Experimental and theoretical study of the secondary equilibrium isotope effect (SEIE) in the proton transfer between the pyridinium-d5 cation and pyridine

Article abstract of DOI:10.1021/jp035109i

In this work we present an experimental and theoretical study of the proton transfer from the pyridinium-d₅ cation to pyridine. FT-ICR measurements yield, at 331 K, an equilibrium constant K = 0.809 (Δ _rG_m^o = 0.58 kJ mol^-1) for the process, favoring the pyridine form. The structural and bonding changes on protonation of pyridine are analyzed by applying the atoms in molecules theory. As a consequence of electronic density redistribution, we found that on protonation the CN and the CC bonds placed farther from the nitrogen weaken. In addition, the CH and the CC bonds closer to the nitrogen increase their strength. Thermostatistical computation of the equilibrium constant from data obtained at the B3LYP/cc-pVTZ level, within the harmonic approximation, predicts a value of 0.827 (Δ_rG_m^o, value of 0.52 kJ mol^-1), in good agreement with the 0.809 ± 0.027 experimental result for a 99.9% confidence level. A simple statistical mechanical model intended to apply under conditions close to the present ones is developed. The model allows for a fine-tuning of the thermodynamic state functions for the equilibrium. This model shows that rather than by translational and rotational variations, the reaction is driven by the changes in zero point energies and in the density of vibrational states. In addition, theoretical analysis of the enthalpic and entropic contributions shows that the Δ_rG_m^o value is determined by the enthalpic part. It is also predicted that the Δ_rG _m^o value decreases with temperature. We found that this effect is due to a higher density of vibrational states in the pyridine-d ₅ form. A new model is developed to correct the vibrational partition function for anharmonicity. This model shows that correction for anharmonicity in the low-frequency modes reduces significantly the difference between calculated and experimental K values.

Full text of DOI:10.1021/jp035109i

6160
J. Phys. Chem. A 2003, 107, 6160-6167
Experimental and Theoretical Study of the Secondary Equilibrium Isotope Effect (SEIE) in
the Proton Transfer between the Pyridinium-d5 Cation and Pyridine†
C. Mun˜oz-Caro* and A. Nin˜o
Grupo de Qu´ımica Computacional, Escuela Superior de Informa´tica, UniVersidad de Castilla-La Mancha,
Paseo de la UniVersidad 4, 13071 Ciudad Real, Spain
J. Z. Da´valos, E. Quintanilla, and J. L. Abboud
Instituto de Qu´ımica F´ısica “Rocasolano”, Consejo Superior de InVestigaciones Cient´ıficas, Serrano 119,
28006 Madrid, Spain
ReceiVed: April 24, 2003; In Final Form: June 18, 2003
In this work we present an experimental and theoretical study of the proton transfer from the pyridinium-d5
cation to pyridine. FT-ICR measurements yield, at 331 K, an equilibrium constant K ) 0.809 (∆rG° ) 0.58
m
kJ mol-1) for the process, favoring the pyridine form. The structural and bonding changes on protonation of
pyridine are analyzed by applying the atoms in molecules theory. As a consequence of electronic density
redistribution, we found that on protonation the CN and the CC bonds placed farther from the nitrogen weaken.
In addition, the CH and the CC bonds closer to the nitrogen increase their strength. Thermostatistical
computation of the equilibrium constant from data obtained at the B3LYP/cc-pVTZ level, within the harmonic
approximation, predicts a value of 0.827 (∆rG° value of 0.52 kJ mol-1), in good agreement with the 0.809
m
( 0.027 experimental result for a 99.9% confidence level. A simple statistical mechanical model intended to
apply under conditions close to the present ones is developed. The model allows for a fine-tuning of the
thermodynamic state functions for the equilibrium. This model shows that rather than by translational and
rotational variations, the reaction is driven by the changes in zero point energies and in the density of vibrational
states. In addition, theoretical analysis of the enthalpic and entropic contributions shows that the ∆rG° value
m
is determined by the enthalpic part. It is also predicted that the ∆rG° value decreases with temperature. We
m
found that this effect is due to a higher density of vibrational states in the pyridine-d5 form. A new model is
developed to correct the vibrational partition function for anharmonicity. This model shows that correction
for anharmonicity in the low-frequency modes reduces significantly the difference between calculated and
experimental K values.
Introduction
Within the framework of the Born-Oppenheimer approxima-
tion, isotopic substitution does not modify the potential energy
hypersurface of the system. The effect of the isotopic substitution
on a chemical system can be accounted for in terms of the
variation of masses. Thus, it is the change in the translational,
rotational, and especially, vibrational behavior that determines
the difference in properties. Therefore, the thermostatistical
determination of the equilibrium constant is the basis for the
treatment of EIE. A similar treatment, within the transition state
theory, represents the basis for the KIE. With simplified
assumptions, the classical Bigeleisen treatment of the isotope
effect is performed in this way.3
The term isotope effect refers to the change in chemical or
physical properties between chemical species differing in their
isotopic composition. Thus, it is the simplest substitution that
we can make in a compound. Despite its simplicity, isotopic
substitution has a clear and observable effect in the molecular
properties. With respect to a chemical reaction, this effect can
be observed in equilibrium conditions, the equilibrium isotope
effect (EIE), or in the kinetics of the reaction, the kinetic isotope
effect (KIE). The first one is measured as the ratio of equilibrium
constants (with and without isotopic substitution), whereas the
second is defined as the ratio of rate constants. In this context,
isotope effects are further classified as primary or secondary.
In primary isotope effects (PIE), the bonds broken or formed
in the reaction involve the isotopic species. Thus, especially
for the pair hydrogen/deuterium, the PIE is clear, corresponding
to a few percent of the observed property.1 On the other hand,
in secondary isotope effects (SIE) the isotopic substituted atoms
are not involved in the breaking of forming of bonds. Therefore,
the SIE has a much smaller effect in the reaction than does
PIE.2
The first experimental proof of a kinetic isotope effect dates
back to the discovery in 1932 by Washburn and Urey that
deuterium is enriched in the liquid phase in the electrolysis of
water.4 Isotope effects can be very large, as for instance the
secondary EIE for the reduction of aromatic hydrocarbons to
their radical anions.5 In this context, theoretical treatments are
found invaluable in the rationale of these phenomena.6 Isotope
effects have also been observed in such reactions as H-atom
transfer and H2-molecule elimination.7 With respect to these
cases, the isotope effects in tunneling are pronounced, because
the difference in mass is important, such as in the hydrogen/
deuterium couple. In some problems of this kind, the variational
† This work is dedicated to the memory of Dr. V. Botella.
* Corresponding author. Fax: 34-926-295354. E-mail: quimcom@uclm.es.
10.1021/jp035109i CCC: $25.00 © 2003 American Chemical Society
Published on Web 07/23/2003

Proton Transfer between Pyridinium-d5 Cation and Pyridine
J. Phys. Chem. A, Vol. 107, No. 32, 2003 6161
transition state theory and the multidimensional semiclassical
tunneling method have been useful.8 Large KIE have also
experimentally detected for the couple 12C/13C in the reaction
of methane with chlorine atoms,9 which is so important in the
chemistry of the atmosphere.
In this work we present the experimental determination and
the theoretical treatment of the secondary equilibrium isotope
effect (SEIE) for the proton transfer between the pyridinium-
d5 cation and pyridine. Thus, the different molecular species
involved are structurally characterized. In addition, the bonding
changes of pyridine on protonation are also taken into account.
Figure 1. Equilibrium for the proton transfer between the pyridinium-
d5 cation and pyridine.
in its (lack of) environment. Thus, we use the correlation
consistent triple-ú plus polarization basis set, cc-pVTZ. This
basis includes as polarization functions, up to d functions on
hydrogens and up to f functions on heavy atoms.13 Correlation
energy is accounted for by means of density functional theory.
Thus, we use the Becke’s three-parameter hybrid functional,
with gradient corrections provided by the LYP correlation
functional: B3LYP.14 The three parameters of the B3LYP
method are adjusted to reproduce the equilibrium data of the
G1 molecule set.15 Thus, although the DFT methology only
partially accounts for the correlation energy, the results for
equilibrium properties can be expected to be very reliable, as
we have previously discussed.16 Also, this basis has been tested
by Szafran and Koput in neutral pyridine and its isotopomers,
obtaining good agreement with experimental data for geometry,
dipole moment, and frequencies.17 All the electronic structure
calculations have been carried out with the Gaussian 98
package.18 The atoms in molecules theory is applied with the
Morphy 98 program.19 Thermodynamic properties are deter-
mined statistically from the geometry and harmonic frequencies
computed at the fully relaxed structures. Thermodynamic
properties and decomposition of the partition functions are
obtained by means of the PARTI program.20
The Gibbs energy variation, ∆rG°, is measured by Fourier
m
Transform Ion Cyclotron Resonance (FT-ICR) spectroscopy.
∆rG° is also thermostatistically determined and analyzed. The
m
variation of the equilibrium constant with the temperature is
determined, as well as the enthalpic and entropic contributions.
Finally, the effect of anharmonicity on the computed thermo-
dynamic values is also taken into account. The small magnitude
of the equilibrium constant makes this problem an interesting
test case to determine the suitability of present theoretical tools
to describe and interpret small physical effects.
Experimental Section
A. FT ICR Spectrometer. The study was performed on a
modified Bruker CMS-47 FT-ICR mass spectrometer10 already
used in previous studies,11 and the spectra were acquired using
an IonSpec Omega Data Station. (IonSpec, Irvine, CA). The
high field strength of the superconducting magnet (4.7 T) easily
allows the monitoring of the ion-molecule reaction for rela-
tively long reaction times.
B. Reagents. Pyridine (Py, purity >99.9%), pyridine-d5 (Py-
d5, isotopic purity >99.6%), and H2S (purity >99.5%) were
purchased from Aldrich and used without further purification.
No impurities were detected in the experiments.
Results and Discussion
The equilibrium reaction for the proton-transfer process is
shown in Figure 1. As written, we have four species; pyridine
(Py) plus pyridinium-d5 ((Py-d5)-H+) cation, to yield pyri-
dinium cation (Py-H+) plus pyridine-d5 (Py-d5). From the
experimental standpoint, the key problem is the accurate
determination of the dimensionless equilibrium constant K for
reaction 1, defined through eq 2.
C. Sample Preparation. Mixtures of both pyridines (Py and
Py-d5), covering the range of molar ratios (nPy-d /nPy) 0.1450-
5
0.36451, were prepared by weight. Typically, samples of these
compounds (masses in the range 5-20 mg) were enclosed and
sealed under nitrogen in separated thin-walled glass ampules.
Masses were determined within (2 µg using a Mettler AT-21
microbalance. The ampules were introduced into a 1-L glass
balloon, and the system was degassed for several hours down
to pressures of ca. 10-7 mbar. At this point, degassing was
stopped and the ampules broken. After 1 h, the gaseous mixture
was allowed to enter the high-vacuum section of the spectrom-
eter.
Py + (Py-d5)-H+ / Py-H+ + Py-d5
K ) [(PPy-H+/P +)][P(Py-d )/PPy]
(1)
(2)
(Py-d5)-H
5
The ratio of the partial pressures of the neutral species, PPy-d
/
5
PPy, is given by their molar ratio, nPy-d /nPy. The latter is known
5
D. Experimental Conditions. The temperature of the cell
was 331 ( 2 K as determined using a calibrated platinum
thermometer (platinum resistor) located close to one of the
repeller plates of the ICR cell. The experimental setup was
similar to the one described by Fridgen et al.12 Nominal total
pressures of the mixtures of pyridines were in the (0.5-5) ×
10-8 mbar range, as indicated by the Bayard-Alpert ionization
gauge (Balzers IMR-132). H2S was added up to nominal
pressures of (3-4) × 10-7 mbar. Electron ionization of H2S
was carried out using nominal energies of 11-12 eV. Under
these conditions, H3S+ was readily generated and used as a clean
proton source. H2S also plays the role of a thermal bath.
When necessary, ion selection was achieved using both broad-
band elimination and single shots.
to better than 4 × 10-4 by using gaseous mixtures of the neutral
species prepared by weight (see Experimental Section). The ratio
of the partial pressures of the ions Py-H+ (m/z ) 80 Da) to
(Py-d5)-H+ (m/z ) 85 Da), PPy-H /P(Py-d )-H , is taken as the
+
+
5
ratio of their relative intensities, I80/I85, as determined by Fourier
transform ion cyclotron spectroscopy (FT ICR).21 ICR has long
been used for the purpose of measuring kinetic22 and thermo-
dynamic23 isotope effects in ion-molecule reactions.
Although full details are given in the Experimental Section,
here we mention the main features of the experiments.
Mixtures of Py-d5, Py, and a large excess of H2S are subject
to electron ionization. The main processes initially taking place
are thus reactions 3a and 3b. H3S+(g) is then isolated using
H2S(g) F H2S•+(g) + e-(g)
(3a)
(3b)
Theory
H2S•+(g) + H2S(g) + F HS•(g) + H3S•+(g)
Because the measurements are made in a vacuum, we use
electronic structure methods to represent the molecular systems

6162 J. Phys. Chem. A, Vol. 107, No. 32, 2003
Mun˜oz-Caro et al.
where the canonical partition function is obtained as Q ) qN/
N!, in terms of the molecular partition function, q. In addition,
and taking into account that the potential energy hypersurface
is the same for the couples (Py, Py-d5) and (Py-H+, (Py-d5)-
H+), the statistical expression for the equilibrium constant adopts
the simple form
K ) [q(Py-d5)‚q(Py-H+)]/[q(Py)‚q((Py-d5)-H+)] (5)
where q represents the molecular partition function obtained as
the product of the translational, qt, rotational, qr, and vibrational,
qv, partition functions. Due to the stoichiometry of the problem,
the equilibrium constant is independent of the pressure.
Equation 5 can be simplified in the following form. In the
framework of the Born-Oppenheimer approximation, the
molecular partition functions in eq 5 are obtained as the product
of the translational, rotational, and vibrational partition functions.
In particular, for the vibrational partition function we have
Figure 2. Example result of the experimental (I80/I85) ratio evolution.
3N-6
3N-6
TABLE 1: Experimental determinination of the equilibrium
exp[-νi/2kT]
exp[-ν (V + 1/ )/kT] )
i i 2
constant, K, for reaction 1 at 331 ( 2 K
q )
∏ ∑
∏
ν
a
a
a
1 - exp[-νi/kT]
i
Vi)0
i
nPy-d5/nPy
Kselected80
Kselected85
Knoselection
(6)
0.1450
0.3036
0.3036
0.3320
0.9919
0.9919
1.8637
2.1794
3.6451
0.7606
0.8283
0.8231
0.7730
0.7812
0.8287
0.7534
0.8474
0.7867
0.7480
0.8098
0.8324
0.7534
0.8350
0.8825
0.7966
0.8445
0.8250
0.7474
0.8045
0.8091
0.7438
0.8281
0.8367
0.7330
0.8488
0.7647
Rather than using the usual closed form, the simplest treatment
considers that for standard conditions only the lowest vibrational
energy level is populated, and thus
3N-6
q )
exp[-ν /2kT]
(7)
∏
ν
i
i
av value, K ) 0.809; sd)0.040
a Defined in the text.
or, in terms of the zero point energy (ZPE):
qν ) exp[-ZPE/2kT]
(8)
ion-selection techniques and allowed to protonate Py(g) and Py-
This treatment is essentially that carried out by Bigeleisen and
Goeppert-Mayer.3a Substituting eq 8 in eq 5 and using the usual
semiclassical expressions for the translational and rotational
partition functions,27 we obtain
d5(g). H3S+(g) is a clean proton source inasmuch as the gas-
phase basicity24 of H2S (673.8 kJ mol-1 25 is 224.3 kJ mol-1
)
lower than that of Py (898.1 kJ mol-1).25 The low hydrogen-
bonding basicity of H2S is most valuable because, even in the
presence of relatively high pressures of this compound, the
formation of species such as Py-H+‚‚‚SH2 is not observed.
Reaction 1 is then monitored for periods of time of up to 120
s. The evolution of the ratio (I80/I85) is examined under three
different experimental conditions: (i) free evolution, (ii) selec-
tion of (Py-d5)-H+, (iii) selection of Py-H+. An example of
such a study is presented in Figure 2.
3/2
1/2
m
mPy-H
(IaIbIc)Py(I I I )
a b c (Py-d5)-H
+
+
Py-d5
K )
m m
[
] [
]
(I I I )
(IaIbIc)Py-H
+
+
Py (Py-d5)-H
a b c Py-d5
exp[(ZPEPy + ZPE(Py-d )-H+ - ZPEPy-H+ - ZPEPy-d )/kT]
5
5
(9)
In the above expression, m represents the molecular mass, and
Ia, Ib, Ic the principal inertial moments. Equations 5 and 9 will
be used and compared to determine our equilibrium constant.
Equation 9 can be decomposed considering each term appearing
to the right of the equal sign. Thus, we can define a translational,
Kt, rotational, Kr, and vibrational, Kv, contribution to the K
constant as
It is clear that equilibrium (1) is reached in all three cases.
Furthermore, these experiments provide limiting values of the
ratio (I80/I85), which agree with a standard deviation of 5%. The
experimental database is presented in Table 1, where as can be
seen, the ratio (nPy-d /nPy) was varied by a factor of 25 and
5
some experiments were repeated using the same ratio (nPy-d
nPy) and different total pressures.
/
5
The average value of K derived therefrom equals 0.809 with
a confidence interval of 0.027 at the 99.9% confidence level26
at 331 ( 2 K.
Because the experimental data are obtained in a vacuum at a
very small pressure, we can use the ideal gas model as a good
approximation. Thus, the statistical expressions for H, S, and
G for our species are given by
3/2
m
mPy-H
+
Py-d5
Kt )
[
mPym(Py-d )-H
]
+
5
1/2
(IaIbIc)Py(I I I )
+
a b c (Py-d5)-H
Kr )
[
]
(I I I )
(IaIbIc)Py-H
+
a b c Py-d5
Kν ) exp[(ZPEPy + ZPE(Py-d )-H+ - ZPEPy-H
-
+
H ) kT2(∂ ln Q/∂T)V + kTV(∂ ln Q/∂V)T
S ) kT(∂ ln Q/∂T)V + k ln Q
5
ZPEPy-d )/kT] (10)
5
The above expressions show that Kt and Kr are independent of
G ) -kT ln Q + kTV(∂ ln Q/∂V)T
(4)
temperature.

Proton Transfer between Pyridinium-d5 Cation and Pyridine
J. Phys. Chem. A, Vol. 107, No. 32, 2003 6163
Figure 3. Numbering convention for pyridine and related ions. The
figure shows the structure of the pyridinium cation obtained at the
B3LYP/cc-pVTZ level.
Figure 4. Electron density difference map, on the molecular plane,
TABLE 2: Structural Data for Pyridine (Py) and
+
between pyridine and the pyridinium cation: ∆F ) FPy - FPy-H
.
Pyridinium Cation (Py-H+)a
Electron density in atomic units (e/bohr3). Interval between isodensity
lines 0.015 e/bohr3. Data obtained from the fully optimized results.
Because the nuclear positions do not match exactly in both molecules,
the results around the nuclear positions must be considered semiquan-
titative. Positive values (shadowed zone) indicate higher electron density
in pyridine, whereas negative values correspond to a higher electron
density in the pyridinum cation.
g
h
Pyb
Pyc,d
Pye Py-H+ c Py-H+ f Fa
Fa
N-C2
1.338
1.394
1.392
1.087
1.083
1.082
1.333
1.390
1.388
1.084
1.081
1.082
1.340
1.392
1.391
1.083
1.081
1.081
1.347
1.377
1.393
1.079
1.079
1.081
1.013
1.337 0.351 0.326
1.371 0.326 0.334
1.390 0.325 0.324
1.073 0.298 0.303
1.072 0.294 0.300
1.075 0.295 0.301
C2-C3
C3-C4
H7-C2
H8-C3
H9-C4
H12-N
1.001
123.1
0.350
shows that, with respect to pyridinium, our results exhibit longer
bond distances than the previous data obtained at the Hartree-
Fock (HF) level. This fact is due to the lack of correlation energy
at the HF level, which is known to lead to shorter bond distances
as a consequence of a too small electron repulsion.
Our results show that on protonation, the C2-C3 (C5-C6)
and the C-H distances decrease, whereas the C3-C4 (C4-
C5) and the N-C distances increase. The reason for these facts
is visualized in Figure 4, which represents the electron density
C6-N-C2 116.9
C5-C4-C3 118.4
C4-C3-C2 118.5
H7-C2-N 116.0
H8-C3-C4 121.4
N-C4-H9 180.0
C2-N-H12
117.9
116.7
123.2
118.6
118.5
116.1
121.2
180.0
118.3
118.7
115.9
120.9
180.0
120.1
119.1
116.8
121.4
180.0
118.4
120.4
118.6
116.8
117.8
180.0
118.5
a Distances in ångstroms and angles in degrees. The table also
includes the value of the electron density, Fa, in atomic units (e/bohr3),
at the bonded critical points. b Experimental data from microwave
spectroscopy.29 c This work, data obtained at the B3LYP/cc-pVTZ
level. d Work of Szafran and Koput,17 data obtained at the B3LYP/cc-
pVTZ level. e Data obtained at the MP2/cc-pVTZ level.17 f Data
obtained at the RHF/6-31G(d,p) level.30 g Pyridine. h Pyridinium cation.
difference between pyridine and pyridinium (∆F ) FPy
-
+
FPy-H ) in the molecular plane. We observe a negative zone
affecting the shortened bonds, i.e., higher electron density in
the pyridinium cation. On the other hand, a positive difference
appears in the elongated bonds, i.e., electron density is smaller
in the pyridinium form.
Nowadays, eq 4 or 5 can be easily computed from the results
of any of the available electronic structure software packages.
In addition, its validity is more general than eq 9, because,
depending on the fundamental frequency and the kT factor,
vibrational energy levels with quantum number V > 0 can be
populated. On the other hand, eq 9 gives a more direct insight
on the mechanisms responsible for the EIE. In particular, it
shows clearly the role played by the change in zero point energy.
Both eqs 5 and 9 should be applied with caution when
anharmonic, large amplitude vibrations do exist. The effect of
these vibrations in the partition function can be, sometimes,
important.20b,28
To compute the partition functions, the structures of pyridine
and the pyridinium cation were fully optimized at the B3LYP/
cc-pVTZ level. For the starting geometries, we have used the
microwave experimental data for pyridine obtained by Villa el
al.29 The numbering convention is shown in Figure 3. After
relaxation of the geometry we found, as expected, a planar (C2V)
structure for each molecule. Because the potential energy
hypersurfaces are identical, the optimized structures have been
used for the deuterated species. Due to the lack of structural
data for pyridinium, we have collected our results in Table 2,
as well as those for pyridine. These results are compared with
the experimental and theoretical data found in the literature.
The optimized geometry determined for pyridine compares
well with the microwave data collected in Table 2. Table 2 also
To quantify these observations, the atoms in molecules (AIM)
theory is applied to the protonated and neutral forms.31 AIM
characterizes a bond by an atomic interaction line (AIL), which
is a line through the electron density, F, along which F is a
maximum with respect to any neighboring line. On the AIL we
found the bond critical point (BCP), a second-order saddle point
where F reaches a minimum. The value of the electronic density
at the BCP for a given bond, Fa, can be correlated to the concept
of bond order,31 with higher values of Fa corresponding to
stronger bonds. Table 2 collects the Fa values for the different
bonds found in pyridine and the pyridinium cation. We observe
that on protonation the Fa value increases for the C2-C3
(C5-C6) and the different C-H distances, which corresponds
to an increase of the bond strength. Accordingly, a decrease,
corresponding to a weakening of the bond, is observed on
protonation for the N-C and C3-C4 (C4-C5) bonds. Thus,
protonation subtracts electron density from the N-C bonds.
Therefore, the electron delocalization in pyridine is weakened.
At the equilibrium position, we obtain the rotational constants
and the harmonic frequencies from a normal modes analysis.
The results are collected in Table 3. The effect of deuteration
is clearly reflected in the value of the frequencies. In all cases,
the harmonic frequencies for the deuterated species are smaller
than for the common ones. This is due to the increase of reduced

6164 J. Phys. Chem. A, Vol. 107, No. 32, 2003
Mun˜oz-Caro et al.
TABLE 3: Molecular Mass (amu), Rotational Constants
(GHz), Harmonic Frequencies (cm-1), and Zero Point
Energies, ZPE (cm-1), for the Protonated and Neutral
TABLE 4: Computed ∆rH°, ∆rS°, ∆rG°, and K Values, at
m
m
m
Different Temperatures, for the Proton-Transfer Reaction
between Pyridine-d5 and Pyridinea
a
Forms of Pyridine and Pyridine-d5
298.15 K
331 Kb
0.57
498.15 K
Py-H+
Py-d5
Py
(Py-d5)-H+
∆rH°c (kJ mol-1
)
)
0.57
0.54
m
∆rS°c (J mol-1
)
0.13
0.53
0.806
0.805
0.13
0.52
0.827
0.822
0.824-0.831
0.813-0.825
0.08
0.50
0.886
0.879
M
A
80.05
5.833
84.0735
5.127
79.0422
6.077
85.0814
5.005
m
∆rG°c (kJ mol-1
m
Kd
Ke
Kf
B
C
5.713
2.886
5.007
2.533
5.863
2.984
4.831
2.458
ν1
393.5
405.9
624.5
647.3
682.7
753.3
859.2
889.9
329.1
380.3
538.8
594.5
641.2
648.2
700.8
786.1
838.6
838.7
839.1
856.4
863.6
908.6
984.4
1027.4
1063.7
1274.9
1331.4
1374.1
1583.6
1584.9
2318.4
2324.9
2341.5
2356.5
2368.5
385.5
421.7
617.2
670.8
721.6
769.1
900.2
964.5
1011.3
1012.3
1023.6
1051.2
1079.5
1096.3
1172.9
1243.7
1283.2
1390.8
1476.8
1517.8
1620.9
1626.3
3142.8
3145.6
3169.4
3184.5
3192.8
350.1
354.2
530.1
601.3
620.4
640.2
693.3
774.2
834.5
841.3
852.9
867.8
876.8
886.7
897.5
984.4
1026.5
1047.7
1247.7
1351.4
1372.8
1505.3
1619.8
1620.6
2367.4
2379.8
2384.5
2397.1
2400.6
3552.6
18939.75
Kg
ν2
ν3
a ∆rH°, ∆rS°, and ∆rG° have only two significant decimal figures.
m
m
The experimenmtal value for the reaction constant at 331 K is 0.809 (
0.027. b Temperature used in the experimental measurements. c Data
obtained with eq 4. d Data obtained with eq 5. e Data obtained with eq
9. f Data obtained with fixed correction factors (0.99-0.95). g Data
obtained with eq 14. Intervals [0.95-1]-[0.99-1].
ν4
ν5
ν6
ν7
ν8
ν9
1004.0
1019.8
1021.9
1052.9
1063.2
1082.0
1085.3
1201.4
1228.9
1291.0
1360.0
1422.2
1524.6
1580.2
1647.2
1668.2
3201.7
3218.8
3220.4
3230.5
3232.0
3552.7
22582.60
ν10
ν11
ν12
ν13
ν14
ν15
ν16
ν17
ν18
ν19
ν20
ν21
ν22
ν23
ν24
ν25
ν26
ν27
ν28
ν29
ν30
ZPE
TABLE 5: Translational, qt, Rotational, qr, and Vibrational,
qv, Contributions to the Molecular Partition Function for the
Different Species Considereda
Py-H+
Py-d5
Py
(Py-d5)-H+
qt 0.2202 × 1032
qv 0.4964 × 10-42 0.3499 × 10-29 0.3969 × 10-36 0.5316 × 10-35
a Data at 331 K.
0.2370 × 1032
0.2160 × 1032
0.2412 × 1032
qr 0.5164 × 105
0.6295 × 105
0.4923 × 105
0.6586 × 105
with the value obtained from eq 5. Thus, the vibrational
contribution to K, in the harmonic approach, seems to be driven
by the ZPE. In addition, the independence of Kt and Kr with
temperature (see eq 10) shows clearly that the variation with
temperature is due solely to the vibrational contribution.
To obtain a more detailed picture of how the change of
molecular properties determines the value of the equilibrium
constant, we have decomposed the molecular partition function
for each molecule. The results are collected in Table 5. We can
see that for the protonated forms the vibrational contribution is
much smaller than for the neutral form. On the other hand, for
the deuterated forms, the vibrational contributions are larger
than in the neutral forms. This fact can be explained as a
consequence of the higher density of states arising from the
decrease in the fundamental frequencies of vibration; see Table
3. Table 5 also shows that the higher vibrational contribution
is associated with pyridine-d5, Py-d5. This fact is a consequence
of the smaller ZPE value of Py-d5, with respect to the other
molecular systems; see Table 3. With a smaller ZPE, we can
expect Py-d5 to exhibit a higher density of states. An increase
of temperature will populate more vibrational states in all the
compounds, resulting in an increase in all the vibrational
partition functions. However, due to its higher density of states,
15849.10
19446.15
a The frequencies are ordered by increasing value. Data obtained at
the B3LYP/cc-pVTZ level.
masses. The phenomenon is specially relevant for the five
(neutral species) or six (protonated species) last frequencies,
which, as determined from the normal modes composition,
correspond to stretching motions of the hydrogens. The last
frequency, ν30, for the protonated species corresponds to the
pure proton stretching, and it does not have a counterpart in
the neutral form. Because the variation of the reduced mass in
deuteration is small, the frequency for this mode, when going
from pyridinium to pyridinium-d5, is almost the same in both
molecules.
From the data in Table 3 and using eqs 4-6, we compute
∆rH°, ∆rS°, and ∆rG° at increasing temperatures. The results
the vibrational partition function of Py-d5, q
will contribute
Py-d5,
m
m
m
more than any other to the variation of the equilibrium constant.
Because q appears in the numerator in the expression for
are shown in Table 4. Our theoretical result for the equilibrium
constant at 331 K, 0.827 (∆rG° ) +0.52 kJ mol-1), compares
Py-d5
m
K (see eq 4), the increase in temperature is reflected in the
increase of K.
well with the average experimental value, 0.809 (∆rG° ) 0.58
m
kJ mol-1). In fact, the theoretical value is within the (0.027
((0.09 for ∆rG°), 99.9% confidence interval of the experi-
It is of interest to consider the factors affecting the agreement
between the theoretical and experimental results within the
confidence interval of the measure. Thus, we apply eq 10 at
331 K. In this form, we can analyze the effect of the
translational, Kt, rotational, Kr, and vibrational, Kv, contributions.
From the data in Table 2, Kt and Kr are found 1.0011 and 1.0025,
respectively. However, Kv amounts to 0.819. These results show
that Kt and Kr predict the inverse trend for our reaction, Figure
1. Kv is the factor responsible for the correction of this trend.
The present values of Kt, Kr, and Kv indicate that some or several
of these contributions must change to predict the observed K
m
mental measure. These values show that in the equilibrium, the
actual reaction (see Figure 1) is displaced to the left. The
evolution with temperature (see Table 4) shows that the increase
of temperature increases the calculated K value. The data in
Table 4 also show that this effect is due to the enthalpic
contributions, because of the small value of the entropic
counterpart. Table 4 also collects the approximate results
obtained using eq 9. In practical terms these results are
equivalent to those obtained with eq 5. In particular, the K value
at 331 K agrees, within the experimental confidence interval,

Proton Transfer between Pyridinium-d5 Cation and Pyridine
J. Phys. Chem. A, Vol. 107, No. 32, 2003 6165
value. Equation 7 shows that the higher influence can be
attributed to the vibrational contribution, due to the effect of
3N - 6 degrees of freedom appearing in an exponential form.
Even using the closed form for the harmonic vibrational partition
function, eq 6, we reach the same conclusion. Thus, the
vibrational contribution must be more realistically described to
account for an effect as small as the SEIE in our reaction.
It is well-known that the harmonic approach leads to
frequencies that are higher than the actual (anharmonic) ones.
For this reason it has become customary to correct (decrease)
the harmonic results obtained by electronic structure methods
by a fixed amount determined by comparison with experimental
data. This amount varies as a function of the level of theory
and basis set used, but for the correlated level, with the B3LYP
density functional, it is usually in the interval [0.99-0.95].32
Applying correction factors of 0.99 and 0.95 to the fundamental
frequencies of vibration in eq 6, we obtain K values of 0.824
and 0.831, respectively. Thus, the equilibrium constant increases
from the original 0.827 value, and the difference with the
average experimental result, 0.809, increases. This increase is
in the direction of the confidence interval upper limit (0.836).
Because the computed results are within experimental error,
drawing any definitive conclusion is difficult. Anyway, it is of
interest to determine the effect of a more reliable correction of
the harmonic model.
Anhamonicity is translated in a set of vibrational energy levels
different from those predicted by the harmonic model. In a pure
stretching mode, a simple Morse oscillator shows that the density
of states increases. The net result is a higher contribution to
the partition function. For other anharmonic, large amplitude
vibrations, such as ring puckerings or torsional motions, the
pattern of energy levels has an effect on the partition function
not so easily interpreted. However, this effect is small for a
high-frequency mode at room temperature, because only the
vibrational ground state is actually populated. The effect will
be more important for low-frequency modes where the separa-
tion between levels would be not far from the kT factor. Thus,
to analyze the effect of anharmonicity on Kv we need to identify
the modes where only the ground state is populated.
Figure 5. Graphical representation of the population of vibrational
energy levels, fv, for a given fundamental vibration frequency, ν0, at a
temperature of 331 K. The isocontour lines represent vibrational energy
levels.
the ground state is populated. In other words, for these modes
we will consider that the contribution to the vibrational partition
function is just given by the ZPE.
To analyze, at least qualitatively, the effect of anharmonicity
on Kv, at 331 K, we will apply the following approach. First,
the set of vibration modes is organized in two blocks, one up
to 1100 cm-1, and another from this limit up. Second, we
compute separately the contribution of each block to the
vibrational partition function. Thus, for the high-frequency
modes we consider only the ZPE. On the other hand, for the
low-frequency modes, we will retain the harmonic model, but
due to the population of several vibrational energy levels, we
will apply the closed formula, eq 6, for the vibrational partition
function. In addition, the effect of anharmonicity on the density
of states is especially important for the low-frequency vibrations.
To include (roughly) this effect, we will apply correction factors
in a given interval [x-1.0] to the low-frequency modes. Here,
x is restricted to the previously considered interval of correction
factors for harmonic frequencies (0.99-0.95). To represent the
(a priori) decreasing density of states with the frequency, we
consider the correction factor, fi, as a function of the frequency,
νi. Thus, expanding in a Taylor series, we have
To have an indication of the last significantly populated
energy level at the temperature, T, we can follow the next line
of reasoning. For a harmonic oscillator with fundamental
frequency ν0, the fraction of molecules, fV, in a given vibrational
energy level, V, is given by
∂nfi
∞
1
f )
νin
(13)
∑
i
exp[-ν0ν/kT]
n
(
)
n!
n
∂νi
fν )
(11)
1 - exp[-ν0/kT]
The simplest approach is to consider up to the lineal term. Thus,
fi ) a + bνi. The a and b coefficients are obtained in each
molecular species considering that fi ) 1.0 for νi g 1100 and
fi ) x (lower limit of the correction factors) for the mode of
lower frequency. The resulting expression for the vibrational
partition function reads
where we have used the closed form for the partition function
and a Boltzmann equilibrium distribution. From eq 11 we can
obtain the vibrational level, V, with a given population, fV, at
the T temperature as
kT
ν0
ν ) - ln[fν(1 - exp[-ν0/kT])]
(12)
limit
3N-6
exp[-νi fi/2kT]
q )
exp[-νj/(2kT)] (14)
∏
∏
ν
Using eq 12, Figure 5 shows the correspondence between
vibrational energy level, population, and ν0 fundamental fre-
quency, at a temperature of 331 K. The diagram shows that the
limit for the V ) 1 vibrational level to be populated less than
1% (fV ) 0.01) corresponds to a ν0 fundamental frequency
around 1100 cm-1. At the considered temperature, vibrations
with a ν0 above that value are only significantly populated in
the vibrational ground state. Thus, we can consider 1100 cm-1
as a practical limit to separate vibrational modes where only
1 - exp[-νi fi/kT] j)limit+1
i
where “limit” corresponds to the first vibration mode where
only the ground state is populated.
Using eq 14 for qv, we compute the equilibrium constant, K,
with eq 5. We found that the K value decreases monotonically
from a value of 0.825 when the correction interval is [0.99-1],
to 0.813 for the interval [0.95-1]. Thus, the original 0.827 value
(see Table 4) decreases, approaching the average experimental

6166 J. Phys. Chem. A, Vol. 107, No. 32, 2003
Mun˜oz-Caro et al.
result of 0.809. The agreement increases as the lower limit of
the correction interval decreases.
02-001), the “Ministerio de Ciencia y Tecnolog´ıa” of Spain
(grant no. BQU2000-1497), and the Universidad de Castilla-
La Mancha.
Conclusions
References and Notes
This work presents an experimental and theoretical study of
the secondary equilibrium isotope effect (SEIE) for the proton
transfer between the pyridinium-d5 cation and pyridine.The
variation in bonding of pyridine, on protonation, is analyzed
from the one-determinantal electron density using the atoms in
molecules (AIM) theory. The results show than on protonation,
electronic density is subtracted from the CN and the CC bonds
further from the nitrogen. Thus these bonds are weakened,
whereas the CH bonds and the CC bonds closer to the nitrogen
experience an increase of electronic density and, consequently,
an increase of bond strength.
The experimental results at a temperature of 331 K and high-
vacuum conditions show that the actual reaction favors the
neutral pyridine against the protonated pyridinium cation. The
theoretical treatment predicts also this trend for the reaction.
The equilibrium constant is thermostatistically computed in the
rigid rotor, harmonic oscillator model. The computed value
(0.827) agrees with the experimental measure (0.809) within
the 99.9% confidence interval ((0.027). Analysis of the
translational, rotational, and vibrational contributions to the
equilibrium constant shows that the vibrational contribution is
the only one responsible for the observed trend of the reaction.
This fact shows that the harmonic oscillator model can yield
good results for the prediction of an effect as small as the SEIE
in our molecules. We also find that the reaction constant
increases with the temperature; i.e., the temperature favors the
formation of pyridinium cation. This fact is as a consequence
of the existence of a higher density of vibrational states in the
pyridine-d5 than in the other molecular species.
We have also considered the factors affecting the agreement
between the observed and experimental results. The analysis
identifies the vibrational contribution as the main one responsible
for discrepancies. Introduction of the usual factors in the interval
[0.99-0.95], to correct the harmonic frequencies for anharmo-
nicity, leads to increasing values of the equilibrium constant.
Thus, the difference with the average observed result increases.
We develop a new model that identifies the normal mode where
the ground state is the only populated level. The zero point
energy is only considered for the levels above this limit. For
the modes below the limit, the full summation of states is
applied. Here, each fundamental frequency is corrected by a
factor of anharmonicity linearly dependent on the frequency
value. This technique simulates an increasing density of states
as the vibrational modes decrease in frequency. With the model,
the computed K value approaches the experimental result. For
correction factors in the interval [0.95-1], K reaches 0.815,
approaching the average experimental 0.809 value. This result
is the consequence of the difference between the energy levels
distribution for anharmonic modes and the distribution predicted
by the harmonic model. Because in the partition function the
energy levels appear in an exponential form, variation in the
distribution of the energy levels in low-frequency vibrations is
translated into important variations of the partition function. In
particular, an increase in the density of states is reflected in an
increase of the partition function value. Our results show that
discrepancies between the experimental and calculated K values
is mainly due to the failure of the harmonic model to describe
low-frequency vibrations.
(1) Sims, L. B.; Lewis, D. E. In Bond Order Methods for Calculating
Isotope Effects in Organic Reactions; Isotopes in Organic Chemistry, Vol.
6; Buncel, E., Lee, C. C., Eds.; Elsevier: Amsterdam, 1984; p 161.
(2) McLennan, D. J. In Model Calculations of Secondary Isotope
Effects; Isotopes in Organic Chemistry, Vol. 7; Buncel, E., Lee, C. C., Eds.;
Elsevier: Amsterdam, 1987; p 393.
(3) (a) Bigeleisen, J.; Goeppert-Mayer, M. J. Chem. Phys. 1947, 15,
261. (b) Bigeleisen, J. J. Chem. Phys. 1949, 17, 675-678. (c) Bigeleisen.
J.; Wolfsberg, M. AdV. Chem. Phys. 1958, 1, 15-76.
(4) Washburn, E. R.; Urey, H. C. Proc. Natl. Acad. Sci. 1932, 18, 496-
498.
(5) (a) Stevenson, C. D.; Reidy, K. A.; Peters, S. J.; Reiter, R. C. J.
Am. Chem. Soc. 1989, 111, 6578-6581. (b) Stevenson, C. D.; Sturgeon,
B. E. J. Org. Chem. 1990, 55, 4090-4093. (c) Stevenson, C. D.; Halvorsen,
T. D.; Reiter, R. C. J. Am. Chem. Soc. 1993, 115, 12405-12408.
(6) Hrovat, D. A.; Hammons, J. H.; Stevenson, C. D.; Borden, W. T.
J. Am. Chem. Soc. 1997, 119, 9523-9526.
(7) (a) Benderskii, V. A.; Makarov, D. E.; Wight, C. A. Chemical
Dynamics at Low Temperatures; J. Wiley & Sons: New York, 1994. (b)
Grigoriev, I.; Trakhtenberg, L. I. Processes in Solid-Phase Theory and
Application; Chemical Rubber Co.: Cleveland, OH, 1996.
(8) Kurosaki, Y.; Takayanagi, T. J. Chem. Phys. 1999, 110, 10830-
10842.
(9) (a) Saueressig, G.; Bergamaschi, P.; Crowley, J. N.; Fischer, H.;
Harris, G. W. Geophys. Res. Lett. 1995, 22, 1225-1228. (b) Crowley, J.
N.; Saueressig, G.; Bergamaschi, P.; Fischer, H.; Harris, G. W. Chem. Phys.
Lett. 1999, 303, 268-274.
(10) Laukien, F. H.; Allemann, M.; Bischofberger, P.; Grossmann, P.;
Kellerhals, P.; Kopfel, P. In Fourier Transform Mass Spectrometry,
EVolution, InnoVation, and Applications; Buchanan, M. V., Ed.; ACS
Symposium Series No. 359; American Chemical Society: Washington, DC,
1987; Chapter 5.
(11) (a) Abboud, J.-L. M.; Castan˜o, O.; Da´valos, J. Z.; Jime´nez, P.;
Gomperts, R.; Mu¨ller, P.; Roux, M. V. J. Org. Chem. 2002, 67, 1057-
1060. (b) Abboud, J.-L. M.; Koppel, I. A.; Alkorta, I.; Della, E. W.; Mu¨ller,
P.; Da´valos, J. Z.; Burk, P.; Koppel, I.; Pihl, V.; Quintanilla, E. Angew.
Chem., Int. Ed. Engl., 2003, 42, 2281-2284.
(12) Fridgen, T. D.; Keller, J. D.; McMahon, T. B. J. Phys. Chem. A
2001, 105, 3816-3824.
(13) Dunning, T. H., Jr. J. Chem. Phys. 1989, 90 (2), 1007-1023.
(14) Becke, A. D. J. Chem. Phys. 1993, 98, 5648-5652.
(15) Pople, J. A.; Head-Gordon, M.; Fox, D. J.; Raghavachari, K.;
Curtiss, L. A. J. Chem. Phys. 1989, 90, 5622-5629.
(16) Mun˜oz-Caro, C.; Nin˜o, A.; Senent, M. L. J. Mol. Struct.
(THEOCHEM) 2000, 530, 291-300.
(17) Szafran, M.; Koput, J. J. Mol. Struct. 2001, 565-566, 439-448.
(18) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb,
M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A., Jr.;
Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Millam, J. M.; Daniels, A.
D.; Kudin, K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi,
M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.;
Ochterski, J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.; Malick,
D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.;
Ortiz, J. V.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi,
I.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.;
Peng, C. Y.; Nanayakkara, A.; Gonzalez, C.; Challacombe, M.; Gill, P. M.
W.; Johnson, B. G.; Chen, W.; Wong, M. W.; Andres, J. L.; Head-Gordon,
M.; Replogle, E. S.; Pople, J. A. Gaussian 98, revision A.7; Gaussian,
Inc.: Pittsburgh, PA, 1998.
(19) MORPHY98, a topological analysis program written by P. L. A.
Popelier with a contribution from R. G. A. Bone; UMIST, Manchester,
England, EU, 1998.
(20) (a) Nin˜o, A.; Mun˜oz-Caro, C. QCPE Bull. 1997, 17, 1. (b) Nin˜o,
A.; Mun˜oz-Caro, C. Comput. Chem. 1997, 21, 143-151.
(21) See, e. g.: Marshall, A. G. Int. J. Mass Spectrom. 2000, 200, 331-
357.
(22) See, e. g.: (a) Bowers, M. T.; Elleman, D. E.; King, Jr. J. Chem.
Phys. 1969, 50, 1840-1845. (b) Wellman, K. M.; Victoriano, M. E.; Isolani,
P. C.; Riveros, J. M. J. Am. Chem. Soc. 1979, 101, 2242-2243. (c) Jasinski,
J. M.; Brauman, J. I. J. Am. Chem. Soc. 1980, 102, 2906-2912.
(23) See, e. g. (a) Wolf, J. F.; Devlin, J. L., III; Taft, R. W.; Wolfsberg,
M.; Hehre, W. J. J. Am. Chem. Soc. 1976, 98, 287-289. (b) Wolf, J. F.;
Devlin, J. L., III; DeFrees, D. J.; Taft, R. W.; Hehre, W. J. J. Am. Chem.
Soc. 1976, 98, 5097-5101.
Acknowledgment. This work has been supported by the
“Junta de Comunidades de Castilla-La Mancha” (grant no. PAI-

Proton Transfer between Pyridinium-d5 Cation and Pyridine
J. Phys. Chem. A, Vol. 107, No. 32, 2003 6167
(24) The gas-phase basicity, GB, of a base B is defined25 as the standard
Gibbs energy change for the reaction:
4135. (b) Nin˜o, A.; Mun˜oz-Caro, C. J. Phys. Chem. A 1998, 102, 1177-
1180. (c) Mun˜oz-Caro, C.; Nin˜o, A.; Senent, M. L. Chem. Phys. Lett. 1997,
273, 135-140. (d) Torres, L.; Moreno, M.; Lluch, J. M. J. Phys. Chem. A
2001, 105, 4676-4681.
BH+(g) f B(g) + H+(g)
∆rG°
m
(29) Villa, E.; Amirav, A.; Lim, E. C. J. Phys. Chem. 1988, 92, 5393-
(25) Hunter, E. P. L.; Lias, S. G. J. Phys. Chem. Ref. Data 1998, 27,
5397.
413-656.
(30) Nguyen, V. Q.; Turecˇek, F. J. Mass Spectrom. 1997, 32, 55-63.
(26) Dowdy, S.; Wearden, S. Statistics for Research, 2nd ed.; Wiley-
Interscience: New York 1991; Chapter 8.
(31) (a) Bader R. F. W. Atoms In Molecules. A Quantum Theory; Oxford
University Press: Oxford, U.K., 1995. (b) Popelier P. Atoms In Molecules.
An Introduction; Prentice Hall: Englewood Cliffs, NJ, 2000.
(27) Lucas K. Applied Statistical Thermodynamics; Springer-Verlag:
Berlin, 1991.
(28) (a) Mun˜oz-Caro, C.; Nin˜o, A. J. Phys. Chem. A 1997, 101, 4128-
(32) Scott, A. P.; Radom, L. J. Phys. Chem. 1996, 100, 16502-16513.