar perturbation
(δAB) is 0, and the highest zwitterionic-state contribution pertains
to the diradical with two amino groups, the lowest value to the
derivative with two nitro groups. The polar perturbation reaches
substantial values in the donor-acceptor substituted, nonsym-
metric derivatives (filled circles), as seen from the displacement
to the left and right of the δAB ) 0 line in Figure 7. Nevertheless,
SOC depends on the covalent perturbation γAB only.
In contrast to these findings, ISC rates do respond to strong
polarizing perturbations δAB in para-substituted singlet phenyl
nitrenes, as was recently reported by Gritsan et al.33 Remarkably,
both observations are consistent34 with Michl’s model of two
electrons in two orbitals.3
Temperature Dependence and Substituent Effects on the
Arrhenius Parameters. The ISC rate constants of diradicals 1
and 2 obeyed Arrhenius relations (eq 1) in the range of 10-50
°C, and the parameters ln A and Ea were determined by linear
regression of ln(kISC) vs 1/T (Table 1). The substituents X and
Y give rise to substantial variation in these parameters, and the
substituent effects on the two parameters are correlated: large
pre-exponential factors tend to be compensated by large
activation energies (Figure 8a).
Such correlation between the Arrhenius parameters Ea and
ln A, or between the equivalent pair of Eyring parameters ∆H‡
and ∆S‡, is a common phenomenon.35 Enthalpy-entropy
compensation (EEC) (eq 7) and rate-equilibrium linear free
Figure 6. Experimental intersystem-crossing rate constants (kISC) of
diradicals 1 (b) and 2 (O) versus the covalent perturbation (γAB) of
the planar diradicals 7. The derivatives substituted with bromine or
iodine (0) were excluded from the regression analysis, since they exhibit
higher ISC rate constants due to the heavy-atom effect.
slightly above the antisymmetric one in the planar diradicals 7
(“reverse” order), in contrast to the “natural” order shown in
Figure 5a. An ab initio calculation (STO-3G)29c,d for diradical
7 (X ) Y ) H, C2V) gave the natural orbital energy sequence.
The AM1 calculations were then repeated for various fixed
values of the dihedral angle Θ (0°-40°, 5°-intervals). The
through-space overlap of the nonbonding orbitals increases with
increasing values of Θ, and as a result, the natural orbital energy
sequence is restored around Θ ≈ 24° for all model compounds
7. At higher values of Θ the energy gap increases monotonically,
and the ratios between the γAB values obtained for different
substituents remained largely constant. The AM1 method is
known to underestimate through-space interactions systemat-
ically.12a,31 Instead of chosing some arbitrary value Θ > 24°
for our calculations, we scaled the AM1 model by deducing a
constant value ∆γ ≈ 0.11 eV32 from the covalent perturbations
calculated at the planar geometry, γAB ) |γAM1 - ∆γ| (Table
1), thereby restoring the natural orbital sequence for all
compounds.
Effect of Polarizing Substitution. Figure 6 shows that the
rate constants kISC increase linearly with γ2AB, as predicted by
the model of two-electrons-in-two-orbitals (eqs 4 and 6). The
polarizing perturbations δAB appear to have no perceptible
influence: nonsymmetric and even push-pull substituted
diradicals obey the same linear relationship with γ2AB as the
symmetric ones. Even for the strong push-pull substitution
pattern in compound 1 (X ) p-OMe, Y ) p-NO2), the polarizing
perturbation, δAB ) 0.58 eV, is well below the critical value,
δcrit ) 3.17 eV, such that it has no noticeable influence on SOC.
To demonstrate the dominance of the covalent perturbation
on the zwitterionic-state contribution (C+), we redraw in Figure
7 that section of the surface showing the variation of C2+ as a
δREa ) RâδR ln(A/s-1)
energy relationships (LFERs) (eq 8) are related,36 so-called
δR ln(k/s-1) ) RδR∆GQ/RT
(7)
(8)
extra-thermodynamic relationships.37 They are encountered
regularly in series of compounds such as 1 and 2, in which aryl
substituents remote from the reaction center are varied. In eqs
(33) Gritsan, N. P.; Tigelaar, D.; Platz, M. S. J. Phys. Chem. A 1999,
103, 4465.
(34) Referring to the interaction matrix, eq 3, two limiting cases of perfect
diradicals may be distinguished: KAB ) 0 (pair diradicals) and K′ ) KAB
(axial diradicals). Diradicals 1 and 2 approach the first case anAdB have a
large value of δcrit (eq 5), as discussed above. Phenylnitrenes, on the other
(31) Comments by Prof. T. Bally, University of Fribourg, in the
discussion following a paper of Rauhut, G.; Clark, T. J. Chem. Soc., Faraday
Trans. 1994, 90, 1783.
hand, are closer to the second case (in methylnitrene the condition K′
)
AB
KAB is imposed by the three-fold axis of symmetry). Therefore, δ is
crit
(32) The precise values of the corrections ∆γ were determined by
nonlinear least-squares fitting of eq 9 (see below), where γAB is substituted
by |γAM1 - ∆γ|. This procedure gave ∆γ ) 0.115 ( 0.003 eV for diradicals
1 and 0.114 ( 0.003 eV for diradicals 2.
expected to be small, and moderate polar perturbations δAB are expected to
increase the ionic contributions to the ground-state singlet wave function
of nonplanar phenylnitrenes by second-order interactions.
(35) Leffler, J. E. Chem. ReV. 1955, 55, 1202.

9272 J. Am. Chem. Soc., Vol. 121, No. 40, 1999
Kita et al.
8b).38 This suggests that the factor A∆ exp(-Ea/RT), which
reflects thermal activation, obeys EEC, δREa ) RâδR ln(A∆/
s-1).
Before we proceed to an interpretation of these results, we
must pause to establish that the apparent EEC exhibited by the
series of compounds 1 and 2 is statistically significant. That
proved to be rather involved, and although the general problem
has received considerable attention in the literature,39-42 we
found that an adequate test for EEC had not been described.
Readers who are willing to accept our assertion that the
correlations depicted in Figure 8b are highly significant and
who are not interested in the intricacies of the statistical analysis
may skip the next section.
Statistical Analysis of the Enthalpy-Entropy Compensa-
tion (EEC). Since Leffler published his classical review article
on enthalpy-entropy relationships,35 the statistical analysis of
such data has been discussed extensively.39-42 The issue is a
delicate one because the experimental errors of the Arrhenius
parameters are strongly correlated: even in the absence of any
correlation between the expectation values (the “true” values)
of Ea and ln A∆, their estimators Eˆa and ln Aˆ∆, which are obtained
by linear regression, will be correlated, and the expectation value
of the correlation coefficient F(Eˆa, lnAˆ∆) will approach unity as
the experimental temperature range becomes small. Moreover,
large experimental errors tend to raise the correlation coefficient
F(Eˆa, ln Aˆ∆) rather than lower it. Therefore, the mere observation
of correlation between estimated Arrhenius parameters of a
reaction series (as in Figure 8b) cannot, by itself, be taken as
evidence for the existence of an IKR. Furthermore, correlation
arising purely from the covariance of the random errors will
Figure 8. Arrhenius parameters log A (upper diagram a) and log A∆
(eq 9, lower diagram b) versus activation energies Ea for ISC of
diradicals 1 (b) and 2 (O).
suggest an apparent isokinetic temperature that is close to the
-1 41
harmonic mean of the experimental temperatures, â ≈ 1/T
,
and one is well advised to be especially suspicious when the
apparent isokinetic temperature lies near the experimental
temperatures, as in the present case.
7 and 8 we use the notation of Leffler and Grunwald:37a the
operator δR denotes the change in the subsequent quantity caused
by the substituent R, for example, δR∆GQ ) ∆GQ[R ) X] -
∆GQ[R ) H]. A series of compounds adhering to eq 7 is also
said to obey an isokinetic relationship (IKR), because eq 7
implies that the dependence of the rate constants on substitution
defined by eq 1, δR ln(k/s-1) ) δR ln(A/s-1) - δREa/RT, vanishes
for T ) â. The parameter â is called the isokinetic temperature,
the temperature at which all compounds of the series react with
the same rate, and where all Arrhenius plots intersect. The slope
R in eq 8 is commonly called the Brønsted parameter.
Two methods have been described to test for the existence
of an IKR,40,41 and it was reported42a that they give essentially
the same results. We have used the method of Exner:40 linear
regressions of experimental Arrhenius plots are done (i)
individually for each compound and (ii) under the IKR constraint
that all the regression lines intersect at some temperature T )
â. As â is not known a priori, the residual sum of squares is
calculated for trial values of â until the minimum, S0, is found.
This is compared to the residual sum of squares that is obtained
when a separate straight line is fitted to the data of each
compound without any restriction, S00. An F-statistic, eq 10, is
Let us return to the extended Arrhenius relation, eq 2. From
eqs 4 and 6 we know that the first factor ASOC is proportional
to γ2 , so we may as well define ASOC/s-1 ≡ (γAB/eV)2 and
AB
(S00 - S0)/f0
leave the proportionality factor to A∆. When ln A∆ ) ln A - 2
ln γAB is plotted against Ea, most of the scatter in Figure 8a is
eliminated and the data separate into two accurately linear
relations, eq 9, for the two sets of diradicals 1 and 2 (Figure
F(f0, f00) ≈
(10)
S00/f00
calculated to test the isokinetic hypothesis, that is, the null
hypothesis H0: the true (error-free) regression lines of all
compounds intersect at one point. It is assumed that the F-test
is not sensitive to the fact that the relation between the two
δR ln(A∆/s-1) ≡ δR ln(A/s-1) - 2δR ln(γAB/eV) ) δREa/Râ
(9)
(38) The ISC rates of heavy-atom substituted derivatives (Br and I) were
substantially faster than predicted by the model function (cf. Figure 6) and
were omitted from this analysis. Derivatives containing cyano substitutents
consistently had ISC rates somewhat less than expected. This is attributed
to the parametrization of the AM1 model, as the calculated orbital energies
also fell out of line with the corresponding Hammett substituent constants
or ionization energies. The cyano-substituted derivatives were, therefore,
also omitted, but their inclusion would not have had a substantial effect on
the overall results. Note that eq 9 implicitly contains ∆γ as an adjustable
parameter (ref 32), which is used to calculate γAB from the AM1 interaction
terms γAM1 given in Table 1.
(36) The existence of a LFER implies the existence of EEC, if the
temperature dependences of the activation enthalpy ∆H‡ and entropy ∆S‡
are negligible in the range covered by the experimental data (ref 37a, pp
155-161). EEC is sometimes used in a qualitative sense. If it implies an
exact linear relation between ∆H‡ and ∆S‡ (or Ea and ln A, eq 7), then
EEC and IKR are synonymous.
(37) (a) Leffler, J. E.; Grunwald, E. Rates and Equilibria of Organic
Reactions; Wiley: New York, 1963. (b) Hammett, L. P. Physical Organic
Chemistry. Reaction Rates, Equilibria, and Mechanisms; McGraw-Hill:
New York, 1970. (c) Grunwald, E. Thermodynamics of Molecular Species;
Wiley: New York, 1997.
(39) Petersen, R. C. J. Org. Chem. 1964, 29, 3133.

1,3-Cyclopentanediyl Diradicals
J. Am. Chem. Soc., Vol. 121, No. 40, 1999 9273
Table 2. Estimators and Credible Intervals for the Isokinetic
Temperature â and for the Correlation Coefficient F between the
Expectation Values of the Arrhenius Parameters ln A∆ and Ea
Determined by Gibbs Sampling (Appendix II)
percentiles of the (re-sampled) posterior distribution
(10000 iterations)
parameter
2.5%
301
-0.9988
300
50% (median)
97.5%
281
-0.9975
281
â(1)/K
r(1)
â(2)/K
r(2)
292
-0.9982
291
-0.9995
-0.9994
-0.9991
even a minuscule difference becomes statistically discernible.
That is, H0 may be rejected although it is practically true.44
The essential first step is to ask the proper question. Above
we asked whether the data are consistent with an exact IKR,
and the answer was No. However, we may still be interested to
know whether the true (error-free) Arrhenius parameters are
correlated, that is, whether EEC, an approximate IKR, exists
and how strong that linear association is. To answer this
question, we need to know whether the observed correlation
between the two parameters, Ea and ln A∆, exceeds that expected
on the basis of purely random errors in the data.
Figure 9. Correlation diagram of the Arrhenius parameters log A∆
and Ea for diradical 2 with 95% confidence regions. Each of the data
points (at the center of the confidence regions, not shown for clarity)
was obtained as the slope and intercept of an Arrhenius plot. The inset
shows the plot for 2 (X ) Y ) NO2) as an example. Note that the long
axes of the elliptic confidence regions are approximately parallel to
the regression line. The ellipses are a pictorial representation of the
random-error contribution to the correlation of the Arrhenius parameters.
The total variation in the data is much larger than that arising from
random error. The plot for diradicals 1 is similar (not shown).
We propose two treatments to answer this question, a quick-
and-dirty method given in Appendix I and a more sophisticated
treatment, which is based on Bayesian inference and uses Gibbs
sampling, in Appendix II. The conclusion from both treatments
is the same: now, the hypothesis cannot be rejected because
we have made a weaker claim. We do not imply the existence
of an exact IKR in the sense that all of the true Arrhenius lines
must intersect at a single point, but we do claim that the
expectation values of Ea and ln A∆ are strongly correlated.
Estimated correlation coefficients between the expectation values
of Ea and ln A∆ obtained by the two methods are similar, and
both are very high (r2 > 0.99). In addition, the second method
provides estimates of the parameters Ea and ln A∆ and of the
isokinetic temperature â with credible intervals (a Bayesian
analogue to confidence intervals: 2.5% and 97.5% percentiles
of the resampled posterior distributions). The values of â
determined in this way for the series 1 and 2 (Table 2), are
nearly the same, â ) 291 ( 10 K, and similar to the â-values
obtained by the method of Exner. However, the interpretation
of â is now different: it is the temperature where we have
maximum, though not perfect, EEC. The 2-dimensional “co-
ordinates” of the “isokinetic” point, â and ln(k0), now represent
the mean of a bivariate normal distribution. The resulting
credible intervals of the Arrhenius parameters are reproduced
in Table S2 of the Supporting Information. They are very similar
to the standard errors given in Table 1 that were obtained by
independent linear regression for each compound.
Toward an Intuitive Understanding of Substituent and
Temperature Effects on Diradical Lifetimes. The π-orbitals
of the sp2 radical centers are parallel in the planar conformation
of cyclopentane-1,3-diyl diradicals, and their conformational
mobility is restricted. According to Salem and Rowland’s first
rule,2 SOC vanishes for parallel π-orbitals, and this is held
responsible for the relatively high persistence of triplet cyclo-
pentane-1,3-diyl diradicals such as 1 and 2, which are stable
for hours at 77 K and decay on the microsecond time scale at
room temperature in solution. The pattern of substituent and
temperature effects on these lifetimes is rather complex.
Examination of the data in Table 1 reveals that activation
models is not accurately linear.43 The degrees of freedom of
the unconstrained regression (i) are f00 ) number of data points
- 2(number of compounds), and those associated with introduc-
ing constraint (ii) are f0 ) number of compounds - 2.
The SOC-adjusted rate constants, k′ ) k/(γAB/eV)2, for the
two reaction series, diradicals 1 (14 compounds, 878 data points)
and 2 (9 compounds, 486 data points), were analyzed by Exner’s
method, which gave â ) 273 K and F(12, 850) ) 9.0 for 1 and
â ) 285 K and F(7, 468) ) 20.9 for 2. Both of these F-statistics
are much larger than the corresponding values of the two-tailed
F-distribution at a significance level of >99% [F(12, 850) )
2.2, F(7, 468) ) 2.66)]. Therefore, H0 must be rejected. The
residual variance of the unconstrained model is lower than that
of the isokinetic model at a confidence level of >99%. We are
forced to conclude that the lines defined by eq 11 do not intersect
ln(k′/s-1) ) ln(k/s-1) - 2 ln(γAB/eV) ) ln(A∆/s-1) - Ea/RT
(11)
at a single point, that is, that the expectation values of the
regression parameters Ea and ln A∆ do not obey the isokinetic
relationship, eq 9.
However, the (elliptic) 95% confidence regions of the data
pairs (Ea, ln A∆) are much smaller than their total variation
(Figure 9), and that indicates that enthalpy-entropy compensa-
tion (EEC) is real. How can these two views be reconciled?
We should not expect an IKR to hold exactly because it is an
extra-thermodynamic relationship that does not have a sound
physical basis. Moreover, large sample sizes will lead one to
reject any null hypothesis, H0, that is not exactly true because
a huge sample will reduce the standard error to the point where
(40) Exner, O. In Progress in Physical Organic Chemistry; Wiley: New
York, 1973; Vol. 10, p 411.
(41) Krug, R. R.; Hunter, W. G.; Grieger, R. A. J. Phys. Chem. 1976,
80, 2335 and 2341.
(42) (a) Linert, W.; Jameson, R. F. Chem. Soc. ReV. 1989, 18, 477. (b)
Linert, W. Chem. Soc. ReV. 1994, 429.
(43) The restricted model (ii) is not simply defined as a submodel of (i)
by the linear eq 11 because â is estimated simultaneously with the
parameters ln(A′) and Ea.
(44) For example, Wonnacott, T. H.; Wonnacott, R. J. Introductory
Statistics, 3rd ed.; Wiley: New York, 1977.

9274 J. Am. Chem. Soc., Vol. 121, No. 40, 1999
Kita et al.
(C+) in the electronic wave function of the diradical’s lowest
singlet state S0. As predicted for nonaxial diradicals,3 SOC is
insensitive to donor-acceptor substitution but is enhanced by
through-bond interaction. The systematic variation of the ISC
rate constants with Brown’s σ+ substituent constants reflects
the through-bond interaction of the nonbonding orbitals, which
increases with the electron-withdrawing propensity of the
substituents.
The temperature coefficients of the ISC rate constants exhibit
strong enthalpy-entropy compensation (EEC). The isokinetic
temperature â of the thermally activated process happens to be
near room temperature in this reaction series. Thus, the effect
of substituents on the activation energy for cyclization is largely
canceled by EEC at room temperature, which results in the
deceptively simple relationship between ISC rate constants and
the calculated magnitude of SOC, kISC C2+.
Figure 10. Singlet and triplet potential-energy surfaces of diradical 7
along the out-of-plane bending mode defined by the dihedral puckering
angle Θ (see structure 7 in the text).
energies and SOC strengths are not related. Both electron-
withdrawing (nitro) and electron-donating (amino) substituents
lower the activation energies in diradicals 1 and 2 compared to
the parent compound. Such behavior is indicative of a radical-
stabilizing effect of the substituents.45 Nevertheless, the lifetimes
determined at ambient temperature exhibit a monotonic increase
with increasing electron-withdrawing properties as expressed
by, for example, Brown’s substituent constants σ+ (Figure 3).
The aryl-stabilized diradicals 1 and 2 are real minima rather
than transition states (Figure 10). An energy barrier of 2-3 kcal
mol-1 has been determined experimentally for ring closure of
the singlet diradical 7(X ) Y ) H) to diphenylhousane.21 The
triplet diradical lies somewhat below the singlet diradical [∆EST
∼0.5 kcal mol-1 by AM1, 3 × 3 CI,29a-c or STO-3G)29c,d],
which further enhances the total activation energy required for
housane formation from the triplet diradical. Aryl substituents
cause various degrees of radical stabilization. Hence, donor and
acceptor substituents stabilize the singlet and triplet diradicals
relative to the housane (vertical arrow in Figure 10) and increase
the activation energies for intersystem crossing.
The simple empirical Ansatz expressed in eq 2 successfully
deals with the present case of a thermally activated reaction
that is inhibited by a spin barrier. The probability of ISC appears
in a transmission factor ASOC that is modeled by SOC strength.
The “standard” Arrhenius term A∆ exp(-Ea/RT) describes
thermal activation in the same way as for a spin-allowed
reaction, and it obeys the structure-reactivity rules expected
for an aryl reaction series. The fact that the lifetimes of diradicals
1 and 2 at room temperature are largely explained by considering
only the factor ASOC now appears to be rather coincidental and
due to the approximate isokinetic relationship obeyed by the
parameters ln A∆ and Ea: the isokinetic temperatures happen
to lie close to room temperature, such that the substituent effects
arising from the term A∆ exp(-Ea/RT) largely cancel.
Appendix I
Let the logarithms of rate constants k′ij, measured at inverse
temperatures θij ) 1/Tij, be linearly related, eq I.1,
log(k′ij) ) Ri + âiθij + ꢀij
(I.1)
where the index i (i ) 1, ..., n) runs over the n substances Bi,
and the index j (j ) 1, ..., ni) runs over the ni measurements
done for compound Bi. These n expressions include Arrhenius’
law for the systematic part, with Ri ) ln Ai′ and âi ) -Ea,i/R,
and error terms ꢀij, which are considered as normally distributed
random variables, ꢀij ≈ N(0,σi2). Estimators for the intercepts
Ri, ai, and for the slopes âi, bi, are obtained by linear regression
of a sample of ni experimental data pairs (k′ , θij) obtained for
ij
each compound Bi. The estimators ai and bi of a given
compound Bi are correlated, as is well-known from the theory
of linear regression, eq I.2.46a That correlation arises from the
cov(ai, bi)
F(ai, bi) ≡
(I.2)
var(a ) var(b )
x
i
i
random error terms ꢀij in the sample data. As an example, the
corresponding 95% ellipsoidal joint confidence regions for the
data pairs (ai, bi) of compounds 2 are shown in Figure 9.
The correlation between the expectation values R and â is
defined by eq. I.3.
cov(R,â)
F(R,â) )
(I.3)
var(R) var(â)
x
Note that eqs I.2 and I.3 address entirely different correlations.
Equation I.3 represents the correlation of the expectation Values
Ri and âi for different substances Bi, the amount of enthalpy-
entropy compensation (EEC) in the reaction series. It is not
contaminated by the correlation of the estimators, eq I.2, which
arises from experimental error. If F(R,â) is high, that is evidence
for a high degree of EEC. Our goal is to obtain an estimate,
r(R,â), for EEC as defined by eq I.3. To this end, we partition
the variances and the covariance of the n estimators, ai and bi,
i ) 1, ... n, into two portions, one due to error correlation, eq
I.2, and another due to the intrinsic correlation F(R,â). As
estimators for the elements of the variance-covariance matrix
of Ri and âi required in eq I.3, we propose eqs I.4-I.6, in which
the contributions arising from error correlation, maxvar(a),
maxvar(b), and maxcov(a,b), are subtracted.
Conclusions
This work represents an extensive study of substituent effects
on a reaction series in which the rate-determining step involves
ISC. The rate constants for ring closure of the triplet 1,3-
cyclopentanediyl diradicals 1 and 2 at room temperature are
affected in a systematic manner by substituent variation.
Analysis within the frame of a two-electrons-in-two-orbitals
active space indicates that the substituent effects on these rate
constants are well described by the magnitude of SOC, which
can be easily calculated from the zwitterionic-state contribution
(45) Adam, W.; Harrer, H. M.; Kita, F.; Nau, W. M. AdV. Photochem.
1998, 24, 205.
(46) Draper, N. R.; Smith, H. Applied Regression Analysis, 3rd ed.;
Wiley: New York, 1998, (a) p 129, (b) p 219.

1,3-Cyclopentanediyl Diradicals
J. Am. Chem. Soc., Vol. 121, No. 40, 1999 9275
Table 3. A Priori Distributions Chosen for Gibbs Sampling
n
1
var(R) )
var(â) )
[a - aj]2 - maxvar(a)
(I.4)
(I.5)
∑
i
prior
distribution
parameter values in
the prior density
n - 1i)1
model parameter
Ri, âi i ) 1, .... n normal
µ ) 0, σ2 ) 104
n
1
[b - bh]2 - maxvar(b)
σi2 i ) 1, .... n
gaussiana
gaussiana
r ) 0.001, µ ) 0.001
r ) 0.001, µ ) 0.001
∑
i
τ2
n - 1i)1
103
0
0
n
(x0, y0)
bivariate normal µ ) (0,0) cov )
1
103
(
)
cov(R, â) )
[a - aj][b - bh] - maxcov(a,b)
i i
∑
n - 1i)1
(I.6)
a f(x | r, µ) ) µr xr-1 e-µx/Γ(r).
for the model parameters (Table 3). Bayesian inference consists
of updating prior knowledge about the model parameters by
the data, resulting in the so-called posterior densities via the
theorem of Bayes: posterior likelihood × prior. All conclu-
sions about the parameters of the model and their functions,
such as the correlation F, are drawn from the corresponding
posterior densities.
To obtain a conservative estimate, we take the terms maxvar-
(a), maxvar(b), and maxcov(a,b) from the largest variance-
covariance matrix of all samples (ai,bi), that is, from that with
the greatest sum of the two eigenvalues (and the largest
ellipsoidal confidence area, cf. Figure 9). If the confidence
interval of r(R,â) does not contain 0, the null hypothesis H0:
F(R,â) ) 0 may be rejected.
The posterior densities given in Table 2 were obtained by
Gibbs sampling48 using the program BUGS.49 Instead of
calculating exact estimates of the posterior distributions, which
would mean performing integrations of high dimension, this
technique generates a stream of simulated values from the
conditional (with respect to the other model parameters)
distributions of each model parameter of interest and of functions
of the model parameters. The process is iterated (10 000
iterations) until the distributions of the estimated parameters
stabilize.
Appendix II
We assume n linear Arrhenius relations, eq II.1, defined as
in eq I.1.
log(k′ ) ) Ri + âiθij + ꢀij, j ) 1, ... ni, ꢀij ≈ N(0,σi2) (II.1)
ij
If an exact isokinetic relationship (IKR) existed, the Arrhenius
parameters Ri and âi were related by the set of eqs II.2, where
Ri ) y0 - x0âi i ) 1, ..., n
(II.2)
Acknowledgment. This work was supported by the Deutsche
Forschungs gemeinschaft, the Volkswagen-Stiftung and the
Schweizerischer Nationalfonds and the German Fonds der
Chemischen Industrie. Dr. H. M. Harrer, Wu¨rzburg, provided
some of the azoalkanes for the laser flash measurements, G.
Persy, Basel, helped with the low-temperature UV spectroscopic
studies, and X. Zhang, Basel, provided the PTOC precursor for
the p-nitrobenzyl radical.
x0 ) 1/â and y0 ) log(k′ij) at T ) â. The set of eqs II.2 is a
recast of eq 11.
As eq II.2 may not hold exactly, in eq II.3 we introduce
“error” terms δi which are assumed to be distributed according
to a normal distribution with expectation 0 and variance τ2. Note
that the errors δi do not represent experimental errors. Rather,
they represent deviations of the expectation values Ri and âi
from the exact IKR (eq II.2). We now proceed to obtain an
estimator r(R,â) for the correlation F(R,â), as defined in eq I.3,
Appendix I.
Supporting Information Available: Experimental spectro-
scopic details, computational procedures, synthesis, and char-
acterization of the four symmetrically substituted azoalkanes 4
(X ) Y ) p-NH2, p-OMe, p-CN, p-NO2) and their correspond-
ing housane derivatives 6, the original kinetic data (Table S1)
and more detailed output of the statistical analyses (Table S2).
This material is available free of charge via the Internet at
http://pubs.acs.org.
Ri ) y0 - x0âi + δi, i ) 1, ..., n, δi ≈ N(0,τ2) (II.3)
The statistical model is defined by eqs II.1 and II.3 simul-
taneously. We treat the model in a Bayesian framework.47 In
this context it is indispensable to make an uninformative choice
of a priori distributions, the so-called prior distributions, for
the model parameters, âi, σi2, τ2, x0, and y0, that are to be
estimated. Diffuse but proper distributions were chosen as priors
JA991362D
(48) Casella, G.; George, E. I. Am. Stat. 1992, 46, 167.
(49) Thomas, A.; Spiegelhalter, D. J.; Gilks, W. R. In Bayesian Statistics;
Bernardo, J. M., Berger, J. O., Dawid, A. P., Smith, A. F. M., Eds.;
Clarendon Press: Oxford, U.K., 1992; Vol. 4, p 837. The program and
manual are available on the Internet, www.mrc-bsu.cam.ac.uk/bugs.
(47) Box, G. E. P.; Tiao, G. C. Bayesian Inference in Statistical Analysis;
Wiley: New York, 1992.