The tightness contribution to the Bronsted α for hydride transfer between NAD+ analogues

Article abstract of DOI:10.1021/ja011855u

It has been shown that the rate of symmetrical hydride transfer reaction varies with the hydride affinity of the (identical) donor and acceptor. In that case, Marcus theory of atom and group transfer predicts that the Bronsted α depends on the location of the substituent, whether it is in the donor or the acceptor, and the tightness of the critical configuration, as well as the resemblance of the critical configuration to reactants or products. This prediction has now been confirmed for hydride transfer reactions between heterocyclic, nitrogen-containing cations, which can be regarded as analogues of the enzyme cofactor, nicotinamide adenine dinucleotide (NAD⁺). A series of reactions with substituents in the donor gives Bronsted α of 0.67 ± 0.03 and a tightness parameter, τ of 0.64 ± 0.06. With substituents in the acceptor α = 0.32 ± 0.03 and τ = 0.68 ± 0.08. The reactions are all spontaneous, with equilibrium constants between 0.4 and 3 x 10⁴, and the two sets span about the same range of equilibrium constants. The two τ values are essentially identical with an average value of 0.66 ± 0.05. These results can be semiquantitatively mimicked by rate constants calculated for a linear, triatomic model of the reaction. Variational transition state theory and a physically motivated but empirically calibrated potential function were used. The computed rate constants generate an α value of 0.56 if the hydride affinity of the acceptor is varied and an α of 0.44 if the hydride affinity of the donor is varied. The calculated kinetic isotope effects are similar to the measured values. A previous error in the Born charging term of the potential function has been corrected. Marcus theory can be successfully fitted to both the experimental and computed rate constants, and appears to be the most compact way to express and compare them. The success of the linear triatomic model in qualitatively reproducing these results encourages the continued use of this easily conceptualized model to think about group, ion, and atom transfer reactions.

Full text of DOI:10.1021/ja011855u

Published on Web 06/07/2002
The Tightness Contribution to the Brønsted r for Hydride
Transfer between NAD+ Analogues
In-Sook Han Lee,* Kim-Hung Chow, and Maurice M. Kreevoy*
Contribution from the Department of Science Education, Kangwon National UniVersity,
Chuncheon 200-701, Korea, and Department of Chemistry, UniVersity of Minnesota,
Minneapolis, Minnesota 55455-0431
Received July 30, 2001. Revised Manuscript Received February 7, 2002
Abstract: It has been shown that the rate of symmetrical hydride transfer reaction varies with the hydride
affinity of the (identical) donor and acceptor. In that case, Marcus theory of atom and group transfer predicts
that the Brønsted R depends on the location of the substituent, whether it is in the donor or the acceptor,
and the tightness of the critical configuration, as well as the resemblance of the critical configuration to
reactants or products. This prediction has now been confirmed for hydride transfer reactions between
heterocyclic, nitrogen-containing cations, which can be regarded as analogues of the enzyme cofactor,
nicotinamide adenine dinucleotide (NAD+). A series of reactions with substituents in the donor gives Brønsted
R of 0.67 ( 0.03 and a tightness parameter, τ, of 0.64 ( 0.06. With substituents in the acceptor R ) 0.32
( 0.03 and τ ) 0.68 ( 0.08. The reactions are all spontaneous, with equilibrium constants between 0.4
and 3 × 104, and the two sets span about the same range of equilibrium constants. The two τ values are
essentially identical with an average value of 0.66 ( 0.05. These results can be semiquantitatively mimicked
by rate constants calculated for a linear, triatomic model of the reaction. Variational transition state theory
and a physically motivated but empirically calibrated potential function were used. The computed rate
constants generate an R value of 0.56 if the hydride affinity of the acceptor is varied and an R of 0.44 if the
hydride affinity of the donor is varied. The calculated kinetic isotope effects are similar to the measured
values. A previous error in the Born charging term of the potential function has been corrected. Marcus
theory can be successfully fitted to both the experimental and computed rate constants, and appears to be
the most compact way to express and compare them. The success of the linear triatomic model in
qualitatively reproducing these results encourages the continued use of this easily conceptualized model
to think about group, ion, and atom transfer reactions.
tightness of the critical configuration.1-4 Tightness is related
to the distance between the end groups, the bond order to the
in-flight atom or group, and the partial charge on the in-flight
atom or group.1-4 The critical configuration has been defined
as the most probable configuration for crossing the hardest to
attain dividing surface that separates products from reactants.5
For a one-step hydride transfer reaction of the type shown in
eq 1, Marcus theory can be formalized in eqs 2 and 3.
1. Introduction
During the last century a number of empirical equations and
semiquantitative relations have been developed to relate rate
constants to one another and to equilibrium constants. These
relations have systematized large areas of chemistry, facilitated
teaching, and facilitated the identification of unusual behavior.
They also have some predictive power. In recent years it has
become possible to calculate the rate constants of specific
reactions quantum mechanically, from first principles, with
useful reliability. While these computations are more transparent
than measured rate constants they do not easily lead to
generalization. The physics underlying the empirical relations
is still imperfectly understood. In this paper we continue our
effort to use Marcus theory of atom and group transfer to provide
a link between empirical correlation and physically motivated,
though highly simplified calculation on a linear, triatomic model
of hydride transfer between nicotinamide adenine dinucleotide
(NAD+) analogues.
Ai+ + AjH f AiH + Aj+
(1)
∆Gij* ) Wr + (1 + ∆Gij°/λij)2λij/4
λij ) (λii + λ jj)/2
(2)
(3)
(4)
(5)
Kij ) exp (-∆Gij°/RT)
kij ) kBT/h exp(-∆Gij*/RT)
Phenomenological Marcus theory, applied to atom or group
transfer reactions, can provide a quantitative measure of the
The subscripts i and j indicate the hydride acceptor and donor,
respectively.
In eq 2, Wr was originally regarded as the free energy required
to form a metastable reaction complex from the separated
* Address correspondence to this author at Kangwon National University.
E-mail: ishl@kangwon.ac.kr.
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J. AM. CHEM. SOC. 2002, 124, 7755-7761
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A R T I C L E S
Lee et al.
reactants.6 However, it appears only to represent that part of
∆Gij* which is insensitive to ∆Gij°.7 In other words, Wr is little
more than an adjustable parameter that permits the theory to
give a useful description of the results. Since the reactants and
products are structurally related and of the same charge type, it
can be assumed that Wr is the same in both directions. In that
case, the standard free energy of reaction, ∆Gij°, is the same as
the standard free energy of reaction within a reactive complex,
∆Gij°′. This simplifies the Marcus formalism to the form shown
as eqs 2 and 3. For the reaction shown as eq 1, Wr has been
taken as -8 kJ/mol.7
When ∆Gij° is zero, λij/4 is equal to (∆Gij* - Wr). This is
the intrinsic barrier. For the nondegenerate reactions λij is the
average of λii and λjj, for the two related degenerate reactions,
Ai+ with AiH, and Aj+ with AjH, respectively.
∆G° and ∆G* can be related to the equilibrium constant Kij
and the rate constants, kij, shown as eqs 4 and 5.
The Brønsted R is defined by eq 6.
Figure 1. A two-dimensional cartoon for hydride transfer between Ai+
and Aj+. The closed circle indicates the illustrative case described in the
text. The open circle indicates the location of the critical configuration for
the present experimental system. The closed square indicates the location
of the critical configuration of the computational results for the linear and
triatomic model.
R ) d(ln kij)/d(ln Kij)
R ) d(∆Gij*)/d(∆Gij°)
(6a)
(6b)
Since this paper will discuss hydride transfer between NAD+
analogues, rate constants, kij, and equilibrium constants, Kij, are
defined with respect to eq 1, but the ideas involved are general,
including also enzyme-catalyzed reactions. The equilibrium
constant for the transfer of hydride from some standard donor
to a series of acceptors, Ai+, is Kij°. In practice, for practical
and historic reasons the standard donor is 10-methylacridan.2
The rate constant for the symmetric transfer for hydride between
one Ai+ and another is kii. The variation of kii with Kij° was
first explicitly discussed by Thornton10 and is called a Thornton
effect. The most striking aspect of this development is that (τ
- 1)/2 is expected to take the positive sign in eq 7 (and the
cross term to take the negative sign) if the structural changes
which generate the changes in kij and Kij are in the acceptor
atom or group, while the opposite signs apply if the structural
changes are in the donor.2,4 This is most easily seen by
considering d(λ/4)/d(∆G°) as we change the hydride affinity
of the donor or the acceptor. To make the reactions more
spontaneous (∆G° more negative) and faster, by changing the
acceptor, we must increase its hydride affinity. To make the
reactions more spontaneous, and faster, by changing the donor,
we must decrease its hydride affinity. But λij is approximately
the mean of the λ values for symmetrical exchange in the donor
and the acceptor, and increases with an increase in hydride
affinity at either site.2,4 Thus d(λ/4)/d(∆G°) is negative, and
makes a negative contribution to d(∆G*)/d(∆G°) if substitution
is in the acceptor, but it is positive if substitution is in the donor.
An intuitive justification of this outcome is given in Figure
1, which is a two-dimensional cartoon.2,9 As mentioned above,
ø is a quantitative measure of the relative resemblance of the
critical configuration to the reactant structure and the product
structure. τ is a quantitative measure of the charge distribution
and reactive bond lengths of the critical configuration. A critical
configuration in the upper left corner is identical to reactants
which are in contact and properly positioned. A critical
configuration in the lower right corner is identical to fully
formed products which have not yet separated or become
From eqs 2-6, and the definitions given in eqs 8 and 9, the
expression shown in eq 7 can be derived for the Brønsted R.2
R ) ø ( (τ - 1)/2 - (RT ln Kij/λij)2(τ - 1)/2
ø ) [1 - (RT ln Kij/λij)]/2
(7)
(8)
(9)
(τ - 1) ) d(ln kii)/d(ln Kij)
Equations 8 and 9 define ø and τ. ø is the Leffler-Hammond
parameter. It gives a quantitative measure of the relative weights
of the reactant structure and the product structure in the structure
of the critical configuration. When the reaction becomes more
spontaneous, K becomes larger, leading to a smaller value of
ø, which indicates a critical configuration closer to the reactant.8
This quantity has the properties formerly ascribed to R. It is
δ(ln kij)/δ(ln Kij) or δ(∆Gij*)/δ(∆Gij°) at constant λ. On the other
hand, the tightness parameter, earlier dubbed τ,9 is expected to
have an important effect on the Brønsted R, as shown in eqs 1
and 7-9. (The last term in eq 7 is a small cross-term that is
insignificant in the present case.) τ implies that the displacement
of the critical configuration toward structures that do not
resemble either reactants or products is needed to describe its
charge distribution and reactive bond lengths.2
(1) (a) Kreevoy, M. M.; Ostovic, D.; Lee, I.-S. H.; Binder, D. A.; King, G. W.
J. Am. Chem. Soc. 1988, 100, 524. (b) Lee, I.-S. H.; Jeoung, E. H.; Kreevoy,
M. M. J. Am. Chem. Soc. 1997, 119, 2722. (c) Lee, I.-S. H.; Jeoung, E. H.
J. Org. Chem. 1998, 63, 7275. (d) Lee, I.-S. H.; Jeoung, E. H.; Kreevoy,
M. M. J. Am. Chem. Soc. 2001, 123, 7492.
(2) Kreevoy, M. M.; Lee, I.-S. H. J. Am. Chem. Soc. 1984, 106, 2550.
(3) Lewis, E. S.; Hu, D. D. J. Am. Chem. Soc. 1984, 106, 3292
(4) Kreevoy, M. M.; Truhlar, D. G. In Rates and Mechanisms of Reactions;
Bernasconi, C. F., Ed.; Wiley: New York, 1986; Chapter 1.
(5) Kreevoy, M. M.; Ostovic, D.; Truhlar, D. G.; Garrett, B. C. J. Phys. Chem.
1987, 90, 3766.
(6) (a) Kreevoy, M. M.; Konasewich, D. E. AdV. Chem. Phys. 1971, 21, 243.
(b) Hassid, A. I.; Kreevoy, M. M.; Liang, T.-M. Symp. Faraday Soc. 1975,
10, 69.
(7) Kim, Y.; Truhlar, D. G.; Kreevoy, M. M. J. Am. Chem. Soc. 1991, 113,
7837.
(8) (a) Leffler, J. E. Science 1953, 117, 340. (b) Hammond, G. S. J. Am. Chem.
Soc. 1955, 77, 334.
(9) Albery, W. J.; Kreevoy, M. M. AdV. Phys. Org. Chem. 1978, 16, 87.
(10) Thornton, E. R. J. Am. Chem. Soc. 1967, 89, 2915.
9
7756 J. AM. CHEM. SOC. VOL. 124, NO. 26, 2002

Hydride Transfer between NAD+ Analogues
A R T I C L E S
disoriented. In the upper right corner there is a hypothetical
critical configuration in which hydride is in contact with both
acceptors but has no covalent bond to either. In the lower left
corner is a hypothetical critical configuration with a hypervalent
hydrogen, which has covalent bonds to both Ai and Aj. In all
critical configurations along the upper right-lower left diagonal
the bonds to donor and acceptor are of equal strength and Kij )
1.0. For all such critical configurations eq 8 shows that ø is 1/2.
We first consider symmetrical rections in which the donor is
AiH, the acceptor is Ai+, and the rate constants are kii. If the
critical configuration for all of these reactions is in the upper
right corner, the original AiH bond is completely broken, but
the new AiH bond is not at all formed. Referring to eq 9, if the
free energy of the AiH bond is increased, Kij° and ln Kij° will
increase, but ln kii will decrease by an equal amount, because
the bond being broken is stronger and the new bond has not
yet begun to form. Thus, [δ(ln kii)/δ(ln Kij°)]j ) -1 in this case
and τ ) 0. At the crossing of the two diagonals the free energies
of the old and new bonds are equal. Therefore, all kii will be
equal, [δ(ln kii)/δ(ln Kij°)]j ) 0, and τ will be 1. In a similar
way, τ will be 2 if the critical configuration is in the lower left
corner. For a single family of reactions it is assumed that τ is
constant for our present discussion, and the critical configuration
is always located in the same place on the cartoon. Using the
two diagonals as coordinates in the cartoon, the upper right-
lower left diagonal gives τ, increasing in that direction, with
the scale established by the values determined above. The value
on the coordinate running from upper left to lower right gives
the other variable, ø.
variational transition-state theory12,13 (ICVT) with the large-
curvature ground-state tunneling approximation.12 These cal-
culations were implemented by the program POLYRATE 3.0.13
The calculations require a family of global potential energy
surfaces. For the present work a previously obtained family of
surface7 was used. However, an error was found and corrected.
These surfaces are based on the good general agreement between
experimental reactivity patterns and rate constants calculated
with use of the surfaces.7 The computations are transparent and
free of experimental error and the idiosyncracies of individual
molecular structures. Unlike experimental results, the origin of
computed effects is generally unambiguous.
We next consider a group of unsymmetrical reactions, in
which the critical configuration is at the closed circle in the
cartoon. If Ai is varied and Aj is constant (constant donor-
variable acceptor) ln kij will increase by less than half the
increase in ln Kij° because the new bond is less than half formed.
On the other hand, if we consider a group of unsymmetrical
reactions in which Ai is constant and Aj is varied (constant
acceptor-variable donor) with the same critical configuration
location, the change in ln kij will be larger than half the change
in ln Kij°, because more than half of the donor bond free energy
is lost in making this critical configuration. However, eq 8 shows
1
2. Experimental Section
that ø should be /2 in both cases, therefore, ø alone cannot be
identified with R. Something must be added if the donor
structure is varied, but subtracted if the acceptor structure is
varied. Mathematical manipulation of Marcus theory yields the
form shown in eq 7.2
Lewis and co-workers have already shown the semiquanti-
tative correctness of this prediction for methyl transfer reac-
tions,3,11 and in this paper, we do so for hydride transfer
reactions.
The 9-aryl-10-methylacridinium ions and their dihydro derivatives,
and the quinoline derivatives, are all previousely reported com-
pounds14,15 and all had acceptable physical and spectroscopic properties.
The water used was distilled and then redistilled in the presence of
a small amount of H2SO4. All other substances used were good quality
commercial materials and were used without further purification. For
reactions which would almost entirely consume one of the reactants,
pseudo-first-order rate constants, k1, were measured spectrophotometri-
cally at 25.0 ( 0.2 °C in the usual way.14 The ions 1a-f all have
The specific series used to illustrate the point are shown in
eqs 10 and 11. In addition the solvolysis equilibria of the cations,
eq 12, were examined, and KR values are reported, for
comparison with the equilibrium constants for hydride transfer.
To take advantage of a large body of information already
available, a 4:1 2-propanol:water mixture was used as solvent.
Therefore, R, in eq 12, may be either H or 2-propyl.
We also present some related computational results which
help to clarify the origin and meaning of these observations.
Rate constants were calculated by using improved canonical
(12) (a) Truhlar, D. G.; Isaacson, A. D.; Garrett, B. C. In Theory of Chemical
Reaction Dynamics; Baer, M., Ed.; CRC Press: Baoca Raton, FL, 1985;
Vol. 4, p 65. (b) Garrett, B. C.; Truhlar, D. G.; Wagner, A. F.; Dunning,
T. W. J. Chem. Phys. 1983, 78, 4400. (c) Garrett, B. C.; Abusabi, N.; Kouri,
D. J.; Truhlar, D. G. J. Chem. Phys. 1985, 83, 2252. (d) Garrett, B. C.;
Joseph, T.; Truong, T. N.; Truhlar, D. G. Chem. Phys. 1989, 136, 271. (e)
Truhlar, D. G.; Garrett, B. C. J. Chem. Phys. 1987, 84, 365. (f) Garrett, B.
C.; Truhlar, D. G.; Grev, R. S.; Magnuson, A. W. J. Phys. Chem. 1980,
84, 1730.
(13) Lu, D.-H.; Truong, T. N.; Melissas. V. S.; Lynch, G. C.; Liu, Y.-P.; Grrett,
B. B.; Steckler, R.; Isaacson, A. D.; Rai, S. N.; Hancock, G. C.; Lauderdale,
J. G.; Joseph, T.; Truhlar, D. G. Quantum Chem. Prog. Exchange Bull.
1992, 12, 35.
(14) Roberts, R. M. G.; Ostovic, D.; Kreevoy, M. M. J. Org. Chem. 1983, 48,
2053.
(15) Kreevoy, M. M.; Lee, I.-S. H. Z. Naturforsch. 1989, 44a, 418.
(11) Lewis, E. S. J. Phys. Chem. 1986, 90, 3756.
9
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A R T I C L E S
Lee et al.
Table 1. Rate Constants and Equilibrium Constants for Reactions Represented by Equations 10-12
eq 10
eq 11
eq 12
Z
k a
K b
k a
K b
KRc,f
ij
ij
ij
ij
4-(CH3)2N
4-NH2
4-CH3O
H
4-CF3
3,5-Cl2
1.38 × 10-2 d
1.10 × 10-2 d
2.07 × 10-3 d
1.13 × 10-3 d
1.35 × 10-4 d
3.30 × 10-5
2.38 × 103 e
1.98 × 103 e
1.21 × 102 e
3.40 × 101 e
2.16f
2.38 × 10-3
2.28 × 10-3
7.04 × 10-3
1.38 × 10-2
2.71 × 10-2
3.33 × 10-2
4.3f
5.7f
5.37 × 10-11
1.00 × 10-10
7.59 × 10-10
2.69 × 10-9
1.95 × 10-8
1.62 × 10-7
8.5 × 101 e
3.0 × 102 e
5.0 × 103 e
2.5 × 104 e
4.07 × 10-1 f
a In units of M-1 s-1 with probable errors of 5-10%, as estimated in the Experimental Section. b Dimensionless. c In units of M. d These values were
taken from ref 15. e These equilibrium constants could not be measured directly, and were estimated by the method described in the text. Their reliability
is discussed in the text. f With probable error of ∼10%17
distinctive electronic spectra in the visible range, and the intensities of
these absorptions were monitored. A pH between 3.5 and 5.5 was
maintained in the reacting solutions by means of an acetic acid-acetate
buffer, with an ionic strength of 0.13.16 The pH of these solutions was
measured before and after k1 was measured, and the two determinations
never differed by more than 0.01.
3. Potential Surfaces
The potential energy surfaces are very similar to those
developed previously.7 They have the general form shown in
eq 13. VELEPS is an extended LEPS function. It, along with the
V ) VELEPS + VW + VCD + Vsolv
(13)
A correction of 0.17 was subtracted from electrometrically measured
pH values in the mixed solvent. The required correction was determined
by measuring the pH values of four HClO4 solutions, of known
concentration between 10-2 and 10-4 M, in the mixed solvent.
At least a 23-fold excess of quinoline derivatives was used for all
k1 measurements. Second-order rate constants, kij, were given by kij )
k1/C2f1f2. The stoichiometric concentration of quinoline derivatives is
C2. The quinolinium ions have solvolysis equilibria analogous to that
represented in eq 12. While the pKR of 3 is above 9 in the 2-propanol/
water solvent, and thus outside our present range of interest, the pKR
of 2 is 5.11.17 The fraction of 2 or 318 actually present as free
quinolinium ion was defined as f2, and calculated by means of the
appropriate equilibrium expression.17 Compounds 1Ha and 1Hb are
the conjugate bases of Brønsted acids, with pKa values of 4.17 and
4.06 in the 2-propanol/water mixture,15 so they were partially converted
to protonated forms which are inactive as reducing agents. The fraction
of 1H in the active form is defined as f1, and calculated from a
conventional equilibrium expression. Each kij measurement was repeated
4-14 times, and the standard errors of the mean values, as judged by
reproducibility, were 2-3%. However there are also systematic errors
and errors due to inaccuracies in KR values and Ka values. If KR and Ka
are uncertain by 10%,17that will introduce an uncertainty of 5-10%
into the kij values.
Several equilibrium constants, Kij, were evaluated by measuring
reaction rate constants in both directions.17 In the less favorable direction
these reactions come to equilibrium with substantial amounts of both
reactants still present. The method to obtain values of k1 for these
reactions was also previously described2 and second-order rate constants
were calculated from them as described above. Some of the equilibrium
constants, which could not be directly measured, were estimated by a
method that is described in the Results section.
Values of KR were determined from the electronic spectra of
compounds 1a-f, in solutions of measured pH, in the same 4:1
2-propanol:water mixture in which rates were measured. The technique
of these measurements has been previously described.17 The KR have
a standard error, based on their reproducibility, of ca. 5%. We are not
aware of any systematic error in the measurement of the present KR
values, so we believe that their accuracy is not much worse than their
statistical standard error.
widening term, expresses the covalent binding between the end
atoms and the in-flight hydride ion. They also express the
repulsion between the end atoms, which is the source of most
of the activation energy. VW is a barrier-widening function. It
augments the covalent binding and end-atom repulsion at an
inter-end-atom distance somewhat greater than that at the saddle
point. It has the effect of reducing the imaginary frequency,
and therefore reducing the effect of tunneling on the calculated
rate constants and isotopes effects. VCD is a term for charge-
induced dipole interaction; and Vsolv is a generalized Born term,
intended to approximate solvation energy, given by eq 14. All
of these terms except Vsolv have been carried over from the
qa2
ra
qaqb
1
1
1
2
Vsolv ) - 1 -
-
1 -
(14)
∑
1 ∑ ∑ r
(
)
(
)
2
ꢀ
ꢀ
a
a
b*a
ab
previous work.7 However, an error has been discovered in Vsolv
and this has been corrected. In Vsolv, qa is the charge on atom
a, ra is the van der Waals radius of atom a, rab is the internuclear
distance between atom a and atom b, and ꢀ is the dielectric
constant. The double summation has the effect of inappropriately
counting each atom-atom interaction twice. However this
double counting is removed when the double summation is
multiplied by 1/2. In the earlier work7 the double counting was
arbitrarily eliminated, but the multiplication by 1/2 was retained,
making the contribution of atom-atom interactions to Vsolv too
small (insufficiently negative). This was then compensated, in
good part, by using too large a ꢀ value (40) for the dielectric
constant. We have now corrected the counting error. The
potential function permits the three-atom model to give a good
representation of a large and diverse body of data for polyatomic
reactants.7 This is the source of its usefulness and the best
evidence that it is physically meaningful. To retain the original
fit, ꢀ, in Vsolv, was reduced from 40 to 1.4. This change leaves
V, and the rate constants calculated from it, almost unchanged,
without requiring us to recalibrate the other terms.
(16) Equations 10-12 all represent reactions of neutrals with cations, to give
cationic products or critical configurations. The acid dissociations of
protonated 1Ha and 1Hb do likewise. Therefore, the effect of ionic strength
on activity cancels out their rate or equilibrium expressions in first
approximation. The reported rate and equilibrium constants should be
approximately valid in infinitely dilute solution.
4. Results and Discussion
Table 1 gives kij and Kij for reactions represented by eqs 10
and 11. Many of the required equilibrium constants could not
be measured directly because the rates of the reactions from
(17) Ostovic, D.; Lee, I.-S. H.; Roberts, R. M. G.; Kreevoy, M. M. J. Org.
Chem. 1985, 50, 4206. The pKR values reported in this paper are all slightly
too large due to a systematic computational error.
(18) The active fraction of 3 was always 1.0 in these experiments.
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Hydride Transfer between NAD+ Analogues
A R T I C L E S
Figure 3. Brønsted plots for the reactions shown in eqs 10 (circles) and
11 (squares). The slopes, which are the values for the Brønsted R, are 0.67
( 0.03 and 0.32 ( 0.03, respectively.
Figure 2. The correlation between Kij, eq 10, and pKR. The least-squares
slope is 0.87 ( 0.04.
with a variety of heterocyclic ring systems, it appears to be valid
for remote substitution in a single ring system.2
right to left are too slow, and side reactions (possibly air
oxidation) intervene before a measurable fraction of the reaction
of interest has occurred. To estimate the rate constants for the
reactions represented by eq 10, right to left (kji), we take
advantage of the similarity of those reactions to the reactions
represented by eq 11, left to right. Two kji were measured for
reactions represented by eq 10. The more reliable of these is
for reaction of 1f with 2H. It has a value of 8.1 × 10-5 M-1
s-1. It was assumed that the effect of altering the 9-aryl
substituents on kji for eq 10 would be the same as the effects
these alterations would have on kij for eq 11. Thus we estimate
kji for 1e in eq 10 as kji (1f, eq 10) × kij (1e, eq 11)/kij (1f, eq
11). The value obtained in this way for kji (1e, eq 10) is 6.55 ×
10-5. When 1.35 × 10-4, kij for this reaction, is divided by kji,
6.55 × 10-5, a value of 2.1 is obtained. This may be compared
with the value of 2.2, obtained experimentally. The other
indirectly estimated Kij values in Table 1 were obtained in the
same way. Those for eq 11, right to left, use the analogy with
eq 10, left to right. The use of these analogies is equivalent to
assuming that eq 10, left to right, has the same Hammett F19 as
eq 11, right to left. And eq 11, left to right, has the same
Hammett F as eq 10, right to left. While this is probably not
exactly true, the two reactions are sufficiently similar that it
seems to be a good approximation. In either case, if the two F
values are different by 0.1, which seems larger than the
difference is actually likely to be for such similar reactions,19
errors up to 35% would be introduced into Kij.
The slight departure of the slope from unity may not be
significant, or it may be due to inaccuracy in the assumption
that was made in estimating Kij, but its direction is intuitively
reasonable. In eq 10 the positive charge replaces a C-H bond
of very little polarity, since the electronegativities of C and H
are similar.21 In eq 12 the positive charge replaces a C-O bond.
Since the oxygen has a much higher electronegativity than
carbon,21 the carbon to which it is attached in 4 has a small
partial positive charge. Thus the latter process would be a little
less sensitive to substituents than the former.
Figure 3 shows plots of ln kij against the ln Kij (Brønsted
plots) for the reactions shown in eqs 10 and 11. For the first of
these reactions the Brønsted plot is linear, as expected, and has
a slope, R10, of 0.67 ( 0.03. The second also should be linear
and it has been treated as such. It has a slope, R11, of 0.32 (
0.03. The slopes and their uncertainties, which are probable
errors, were evaluated by the method of least squares.22
For the reactions described by eq 10, a ø10 value, 0.49, can
be obtained from eq 8, and similarly, the ø11 value is 0.48.23
Combining these ø values with the R values based on all the
data, τ values of 0.64 ( 0.06 and 0.68 ( 0.08 are obtained.
These differ by much less than their combined probable error24
and give a mean value of 0.66 ( 0.05. The near identity of the
two τ values strongly supports eq 7, and the Marcus formalism
from which it is derived.
The reactions described by eq 10 have ln Kij values ranging
from -0.8 to 7.8, with an average of 3.5. Those described by
Table 1 also gives the measured values of KR. Figure 2 shows
the correlation between the log of Kij for reactions represented
by eq 10 and pKR. The good linear correlation, with a slope of
0.87 ( 0.04, provides additional support for the estimated values
of Kij. Because of the way they were calculated the log Kij values
for reactions represented by eq 11 must correlate with pKR
exactly as well as those for reactions represented by eq 10.
Values of KR have been used as a model for the equilibrium
constants of reactions such as those represented by eqs 10 and
11.2,20 While the model does not hold for a series of acceptors
(19) (a) Hammett, L. P. Physical Organic Chemistry, 2nd ed.; McGraw-Hill:
New York, 1970; Chapter 11. (b) Wells, P. R. Chem. ReV. 1963, 63, 171.
(20) Bunting, J. W.; Bolton, J. L. Tetrahedron 1986, 42, 1007.
(21) Levine, I. N. Physical Chemistry, 3rd ed.; McGraw-Hill: New York, 1988;
p 655.
(22) Topping, J. Errors of ObserVation and Their Treatment; Chapman and
Hall: London, 1962; pp 101-109.
(23) For each series of reactions its average Kij value was used, A typical λ
value of 88.5 kcal mol-1 was used for both. These approximations are quite
adequate for evaluating the small second term in eq 8 to the required
accuracy.
(24) Livingston, R. Physico Chemical Experiments, 3rd ed.; Macmillan: New
York, 1956; pp 22-30.
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J. AM. CHEM. SOC. VOL. 124, NO. 26, 2002 7759

A R T I C L E S
Lee et al.
eq 11 have ln Kij values ranging from 1.5 to 10.1, averaging
5.8. As shown above, their ø values are similar. Despite their
similar ranges of equilibrium constants (and the presumable
similarity of their critical configurations) the two series give
quite different R values. This result, which is entirely consistent
with Marcus theory, emphasizes the shortcomings of a one-
dimensional interpretation of R, which would identify a large
value with a critical configuration similar to the product structure
and a small value with a critical configuration similar to the
reactant structure. As described in the Introduction section, the
critical configuration for the present system is at the open cicle
in Figure 1. The distance in the critical configuration between
the carbon donating the hydride and the carbon accepting the
hydride increases as τ becomes smaller. In the present case it
is larger than it would be if τ were unity. Also, the in-flight H
takes on a partial negative charge as τ becomes less than unity.
Bernasconi and co-workers have provided many examples
of reactions in which the introduction of substituents in any
place in the reactants leads to a Brønsted relation (eq 6) but the
value of R varies, depending on the location of the substitution.25
Such observations have been identified with the imperfect
synchronization of the various changes leading from reactant
to product.25 The present observations can be regarded as another
example of this phenomenon; however, our treatment of them
is somewhat different from that developed by Bernasconi. We
now show how the two treatments are related, and attempt to
illuminate both of them.
Marcus theory, eq 2, has been shown to be generally
consistent with a linear triatomic model of the critical complex.5,7
Such a model has only two, orthogonal, internal coordinates:
the end atom breathing coordinate and the central atom
(hydrogen) displacement coordinate. After the separation of the
work term, Wr, the two variables of the Marcus theory are ln
Kij, or its energetic equivalent, and λij, both of which are related
to both coordinates of the model. Equations 8 and 9 define the
two new coordinates, ø and τ, in terms of ln Kij and λij. Each of
these depends, to a good approximation, on only one of the
model coordinates: τ on the distance between the end atoms
and ø on the donor-hydrogen distance. Equation 7 follows from
the Marcus equation and the definitions of ø and τ, eqs 8 and
9, without mathematical approximations.2-4 Equations 8 and 9
permit ø and τ to be evaluated from experimentally measurable
quantities, and τ can be evaluated in two, independent ways:
directly, using eq 9, or from eqs 7 and 8.
Figure 4. Brønsted plots of computed rate constants for Ai+ + AjH f
AiH + Aj+. Circle points were generated by varying the AjH bond strength,
using a bond strength of 73 kcal mol-1 in AiH. Square points were generated
by varying the AiH bond strength and using a bond strength of 73 kcal
mol-1 in AjH. The lower left-hand section of the figure corresponds to Figure
3.
of a model, short of very extensive (and expensive) quantum
mechanical and dynamic calculations.
The mean value of τ obtained in the present work, using eqs
7 and 8, is 0.66 ( 0.05. This is significantly less than the value,
0.81 ( 0.04, that was previously obtained by applying eq 9 to
hydride transfer reactions of structurally similar but simpler
classes of reactants.1,15 We now attempt to find the origin of
this discrepancy by examining computational results.
Three series were studied: one in which the donor C-H bond
strength was varied; one in which the acceptor C-H bond
strength was varied; and a series of symmetrical reactions in
which both bond strengths were varied. A τ value of 0.87 was
obtained by applying eq 9 to computational rate constants.7
However, τ depends slightly on the range of Kij° values used.
Using a range of Kij° values centering on 1.0, a value of 0.88
is obtained for τ. With these values, the critical configuration
of the triatomic model is located at the closed square in Figure
1. Also, Figure 4 shows Brønsted plots made with computed
rate constants, taken from ref 7. They are very nearly linear.
The R value is 0.56 if the donor bond strength is varied and
0.44 if the acceptor bond strength is varied. Since these Brønsted
plots are both centered on Kij ) 1.0 (a luxury not available for
the experimental results), ø is exactly 0.5 and the cross term in
eq 7 is exactly zero. This permits τ to be evaluated from each
R value, using eq 7. The two values are each 0.88, exactly the
same as the value obtained with eq 9. Linearized Marcus theory
applies very well to the computed rate constants. We conclude
that the discrepancy between the present and earlier values of
τ is not due to the use of eq 9 in one case and eqs 7 and 8 in
the other.
In a similar way τ and R can be evaluated from rate constants
computed for the model system. This permits the use of the
transparent, easily manipulated model as a surrogate for the
opaque experimental system. The price of this is loss of
flexibility. In effect, the electron distribution and geometry of
the model are completely specified when the values of the two
coordinates are specified, leaving no way to account for
redistribution of electrons due to structure changes within the
real, polyatomic donor and/or acceptor.
Bernasconi’s treatment always provides an evaluation of the
“transition state imbalance”. This quantity can be qualitatively
related to ideas such as resonance and solvation, which are
widely used to systematize organic chemistry.25 However it does
not appear possible to relate these imbalances to the properties
The τ given by computed rate constants, k, was shown to
decrease with decreasing donor and acceptor C-H bond
energy,7 but the effect is not large if the bond strength is kept
within reasonable limits. For example, τ decreases from 0.88
to 0.85 if the bond energies of donor and acceptor in sym-
metrical reactions are decreased from 73 kcal mol-1 to 58 kcal
(25) (a) Bernasconi, C. F. Acc. Chem. Res. 1992, 25, 9. (b) Bernasconi, C. F.
AdV. Phys. Org. Chem. 1992, 27, 119.
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7760 J. AM. CHEM. SOC. VOL. 124, NO. 26, 2002

Hydride Transfer between NAD+ Analogues
A R T I C L E S
Table 2. The Effect of Equilibrium C-C Distance on Computed
Rate Constants and τ Values for Symmetrical Hydride Transfers
the τ values obtained with the two different, Marcus-theory-
derived equations are identical, so the difference between the
two experimental values does not arise from the difference in
the method of evaluation. A change in the hydride affinity of
the acceptors and a steric effect, leading to an increase in the
donor-acceptor distance in the critical configuration, were also
mimicked computationally. Neither seems able to reproduce the
observed change in τ.
rCC(e) Å
ln k
ii
τ
KIE
rCC(‡),a Å
3.228
3.254
3.360
3.994
-5.18
-5.57
-7.52
-18.94
0.87
0.87
0.84
0.70
6.0
6.9
2.92
2.94
3.04
3.60
39.2
106
a The C-C distance at the potential energy saddle point.
Values of τ obtained from the model-calculated rate constants
appear to be consistently larger than those obtained from
experimental data, and more nearly constant. There does not
appear to be any acceptable explanation for this discrepancy
within the limits of the model and the framework of Marcus
theory. It seems likely to us that the source of the smaller, and
more variable, τ values is the possibility of delocalizing charge
within the real, polyatomic donors and acceptors. The 9-phe-
nylacridinium ions of the present study have an additional phenyl
group into which charge is delocalized. The τ ) 0.81 value
was obtained with all secondary carbocations.1 This difference
cannot be mimicked in the three-body model.
mol-1.7 This decrease in τ is accompanied by an increase of a
factor of 29 in rate constant and an increase in KIE from 6.2 to
10.3.7 Judged by the Kij values shown in Table 1, the hydride
affinities of the donors used in the present studies are not
strikingly different from those of previously used acceptors. And
no increase in the isotope effect was noted.26 We conclude that
the decreased value of τ cannot be attributed to decreased C-H
bond strength in the present examples. We now find that τ also
decreases gradually as the equilibrium C-C distance, rcc(e),
increases, but the calculated rate constant falls and the primary
kinetic isotope effect (KIE) rises very rapidly as the equilibrium
C-C distance is increased.26 To decrease the computed τ to
∼0.7 by increasing rcc(e), that distance would have to be
increased from 3.228 Å to ∼4.0 Å, which would reduce the
computed rate constant by a factor of ∼106 and increase the
KIE to 106! These results are shown in Table 2. The explosion
of the KIE at larger equilibrium C-C distance is mainly due to
tunneling.26 The KIE also increases substantially as the C-H
bond energy decreases.7
5. Conclusion
As indicated by Marcus theory of atom and group transfer,
the tightness of the critical configuration either increases or
decreases the Brønsted R for hydride transfer, depending on
the location of the structure change, in the donor or in the
acceptor. The size of the contributions differs somewhat from
expectations based on related measurements using somewhat
different compounds.
The discrepancy is not too large and oversimplification in
the theory seems to be responsible for the discrepancy. The
qualitative applicability of the theory is confirmed. The Brønsted
R is not a simple indicator of the similarity of the critical
configuration to reactant or product.
The decrease in the computed kij is due to the energetic cost
of the additional bond stretching that the longer donor-acceptor
distance requires.
The reactions shown in eqs 10 and 11 appear to be more
sterically hindered than those that were studied previously,1,2
which might lead to a longer donor-acceptor distance in the
critical configuration.5 However, the rate constants recorded in
Table 1 are only 1-2 orders of magnitude smaller than those
previously reported for reactions with similar Kij values. And
the primary kinetic isotope effects exhibited by the variants of
eq 10 are in the range 4.9-5.8,15 not significantly higher than
the isotope effects observed in previously studied systems.27
We conclude that steric hindrance, lengthening the donor-
acceptor distance in the critical configuration, also cannot
account for the lower values of τ we have now found.
Our model attempts to relate potential energy to the spacial
relations of a small number of atoms undergoing covalency
change. Trends in computed rate constants for hydride transfer
in a linear, triatomic system can be made to resemble the trends
observed in experimental rate constants for solution reactions.7
The computed rate constants are very well reproduced by
Marcus theory.7 Using the same sets of computed rate constants,
We believe that these conclusions are general, and this view
is supported by the previous observation that they apply to
methyl transfer reactions.3,11 There is no evident reason that
they should not also apply to enzyme to catalyzed reactions.
Acknowledgment. This work was supported by grant No.
R01-1999-00047 from the Interdisciplinary Research Program
of the KOSEF.
Note Added in Proof. Mayr and co-workers have applied
our theory successfully to hydride transfer reactions between
carbocations.28 Their τ value is larger than 1.0 (actually 1.24)
that leads to different values of R for substitution in the donor
and substitution in the acceptor. Their critical configuration
represents the tighter critical complex between the donor and
the acceptor and the positive charge on the in-flight hydrogen
as confirmed by their electronic structure calculation.
JA011855U
(26) Kim, Y.; Kreevoy, M. M. J. Am. Chem. Soc. 1992, 114, 7116.
(27) Ostovic, D.; Roberts, R. M. G.; Kreevoy, M. M. J. Am. Chem. Soc. 1983,
105, 7629.
(28) Wu¨rthwein, E.-U.; Lang, G.; Schappele, L. H.; Mayr, H. J. Am. Chem.
Soc. 2002, 124, 4084.
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J. AM. CHEM. SOC. VOL. 124, NO. 26, 2002 7761