Russian Chemical Reviews 72 (1) 1 ± 33 (2003)
# 2003 Russian Academy of Sciences and Turpion Ltd
DOI 10.1070/RC2003v072n01ABEH000774
Theoretical investigations of proton and hydrogen atom transfer in the
condensed phase
M V Basilevsky, M V Vener
Contents
I.
II.
III.
IV.
V.
VI.
Introduction
Methods for the theoretical description of proton and hydrogen atom transfer
Calculations for particular systems
Experimental data and summary of calculations of the potential and free energy surfaces
Supplement I. Generalised Langevin equation
Supplement II. Spectral density
Abstract. Theoretical studies of the dynamics and kinetics of
proton and hydrogen atom transfer processes occupy a special
place in the kinetics of chemical reactions. The transition state
theory is often inapplicable to these processes due to substantial
quantum effects. Different approaches to the description of these
reactions are discussed and compared. Calculations for a number
of particular condensed-phase reactions involving proton or
hydrogen atom transfer are analysed. Data of calculations of
potential energy surfaces for the considered systems and the
results of experimental kinetic and spectroscopic studies are
presented. The bibliography includes 469 references.
references.
I. Introduction
Proton transfer (PT) and hydrogen atom transfer (HAT) deserve
special consideration in the study of chemical reactions. Due to
substantial quantum effects, the transition state (TS) theory Ð
which today forms the basis for investigations of reaction kinetics
and mechanisms Ð is often inapplicable in these cases. The
existing numerous theories of PT and HAT are complicated and
diverse. This review is an attempt to consider and compare, from a
unified standpoint, various approaches to the theoretical description of PT and HAT and to identify the common points and the
scopes of applicability of these approaches. Without pretending to
an exhaustive consideration, we endeavoured to provide a concerned reader with the main ideas and and trends in the development of the research in this field. More comprehensive
formulation of the outlined approaches can be found in the key
references.
It should be noted that the difference between PT and HAT is
largely conventional. In the limiting case, PT is an ionic heterolytic
process in which the influence of the medium is crucial, whereas
HAT is a radical homolytic process depending only slightly on the
medium. In reality, all types of intermediate situations are
encountered and it is difficult to adhere to a consistent terminology. In some cases, the use of a particular term can be contested;
M V Basilevsky, M V Vener State Scientific Centre of Russian Federation
`L Ya Karpov Institute of Physical Chemistry',
ul. Vorontsovo Pole 10, 105064 Moscow, Russian Federation.
Fax (7 095) 975 24 50. Tel. (7 095) 917 39 03.
Received 3 October 2002
Uspekhi Khimii 72 (1) 3 ± 39 (2003); translated by Z P Bobkova
1
2
16
20
27
28
however, in our opinion, this discussion does not deserve much
attention.
Two factors, namely, tunelling and effect of the medium,
restrict the applicability of the TS theory and account for the
exceptional position of the PT and HAT processes in the theory of
chemical reactions. For consistent treatment of tunnelling, one
should consider (or, more precisely, calculate) the quantumdynamic evolution of the reaction subsystem on a multidimensional potential energy surface (PES). The reaction rate constant is
obtained by statistical averaging of the results of this calculation.
An adequate allowance for the medium implies that a very large
number (hundreds or even thousands) of degrees of freedom for
particles surrounding the reaction subsystem should be included
in this dynamic investigation. Taking into account simultaneously
both factors mentioned above constitutes the essence of the
dynamic theory of chemical reactions in the condensed phase as
a modern line of research in theoretical chemical physics.
In practice, the stringent requirements outlined above can be
markedly released, in particular, by using simplified models for
tunnelling transitions and by decreasing the number of additional
degrees of freedom interacting with the tunnelling coordinate to
tens or even less. Nevertheless, in recent years, record-breaking
computations have been carried out (they are described in this
review) which attempted to minimise the simplifying assumptions
and their consequences. However, real objects (e.g., reactions in
enzymes) are so complicated that even these calculations are
simplified models.
In any case, the major requirement of explicit inclusion of the
tunnelling dynamics into the mechanism of the elementary process
remains valid. This feature distinguishes the studies discussed in
this review from the numerous studies whose authors attempted to
consider the PT and HAT reactions on the basis of the traditional
TS theory, i.e., using a quasi-equilibrium statistical rate constant.
As the premise, we accepted the statement that interpretation of the
kinetics and the mechanism of these reactions based only on the
language of the energetic and topographic characteristics of PES
such as the height of the potential barrier and the reaction
coordinate is inadequate and recedes to the background.
The material presented here is arranged in the following way.
First, the main concepts of the PT and HAT dynamics in the
condensed phase are considered. They were formulated in the
1970s ± 1980s and they are still being developed and refined. This
is followed by applications of the theory to calculations of
particular systems carried out during the last 10 ± 15 years.
Many of the publications cited present also original theoretical
2
M V Basilevsky, M V Vener
developments; therefore, the subdivision into `theory' and `applications' is not fully consistent.
Simpler gas-phase reactions have been investigated much more
extensively than reactions in the condensed phase. The relevant
studies are not covered systematically but references to earlier
reviews and monographs are given. However, since the understanding of the tunnelling dynamics of gas-phase reactions should precede
any discussion of tunnelling in the liquid or solid phase or in
biological objects, the key studies dealing with the theory of gasphase PT and HAT are considered as examples.
Since we selected studies focused on the dynamics of an
elementary process, the quantum-chemical calculations of PES
(even the highest-level ones) are not of interest by themselves in this
context. A summary of such calculations (the most significant in
our opinion) is included as reference data together with experimental results. The experimental works presented in brief represent
mainly investigations of the kinetics and mechanisms of PT and
HAT by spectroscopy (NMR, UV, IR, radiospectroscopy, etc.).
These studies provide the factual information necessary for the
development and testing of the theoretical models discussed in the
review.
a
R
Re
1
4
4
2
R6
3
3
2
Rc
1
s
s=0
b
II. Methods for the theoretical description of
proton and hydrogen atom transfer
R
R = Re
1. Potential energy surfaces
a. Reaction centre
Let us consider the reaction centre for PT or HAT in a simple twomode linear model.
A1
H A2
(1)
R = R6
Here A1 and A2 are `heavy' molecular fragments; their internal
structure is not specified in detail now. The centres of gravity of
particles A1, A2 and H lie on the same straight line; this type of
reaction is called collinear.
The dynamic behaviour of system (1) is fully described by
variations of two coordinates, namely, the proton coordinate (s)
measured from the centre of gravity of the whole complex (1) and
the distance (R) between the heavy fragments. Figure 1 presents a
typical PES V(s,R) in these coordinates and cross-sections V(s jR)
of this surface along the coordinate s, which parametrically
depend on R. For a symmetrical system (A1 = A2), the crosssections are symmetrical. As R decreases, the double-well profiles
become more gently sloping, which is accompanied by equalisation of the proton (hydrogen atom) bonds with fragments A1
and A2. At the critical point Rc, the double-well potential profile
turns into a single-well one. This PES topography determines the
key characteristics of a PT or HAT reaction. Since one of the two
equilibrium configurations of complex (1) is the initial state, either
a PES with potential barriers with different heights (if Re > Rc) or
a barrierless PES (if Re < Rc) appear, depending on the equilibrium distance Re. In the former case, the initial equilibrium
configuration is markedly asymmetrical (A17H_A2), while in
the latter case, it is nearly symmetrical (A1_H_A2), even when
the fragments A1 and A2 are different. In symmetrical systems, a
combination of asymmetrical configurations of the reactants and
products gives rise to a symmetrical double-well profile. The
position of the top of the barrier, i.e., TS (designated by R6 in
the axis R) cannot be smaller than Rc (R6 5 Rc). The Re and Rc
values substantially depend on the nature of fragments A1 and A2,
most of all, their terminal atoms, which are linked to the proton or
hydrogen atom that is transferred.1 The distance between the
terminal atoms (rather than between the centres of gravity of
fragments A1 and A2) is usually taken as the coordinate R. For
bimolecular reactions of non-polar reactants in the gas phase (for
example, C_H_C), the Re distance is great, Re 4
4 Rc (which
corresponds to a van der Waals complex) and the barrier is usually
R = Rc
s
R
R < Rc
s
s=0
Figure 1. Typical symmetrical PES for PT or HAT reaction in the
simplest two-mode collinear model;
(a) general view of the PES with isoenergy contour lines; closed loops
(1 ± 4) are isoenergy contour lines in arbitrary units; (b) PES cross-sections
along the coordinate s for different R values; R = Re (reactants and
products), R = R6 (transition state) and R = Rc (critical point). In the
given case, Rc < R6. The projections of the cross-sections on the R, s plane
are shown by a dashed line in Fig. 1 a.
rather high. Conversely, in the case of systems which tend to form
strong and medium-strength hydrogen bonds (for example, the
O_H_O bonds in the condensed phase), the inequality Re < Rc
usually holds, and the PES is often a single two-dimensional
potential well. In this case, one cannot speak about a PT or HAT
reaction. Particular examples of different possible situations are
presented in Table 1. They mainly refer to electrically neutral
systems; charged systems (FHF7, H5O
2 , etc.) with hydrogen
bonds normally tend to form single-well PES.11
In unimolecular processes, the reaction centre (1) is a fragment
of an integrated, more or less rigid molecular structure. The
equilibrium distance Re stipulated by the structure is usually
shorter than the Re value between the reaction centres of the
same type in bimolecular reactions. This is due to the fact that the
A1_H_A2 fragment in systems with intramolecular hydrogen
bonds is non-linear. A typical example of the latter type is a cisenol form of a b-dicarbonyl compound, for example, malonaldehyde or acetylacetone,12, 13 which are six-membered chelate rings
with an intramolecular hydrogen bond. The Re values for these
Theoretical investigations of proton and hydrogen atom transfer in the condensed phase
3
Table 1. Typical Re and Rc values in PT and HAT reactions for the A17H_A2 reaction sites.
Fragment
Process
Re /AÊ
Determination method
Rc & R6 /AÊ
(see a)
Note
Ref.
(see b)
O7H_O
PT and double PT in
a symmetrical double-well
potential
2.64
X-ray diffraction analysis
of the benzoic acid crystal
2.40
the O_O distance in the TS for
systems with a symmetrical
two-well potential
2, 3
N7H_O
double PT in an asymmetrical double-well
potential
2.95
X-ray diffraction analysis
of the formamide cyclic
dimer
2.55
the distance in the TS structure
[B-LYP/6-311++G(d,p)
calculations]
4, 5
N7H_N
double PT in a symmetrical
double-well potential
2.93
the calculation (HF/3-21G)
for the amidine dimer
2.53
the N_N distance in the TS
structure for systems with a
symmetrical double-well potential
6, 7
3.71
the calculation [MP2/6-31+
G(d,p)] for the CH4/CH3
ion ± molecular complex
2.86
the C_C distance in the TS
structure [MP2/6-31+G(d,p)
calculation]
8
3.35
spectroscopic and structural
data on H abstraction from
ethanol by the methyl radical
in glasses
2.68
the C_C distance in the TS
.
structure of the CH3 /CH3OH
system (MP2/6-31G calculation)
9, 10
{C7H/C}7 PT in a symmetrical
double-well potential
{C7H/C}
.
HAT in an asymmetrical
double-well potential
Note. Systems with intramolecular PT or HAT were not considered because in this case, the triatomic A17H_A2 fragment is essentially non-linear. a No
direct estimate for the Rc value is known; therefore, the corresponding R6 values are given, which are close to Rc. b The Re value was taken from the first
reference and Rc is from the second reference.
systems are much shorter than the corresponding value for the
O7H_O fragment presented in Table 1. The rigid structure of
the environment (`cage effect') is probably responsible for acceleration of enzymic PT reactions.14, 15 In this case, the structure of
the reaction centre is dictated by the architecture of the protein
globule of the enzyme, while Re is sufficiently small to lower the
reaction barrier to a plausible value.
Thus, the potential relief depends appreciably on the distance
between the chemically non-bonded heavy atoms, which accounts
for stringent requirements to the accuracy of quantum-chemical
methods used to calculate the PES. Slight errors in the estimation
of Re can change the calculated height of the potential barrier
(defined as the difference between the energies of the saddle point
and the reactant minimum on the PES) and can appreciably
distort the potential relief that controls the reaction event.16 On
the other hand, in constructing semiempirical PES, it is expedient
to consider the Re value as a control parameter whose variation
allows one to reproduce a required potential relief.17
b. Rearrangement of the medium and classification of modes
Intramolecular and intermolecular relaxation. In most cases, the
simplest two-mode model (1) does not provide an adequate
description of PT. As an example, we consider the PT in acetylacetone molecule.
H
O
Me
H
O
O
Me
Me
H
O
O
Me
Me
O
Me
A (see Ref. 18). According to the above simplest
Here Re = 2.535
model, this geometry is expected to produce a single-well symmetrical equilibrium configuration of the O_H_O hydrogen
bond. In reality, acetylacetone is an example of a system having
a potential barrier, although it is relatively low: DV6 =
2.5 kcal mol71 (see Ref. 19). This `classical' barrier disappears
when the quantum zero-point vibrations of the H atom are taken
into account.19 The tendency for the formation of an asymmetrical structure by the molecule is enhanced by rearrangement of its
bond lengths and bond angles around the reaction centre caused
by the variation of the proton coordinate s (Table 2). In the
literature, this effect is often referred to as `environment reorgan-
Table 2. Bond lengths (
A) and O_H7O angles (deg) for a fragment of the
cyclic acetylacetone structure and the structure corresponding to the TS of
the intramolecular PT reaction found by the MoÈller ± Plesset ab initio
calculations 19 (MP2/D95++**).
Parameter
Cyclic enol form a
TS
C=C
C7C
C7O
C=O
O7H
O_H
O_O
O_H7O
1.376 (1.382)
1.452 (1.430)
1.338 (1.319)
1.259 (1.243)
1.004 (1.049)
1.626 (1.626)
2.549 (2.512)
150.6 (137.0)
1.410
1.410
1.295
1.295
1.201
1.201
2.363
159.1
a The
values in parentheses are experimental data.18
isation'; this term has been borrowed from electron transfer
theory. In this particular case, an intramolecular reorganisation
is involved.
When the PT process is accompanied by a substantial change
in the charge distribution and takes place in a polar solvent, there
exists an alternative mechanism for stabilisation of asymmetrical
structures due to solvent reorganisation: the solvation shell is
adjusted to the charge distributions, which are different for the
reactants and the reaction products. This situation is referred to as
the `medium reorganisation'. Proton transfer in electrically neutral systems, for example, in a conjugate acid ± base system usually
does not occur in the gas phase,20 but readily proceeds in polar
solvents,21 which is due to medium reorganisation.
It can be seen from the above examples that reorganisation of
the surrounding of the reaction centre (both intramolecular
reorganisation and that of the environment) includes simultaneous change of many degrees of freedom, i.e., it is a multimodal
process.
Harmonic double-well potential energy surface. The simplest
pattern of multimode reorganisation corresponds to the model
harmonic PES formed by two multi-dimensional paraboloids
shifted relative to each other. The two-dimensional analogue of
this PES is shown in Fig. 2. The reorganising modes Q are
4
M V Basilevsky, M V Vener
a
DV = Vf 7 Vi ,
Energy
(4)
Er is the reorganisation energy characterising the energy consumption for transition from the reactant minimum to the product
minimum; Er is an additive relative to the contributions of
separate modes characterised by masses mn and frequencies on:
Er
Q2
Ern ,
n
(5)
m o2
Ern n n Qfn
2
Q(l)
Qf
X
Q1
2
Qin .
The height of the potential barrier in this PES obeys the
Marcus formula
Qi
DV6
b
Er DV 2
.
4Er
(6)
Energy
Q(l)
Qi
Q6 Qf
Figure 2. Two-dimensional scheme of the reaction transition between
two paraboloids;
(a) two-dimensional analogue of the model harmonic PES; the paraboloid
minima are located at points Qi and Qf; (b) energy profile of the reactive
transition (shown by a dashed line in the intersection area of two parabolic
potential curves corresponding to the PES cross-section along the straight
line QiQf). The continuous lines show splitting of two energy profiles due
to anharmonic interactions.
numbered by using subscripts n = 1,...,N; for Fig. 2 a, N is equal
to 2. Their equilibrium positions for the reactants (subscript i) and
products (subscript f ) will be denoted by Qin and Qfn . The
description is simplified if the paraboloids are identical, i.e., the
reactants and the products are characterised by identical systems
of vibration frequencies and forms of normal vibrations (modes).
Let us denote the vectors with the components Qin and Qfn by
(Qi) and (Qf ) and then form the vector
[Q(l)] = (Qi) + l[(Qf ) 7 (Qi)]; 0 4 l 4 1,
1
DV
,
1
2
Er
a
b
Energy
Energy
(2)
which describes the change of the modes along the straight line
connecting the projections (Qi and Qf) of the minima of the two
paraboloids onto the Q1Q2 plane (Fig. 2 a). Then the parameter l
can be considered as the `reaction coordinate'. If the paraboloids
are identical, they intersect along some straight line in the
coordinate plane. In this straight line (or in a planar hypersurface
if N > 2), the PES cross-section passes through a minimum, which
6
is characterised by a set of coordinates: Q6 = (Q6
1 ),...,(QN ), the
6
point Q lies in straight line (2). The cross-section of the PES
along the coordinate (2) is shown in Fig. 2 b. At the intersection
point of the parabolic potential curves, the reaction coordinate is
found as
l
This PES model underlies most of the theories of multimode
processes. Although the model is idealised, it can be useful for a
qualitative discussion. Real PES are highly anharmonic,22 ± 24 and
the contributions of separate modes to the potential barrier
cannot be distinguished unambiguously, as would be the case
according to relations (5) and (6). There exist semiempirical
interpolation-type analytical formulae, which allow one to
describe the anharmonic PES and the free energy surface (FES)
for PT in terms of a set of characteristic parameters.25, 26
Combination of reorganisation and promoting modes. It is of
interest to analyse how the allowance for reorganisation modes
modifies the two-mode model described in Section II.1.a.27 ± 31 For
this purpose, we consider three points, (Qi), (Q6) and (Qf ), on the
reaction coordinate Q. Even for a symmetrical system (A1 = A2),
by fixing (Q) at the points (Qi) and (Qf ), we obtain asymmetrical
potential energy profiles along the proton coordinate s (here we do
not consider the dependence on R). This is shown schematically in
Fig. 3. When (Q) = (Q6), the V(s) profile is symmetrical or nearly
symmetrical even for the systems with dissimilar heavy fragments
A1 and A2. As noted above, in a two-mode double-well symmetrical system, the potential barrier is often relatively low and
s
c
Energy
Energy
s
(3)
where DV is the difference between the energies corresponding to
the reactant and product minima
s
d
s
Figure 3. Potential curves of the reaction system along the proton
coordinate s with allowance for reorganisation modes;
(a, d ) reactant and product regions, respectively; (b, c) TS region.
Theoretical investigations of proton and hydrogen atom transfer in the condensed phase
transitions between the wells occur rapidly. The asymmetry
brought about by the relaxation of reorganisation modes hinders
the transitions and thus contributes to fixing of the reactant and
product configurations. In the case of bimolecular gas-phase
reactions, this fixing is possible even in a two-mode model due to
the spatial separation of the reactants and products along the
reaction coordinate R. In the condensed phase, the reactants and
products are in relatively close contact [the Re distance is about the
van der Waals distance in complex (1)]. Therefore, stabilisation of
the initial and final states owing to relaxation of the environment
modes acquires a crucial role.
In view of the foregoing, the following, frequently used
classification of inter- and intramolecular degrees of freedom
involved in PT processes can be proposed. First, the proton
coordinate s is considered, which is called the reaction mode.
Second, those modes are considered that do not change (or almost
do not change) in the initial and final states but appreciably
change in the TS and thus modify the height of the potential
barrier. There modes are called promoting. A typical example is
the R coordinate; other promoting modes are discussed below.
Finally, some modes change their equilibrium positions on passing from the reactant configuration to the product configuration.
They are referred to as reorganisation modes.
Naturally, this classification is not universal. However, it
often reflects adequately the essential features of the PT process
and is widely used in model considerations. Note that the term
`reaction mode', introduced above for the coordinate s does not
necessarily mean the `reaction coordinate'. For real anharmonic
PES, the mode s interacts with other reaction modes and,
generally, it is not an independent motion in the region of the
TS, as it would be the case for a strictly defined reaction
coordinate. The mode s may be identified with the reaction
coordinate only in terms of the simplest models; this terminology
is often encountered in the literature and below it is used in some
cases without special reservations.
In reality, the reaction mode for PT and HAT is always a
combination of several simple modes. This is illustrated, in
particular, by the following examples. According to the data of
femtosecond spectroscopy, the rate of PT for the lower singlet
electronically excited state of methyl salicylate, 2-(20 -hydroxyphenyl)benzothiazole
and
1,8-dihydroxyanthraquinone
is
50 ± 60 fs71 in the gas phase 32 ± 34 and 170 fs71 in aprotic solvents.35 The reaction coordinate for the PT includes, as a first
approximation, the coordinate of the OH stretching vibrations
and the coordinate of the O_H_O out-of-plane vibration.32 The
periods of vibration are 13 fs for the OH bond and 190 fs for the
O_H_O fragment. These vibration modes interact appreciably
with one another, their relative contributions to the reaction mode
being dependent on the process conditions. In systems with
equivalent initial and final states, PT is manifested as tunnel
splitting of the ground vibration state.36 ± 38 The tunnel splittings
have been measured for not only the ground state but for a
number of vibrationally excited states for some systems in the
lower singlet electronically excited state.39 ± 42 A sharp increase in
the tunnel splitting was observed upon selective excitation of
particular low-frequency vibrations.39, 40 In the case of tropolone,
this active vibrational mode was the displacement of heavy atoms
of the O_H_O fragment,41 and in the case of azaindole dimer,
an intermolecular stretching vibration.39 Data of ab initio calculations of the PES for intra- and intermolecular PT in various
systems 22 ± 24, 43 confirm the experimental results.
5
theless, the classical approach can be used in the modified form if
the height of the barrier DV6 on the PES is sufficiently low for the
reactant well to contain only one quantum level with a frequency
close to o0:
ho0
9 1.
DV 6
The quantum motion along s occurs much faster than along
4 oi , where oi are other eigenfrequencies of
other modes (o0 4
the system); this allows one to use the double (along both the
electronic and proton coordinates) adiabatic approximation.
Different modifications of this model with a classical description
of the promoting mode R have been discussed.29 ± 31 The SchroÈdinger equation for the mode s has the form
hsjn(s) = enjn(s),
(8)
where hs is a Hamiltonian,
hs
2 q2
h
V sjR; Q:
2m0 qs2
(9)
Here, m0 is the corresponding mass, and the potential coincides
with the full PES V(s,R,Q). The notation of this function in Eqn
(9) emphasises that it follows a parametric dependence on the R,Q
coordinates of the heavy atoms. The eigenvalues and eigenfunctions in Eqn (8) depend on R and Q. In particular, the energies
en(R,Q) are the PES (electron ± proton PES) for the coordinates
R,Q. In the adiabatic limit, it is sufficient to consider the ground
state (n = 1), i.e., the PES e1(R,Q). In many cases, the motion of
heavy atoms in such a potential can be regarded to be classical and
described using the TS theory in one or another variant.
A similar approximation is widely used in hydrogen bond
spectroscopy.44, 45 It is called the Born ± Oppenheimer separation
of motions of light and heavy nuclei 46 or the adiabatic separation
of vibrational variables.47 A significant feature of the calculations
of the vibrational spectrum of the A17H_A2 fragment is the
quantum description of the promoting mode R.48 ± 53
Evolution of heavy modes. The specificity of using the double
adiabatic approximation lies in the fact that the reaction coordinate s, natural for a chemist, is excluded from dynamic consideration. The role of the reaction coordinate is played by one of the
reorganisation modes Q. For example, for low-barrier PT in polar
media, the collective variable Q = y, describing the behaviour of
the medium, is used as the new reaction coordinate. According to
this model, the motion along y determines the reaction dynamics
and kinetics. In the simplest case, the mode R is excluded from
consideration, and the equation of motion reduces to the diffusion
equation for the distribution function n(y,t), which determines the
probability density for realisation of various y values, depending
on time t
qn
q2 n
D 2
qt
qy
D q
qU
n a ,
qy
kB T qy
(10)
where D is the corresponding diffusion coefficient, and the
potential profile Ua(y) should be regarded as the cross-section of
the electron ± proton PES (Fig. 4).{ The subscript `a' means that
the potential is adiabatic, i.e., double-well. (Further, an alternative description using diabatic, single-well potentials is also
discussed.) The rate constant for PT can be estimated in terms of
the Kramers approximation.54, 55
2. Dynamics and kinetics
a. Reactions with a low potential barrier
Quantisation of the reaction mode. A key feature of PT reactions is
a quantal nature of, at least, one essential mode, namely, the
reaction mode s. Let o0 be the characteristic frequency of the
mode s and ho0/kBT 4
4 1. In this case, the traditional description
of the kinetics using the classical TS theory is inapplicable. Due to
this fact, theoretical description of PT is a difficult task. Never-
(7)
k A exp
DU6
,
kB T
(11)
{ From here on, one-dimensional potential profiles present in the diffusion
equations are denoted by the letter U.
6
M V Basilevsky, M V Vener
y /(kcal mol71)1/2
Ua(y)
1.20
7m6
7107
0.60
y
Figure 4. Adiabatic potential profile Ua(y) obtained as a cross-section of
the electron-proton PES and used in the Kramers equation.
where
A
m0 m
pg
(12)
The Arrhenius dependence (11) with an activation energy
equal to the height of the potential barrier is an immutable
indication of the classical equilibrium kinetics. In more complex
cases, the expression for the pre-exponential factor is modified
depending on the particular dynamic model accepted for description of heavy modes instead of the simplest diffusion equation
(10). For example, the common one-dimensional Kramers stochastic model proceeds from the Langevin equation
qUa
GRF,
qy
(13)
where my is the mass corresponding to the collective mode of the
medium, and GRF stands for the Gaussian random force, which
gives rise to fluctuations of the variable y and is responsible for its
diffusion motion. If the g/my ratio is not very low, Eqn (13)
corresponds to the Arrhenius rate constant (11) with the preexponential factor
A
g
2my
710
7
,
kB T
.
g
my y gy_
798
7104
70.60
6 1=2
DU6 is the height of the potential barrier on the electron ± proton
PES, m0 and 7m6 are the force constants of the potential Ua(y) in
the bottom of the reactant well and on the top of the barrier. The
friction coefficient g is related to the diffusion coefficient D by the
Einstein relation
D
0.00
7101
7101
m0
4
710
DU6
2
o6 2
1=2
g
2my
m0
m6
1=2
,
(14)
where o6 is a characteristic frequency on the top of the barrier
[that is, io6 is the decay frequency, my(o6)2 = m6]. This wellknown Kramers formula provides the pre-exponential factor in
Eqn (11) if g/(2myo6) 4
4 1, i.e., in the limiting case of great
friction. Further correction of the dynamic description results in a
stochastic equation with a time-dependent friction coefficient
(generalised Langevin equation). This equation is applicable to
multimodal systems. The rate constant is calculated in terms of the
Kramers ± Grote ± Hynes approximation (KGH).54 ± 56 This procedure is briefly described in Supplement I. The standard multidimensional TS theory can also be used; with allowance made
consistently for the medium modes, it is equivalent to the KGH
theory.54 ± 60 Finally, direct calculation of the reaction rate constant by classical molecular dynamics (MD) is also possible,61, 62
for example, by the reaction flux method.55
Electron ± proton PES.63 ± 65 A typical electron ± proton PES
depending on two heavy coordinates, the medium coordinate y
and the promoting mode R, is shown in Fig. 5. The direction of the
reaction coordinate at the saddle point coincides with the direction of the mode y. The rate constant can be estimated within the
framework of the multi-dimensional KGH theory. Estimation of
71.20
2.90
3.50
4.10 R /AÊ
Figure 5. Electron-proton PES e1(y,R) for the ground state of the
reaction system ACH3 + 7CH2A ? ACH2 + CH3A; A = C6H5 .64
Isoenergy contour lines are spaced by 1 kcal mol71. The reactant and
product minima correspond to the energy 7107.3 kcal mol71.
the rate constant in terms of the stochastic (or the equivalent
diffusion) theory requires the knowledge of dynamic characteristics of the solvent collective mode y such as the mass my, the
friction or diffusion coefficient, etc. They can be found using
semiempirical continuum models of the solvent.65 ± 69 The most
consistent and direct way of calculating these values is based on
MD.61 ± 63
To conclude, note that the classical description of the promoting mode R accepted in the multidimensional TS theory, KGH
theory and classical MD calculations is an approximation which
often does not correspond to the real situation.70 ± 77 The quantum
treatment of R is discussed in Section II.4.c.
b. Proton tunnelling as a non-adiabatic transition
Adiabatic and non-adiabatic models of reaction transitions. Section
II.2.a considered the adiabatic PT model. In this context, the term
`adiabatic' means that a double-well potential V appears in the
SchroÈdinger equation (8). It corresponds to the lower electronic
adiabatic PES; in more detailed notation,
V = Va1(s,R,Q).
The next PES Va2(s,R,Q) corresponds to the first electronically
excited state of the system. This is a single-well surface, its
minimum being close to the top of the barrier of the lower PES,
which corresponds to the ground state. The profiles of these two
adiabatic PES are shown in Fig. 2 b by continuous lines.
In an alternative description, the potentials Va1 and Va2 are
considered as combinations of the potentials Vd1 and Vd2,
corresponding to single-well surfaces, which are called `diabatic'.
The theory of this transformation was outlined in several publications.78 ± 81 The diabatic and adiabatic PES are sketched in
Fig. 2 b. They almost coincide near the minima. The error of the
diabatic description is the highest near the TS, i.e., near the barrier
top of the lower adiabatic surface or, what is the same, around the
intersection of the diabatic surfaces (shown by dashed lines).
Equations (8) and (9) describe the transition from the electronic PES Va1(s,R,Q) to the electron ± proton PES e1(R,Q). In the
same way, by substituting Va2(s,R,Q) into Hamiltonian (9), one
can obtain the electron ± proton PES e2(R,Q) for the electron ±
Theoretical investigations of proton and hydrogen atom transfer in the condensed phase
proton excited state. Like the initial electronic PES, e1 has two
wells, while e2 has one well. These surfaces are adiabatic, and they
can be transformed into two single-well diabatic electron ± proton
PES, which we designated by e1(R,Q) and e2(R,Q). By substituting
diabatic electronic potentials, instead of V, into Hamiltonian (9),
we obtain two diabatic SchroÈdinger equations similar to Eqn (8).
The surfaces e1 and e2 correspond to their eigenvalues. The
corresponding eigenfunctions, which we designate by w1(s) and
w2(s), are referred to as diabatic states. Unlike adiabatic states
jn(s) (n = 1,2) in Eqn (8), these are always localised in their
diabatic potential wells. The difference between the lower adiabatic potential and the combination of low-energy fragments of
diabatic potentials (which is substantial near the TS) is considered
to be responsible for the transitions between the diabatic states w1
and w2. These transitions, called non-adiabatic, are identified with
a PT or HAT process. For reactions with high barriers, splitting of
a
Energy /kcal mol71
Ua2
7100
D
U6
Ua1
7102
0.1
0.0
70.1
y /(kcal mol71)1/2
b
Energy /kcal mol71
Ud2
Ud1
788
the adiabatic potentials is extremely small (Fig. 6) and the
intensity of the transitions is very low.
Two-level kinetic equation. The scenario of a PT reaction
described in Section II.2.a corresponds to the situation shown in
Fig. 6 a. For a low-barrier potential Va1, the lower adiabatic level
e1 is located near the top of the barrier and the next adiabatic level
e2 is located much higher [D > 1000 cm71 (see Ref. 82)], therefore,
its population is negligibly low. In this case, it is sufficient to
consider the PES e1(R,Q).
The alternative situation shown in Fig. 6 b with a high barrier
and a clearly defined double-well-potential relief requires a different description. Despite the fact that the proton motion in the
wells remains fast, the probability of the inter-well tunnelling
transition P12 is low: P12 5
5 1. If it is low enough, the transition
between the diabatic states w1(s) and w2(s) becomes the ratedetermining step of the process. In the two-level approximation,
the localised diabatic wave functions w1(s) and w2(s) are linear
combinations of the adiabatic functions j1, j2. The variable s
becomes the reaction coordinate, although of the quantum type. It
is represented by two diabatic states with the PES e1(R,y) and
e2(R,y) (we have replaced Q by y) and their cross-sections Ud1(y)
and Ud2(y).
The limiting cases illustrated in Fig. 6 correspond to two
limiting mechanisms of PT, namely, the classical motion of
heavy atoms over the adiabatic surface e1(R,y) with the crosssection Ua1(y) and a quantum non-adiabatic transition between
the PES e1(R,y) and e2(R,y) with the cross-sections Ud1(y) and
Ud2(y). Now we consider the competition between these reaction
mechanisms at a qualitative level using a simple one-dimensional
model (Q = y is the medium coordinate, and promoting mode R is
excluded from consideration). The equations of motion for the
populations n1(y,t) and n2(y,t) of the diabatic states have the form
2
qn1
q n1
D
qt
qy2
1 q
qU
hoi n2
n1 d1
qy
kB T qy
n1 ,
2
qn2
q n2
D
qt
qy2
1 q
qU
n2 d2
hoi n1
qy
kB T qy
n2 ,
Uc
790
791
Ud2
Ud1
70.1
0.0
0.1
y /(kcal mol71)1/2
Figure 6. Double-well potential profiles for the two limiting mechanisms
of PT and HAT reactions; only the TS region is given;
(a) adiabatic energy profiles for a low-barrier reaction, see Fig. 5
(A = C6H5); U6 is the saddle-point energy for the ground state of the
reaction system, D is the splitting of PES in the saddle point; (b) diabatic
energy profiles for a high-barrier reaction, A = C13H9 (fluorenyl); 65 Uc is
the level intersection point; the abscissa y0 of this point is taken to have the
coordinate 0.0. The lower and upper broken lines consisting of two
diabatic `rays' can be regarded as two adiabatic potential profiles (shown
by dots). The splitting of PES is very small; therefore, Uc = U6.
(15)
where hoi is the average frequency of interlevel transitions. In the
standard theory of non-adiabatic transitions with a linear approximation of the diabatic energy levels (terms) in the region of their
intersection (see Fig. 6 b), it is usually assumed that 83 ± 87
hoi
789
7
2pV212
dy
hjDFj
y0 ,
(16)
where V12 is the matrix element of the interaction between terms;
DF = F2 7 F1, where Fn = 7qUdn =qy, (n = 1, 2) are forces acting
on the system at the intersection point y = y0; d(y 7 y0) designates
the Dirac d-function.
According to this (local) transition model, the expression for
the probability of transition between the terms is formally similar
to the Landau ±Zener approximation 88
P12
2pV212
.
hjDF j y_
(17)
The value
jna
2pV212
exp
hjDFj
Uc Ue
kB T
j yjP
_ 12 exp
Uc Ue
kB T
(18)
is a non-adiabatic diffusion flux at the point y0,89 Uc =
Ud1(y0) = Ud2(y0), and Ue is the energy in the reactant minimum.
Turning back to the adiabatic description in terms of the
diffusion equation (10), we can express an analogous adiabatic
flux near the top of the barrier of the lower PES Ua1 (see Fig. 4).
For the case where the barrier DU6 on the initial electronic PES is
8
M V Basilevsky, M V Vener
high and the splitting D is small (see Fig. 6 b), this potential profile
corresponds to a triangular potential barrier (shown by dots). For
such PES, the adiabatic flux equals 89
ja
D jF1 jjF2 j
exp
kB T jF1 j jF2 j
Uc Ue
.
kB T
(19)
A different adiabatic flux corresponds to Kramers' result (11),
(14). The difference is due to the fact that the potential shown by
dots in Fig 6 b has a singularity (cusp) on the top and the potential
in Fig. 4 is smooth. Fluxes (18) and (19) are found by solution of
diffusion equation (15) in the limiting cases of weak and strong
coupling between the levels. In the intermediate kinetic regime, the
flux j can be expressed as the ratio of the non-adiabatic to
adiabatic fluxes 89
The inclusion of the promoting mode R substantially complicates the PT description. In this case, the cross-section profile of
the reaction PES depends on R (see Fig. 1 a): the barrier is
relatively low in the TS region (R = R6); when R 4
4 R6, the
product and reactant wells become deeper, which disturbs the
adiabaticity criterion (7). Hence, according to criterion (23), the
transition is either adiabatic (activated, jna/ja 4
4 1) or non-adiabatic (tunnelling, jna/ja 5
5 1), depending on R and the temperature. In many cases, both mechanisms occur simultaneously on
the same PES.90 If the equilibrium position Re is close to R6 and
(or) the temperature is relatively high, the reaction dynamics will
mainly be classical. In the case of a sharp relief of the PES crosssection along the coordinate s and (or) low temperature, the
reaction mainly follows the tunnelling mechanism.
3. The principles of description of the tunnelling kinetics
jna
j
.
1 jna =ja
(20)
The corresponding rate constant is found from Eqn (20) by
division by the partition function of the reaction system calculated
within the reactant well.
The ratio jna/ja can be estimated by taking into account the fact
that, within the model of non-adiabatic transition with linear
intersecting diabatic energy levels,65, 79 ± 81
V12
1
D;
2
(21)
a. One-dimensional and quasi-one-dimensional tunnelling
Bimolecular rate constant. The character of tunnelling transitions
is substantially different for the three different types of PES crosssections V(s) (Fig. 7). In the one-dimensional model of the
bimolecular reaction, the states in the reactant and product
regions belong to a continuum spectrum (Fig. 7 a). In a quasiclassical approximation, the rate constant for this reaction is as
follows: 55, 91 ± 94
?
1
k T
2p
hZ T
0
E
P EdE ,
kB T
(24)
where E is the energy referred to the threshold (a minimum in the
reactant region) and P(E) is the quasi-classical coefficient for
barrier penetration
jDF j jJ jD.
Here D and J are characteristics of adiabatic levels:
D e2 y0 e1 y0 ,
q
qe
J j2
D 1 j2 1 j1
,
j1
qy
qR
y0
y0
exp
(22)
a
V(s)
V6
i.e., D is the minimum splitting of the adiabatic levels and J is the
measure of their interaction calculated for y = y0. If the signs of
the forces F1 and F2 are opposite, then
E
DV6
jDF j = jF2j +jF1j.
Thus,
a2
2
jna 2p D
kB T
2p kB T
.
^
ja
h jF1 jjF2 j 4D
DJ2
h
(23)
The above approximate equality corresponds to the symmetrical case
a1
V(s)
s
b
E
ei
1
jF1j = jF2j = jF1 7F2j.
2
Competition between adiabatic and non-adiabatic mechanisms
of proton transfer. It follows from Eqns (20) ± (23) than the nonadiabatic kinetic mechanism of PT takes place when the splitting
of the adiabatic levels is small (in this case, Uc = U6). Small D
values are typical of double-well potential profiles of the ground
electronic state with a sharp relief. They give rise to electron ±
proton profiles with a cusp at the intersection (see Fig. 6 b). For
these electronic PES, tunnelling is observed, the model of nonadiabatic transitions on the electron ± proton PES being a way of
describing the tunnelling phenomenon. Equation (23) expresses
the relative intensities of the tunnelling (non-adiabatic flux jna)
and activation (adiabatic flux ja) mechanisms of proton transition.
According to (23), they depend on the temperature. At the critical
temperature T = Tc where jna/ja = 1, the reaction mechanism
switches. Thus, the flux ratio (23) is the key control parameter of
the simplest expression for the rate constant taking into account
both PT mechanisms.87
a1
a2
a3
V(s)
s
c
ei
D
ef
s
Figure 7. Three different types (a ± c) of PES cross-sections V(s) for the
one-dimensional reaction scheme.
The turning points a2, a1 bound the classically inaccessible region, while
points a3, a2, confine the classically accessible region; ei is the reactant
energy level, ef is the product energy level, D is the resonance misfit.
Theoretical investigations of proton and hydrogen atom transfer in the condensed phase
P E
1
,
1 exp2W E
(25)
a1
1
f2m0 V s
W E
h
Eg1=2 ds.
where V6 is the height of the potential barrier. In this case, relation
(24) provides the standard result of the one-dimensional classical
transition state theory, TST.
DV 6
.
kB T
(27)
Relations (24) ± (27) can be recommended for application to
real three-dimensional PT processes if a `one-dimensional' transmission coefficient
k T
k
kTST T
(28)
is introduced and used as a correction factor, i.e., the rate constant
for a real three-dimensional reaction obtained by the standard TS
method is multipled by this factor.
Decay kinetics of single-well systems. In the decomposition
reaction with a potential profile shown in Fig 7 b, the states of
reactants are quantised, while those in the product region belong
to a continuum spectrum. By designating the energies of the
reactant level by en and the partial decay rates by (1/
h)Gn (Gn is
the width of the level), we obtain the rate constant
k T
X exp
n
en =kB T G en
.
Z T
h
(29)
In the quasi-classical approximation,95
G E
ho E
exp 2W E,
2p
(30)
where the under-barrier action integral W(E) is defined in the
same way as in Eqn (25); however, now V(s) has the form shown in
Fig. 7 b. The decay frequency o(E) is expressed in terms of the
action integral y(E), which is calculated in the region of classical
motion confined by the turning points a3 and a2,
1
y E
h
o E
a2
a3
ds 2m0 E
p
.
h qy=qE
1=2
V s ;
(31)
When the interaction of the coordinate s with the environment
modes is taken into account, these relations are modified in the
following way. The interaction makes the levels en diffuse; in this
case, it is expedient to introduce the distribution function rn(E) for
a level n
?
rn E dE 1,
0
(33)
The N(E ) value, called the `cumulative transition probability',92 ± 94 is defined as follows:
The turning points a1, a2 depend on the energy E and confine
the under-barrier area inaccessible for classical motion. In this
case, the partition function Z(T ) corresponds to one-dimensional
translational motion; as a result, the bimolecular rate constant has
the dimensionality (time)71(length)71. In the classical limit, P(E)
is replaced by a stepwise function
(
0; E < DV 6
P E
(26)
1; E > DV 6 ;
1
exp
2phZ T
and the density of states
X
r E
rn E .
n
a2
kTST T
9
(32)
N E 2pr E G E .
(34)
Then the expression for rate constant (29) is reduced to the
form
?
1
E
k T
exp
N EdE .
(35)
2p
hZ T
kB T
0
This expression formally coincides with expression (24) for the
rate constant of the bimolecular reaction, the difference being in
the interpretation of the partition function. The Z(T ) value now
includes the states located in the reactant well; taking account of
the level broadening due to interaction, it is found as
?
Z T
0
exp
E
r EdE .
kB T
(36)
The quasi-classical expression for the cumulative probability
(34) is similar to expression (25) 55, 92 ± 94
N E
1
1 exp2W E
(37)
with an action integral defined in the under-barrier region (see
Fig. 7 b). This description of decay processes is almost identical to
the quasi-one-dimensional theory of bimolecular reactions. However, a consistent definition of the decay model should take into
account the interaction of the reaction centre with its environment
not only in the reactant region but also for tunnelling under the
barrier.55, 96 The calculation of the action integral is modified by
introduction of a non-local potential brought about by the
interaction. The corresponding theory 55, 96 ± 98 is a quantum
generalisation of the classical stochastic scheme based on the
Langevin equation (13) for the tunnelling coordinate s. The
friction coefficient g is a parameter of the interaction intensity; it
is present in the final quantum expression for the rate constant.
The activation and tunnelling mechanisms. The classical and
quantum dynamic effects are manifested in the chemical reaction
kinetics as a competition between the activation (classical) and
tunnelling (quantum) mechanisms of the reactive transition. This
was described above within the framework of a non-adiabatic
diffusion model. Now it will be considered using an alternative
tunnelling dynamics language.
As in the previous case, in the simplest one-dimensional
scheme, there exists a critical temperature Tc (see several studies 55, 97 ± 99) such that the activation mechanism with the Arrhenius temperature dependence of the rate constant predominates
above this temperature. At temperatures below Tc, the reaction
5 1,
mainly follows a tunnelling mechanism and in the limit T/Tc 5
the reaction rate does not depend on the temperature. This lowtemperature limit differs from zero when the reactant region is a
potential well; it corresponds to transition from the ground
vibrational level of the reactants. Systematic analysis of the ratio
of the tunnelling and activated transitions is reduced to investigation of the integral over energy in Eqns (24) and (35). The
theory can be reformulated using the language of classical
dynamics, which describes the motion in the `inverted' potential
V s V s V s for the total energy E.55, 97 In this potential, a
cyclic classical trajectory with the period t(E ) is formed. The
energy corresponding to the highest contribution of the integrand
to integral (35) satisfies the equation
10
M V Basilevsky, M V Vener
t E
h
.
kB T
(38)
Its solution determines the energy corresponding to the cyclic
trajectory as a function of temperature. This consideration underlies the calculation of integral (35) by the stationary phase
method.97, 100, 101 The temperature-dependent extremal trajectory
that satisfies condition (38) is called instanton. There exists a
critical temperature above which Eqn (38) has no solution. It can
be identified with the above-mentioned Tc value. Then the
maximum contribution to the integral is determined by the energy
near the top of the barrier, and formula (35) reduces to the result
of the TS theory with slight quantum corrections. In the stochastic
tunnelling model, which takes into account the interaction with
the environment based on the Langevin equation, the Kramers
equation for the rate constant, similar to relation (11) but with
allowance for (14), is obtained in the classical limit.102 ± 104 Thus,
the relationship between two theories, the non-adiabatic and the
tunnelling, is established; the tunnelling probability is determined
in the latter case by the value of action on the instanton trajectory.
The simplest equation for the critical temperature obtained for a
parabolic barrier V(s) with the decay frequency o6 has the
form 99, 105
Tc
ho6
.
2pkB
(39)
Tunnelling in a double-well potential. The calculation of
tunnelling is complicated on passing to a double-well potential
where it is necessary to take account of the quantisation of energy
levels in both wells. Transitions between two levels take place only
in the presence of interaction between the coordinate s and other
degrees of freedom. Even where the levels are in resonance, i.e.,
very close or coinciding, as in symmetrical systems, the interaction
with the environment is required to ensure energy dissipation and
destruction of the transition coherence, which are necessary
conditions for irreversibility. Only due to these factors, can the
system be fixed in the product region and a stationary flux for the
two wells can be formed. In asymmetric systems, the interaction
with the environment is also necessary in order to compensate for
the resonance misfit (i.e., the energy difference between the levels
between which the transition takes place) by means of the
intermode energy exchange.
Thus, double-well transitions are, in essence, multidimensional even in the case where the PES structure seems to allow
the reaction mode to be considered as an independent variable.
The intermode interactions, which ensure the energy exchange
and phase relaxation (the latter term is equivalent to the term
`coherence desctruction'), can be small within the scale of the
energy variation on the PES, but only consistent allowance for
these interactions provides the possibility of correct description of
the tunnelling dynamics.
b. Collinear model of simple gas-phase reactions
Quantum computation scheme. Let us consider a two-dimensional
quantum problem for collinear model (1) of a PT with two
coordinates s, R and the corresponding reduced masses m0 and
M. After conversion to a mass-weighted coordinates s 0 = s(m0)1/2
and R 0 = R(M)1/2, it is expedient to introduce polar coordinates
1=2
,
(40)
y s0 2 R0 2
s0
.
R0
The potential energy surface in these coordinates is shown in
Fig. 8. The two valleys stretched along y correspond to the
translational motion in the asymptotic regions, as we consider a
gas-phase bimolecular reaction. This PES is confined within the
angle
a arctan
s0
y
y
R0
a
y
M
Figure 8. Potential energy surface of the gas-phase PT and HAT reaction
in the polar coordinates.
The isoenergy contour lines of PES are shown only for the product valley
(a > 0). The potential energy in the reactant valley (a < 0) has the same
form. The point M has the polar coordinates y, a.
mA1 mA2 mH 1=2
2y arctan mH
,
mA1 mA2
(41)
where mH, mA1 , mA2 are the masses of the H atom and the heavy
fragments A1 and A2 , respectively. The reduced masses m0 and M
are related to these masses in the usual way. The Hamiltonian for
this problem has the form
h2 q2
1 q
(42)
H
ha ,
2
2 qy
y qy
ha
2 q2
h
V ajy,
2y2 qa2
where the operator ha depends parametrically on y. When y is
constant, variation of the mode a is described by the SchroÈdinger
equation
ha jn ajy en yjn ajy.
(43)
Equations (42) and (43) are similar to Eqns (8), (9) for the
coordinate s and are converted into the latter (to within a mass
factor) after the substitution
s = ya .
(44)
As a result of the coordinate scaling performed, the major
dynamic feature of PT reactions, i.e., the fact that the ratios
mH
m
5
51 , H 5
51
mA1
mA2
are small, is manifested in the PES relief, in particular, the angle y
becomes a small parameter in this description
y5
5 1.
(45)
It can be seen in Fig. 8 that in this case, the reactant and
product valleys are nearly parallel, the barrier separating them
being narrow. This accounts for the specific features of process
dynamics (1), the possibility of tunnelling along the variable a
being the most important of them.
In numerical dynamic studies of system (1), the solutions jn(a)
of Eqn (43) are used as the basis functions. The two-dimensional
SchroÈdinger equation (43) is reduced to a set of one-dimensional
coupled equations for the motion along the translational coordinate y. The solution of this set of equations provides the
probabilities Pmn(E) of reactive transitions from the reactant
valley to the product valley and the probabilities Rmn(E) of
reflection in the reactant valley. The transition takes place
between the states jm and jn at a given total energy E. The
Theoretical investigations of proton and hydrogen atom transfer in the condensed phase
procedure of such calculations has been well developed 106, 107 and
extended to the description of reaction in a real 3D space, which
takes into account the interaction of the internal degrees of
freedom with the rotational motion.108 The results of numerical
studies are reported in several reviews.109 ± 112 The potential in Eqn
(43) has a complex form. Depending on y, it can be either doublewell (large y) or single-well (small y). For non-symmetric reactions, the potential is asymmetric. It should be emphasized that
the use of eigenfunctions of the Hamiltonian ha with this complex
potential as the basis functions has a crucial impact on the success
of calculations.106, 107 The convergence of the solution of Eqn (43)
with oscillatory basis functions proved to be unsatisfactory.
The results of calculations showed that the reactions in symmetrical systems are adiabatic, i.e.,
Pmn E
Pmm E Pnn E 1=2
5
5 1.
(46)
11
a
Energy
Vdf (a)
Vdi (a)
V(a)
ei
D
a2
ef
a
a1
b
Energy
In the case of asymmetric reactions, transitions between the
vibrational levels of the reactants eim and the products efn
characterised by the smallest in magnitude resonance misfits
E
Dmn = efn 7 eim
sharply predominate. This is a natural consequence of inequality
(45). The small parameter y is a measure of dynamic interaction
between the coordinates a and y. Since the interaction is weak,
transformation of the resonance defect energy into the energy of
translational motion is unlikely and resonance or nearly resonance processes predominate.
Study of the collinear dynamics at a qualitative level. When
analysing studies of the collinear dynamics at a qualitative level,
we restrict ourselves to the discussion of HAT reactions that have
rather high potential barriers (V6 5 10 kcal mol71) and can be
expected to proceed in the tunnelling regime even at high temperature (T 5 300 K). The early studies of these systems 113, 114 were
performed using a multi-dimensional quasi-classical approximation,115, 116 whose one-dimensional version is considered in Section II.3.a. Extension to multidimensional (two-dimensional, for a
collinear reaction) case brings about a procedural difficulty: it is
necessary to match the trajectories of a multidimensional classical
motion at the boundary of classically allowed (reactant and
product valleys) and classically forbidden (under-barrier region)
PES sections. A rigorous solution of this problem is extremely
unwieldy. It is more convenient to proceed directly from quantum
equations of motion which do not face the matching problem.
The quantum interpretation becomes simple for those regions
of PES where the cross-sections along the coordinate a have a
rather sharp double-well profile. This is not the case in the vicinity
of the saddle point, which imposes restrictions on the conditions
under which the HAT process can be calculated; it is assumed to
proceed in the tunnelling regime, the classical activated transitions
being suppressed. In this situation, there exist diabatic states well
localised in the reactant wim(a) and product wfn (a) regions. They
can be regarded as eigenfunctions of two different one-dimensional Hamiltonians hi and hf :
hi wim a eim wim a;
(47)
hf wfn a efn wfn a:
Single-well potentials in the Hamiltonians hi and hf are
distorted with respect to the true double-well profile V(ajy) [see
Eqn (42)] in such a way as to ensure localisation of eigenfunctions.
In Fig. 9 a, the continuous line shows the double-well adiabatic
profile V(a) for a given value y. The diabatic potentials coincide
with it in the region of one of the wells and are continued by dots to
the region of the neighbouring well. As noted above, it is
legitimate for considering the transitions between two diabatic
states closest in energy. Let energies of these states be eim and efn.
This corresponds to the two-level model in which the designations
can be simplified: wim ? wi, eim ? ei, wfn ? wf, efn ? ef. The solu-
D
ei (y)
ef (y)
yf yi
y
Figure 9. Model of a two-dimensional tunnelling transition for the PES
shown in Fig. 8 (asymmetric case);
(a) adiabatic [V(a)] and diabatic [Vdi (a) and Vdf (a)] energy profiles along
the coordinate a; (b) pattern of the translational motion along the
coordinate y in the reactant and product valleys. The difference in the
diabatic potentials ei (y) 7 ef (y) = D is constant. The turning points yi , yf
correspond to the energy E.
tions of the complete SchroÈdinger equation (43) designated by jn
are adiabatic states; in the two-level model, they are linear
combinations of two functions, wi and wf. For symmetrical
systems, ei = ef = e and adiabatic functions possess definite
symmetry properties, one of them being symmetric and the
other, antisymmetric
js a
ja a
1
2 1 s1=2
1
2 1
s1=2
wi a wf a,
(48)
wi a
wf a;
where s is the overlap integral of diabatic functions.
When the potential V(a,y) is substantially asymmetric, one of
the adiabatic functions is mainly composed of function wi with a
small portion of function wf, while the other, mainly of the
function wf. Then the adiabatic states are labelled by the same
subscripts i and f. In the tunnelling regime, the adiabatic energy
levels ei, ef differ little from the diabatic energies ei, ef. In this case,
the resonance misfit
Dif ef
ei & ef 7ei
(49)
becomes an important dynamic parameter.
Within the framework of the model with two coordinates a
and y, the potential profiles shown in Fig. 9 a depend on y.
Diabatic levels become the functions of y and their energies play
the role of repulsive potentials ei(y) and ef (y) in the reactant and
product channels, as shown in Fig. 9 b. More complex potential
profiles along y (with a well near the saddle point) can arise in
special cases;106, 107, 117 however, for reactions with high potential
barriers considered here, these situations are unlikely. Each vibrational state of the reaction system or `reaction channel' (i and f, or
s and a for a symmetrical reaction) is matched by its own potential.
12
M V Basilevsky, M V Vener
Due to the small magnitude of the y parameter [see inequality
(45)], the interaction of a given pair of closest channels with other
channels is negligibly small. This simplification has been suggested
by the experience of full multichannel computations. In an
approximation that takes into account only transitions between
two reactant and product channels closest in energy, we will
consider analytical expressions for the probability of reactive
transitions for different limiting situations.
In the case of a symmetrical system, channels s and a also split.
By solving a one-dimensional SchroÈdinger equation for the
coordinate y for each of them, one can get the following expression
for the probability of the reaction transition between the diabatic
states i and f:118
Pif E sin2 xs
xa ,
(50)
where xs and xa are the shifts of scattering phases in the channels s
and a. Quasi-classical estimate gives
?
sin xs
xa & xs
2
p
xa &
y0
ei y
exp Wi ydy.
pi y
(51)
Here
ei (y) = ef (y),
pi (y) is the momentum for the translational motion along the
coordinate y
Wi (y) is the under-barrier action integral calculated between the
turning points a2 and a1 [the symmetrical variant is considered (see
Fig. 9 a)].
Expression (51) can be simplified by expanding in a series near
the y0(E ) turning point in which pi (y) = 0 (see
Fig. 9 b).113, 114, 119 ± 121
In the case of an asymmetric potential profile, the transition
probabilities Pif (E) can be estimated only in terms of the perturbation theory (`golden rule'), whose applicability is ensured by low
values of these probabilities (Pif 5
5 1) and, finally, by the smallness of the PES parameter y. It is convenient to accept an
exponential form of diabatic repulsive potentials in channels i
and f (see Fig. 9 b):
ei (y) = e0 exp(7cy),
(52)
where c is the potential parameter, D = Dif is the resonance misfit
[see Eqn (49)].
For energy E, the potentials have turning points yi (E) and
yf (E). When the resonance misfit jDj is small, the following
expression was obtained:119 ± 121
Pif E Psif Eexp
D2
,
2cF
(53)
where Psif (E ) is the probability of non-adiabatic transition [see
Eqn (50)] for a symmetrical potential profile (see Fig. 9 a),
F
qe
qy
Pif E
6exp
6
p2 cjDj E D1=2
2p
E D1=2
c
E
E1=2
1=2
2
d lnA
A 6
dy
(54)
.
Here A = A(y) is a matrix element for the transition along the
vibrational coordinate (the tunnelling transition amplitude)
A y wi ajy H
1
ei ef wf ajyda,
2
(55)
where H is a two-dimensional Hamiltonian [see Eqn (42)]. The
amplitude A(y) is calculated at the turning point for function wi.
The dependence of the matrix transition element along the translational coordinate y plays an important role in HAT reactions
[the probability Pif (E) vanishes if this dependence is missing].
Equations (53) ± (55) reflect the main features of energy transfer between vibrational and translational degrees of freedom in
collinear HAT reactions. The main parameters that determine the
probability of transfer include the resonance misfit (D), characteristics of adiabatic translational potentials (c, F ), the amplitude of
the tunnelling transition, and its dependence on the translational
coordinate.
4. Tunnelling in a multidimensional double-well potential
pi (y) = {2[E 7 ei (y)]}1/2,
ef (y) = e0 exp(7cy) + D,
If jD/E j 4
4 1, a different asymptotic estimate follows 122
,
y0 E
F is the force for the average potential
1
e ei y ef y
2
at the turning point y0(E).
Relation (53) is valid when jD/E j 5
5 1 and some additional
conditions are met.
a. Specific features of the reactive transition dynamics in double-well
systems
Continuum spectrum of vibration frequencies. Up to here, the
discussion has been concerned with reactions for which the states
of the reactants or the products (or both) have a continuum
spectrum of eigenvalues. The reactions in double-well systems
where all the energy levels are discrete require a specific approach.
They can proceed only due to interactions with the environment,
which is assumed to be a macroscopic system and, hence, has a
continuum energy spectrum. However, the continuum spectrum
of the medium differs from the one-dimensional continuum
spectrum inherent in the translational degrees of freedom of the
reaction system. The medium can consist of subsystems, each
having a discrete energy spectrum: in the simplest case, the
medium can be considered as an ensemble (`reservoir') of harmonic oscillators. The continuous nature of the general spectrum
of the medium is ensured by the fact that the number of its degrees
of freedom is infinite, i.e., we are dealing with a continuum
spectrum of vibration frequencies.
The interaction of the reaction mode with the oscillator
reservoir having a continuum spectrum of frequencies, as well as
the interaction with the translational degree of freedom (see
Section II.3.b) ensures transfer of the resonance misfit energy
and destruction of the transition coherence Ð i.e., two conditions
needed for a chemical reaction to proceed. As a result of this
interaction, the reservoir oscillators rearrange (i.e., change equilibrium positions) and are involved in the reaction. Thus, the
chemical reaction in a double-well reaction system is an essentially
multimode process that includes rearrangement of the states of the
medium with a continuum spectrum of frequencies.
The simplest pattern of quantum rearrangement. The theory of
quantum rearrangement processes has been developed for a model
in which the reaction transition occurs between two multidimensional paraboloids with shifted minima which describe the PES of
the reactants and the products (see Fig. 2).123 ± 129 We restrict
ourselves to the case where the normal modes of the reaction
system and their frequencies do not change during the transition
(a more general theory is extremely cumbersome, although it still
can be formulated 123, 127). The straight line connecting the minima of the paraboloids (see Fig. 2 a), i.e., the reaction coordinate,
is a linear combination of the normal modes of the reactants and
the products. The change in the distance along this straight line is
Theoretical investigations of proton and hydrogen atom transfer in the condensed phase
measured using the scalar parameter l [see Eqn (3)]. Let one
normal mode with the frequency o0 be quantum
ho0
4
4 1.
kB T
(56)
First, this is the high-frequency proton mode; as in
Sections II.1 and II.2, we designate it by s. On the harmonic PES
that we consider, each mode makes its individual contribution to
the height of the potential barrier; the contribution of the mode s
equals
DVs6
m0 o20 6 2
l sf
2
si 2 ,
(57)
where m0 is the mass corresponding to this coordinate, l6 is the
value of the reaction coordinate at the point of quasi-intersection
of diabatic PES (this is a minimum on their intersection hypersurface),
l6
1
DV
,
1
2
Er
DV Vf Vi is the energy difference between the minima of the
diabatic paraboloids, i.e., the heat of the reaction; Er is the
reorganisation energy, sf and si are the values of the coordinate s
at the minima. The other modes on will be considered classical
hon
5
5 1.
kB T
(58)
The frequencies on form continuum spectra. Then one can
expect that the reaction rate constant would have the form
k ks exp
DV16
,
kB T
(59)
where ks is the one-dimensional tunnelling rate constant in a
double-well potential profile that corresponds to the PES crosssection along the mode s, DV6
1 is the contribution of classical
modes to the height of the potential barrier. It is significant that
the partial height of the barrier determined by formula (57) is
much smaller than the total value V6, which is given by the
Marcus formula (6) with allowance for (5).
This qualitative discussion has been extended to the case of a
larger number of quantum modes.126 ± 129 The quantum modes
that satisfy criterion (56) form altogether the tunnelling preexponential factor of the rate constant, which depends only
slightly on the temperature. Classical modes are responsible for
the Arrhenius exponential factor in which the activation energy is
equal to their contributions to the barrier height.
Turning back to the classification of modes introduced in
Section II.1.b, one can state that formula (59) reflects the situation
where all the rearranging modes Qn (n = 1, 2...) are classical in
conformity with criterion (58). As the temperature decreases,
condition (58) starts to be violated for some of these modes.
They become quantum modes and form the tunnelling preexponential factor, whereas those modes that still remain classical
form the ever decreasing reaction barrier. This idealised pattern is
valid only for a double-well harmonic potential. In PT reactions,
the PES anharmonicity is substantial and the separation of
variables throughout the whole path along the reaction coordinate (i.e., along the curve connecting the potential minima via the
saddle point) is impossible. However, at a qualitative level, the
idea of more or less independent contributions of the proton (s)
and rearrangement (Qn) modes to the potential barrier remains
valid.
Comparison of different ways of description of the reaction
transition. Let us discuss in more detail how the theory formulated
in terms of normal modes (non-interacting degrees of free-
13
dom) 124 ± 129 can be correlated with the statement according to
which one or several reaction modes interact with an ensemble of
harmonic oscillators of the medium.130, 131 We have in view the
following. The discrete coordinates of the reaction system are
considered oscillatory in each of the two diabatic regions (those of
the reactants and the products). In each of these regions, their
interaction with the medium oscillators is assumed to be bilinear
and is excluded by diagonalisation of the corresponding diabatic
oscillatory Hamiltonian. The normal modes thus found are those
used in the double-well transition model. The allowance for the
interaction results in renormalisation of the initial oscillator
frequencies (both discrete and medium oscillators), shift of their
equilibrium positions and the change in the form of vibrations.
The assumption that normal vibration frequencies and forms are
the same in both diabatic regions is actually made for the system of
modes resulting from the diagonalisation mentioned above. The
oscillator modes of the medium change most appreciably. Their
interaction with the reaction system leads to a shift in the
equilibrium of the reservoir oscillators, which are different for
the reactants and the products.
Thus, the chemical reaction is accompanied by reorganisation
of the medium, which is a necessary item of the theory we consider.
Two limiting cases are analysed below. In the first case, the
promoting mode R is ignored and the reaction dynamics is entirely
determined by the medium reorganisation (electron transfer).
Conversely, in the second case, the role of the medium is
minimised and the main dynamic features of the reaction system
are determined by the interaction between the proton (s) and
promoting (R) modes (HAT in non-polar media). The promoting
mode is usually assumed to be classical.
b. Electron transfer as an example of multimode tunnelling
Calculation technique. We will consider multimode transitions
between the reactant and product states in which the quantum
numbers of oscillators in the reactant well (im1 , im2 , ...) are replaced
by oscillator quantum numbers in the product well (fn1 , fn2 , ...); the
first character implies the belonging to either the reactants or the
products, while the second one is the number of the level of the
corresponding oscillator. To simplify the designations, we replace
the set of characters in the parentheses by a single letter (a or b) and
write down the transition probability per unit time as Tab (it is
measured in s71). In a deep tunnelling regime, the probabilities of
individual transitions are low. They can be calculated using the
perturbation theory (in terms of the golden rule)
Tab
2p 2
A d Eb
h ab
Ea .
(60)
Here, Aab is the transition amplitude
Aab = hwb jH
Ea jwa i,
(61)
where wa and wb are the wave functions of multimode diabatic
states, which are the products of oscillatory functions; H is the
total double-well Hamiltonian; Ea and Eb are the energies of
diabatic multimode states in the reactant a and product b wells,
respectively; they are the sums of energies of the levels em1 , em2 , ...
and en1 , en2 , ... for separate oscillators
Ea = em1 + em2 + ... ;
Eb = en1 + en2 + ...
(62)
As in the one-dimensional formulation (see Section II.3.b), wa
and wb are not eigenstates of H: the oscillatory functions corresponding to them are localised in the reactant and product wells
and, hence, they are the eigenfunctions of two different diabatic
Hamiltonians, so that the corresponding off-diagonal matrix
element Hab does not vanish. According to the perturbation
theory, the probabilities Tab are additive, and the resultant
thermal rate constant for the reaction equals 99, 124, 125
14
M V Basilevsky, M V Vener
k
2p X
Ea
exp
A2 d Eb
kB T ab
h a;b
Ea .
(63)
In Eqns (60) and (63), the delta-function expresses the energy
conservation law for the multimode transition, which is accompanied by the intermode energy exchange. The summation is
carried out over all combinations of the oscillatory functions, i.e.,
multipliers in the multimode product functions wa , wb . The use of
the delta-function provides the possibility of counting all combinations that ensure energy conservation.
Like any other relation containing the delta-function, expression (63) acquires a definite meaning and can be really calculated
only if the sums appearing in it contain a continuum frequency
spectrum and can be reduced to integrals. Reactions in the
condensed phase represent an appropriate object for the application of this calculation procedure. In the harmonic model of the
medium, a continuum frequency spectrum is specified using the
spectral function J(o) (spectral density, see Supplement II).96, 130 ± 132
Let us consider the case where there is only one quantum
reaction mode (with the frequency o0), while the other modes
(modes of the medium) are reorganisation modes with a continuum frequency spectrum. Then the function J(o) gives the
frequency distribution for the intensity of interaction between
the reaction mode and the medium modes
J o const DQ or o,
(64)
where DQ(o) is the shift of the equilibrium position of the medium
reorganisation mode Q with the frequency o, which is induced by
this interaction and can serve as its measure; the function r(o)
characterises the density of oscillator levels with the frequency o;
it is different for different condensed media; the constant in the
equation does not depend on o.
The description of the continuum frequency spectrum by
means of the spectral density is considered in more detail in
Supplement II.
Expression for the rate constant.124, 125, 127, 131 Now we list once
again the key assumptions made in the model of multidimensional
tunnelling transition in a medium consisting of a continuum
ensemble of harmonic oscillators.
First, the reaction system is described by a single reaction
mode s with the frequency o0. This is a quantum mode [see
criterion (56)] related to the medium modes Qn, which are
characterised by frequencies on and masses mn. All modes of the
medium are reorganisation modes. In the continuum limit,
Qn ? Q(o). This limiting transition gives rise to the spectral
function J(o) as a characteristic of the interaction between the
reaction mode and the medium.
Second, the tunnelling reaction does not change the frequency
spectrum of the medium [i.e., the function J(o)] or the forms of
normal vibrations Qn [or Q(o)]. This assumption is not fundamental but it allows one to represent the results of the theory in a
rather simple form.
Finally, one special assumption concerning the form of the
amplitude of the tunnelling transition is also required
Aab A0 hwa jwb i;
A0
wi H
(65)
1
ei ef wf .
2
Here, the diabatic functions are represented by the products
wa wi wa , wb wf wb ,
where wi and wf are the eigenfunctions for the reaction mode s,
corresponding to the energies ei and ef, while wa and wb are the
products of the wave functions of the medium oscillators. The
functions wa and wb correspond to the energies
7
7
Ea=Ea 7ei, Eb=Eb 7ef .
These relations follow from the energy conservation law. The
overlap integral hwa jwb i is equal to the product of the overlap
integrals for the shifted oscillators of each of the medium modes; it
is called the multidimensional Frank ± Condon factor.
Equations (65) represent the Condon approximation according to which the dependence of the matrix element A0 on the
oscillatory coordinates is not taken into account. This approximation is justified provided that the frequency o0 is much greater
than the medium frequencies. In this case, the A0 value is
calculated for those medium coordinates that make the greatest
contribution to the full transition amplitude.
When these assumptions are taken into account, the reaction
rate constant given by relation (63) is transformed in the following
way: 127, 133 ± 137
A20
h2
k
?
C texp
?
C t exp
?
6
0
itDV
dt,
h
1
6
p
h
(66)
J o ch
ho=2kB T 1 cosot ish
ho=2kB Tsinot
do
.
sh
ho=2kB T
o2
Here, t is time; DV = Vf 7 Vi is the difference between the energies
of the diabatic PES at their minima [see Eqn (4)].
Relation (66) is a typical result of the multimode oscillatory
perturbation theory. It is widely used in the theory of electron
transfer in polar solvents.127, 129, 133 ± 137 In this case, s is the
electron coordinate and the function J(o) is proportional to the
reorganisation energy of the modes of the medium, i.e., the energy
evolved upon the shift of the equilibrium positions of the
oscillatory modes. The frequency dependence of J(o) is determined by the function Im{E(o)/jE(o)j2}, where E(o) is the
frequency-dependent complex dielectric constant, which is the
key phenomenological characteristic of the dynamic properties of
the dielectric medium. This representation of the spectral density
is valid for any reactions with charge redistribution in polar
disordered media. The parametrisation of J(o) can be quite
different for reactions in non-polar media or in molecular crystals
(see Supplement II).
Equation (66), which seems to be complicated, holds for a
broad range of temperatures including the lowest ones and
describes the reorganisation dynamics for both classical and
quantum modes of the medium. More precisely, this theory is
valid as long as the key expression of the golden rule (60) holds.
There is always a lower temperature limit depending on the
resonance misfit and on the spectral density,130, 131 beyond which
the steady-state regime (which ensures the existence of exponential
decay and the rate constant) is violated and quantum beats
appear, i.e., coherent periodic transitions of the reaction system
from the reactant well to the product well and back. This lower
temperature limit is commonly considered to arise in the extremely
low-temperature region.
If all modes of the medium are considered to be classical in
conformity with criterion (58), expression (66) is reduced to the
usual Arrhenius law (59), which was postulated above relying on
intuitive considerations:
k
1=2
A20
p
exp
h E r kB T
Er DV2
.
4Er kB T
(67)
Here, the activation energy coincides with the potential barrier (6).
Thus, in the high-temperature limit, the electron transfer
kinetics is mainly determined by three key reaction parameters,
namely, the transition amplitude for the reaction mode, the
medium reorganisation energy Er, and the heat of the reaction
DV. The dependence of the rate constant on details of the
frequency spectrum of the medium is manifested only in the low-
Theoretical investigations of proton and hydrogen atom transfer in the condensed phase
temperature quantum regime. Various approximations of expression (66) are possible, which are valid in more or less broad
temperature ranges.99, 138 ± 141 One of them is given below.
c. Hydrogen atom transfer in the liquid and solid phase
Reaction model. The PT and HAT reactions in the condensed
phase have been considered in a number of early studies using
relation (66).27 ± 31, 142 ± 147 The proton coordinate was used as the
only quantum coordinate s. It can be seen from the above
consideration that this approximation is inadequate and that the
promoting mode R is to be taken into account.148 The simplest
plausible model should explicitly include the modes s, R and the
medium modes Qn.30, 148 ± 151 This model is formulated as follows.99, 148 The interaction of modes s and R is introduced through
the dependence of the proton transition amplitude Aif on R (recall
that this dependence plays a crucial role in the energy exchange in
gas-phase hydrogen atom abstraction). A convenient approximation is
7
(68)
Aif R A0 exp a R Re g R Re 2 ,
where Re is the equilibrium R value.
The second term in brackets rapidly decreases both for
positive and negative values of the R 7 Re difference. It is
necessary for providing the convergence of the perturbation
theory integrals with the continuum spectrum. Then it is sufficient
to assume that only the low-frequency mode R interacts with the
medium oscillators. The corresponding frequency and mass will
be designated by oR and mR. This mode is not a reorganisation
mode, i.e., its equilibrium position does not change upon the
reaction. Correspondingly, the modes of the medium do not
change equilibrium positions and remain exclusively promoting
modes. The full expression for the rate constant in this model 99 is
cumbersome and unwieldy. A rather simple expression can be
obtained by neglecting the squared dependence on R in the
exponent in Eqn (68) and taking that g = 0. Then
2
A
k = 02
h
C t exp
?
6
J o
0
?
?
itDV
C texp
dt ,
h
(69)
ha2
6
ph
ch ho=2kB T 1 cosot ish
ho=2kB T sinot
do .
sh
ho=2kB T
Here the spectral density J(o) describes the expansion of the
promoting mode R over the normal modes of the full system
comprising the reaction subsystem and the medium. In the
discrete representation,
X
Cn Qn .
R Re
n
The methods for the transition to the continual representation
that allow one to obtain expressions (66) and (69) are considered
in Supplement II.
Temperature dependence of the rate constant. One can expect
that there exists, at least, one localised normal mode that makes a
substantial (perhaps, the major) contribution to the coordinate R.
Therefore, the expansion
J o const d o
oR Jcont o,
(70)
where oR is the frequency of the localised normal mode, can be an
appropriate approximation.
The second (continuum) term remains, as usual, a
smooth spectral function. Therefore, using relation (70), the
factor 99, 148, 152
ha2
hoR
k * exp const
cth
,
(71)
p
h
2kB T
15
which determines the temperature dependence of the rate constant
in the temperature range where
hoR
9 1,
2kB T
(72)
can be separated in the rate constant.
This dependence is well fulfilled for photochemical elimination of a hydrogen atom by the singlet excited acridine impurity in
the molecular crystal of fluorene (Fig. 10). Condition (72) determines the position of inflection point on the temperature kinetic
curve; this is used to determine the resonance frequency oR. The
value found in this way coincided with the frequency of the
promoting mode (oR = 130 cm71) found by an independent
procedure using the vibrational spectrum of the fluorene crystal.153, 154
ln[k(T )7k(0)] (s71)
8
4
0
74
50
100
150
200
250
T /K
Figure 10. Temperature dependence of the rate constant for a photochemical reaction of HAT in a fluorene molecular crystal doped with
acridine.
Experimental data are shown by dots.153 The continuous line was
calculated using relation (71) for oR = 131 cm71 (see Ref. 152).
The factor (71) can also be distinguished in the equation for
the rate constant of the reaction with medium reorganisation [see
Eqn (66)]. The correlation (71) cannot be extended to a too broad
temperature range. With an increase in temperature, one should
take into account the Gaussian exponential component in the
transition amplitude (68),99 which is ignored in expression (69).
The corresponding temperature factor for
hoR =kB T 5
5 1 (when
the promoting mode becomes classical) has the form
1
a2 l 2R
k * exp
,
2 2l 2R g
hoR =4kB T
(73)
where lR is the amplitude of zero-point vibrations along the
coordinate R
lR
h
mR oR
1=2
.
This factor can be derived using simple quasi-classical consideration.148 Provided that
5
2l 2R g 5
hoR
4kB T
this leads to a linear dependence of lnk on T, which was observed
in many experiments. As the temperature further increases, the
dependence becomes more gentle. The linear section can be clearly
seen in Fig. 10. At very low temperatures (
hoR =kB T 4
4 1), the
continuum spectrum component J(o) should be taken into
account. The factor (71) becomes constant at this temperature.
The corresponding temperature dependence, which follows from
relation (69), has the form
lnk * T n,
16
M V Basilevsky, M V Vener
the power n (2 < n < 4) being determined by the spectral density
J(o) of the condensed medium.99, 155 This conclusion, however, is
not fully unambiguous due to the presence of a lower temperature
limit below which the concept of rate constant becomes invalid in
terms of the harmonic oscillatory model of the medium, and the
energy exchange proceeds by an anharmonic mechanism.
The above analysis of the temperature dependence of the rate
constant is wholly determined by the dynamics of the promoting
mode, which interacts with the modes of the continuous spectrum,
according to the accepted model [in the derivation of formula (69),
it is expanded over normal vibrations of the overall system
composed of the promoting mode and the medium coordinates].
The coordinate R does not pass across a potential barrier; therefore, the usual Arrhenius dependence such as (59) is not observed
at all. Within the framework of a classical consideration of the
coordinate R, this dependence could arise only due to the reaction
mode s, but this is a quantum mode at any temperature that may
be of interest and, hence, it does not make a significant contribution to the temperature dependence of the reaction rate.
III. Calculations for particular systems
1. Multidimensional quasi-classical calculations
a. Problem definition
Quasi-classical calculations are based on the quasi-one-dimensional model of tunnelling (see Section II.3.a). In the space of
active coordinates of the reaction system, the reaction path curve
connects the regions of reactants and products. Let us consider
one-dimensional tunnelling on the energy profile along this curve
and calculate the quasi-classical action W(E) [see Eqn (25)] in the
region under the potential barrier, which determines the amplitude of the tunnelling transition. Selection of the reaction path is
the key problem in this procedure. For a rigorous definition of the
problem, the curve should ensure a classical action extremum for
motion of the system in the inverted potential (the potential
barrier becomes a potential well, and the reactant and product
minima become top points of the potential relief). The classical
trajectory with specified initial and end points is calculated in the
inverted potential. The trajectory depends on the initial and final
quantum states between which the transition takes place. Boltzmann averaging of the initial states gives rise to a dependence of
the reaction path on temperature. The paths are different for lowtemperature (tunnelling predominates) and high-temperature
(tunnelling effects are slight) processes; its optimisation ensures a
smooth transition between these kinetic regimes. However, in
practice, such advanced scheme has never been applied to particular HAT or PT reactions. Actually, calculations are always
carried out for a fixed trajectory chosen by some procedure.
The equation for the rate constant in a multidimensional case
has the form:156, 157
k = A exp[7W(E )].
The pre-exponential factor A can be conveniently calculated
in a special coordinate system, which is usually referred to the
reaction path curve, also called the reference curve. One of the
coordinates (the reaction coordinate) is measured as an arc length
along the reference curve. A local system of Cartesian transverse
coordinates is specified at every point of the reference curve;
usually, the coordinates are modelled as harmonic oscillators. In
these `natural reaction coordinates,'158, 159 the PES is a multidimensional valley stretched along the reaction path and the
expression for kinetic energy has a complex form depending on
the curvature of the reference curve.158 ± 161 The corresponding
multidimensional Hamiltonian is called the reaction path Hamiltonian.160 The mimimum potential energy route is normally used
as the reference curve: in this curve, the PES cross-sections along
the transverse coordinates have minima.161 This path connects the
TS, i.e., the saddle point on the PES, with the reactant and product
minima, although the action for transverse coordinates rather
than the potential cross-section profile should have a minimum on
the reference curve in a consistent theory.157
b. The variational transition state theory
In the variational approach, the standard minimum potential
energy route on the PES is chosen as the reaction path (in some
cases, insignificant modifications are made).162 ± 164 The theory
remains classical in this respect. The expression for the rate
constant derived in terms of the classical TS theory by the
variational method is used.91, 165 The critical point in the reaction
coordinate at which the reaction flux is calculated (i.e., the TS)
does not coincide with the top of the barrier. According to the
variational theory, it is determined by minimising the expression
for the rate constant with respect to the position of the TS in the
reaction coordinate. At the final stage, a transmission factor (28),
determined by tunnelling along a curvilinear reaction trajectory, is
introduced to the rate equation.
This method was used to study HAT reactions in the gas
phase 166 ± 168 and in solutions 169 ± 172 and PT and hydride ion
transfer in enzymes.173, 174 The calculation of a model collinear
reaction (see Section II.3.a) in the presence of 250 solvent (methyl
chloride) molecules is notable from the procedural standpoint.175
The classical minimum energy path was calculated taking into
account all degrees of freedom of the system. As noted above, the
variational TS theory was implemented to calculate the transmission factor.
It is clear that this approach cannot be extended to the region
of low-temperature kinetics.
c. Low-temperature instanton method
In the low-temperature instanton approach,176 the reaction path
corresponds to a classical action minimum on the under-barrier
trajectory in the inverted potential. The start of the trajectory is
chosen in the reactant minimum on the real (rather than inverted)
PES. This is valid for T = 0 K. The classical action is calculated
along such trajectory and a correction for the interaction with
transverse vibrational degrees of freedom is introduced into the
resulting expression (in terms of the perturbation theory).177 ± 180
The use of this method is currently restricted to the study of
tunnel splittings in the microwave spectra of non-rigid molecules.
The calculations of splittings have been carried out for both the
ground state and the vibrationally excited states. Systems with
intra- and intermolecular hydrogen bonds, namely, malondialdehyde and formic acid dimer, have been considered. Two-proton
tunnelling transfer in the porphyrin molecule has been discussed.181 A study of splittings in the malondialdehyde molecule
with 21 degrees of freedom is the utmost achievement in this series
of studies.182
d. The model instanton method
The model instanton approach 151 employes the classical reaction
path with a classical TS at the PES saddle point. This is an attempt
to take into account the influence of all the other (transverse)
vibration modes without resorting to the perturbation theory.
This is indeed possible for a model two-dimensional PES formed
by two paraboloids (see Fig. 2 a).183 Extrapolation of this model
to a multidimensional situation gave the equation 184, 185
W
X
W
X0
da T ,
as
ds T
1
a
(74)
s
where W is the dimensional action, unlike the dimensionless
action (1/
h)W [see, for example, relation (25)]. The action W0 is
calculated along the standard classical reaction path. The transverse degrees of freedom are separated into symmetrical (the
subscript s) and antisymmetrical (the subscript a); the symmetry
is determined at the point of classical TS. `Symmetrical' and
`antisymmetrical' are synonyms for `reorganisation' and `promoting' modes. The corrections they introduce to the action are
additive. The ds, as and da values can be explicitly calculated
Theoretical investigations of proton and hydrogen atom transfer in the condensed phase
from a specified multidimensional PES. Expression (74) is an
empirical prescription, which provides physically reasonable
estimates in some limiting situations. Of interest is the appearance
of a typical temperature dependence of the action
ds T ds
cth
T0
hos
,
2kB T
As in standard instanton methods, the pre-exponential factor
is replaced by an effective frequency, which characterises the
system vibrations in the reactant potential well.
This method was applied to study the tunnel splittings for
intramolecular PT (malondialdehyde,151 tropolone 186 and
9-hydroxyphenalenone 185). The rate constants have been calculated for many reactions 187 ± 191 including two-proton transfer in
porphins.185 The temperature dependences of the rate constants
and the isotope effects were compared with experimental data.
The tunnel splittings for malondialdehyde have been calculated taking into account all the vibrational degrees of freedom
and using various modifications of the instanton method.192 ± 194
2. Two-state model with the molecular description of the
medium
a. Collective coordinate of the medium
The quantum theory of PT reactions with allowance for two
electron ± proton states is used fairly often (see Section II.2).
A Hamiltonian depending on the variables s (proton coordinate)
and R (symmetrical vibration of the A1 and A2 fragments,
promoting mode) is written down for a three-particle
A1_H_A2 model system. An ensemble of modes of the medium
Qn, which interact with s and R, is added to the system. The
complete Hamiltonian depends on each of the variables. Quantisation of the coordinate s in the basis set of two diabatic functions
jn(sjR,Qn), n = 1, 2, gives the matrix Hamiltonian
H11
H21
H12
, H12 H21 .
H22
(76)
The diagional elements are the classical Hamiltonians for two
electron ± proton states
H11 TR TQ e1 R; Qn ;
(77)
H22 TR TQ e2 R; Qn ;
where TR and TQ are the kinetic energies for the mode R and the
medium coordinate Q; e1 and e2 are the corresponding FES, which
are found by solving the SchroÈdinger equation for the coordinate s
hs jn sjR; Qn en R; Qjn sjR; Qn ; n 1; 2 .
It should be noted that Hamiltonian (76) can be represented in
the form
0
H12
H
H11 ,
(79)
H21 De
De H22
where os is the frequency of the corresponding symmetrical
vibration. The reaction rate constant is calculated in terms of the
standard instanton procedure
1
k T A T exp
W T .
(75)
h
H
17
H11 e2 R; Qn
e1 R; Qn .
The dynamics and kinetics of the interlevel transition do not
depend on the scalar (non-matrix) value H11. The matrix element
of the transition H12 (equal to H21) is specified empirically. The
energy gap De is the only significant variable; as can be seen from
Eqn (79), the dependence on the particle velocity, present in Eqn
(77), has disappeared because the classical kinetic energy is the
same in the states j1 and j2. The variable De is a typical collective
coordinate; calculation procedures with the use of this variable are
well developed (numerical modelling by the MD or Monte-Carlo
methods). This investigation scheme was first applied by Warshel
and coworkers;195 ± 197 subsequently, various modifications of this
scheme were applied by many authors.
b. Triatomic model of a proton transfer reaction in a model solvent.
Calculations by molecular dynamics
A popular investigation object is the PT reaction (acid ± base
equilibrium) in the molecular complex
A7H_B
A7_H7B+,
(80)
where PT is accompanied by pronounced polarisation of the
heavy fragments A and B. In the calculations described above,
the fragment A is represented by phenoxide C6H5O7, while the
fragment B is the amino group [trimethylamine N(CH3)3 is a
typical example]. The reactants and products in reaction (80),
described by the functions jn, which have served as the basis for
constructing Hamiltonian (76), have dipole moments m1 & 2 D
(covalent complex) and m2 & (10 ± 15) D (ion pair), respectively.
The solvent particles are simulated in MD calculations by methyl
chloride (CH3Cl, m = 2.15 D).
The calculation starts with explicit solution of the SchroÈdinger
equation (78) for the diabatic proton states. The single-well
diabatic potentials V(sjR,Qn) appearing in the proton Hamiltonian hs are modelled empirically by combining the LEPS (London ± Eyring ± Polanyi ± Sato) type schemes for the internal
variables s, R in the gas-phase reactions 198 with the MD parametrisation of the interactions involving the solvent particles.63
With respect to the proton coordinate s, the functions V(sjR,Qn)
are highly asymmetric single-well potentials (see Fig. 3). With
these potentials, Eqn (78) can be solved numerically. This demonstrates the unique advantage of introducing the collective
variable De(R,Qn) [see Eqn (79)]. Equation (78) is solved for a
one-dimensional set of values of the coordinate De rather than for
the whole set of combinations of R and Qn (which arise, for
example, in a calculation with 256 solvent particles 63). This gives
two diabatic FES, e1(R,Qn) and e2(R,Qn), in which the explicit
dependence on the medium coordinate is included through De.
Diagonalisation of the Hamiltonian (79) gives two adiabatic FES
(78)
The proton Hamiltonian hs is specified by Eqn (9). The
potential in this case is the full PES in the coordinates s, R and
Qn, in which R and Qn are considered as parameters. The
corresponding parametric dependence appears in the solution of
Eqn (78), i.e., in the eigenvalues en and eigenfunctions jn.
A remark concerning the designations is pertinent. In the key
relations (8), (9) for quantisation of the proton coordinate s, the
type of representation (either adiabatic or diabatic) is not specified; therefore, the potentials V(s) and the eigenfunctions jn(s) can
be both adiabatic and non-adiabatic. In further consideration in
Section II.2, the designation jn is used for functions of the
adiabatic basis, whereas diabatic functions were designated by
wn. Here, it is more convenient to use the common designation jn
for both diabatic and adiabatic functions.
E1;2
1
2 1=2
.
e1 e2 De2 4H12
2
(81)
The free energy surface for the ground adiabatic state E1 is
double-well (see Fig. 2 b). It is significant that in the TS region, the
medium coordinate De is the reaction coordinate: the FES profile
along De has a maximum at the saddle point.
The way for calculating the kinetics of reaction (80) is obvious
in the adiabatic approximation where the transitions between
surfaces E1 and E2 are ignored. In the simplest version, the
coordinate R is considered to be classical (the medium coordinates
Qn are always classical in this model). The rate constant is then
calculated using a standard MD procedure, namely, the steadystate reaction flux method.55 It is reduced to the calculation of
18
M V Basilevsky, M V Vener
kTST, i.e., the rate constant in the TS approximation, and the
correcting classical transmission factor kc :
k = kc kTST.
(82)
For determining kc, the correlation functions for the coordinate De and its derivative with respect to time (D_e) have to be
calculated (by MD methods).63, 199, 200 When these functions are
known, one can also determine the friction kernel in the generalised Langevin equation (see Supplement I) and calculate kc within
the framework of the KGH stochastic theory.63
In earlier studies of reaction (80), the promoting mode R was
not taken into account.29, 200 ± 203 The most comprehensive study
was carried out for a two-dimensional model (R,De) with allowance for quantisation of R.63 The transmission factor varied in the
range of 0.4 < kc < 0.8 depending on the parametrisation. The
attempts to take into account the non-adiabatic transitions within
the framework of this scheme are not quite consistent.63, 200 The
transition probability for a one-dimensional potential profile
along the coordinate De has been calculated by the Landau ±
Zener method.88 Then it was averaged with allowance for the
statistical distribution of the De values using the MD technique.
The choice of parametrisation for the transition matrix element
H12 is significant at this stage.
c. Quantum effects
According to the accepted terminology (see Section II.2.a),
reaction (80) has a low barrier. The quantum effects involved are
insignificant. Conversely, HAT reactions in systems containing
hydrocarbon molecules have high barriers. The quantum transmission factors for them are low (kq < 1074 or lower 166 ± 168). The
golden rule procedure for their calculation is considered in
Sections II.4.a and II.4.c. This line of research was initiated by
early studies performed before the mid-1980s.126, 143, 148 Subsequently, this approach was modified by using advanced computer
simulation procedures.27, 28, 30, 149 However, no calculations of
this type for particular systems taking account of quantum effects
have been reported so far.
A complete and consistent theory for describing the kinetics of
reaction systems with multidimensional double-well FES taking
into account the tunnelling and non-adiabatic transitions should
cover the two above-mentioned limiting cases and intermediate
kinetic regimes. No theory of this type has yet been developed.
Some approximate approaches are considered in Section II.2.b.
Calculations of the probabilities of non-adiabatic transitions in
model one-dimensional multilevel systems are documented.204 ± 206 In terms of the principles inherent in them, these
calculations are related to calculations of two-level systems
considered in this Section and to quantum-classical calculations
of PT (see below, Section III.4).
3. Continum models of the medium
a. Definition of the problem in the adiabatic representation
A two-state model of the reaction subsystem with explicit inclusion of the surrounding solvent particles is considered in
Section III.2. An alternative problem formulation is possible on
the basis of a continuum description of the medium.64, 65, 69
The first stage is to construct the FES for the ground state of
the system spanned by the coordinates s, R, X, where X is the
collective coordinate of the medium. Its definition was borrowed
from the theory of electron transfer.66, 87, 207, 208 The corresponding Hamiltonian has the form
H Ts TR S W s; R; X,
(83)
where Ts and TR are the kinetic energies of modes s and R; in the
general case, these are quantum-mechanical operators; the value
S = X2/2Er is the self-energy of the polarisation field of the
medium. Polarisation is induced by the electric charges of the
chemical subsystem, which are calculated by quantum-chemical
methods. The reorganisation energy Er appearing in the expression for the self-energy is the key parameter of the electron transfer
theory (see Sections II.1.b and II.4.b); it is calculated within the
framework of continuum models of the medium.209 ± 212 The threedimensional FES W(s,R,X) is constructed in terms of the semiempirical scheme combining the quantum-chemical calculation
with calculation of the polarisation effects (dependence on X) in
terms of continuum models.213, 214 This FES is a double-well
surface on which the reaction transition takes place.
The proton coordinate s is excluded at the next stage. The
numerical solution of the SchroÈdinger equation
[Ts + W(s,R,X)] jn(sjX,R) = en(X,R) jn(sjX,R)
(84)
for different fixed X and R values gives the eigenvalues en(X,R)
and the eigenfunctions jn(sjX,R); n = 1, 2. They correspond to
the adiabatic representation, because W(s,R,X) in Eqn (84) is the
total potential of a two-level system. (In this respect, this scheme
differs from the diabatic one considered in Section III.2.b,
although the diabatic definition is also possible in the continuum
model of the medium.215) Two two-dimensional FES, e1(X,R) and
e2(X,R), obtained from Eqn (84) correspond to the ground and
excited states of the proton subsystem. The dynamic description of
the continuum variable X is performed using stochastic or
diffusion models. Two-dimensional and two-level stochastic (or
diffusion) equations of motion, whose one-dimensional versions
are considered in Section II.2.b and in Supplement I, are found for
these surfaces.
b. Non-adiabatic transitions
The matrix element which determines non-adiabatic transitions
between the levels in the adiabatic representation is an imaginary
quantity 65, 69
H12
h
u J uX JX ,
i R R
(85)
where uR and uX are the velocities corresponding to the coordinates R, X
q
JR j2
j1 ,
qR
(86)
q
JX j2
j .
qX 1
The intensity of non-adiabatic transitions is determined by the
parameter x
jhH12 iT j
,
e2 e1
2kB T 1=2 JR
JX
hH12 iT
h
p p
.
mX
mR
p
x
(87)
Here, the averaging h:::iT takes place over the Maxwell
velocity distribution; mR and mX are the masses corresponding
to the coordinates R and X (mX is estimated on the basis of
generalised Langevin equation 64, 69). The x value is calculated at
the point where the surfaces e2 and e1 approach each other most
closely and the energy gap e27e1 is the smallest.
c. Systems with the C7H_C reaction centre
The free energy surface e1(X,R) for the ground (proton) state of
the reaction system
C6H5CH3 + 7CH27C6H5
C6H5CH2 + CH3C6H5
(88)
is shown in Fig. 5. It bears a strong resemblance to a similar
double-well FES calculated in the molecular description of the
medium.63 The calculation of reaction (88) showed 64 that x 5
5 1,
i.e., the reaction is adiabatic. This is a reaction with a low barrier.
If non-adiabatic transitions are neglected, the classical reaction
kinetics can be derived using the KGH theory.
The opposite limiting case is represented by the reaction
Flu7 + HFlu
FluH + Flu7,
(89)
Theoretical investigations of proton and hydrogen atom transfer in the condensed phase
where FluH is the fluorene molecule. This is a reaction with a high
barrier, a very small energy gap e2 7 e1 and x 4
4 1, which
complies with the non-adiabatic kinetics. The quantum transmission factor is estimated as kq & 1075 ± 1077 (see Ref. 65). This
exceptionally low value corresponds to the deep tunnelling regime
and is at variance with the insignificant isotope effect
(kH/kD & 10) observed in the experiment. Apparently, this contradiction can be eliminated by noting that two internal coordinates s
and R are inadequate for full description of the PES for reaction
(89). It is necessary to take into account the dependence of PES on
the reorganisation modes describing the change in the environment of the reaction centre. Their role is described in Section
II.1.b. In view of this circumstance, it is possible to decrease the
effective barrier to tunnelling 216 and to increase the quantum
transmission factor by several orders of magnitude.
d. Coupled electron and proton transfer
The model described above was developed and applied for
interpreting the features of coupled electron and proton transfer.215, 217 ± 219 This type of process takes place in the system
N
R3Ru N
N H
O
+
7
N H
NO2
,
(90)
O
NO2
0
0
0
where R is 3,3 ,4,4 -tetramethyl-2,2 -bipyridine. The synchronous
transition of two protons is accompanied by substantial charge
redistribution between fragments of system (90). For describing
these effects one should circumvent the limitations of the two-level
model of the reaction subsystem. A large number (4 or 8 215, 218) of
electron ± proton states corresponding to various basis structures
of the valence bond method should be explicitly included into
consideration. In the case of N states, a matrix (N6N) Hamiltonian and N coordinates of the medium (X1, ...,XN) appear.
Before constructing the matrix Hamiltonian (i.e., before excluding
the coordinate s), one should proceed from the ground state
Hamiltonian, similar to Hamiltonian (83) but depending on all
the medium coordinates Xi, i = 1,...,N. The self-energy S of the
medium polarisation field is 208, 209
S
1X
T
2 i;j
1
ij Xi Xj ,
(91)
where T is the reorganisation matrix, which is calculated using
techniques of the continuum theory of the medium.
When dealing with the given multilevel theory, non-adiabatic
transitions between various pairs of basis states arise. They can be
classified into transitions with electron transfer and transitions
with PT. This calculation procedure has been developed in
detail 215, 220 and brought to the calculation of the rate constant
in a kinetic regime such that the non-adiabatic electron transfer is
treated in the usual way in terms of the golden rule and PT takes
place on adiabatic FES without transitions between the
FES.215, 219 The results provided an explanation for some features
of the coupled electron and proton transfer for system (90) and
other related systems. An alternative interpretation has been
proposed in earlier theoretical studies,221, 222 but they appear less
convincing.
e. Some conclusions
Little experience in using continuum models of the medium to
calculate PT reactions has been accumulated so far. Apart from
the studies described above, calculations of bimolecular and
intramolecular PT in water should be mentioned.223, 224 The
available material allows one to draw some conclusions concerning the prospects of the continuum method as applied to PT
reactions.
19
First, the use of continuum models greatly facilitates the
calculations compared to those done in terms of similar molecular
models of the medium.
Second, description of the rearrangement of chemical subsystem in the basis set of two electron-proton states restricts the
scope of the method. This is true not only for coupled electron and
proton transfer where extension of the basis of quantum states is
the key item of the theory.215, 218, 219 The calculations of FES for
usual PT systems revealed characteristic situations in which the
limitations of the two-level model result in serious inconsistencies.69 The two-level approximation equally restricts the validity
of calculations with the molecular description of the solvent.
Third, a substantial restriction follows from the usual twocoordinate model of PT, which takes into account only two
internal degrees of freedom (coordinates s and R) of the reaction
subsystem. The reorganisation of modes associated with the
change in the close environment of the reaction centre is an
important factor for many PT and HAT reactions with a high
barrier.216 It has a crucial influence on the kinetics of HAT
reactions; when the changes in these modes are ignored, the
tunnelling effect is substantially overestimated.
Fourth, it is necessary to note a serious problem of the
continuum theory, which restricts its practical value. The problem
lies in the basic concept of the cavity which incorporates the
reaction subsystem in the continuum medium. The change in the
cavity during the reaction is supposed to induce no substantial
energy or dynamic effects. This hypothesis is applicable to intramolecular processes for which the change of the cavity can indeed
be considered insignificant. In bimolecular systems with PT and
HAT, the cavity changes during the reaction but no procedures
for calculation of the concomitant effects are suggested by the
continuum model of the medium. This problem does not arise in
the case of molecular models of the medium; this forms the
strategic advantage of the latter.
Thus, the existing continuum theories of the solvent are barely
applicable to a detailed quantitative description of PT and HAT
reactions, although they are attractive for a qualitative description
due to their simplicity and clarity.
4. Numerical modelling of proton transfer reactions by
molecular dynamics
a. Quantum-classical methods
The use of simplified structural and dynamic models for a reacting
subsystem (see Sections III.2 and III.3) restricts the scope of
investigation of real chemical objects. In this respect, of interest
are attempts at direct computer simulation of PT dynamics based
on microscopic (molecular-dynamic) description of the solvent
without rigorous constraints on the form of quantum electron ±
proton states. These approaches consider joint dynamic evolution
of quantum and classical degrees of freedom. The number of
quantum variables is limited; usually, only a single proton
coordinate s is considered. The list of classical variables includes
several selected kinetically active coordinates of the reacting
subsystem and the coordinates of several hundred of solvent
particles. The general strategy for deriving the quantum-classical
equations of motion is standard (the main principles are outlined
in Sections II.2.a, II.2.b and III.2). The equations of motion along
quantum coordinates (the SchroÈdinger equation for the wave
function or the von Neumann equation for the density matrix)
involve a parameteric dependence on all the classical variables (via
the interaction potential).
Further development of this approach includes explicit derivation of the time dependence of the classical variables by MD
computation of their evolution. Substituting the resulting multidimensional classical trajectory into the quantum equation of
motion, i.e., expressing the parameters of the potential of a
quantum variable as functions of time, gives a time-dependent
Hamiltonian. It controls the quantum evolution, which is accompanied by transitions between the levels. This non-adiabatic
multilevel dynamic set of equations contains, in principle, tunnel-
20
M V Basilevsky, M V Vener
ling effects. The back influence of the quantum subsystem on the
classical variables is taken into account using the generalised
Helmann ± Feynman theorem; the corresponding classical forces
are determined by differentiating the electron ± proton PES along
the classical coordinates. Since the total quantum state is timedependent, i.e., it contains several stationary adiabatic components with time-dependent coefficients, their contributions are
averaged with time-dependent weights.
Thus, a classical MD computation gives non-pairwise multiparticle forces, which are added to the pairwise Lennard-Jones
forces acting between the solvent molecules. As a result, one can
obtain a closed self-consistent scheme for the calculation of
combined dynamic evolution of quantum and classical coordinates adapted for the use of well-developed MD procedures. This
is the main advantage of this scheme; however, the scope of its
applicability is not entirely clear. The completeness and selfconsistency is attained at the expense of several simplifying
assumptions having no convincing substantiation. Moreover, no
consistent algorithms for the calculation of reaction rate constants
(similar to the reaction flux method in usual classical MD 55) have
yet been formulated within the framework of this theory.
b. Some applications
Without further dwelling on the principles of quantum-classical
dynamics, we would like to note that there are several versions of
its computational implementation, which have been described in
reviews.225, 226 The calculations are rather laborious; most often,
they are applied to oversimplified model objects. We restrict our
consideration to the studies attempting to apply these procedures
to real chemical or even biochemical systems.
Ester hydrolysis. Hydrolysis of esters in a neutral aqueous
medium has been studied.227 The reaction complex of a dichloroacetic acid ester includes two water molecules. The following
mechanism was postulated:
O
O
7
Cl2CH
C
O
d3
O d1
H
H
d2
O
R
Cl2CH
H
H
C
O
R
H
O
O
+
H,
H
H
where R is the methoxyphenyl group. The distances d1 (O7H,
quantum variable), d2 and d3 (O_O and C_O, two classical
variables) were included in the dynamic calculation, while the
other coordinates within the complex were optimised. The PES
was calculated by the AM1 semiempirical quantum-chemical
method with the addition of the reactive field of the solvent to
the electronic Hamiltonian. This system, together with 510 solvent
(water) molecules, was included in the MD calculation. For the
estimation of the rate constant, the procedure described in
Section III.4.a was markedly simplified. A mechanism was postulated according to which the most probable pre-reaction configuration of the complex is formed initially and then fast PT takes
place in this configuration. The structure of the pre-reaction
configuration was also postulated; its formation was considered
as a purely classical equilibrium process (as in the TS theory). The
free energy change was calculated by the thermodynamic integration method, standard for MD. A temperature dependence of the
rate constant (free activation energy is 10.7 kcal mol71) appears
at this stage. The frequency of the quantum transition that
contributes to the pre-exponential factor for the PT was then
determined by solving the time-dependent SchroÈdinger equation.228 The primary result is the value of kinetic H/D isotope
effect, equal to 3.9, which is consistent with experimental data.
Enzymic hydrolysis.229, 230 The active site of the catalytic
system of an enzyme (phospholipase A2) can be represented in
the form
H2 C
H
H
O
O
C
O
CH2
H1
N
N
H
H
H
During enzymic hydrolysis, dissociation of water molecules
takes place
H2O
OH7 + H+.
As a consequence, the H1 atom moves to the imidazole fragment
(shown by an arrow).
The system fragment represented in the picture was calculated
by quantum-chemical methods (valence bond method calibrated
against the data of ab initio calculations using the density functional theory). This fragment was inserted into the remaining part
of the enzyme structure, which was calculated by molecular
mechanics, while its dynamics was simulated by MD techniques.
The coordinate of the H1 proton was assumed to be a quantum
one, while the other reaction centre coordinates were taken to be
classical and were included in the MD calculation. The combined
dynamic calculation of the evolution of the quantum variable and
its classical environment was described in Section III.4.a; as has
been noted, the reaction rate constant cannot be calculated in this
way. Therefore, auxiliary characteristics of the reaction were
extracted from the calculation which provided useful information
for the understanding of the mechanism of catalysis.
Tentative conclusions. It can be seen from the above examples
that model-free quantum-classical calculations still cannot be
regarded as a routine tool for chemists. However, good prospects
of this approach are beyond doubt. Its practical implementation
requires higher computational facilities and development of new,
more efficient and reliable calculation algorithms.
Note that in all systems studied by this method, the tunnelling
effects were insignificant. They were also insignificant in the test
calculations for simple systems that preceded the calculations
described above {PT reactions in the [NH3_H_NH3]+ system,229, 231 in protonated malonate ion,228 and in deprotonated
water dimer [H7O7H_OH]7 (see Ref. 232) were studied}. The
question of the possibility of reliable description of PT kinetics in a
deep tunnelling regime in terms of the quantum-classical
approach remains open. More promising in this respect is the
theory in which the PT or HAT kinetics is considered on the basis
of the quantum-mechanical perturbation theory, while the
dynamic effect of the medium can be taken into account within
the framework of classical MD.233 This advanced modification of
the multimodal harmonic PT theory (see Section II.4) fully takes
into account the anharmonic effects and removes restrictions on
the shape of diabatic potentials of the reactants and products.
However, practical application of this strategy to PT or HAT
processes is still unknown.
IV. Experimental data and summary of
calculations of the potential and free energy
surfaces
Detailed description of PT and HAT kinetics in the gas phase and
condensed media requires determination of the reaction rate
constants and the isotope effects over a broad temperature
range. These experimental data are, however, available for a
small number of reactions, because the temperature range in a
particular experiment is often limited due to the destruction of
organic molecules and/or crystal melting. In the liquid phase, the
temperature variation range is determined by the melting and
boiling points of the solvent; in addition, the solubility of many
substances sharply decreases with a decrease in temperature.
Moreover, the reaction mechanism can change on passing from
Theoretical investigations of proton and hydrogen atom transfer in the condensed phase
one aggregate state to another. For example, many processes
involving acids and bases do not proceed in the gas phase but
easily occur in polar media;21 bimolecular HAT reactions which
are barrierless in the gas phase are characterised by high barriers in
organic crystals,234, 235 and crystal field effects can result in
asymmetric double-well PES for systems with symmetrical
O7H_O fragments.236 Therefore, experimental data often cannot be interpreted unambiguously without invoking quantumchemical calculations. In what follows, the conclusions drawn in
experimental works are compared, whenever possible, with the
data of calculations. Note in this connection that the applicability
of quantum-chemical methods is restricted by many factors: the
size of the hydrogen-bonded system (not more than 10 heavy
atoms), the occurrence of PT or HAT in electronically excited
states and the influence of solvation or crystal environment.
tion of reaction (92) requires that at least two PES be considered.238
In the general case, HAT in the gas phase can be represented as
follows:
.
Y + HX
.
YH + X ,
In descriptions of intramolecular PT in the lower singlet or triplet
electronically excited states, the terms of PT and HAT are often
used as synonyms. When intra- and intermolecular reactions are
considered together and the principles outlined in the Introduction are taken into account, it is expedient to distinguish between
these terms. Proton transfer usually involves a step of hydrogen
bond formation.237 The PES for this process is characterised by
the presence of a potential well (or two wells), whose depth can
exceed the hydrogen bond energy (5 ± 10 kcal mol71). Hydrogen
atom transfer is a radical process; therefore only shallow minima
can appear on the PES (about kBT).
Let us consider, for example, two exothermic reactions 238 for
which the motion in the reactant and product valleys can be
considered translational as a first approximation (see Section
II.3.b). One of these reactions is HAT involving neutral molecules
and radicals
.
FH + Cl .
(92)
The other process is an ion ± molecular PT reaction
F7 + HCl
(93)
FH + Cl7.
The PES profiles for these reactions are essentially dissimilar
(Fig. 11). The PES (93) has a deep minimum [*14 kcal mol71
(see Ref. 238)], which corresponds to the formation of the FHCl7
complex. In addition, reaction (93) involves reactants with closed
electronic shells and their electronically excited states lie very high
(in kBT units). The radicals in reaction (92) can exist in different
electronic states, F(2P3/2) and F(2P1/2); therefore, correct descrip-
.
F + HCl
1
33 kcal mol71
79 kcal mol71
.
Cl + HF
2
F7 + HCl
38 kcal mol71
*14 kcal mol71
Cl7 + HF
FHCl7
Figure 11. PES profiles for two bimolecular gas-phase reactions.
(1) radical HAT reaction, (2) ion ± molecular PT reaction.
(94)
etc.239, 240
where Y = H, D, F, Cl, O, CN; X = H, D, Cl, Br, I,
The PT reaction in the gas phase can be described by several
schemes:
Y7 + HX
(95)
YH + X7,
(Y and X are the same as in the previous
Y + H+
case 239);
(96)
YH+,
(Y is water, ammonia, etc.241);
YH+ + X
1. Gas phase
.
F + HCl
21
(YH+ is
H3O+,
AH_B
(97)
Y + XH+,
X is
methylamine 242);
(98)
A_HB,
(A and B are either heteroatoms incorporated in the same
molecule and linked by an intramolecular H-bond or terminal
atoms in hydrogen-bonded dimers, trimers, etc.);
(99)
A7 + BH+,
AH + B
(AH and B designate a conjugate acid ± base system).
Many HAT reactions [see Eqn (94)] and ion ± molecular PT
reactions [see Eqns (95) ± (97)] are highly exothermic and readily
proceed in the gas phase. The intramolecular PT process (98),
where A = B (or the synchronous transfer of two protons in
dimers with two hydrogen bonds), proceeds in a symmetrical
double-well potential and is accompanied by simultaneous rearrangement of the system of double and single bonds. Finally, the
gas-phase PT reaction between neutral molecules [see reaction
(99)] is endothermic, the equilibrium being often appreciably
shifted to the left.20, 21
A special type of reaction is represented by hydride ion H7
transfer, for example
[R1]+ + R2H
R1H + [R2]+,
[R1]+ = C
2 H5 ,
tert-C4 H
9
(100)
R 2H
and so on, and
are hydrowhere
carbons with 4 ± 8 carbon atoms containing at least one quarternary carbon atom.243
a. Intermolecular hydrogen atom transfer
A large number of experimental and theoretical works on the
intermolecular HAT in the gas phase have been published.109 ± 112
The corresponding rate constants and their temperature dependences are systematically summarised and discussed in monographs and reviews.244 ± 246 It has been noted above that this
reaction follows a radical mechanism and that the pre- and postreaction states in the gas phase have shallow minima on the PES
(depths of about kBT). Therefore, the scattering cross-section is a
fundamental characteristic of the process and the methods of
process investigation are based on the scattering theory, which
describes transition from an unbound state of the reactants into an
unbound state of the product (see Section II.3.b). An adequate
description of HAT dynamics requires that at least two coordinates be taken into account. In the simplest case, these are the
lengths of the breaking and forming bonds of the hydrogen atom.
Considerable attention has been devoted to the reaction H2 + X,
where X = H, D, F, Cl, etc.247 ± 253
The PES of reaction (92) has a relatively low potential barrier
[*1 kcal mol71 (Ref. 254)].
The scattering in a system comprising three atoms, `heavy ±
light ± heavy', for example,
.
I + HI
.
IH + I ,
(101)
requires particular attention because many theoretical methods
developed to describe this reaction 49, 255 are applicable to the
22
M V Basilevsky, M V Vener
description of intramolecular PT or synchronous transfer of two
protons in dimers (98). The main conclusion of the theory is the
possibility of formation of the metastable IHI complex in the PES
region corresponding to the TS. The complex is of a dynamic
nature. The minimum on the PES is not very deep, about
2 kcal mol71 (see Ref. 49). However, calculations show that
several levels corresponding to symmetric vibrations of IHI may
occur in the potential well.256 Experimental studies confirmed the
existence of a metastable IHI complex.257, 258
The bimolecular elimination of a hydrogen atom from
methanol and dimethyl ether
.
H + CH3X
.
H2 + CH2X, X = OCH3, OH
(102)
has been studied both in the gas phase and in aqueous solutions. In
the gas phase, this exothermic reaction (DH * 710 kcal mol71)
was studied over a broad temperature range (298 to 575 8C).259
The H/D-isotope effect is very small; the temperature dependence
of the rate constant is well described by the Arrhenius equation.
Therefore, it was concluded that no tunnelling effects are involved
in reaction (102).259
b. Proton transfer in ion ± molecular reactions
Gas-phase PT in ion ± molecular reactions of type (95) ± (97) have
been extensively studied (see reviews 239, 241, 242, 260). Ion cyclotron
resonance, high-pressure mass spectrometry and flowing afterglow technique were used most often in these studies. The
principal results of these investigations include elucidation of
gas-phase acidity scales,260 ± 264 basicity and proton affinity
scales 242, 264 ± 266 and a detailed description of solvation of the
H3O+ ion in the gas phase.267
A large array of kinetic data has been obtained for PT reaction
(95) (see Table III in Ref. 239). Depending on the nature of the
anion and the `acid' molecule participating in the reaction, its
exothermic effect varies from 70.68 to 72.82 eV and the rate
constant ranges from 1.5610710 to 3.7610714 cm3 (molecule)71 s71. Comparison of the kinetic parameters of this reaction with similar data for HAT reactions [see reaction (94)] shows
that, with exothermic effects being similar, the rate constant for
PT is two or three orders of magnitude higher than the rate
constant for the corresponding HAT reaction, which is due to the
presence of a deep minimum on the PES of ion ± molecular
reactions (see Fig. 11).
c. Intramolecular proton transfer
The intramolecular PT and synchronous transfer of two protons
in carboxylic acid dimers in the electronic ground state have been
studied by radiospectroscopy and vibrational spectroscopy. Symmetrical reactions, i.e., those with equivalent initial and final
states, are usually considered. Proton transfer in these reactions
shows itself as tunnel splitting of the ground vibrational state. The
experimental value of splitting is *21 cm71 for malondialdehyde,37 varies from 1 to 5 cm71 for mixed carboxylic acid
dimers,36 or equals 1 cm71 for tropolone.38 Due to procedural
difficulties, virtually no data on the tunnel splitting for vibrationally excited states are available.36, 268, 269
The influence of H/D isotope substitution on the tunnel
splitting and on the geometric structure of H-bonded systems
has been studied. The tunnel splitting for deuteron was found to
be strongly reduced (by an order of magnitude). The equilibrium
distance between hydrogen-bonded atoms usually increases by
*0.02
A. This is the so-called Ubbelohde effect,270 whose theoretical interpretation is based on the assumption of strong
interaction between the coordinate of the tunnelling particle and
low-frequency vibrations of hydrogen-bonded heteroatoms.3, 78
Theoretical estimates of tunnel splitting are discussed in Sections III.1.c and III.1.d.
Vibration-rotational tunnelling spectroscopy seems to be a
promising method for the study of systems with H-bonds in the
ground electronic state.271 However, this method has not yet been
applied to PT reactions.
Numerous publications describe the use of spectroscopy to
study PT in the lower singlet electronically excited state S1 (see
Refs 90 and 272). This is due to fast development of spectral
methods with high temporal or space resolution based on the use
of frequency-controlled lasers.42, 273, 274 The studies are usually
carried out for large aromatic (pseudo-aromatic) molecules containing one or several functional groups which are linked by an
intra- or intermolecular H-bond. The type of PES in the ground
electronic state S0 of the H-bond varies from a symmetric or
asymmetric double-well surface to a highly asymmetric single-well
one. Upon excitation to the S1 state of the pp* type, the H-bond is
usually retained; however, its potential markedly changes.
For systems with a symmetric double-well PES, the potential
barrier separating two minima normally decreases along the
effective PT coordinate (9-hydroxyphenalenone, tropolone, benzoic acid dimer), which results in a sharp increase in the tunnel
splitting of the ground vibrational state.40, 275, 276 This is due to
substantial shortening of the distance between the atoms linked by
the hydrogen bond as a result of electronic excitation and is
confirmed by ab initio calculations for malondialdehyde and
tropolone in the S0 and S1 states.24, 277, 278
Electronic excitation of systems with highly asymmetric intramolecular hydrogen bonds often results in an inversion of the
H-bond potential.272 The excitation causes a sharp change in the
acid ± base properties of the functional groups that form the
H-bond.279 Proton transfer is accompanied by very fast rearrangement of the system of double and single bonds; therefore, the
`classical' keto form is formed in the S1 state, instead of the
zwitter-ion.90 This is confirmed by the data of ab initio calculations of o-hydroxyacetophenone, 2-(20 -hydrophenyl)benzoimidazole and salicylic acid in the S0 and S1 states.52, 280, 281 The
intramolecular PT in the S1 state usually proceeds in such systems
without a barrier 32 and can be regarded as intramolecular vibrational relaxation. This accounts for the very high rate of PT in the
S1 state (more than 1012 s71),90 typical of systems with hydrogen
bonds. The absence of H/D isotope effect for the rate of PT
supports the conclusion that the reaction is barrierless. 1,5-Dihydroxyanthraquinone is, apparently, the only system with a
highly asymmetric H-bond for which the intramolecular PT was
proved to proceed through tunnelling.282
Transfer of several protons in various hydrogen-bonded
complexes in the S0 state has been studied in a number of
theoretical works. Carboxylic acid dimers,43, 283 ± 286 hydrogen
fluoride oligomers,287 water tetramers,288 methanol tetramers 289
and mixed dimers191 have been considered. A synchronous PT
mechanism was assumed for the most stable cyclic structures.290 ± 292 Various dynamic methods were used to calculate
the tunnel splittings on the basis of PES determined by ab initio
calculations. Note that the magnitude of tunnel splitting is very
sensitive to calculation details. Slight variations of the PES caused
by the use of different basis sets 283 or insignificant modifications
of the dynamic calculation procedure 288 can change the resulting
value of tunnel splitting by two orders of magnitude. Experimental determination of the tunnel splitting in cyclic molecular
complexes by microwave spectroscopy is difficult because of the
absence of a permanent dipole moment. It was suggested that
synchronous transfer of protons in methanol-based complexes
takes place only in cyclic structures with an even number of
molecules.289
d. Proton transfer in molecular clusters in the S0 and S1 states
Proton transfer in isolated 1 : 1 asymmetric complexes with one
hydrogen bond in the S0 state is usually not observed in the gas
phase.293 ± 296 In neutral weak acid ± base complexes (FH : NH3
and PhOH : NH3), the proton is localised near the `acid' heteroatom. The potential profile along the proton coordinate corresponding to stretching of the O7H bond is single-well and highly
asymmetric.297, 298 As the strength of the acid increases, the proton
moves to the base; the profile remains single-well but becomes
Theoretical investigations of proton and hydrogen atom transfer in the condensed phase
very smooth.298 The potential energy surface for charged 1 : 1
complexes also has one minimum.299, 300
When considering ROH : Xn type clusters where ROH is an
`acid' molecule (phenol, 1-naphthol, etc.) and Xn are n solvent
molecules that interact with the `acid' (and with each other)
through H-bonds, note that most of experimental data correspond to the electronic state S1. Ammonia or water are used most
often as solvents. In the case where Xn = (H2O)n for n from 1 to
21, no PT was detected.301 For Xn = (NH3)n, PT does take place in
the S1 state when n = 3 for 1-naphthol and n = 5 for phenol,301, 302
because ammonia is a stronger base than water. Clusters consisting of several ammonia molecules, for example (NH3)5, are known
to have a fairly high proton affinity.20 Thus, the PES of phenol or
1-napthol clusters with ammonia in the S1 state can have two
minima.
Experimental and theoretical data allow one to draw an
important conclusion, namely, there exists a threshold number
of solvent (ammonia) molecules starting from which PT is
possible 301 ± 308 and an adequate description of the PT dynamics
requires explicit consideration of the collective coordinate, which
describes synchronous motion of molecules forming the first
solvation shell 309, 310 and ensures favourable energy characteristics for the PT process.311
2. Liquid phase
The structure of the reaction site (and the whole reaction system)
in the liquid phase depends on the properties of the solvent
(polarity, proton-donor or proton-acceptor characteristics), reactant concentrations, temperature and impurities. Proton or
hydrogen atom transfer is accompanied by the rearrangement of
the solvation shell of the system,312 ± 314 which sharply complicates
the process description at the molecular level. During the last five
decades, enormous experimental material on liquid-phase reactions has been accumulated; however, interpretation of the results
is the underbelly of most studies. The key problem is the difficulty
of experimental determination of the geometric structures of
reaction products in the liquid phase. In many cases, the product
structures are postulated in order to bring the experimental data in
the correspondence with the model used. The model is usually
based on representing the isolated reaction complex as A7H_B
or on placing the reaction complex in a cavity surrounded by a
dielectric.
Thus, the molecular structure of the solvent and the problems
caused by the presence of impurities (traces of water), counter-ion,
complexation, etc. are usually neglected in the interpretation of
experimental data.
a. Proton transfer between neutral molecules in aprotic solvents
A number of terms are used in the literature to describe the PT in
organic aprotic solvents:315 proton exchange reactions, molecular
exchange between H-bonded complexes, ion ± molecular PT, and
reversible PT in H-bonded complexes. The last-mentioned term
appears the most appropriate for describing the PT between
neutral molecules [see Eqn (99)]. Phenol and carboxylic acid
derivatives are usually considered as acids, and pyridine derivatives, pyridine N-oxides and aliphatic amines are taken as bases.
Halogen-substituted hydrocarbons and acetonitrile are the most
typical solvents.
The proton transfer in H-bonded complexes is accompanied
most often by a substantial increase in the dipole moment (to
10 D),316 pointing to an appreciable change in the geometric and
electronic structures of the complex induced by PT. The changes
are detected by NMR, IR and UV spectroscopy and by measurement of dielectric properties. It is clear that in a `strong base ±
weak acid' system, the equilibrium is shifted almost entirely to the
left, while in a `weak base ± strong acid' system, it is shifted to the
right. The degree of shifting of equilibrium (99) can be found from
a plot where the value
DpKa = pKa(HB+) 7 pKa(AH),
23
where pKa (the proton affinity), is laid off along the x-axis, while
the y-coordinate is an observed quantity (the change in the dipole
moment,317 the 1H, 13C or 15N NMR chemical shift,318 integral
intensity of the A7H stretching vibration 319 or the centre of
gravity of this band,320 etc.). Usually, this gives an S-shaped curve
whose inflection point corresponds to a 50% degree of PT.
A feature of these dependences is sharp change in the spectral
characteristic over a small range of variation of DpKa. The proton
affinity,320 the enthalpy of formation of the H-bonded complex,321 or another quantity can be used instead of DpKa. The
plots obtained in this case will have a break. A unified theory for
explaining these dependences is missing. It was suggested 322 that a
mobile equilibrium of the considered type exists in the liquid phase
but this hypothesis provoked a lot of debate. According to
another publication,323 only one complex is formed in each case
in which the proton shifts from A7 to B+ as the base becomes
stronger and the acid becomes weaker. Nevertheless, in some
cases, mobile equilibrium still does exist because the dependence
under discussion does not have any break in systems without
PT.324
Equilibrium (99) has been repeatedly studied by UV spectroscopy.21, 325, 326 It was shown that depending on the dielectric
constant of the solvent 327 or the solution pH,328 different complexes with H-bonds are formed, in particular the A7_H7B+
complex with the transferred proton. The UV absorption bands of
these complexes markedly differ in frequency and intensity.
Most of the studies mentioned above give only qualitative
characteristics of PT: the structure with the transferred proton is
detected and the degree of PT is estimated. The reported equilibrium constants of the PT process (see, for example Refs 317 and
329) are often phenomenological (i.e., they characterise a complex
multistep process whose detailed mechanism is unknown). In the
case of relatively weak acids and bases, the following reactions are
most probable:330 ± 333 dimerisation of carboxylic acids or polymerisation of phenols; dissociation of the structure with a transferred proton to give free ions; and self-association of complexes
with a transferred proton.
With excess free acid or base, other side reactions can also
proceed to give ions like (BHB)+, (AHA)7 , etc.334 ± 336 A review
of the literature published before 1968 dealing with the secondary
processes that accompany PT in solutions can be found in Chapter
8 of the monograph (Ref. 337).
It follows from the foregoing that experimental determination
of the rate constant for a PT reaction in the conjugate acid ± base
system in an aprotic solvent in the S0 state is a rather complicated
task. Quantitative estimates of the PT kinetics in aprotic solvents
were obtained in a number of the most recent studies.338, 339 The
kinetic H/D-isotope effect in the reaction between 2,4,6-trinitrotoluene and 1,8-diazabicyclo[5.4.0]undec-7-ene in benzonitrile
was found to change from 18.3 to 14.5 as the temperature rises
from 288 to 308 K.338 The ratios of the pre-exponential factors
(*2.5) and the difference between the activation energies
(*2.1 kcal mol71) are also reported. In is suggested that these
values point to a tunnelling mechanism of PT.
In a study of the PT in the complex of 2,4-dichlorophenol with
triethylamine in chloroethane,339 measurement of the temperature
dependence of the chemical shift of the bridging proton made it
possible to establish thermodynamic and kinetic characteristics of
the PT process. It was concluded that in this case, the FES is
almost symmetrical and has two minima. The free activation
energy of PT at 195 K is only 8.9 kcal mol71 and the heat of the
reaction is 0.2 kcal mol71. The rate constant of the forward PT
reaction is 605 s71. The concentration dependences of these
values were not studied in this publication;339 therefore, the
assignment of these characteristics to the PT process is not
validated. Rather high reactant concentrations were used,
0.2 mol litre71 for phenol and 0.4 ± 0.6 mol litre71 for amine.
At these concentrations, dissociation of aggregates with highmolecular masses to give kinetically active components rather
than the PT is the rate-determining step of the process.340
24
Subsequently, the occurrence of a competing process, namely, the
formation of a complex consisting of two 2,4-dichlorophenol
molecules and one triethylamine molecule was discovered.331
The PT kinetics in the S1 state has been characterised more
comprehensively than that for the S0 state.341 ± 346 Depending on
experimental conditions (temperature and solvent) and the nature
of the reactants, the rate of this reaction can be limited by either
non-adiabatic transitions or diffusion processes.347
The results of some other studies are briefly considered below.
Study of H-bonded complexes of 7-hydroxyquinoline and
methanol by UV spectroscopy 348 showed that PT takes place in
the S1 state of the 1 : 2 complex. The activation energy of this
process equals 0.54 kcal mol71. The activation energy of the
`back' PT in the S0 state is 4.2 kcal mol71 in CH3OH and
5.5 kcal mol71 in CH3OD.
The PT kinetics in the S1 state of 2-(20 -hydroxyphenyl)benzothiazole in aprotic non-polar solvents has been studied by femtosecond spectroscopy.35, 349, 350 The formation of the keto-form
upon PT was confirmed by the appearance of a spectral band
corresponding to the N7H stretching vibrations.349 The characteristic time of the PT is equal to 170 fs.35 In 2-(20 -hydroxyphenyl)benzothiazole derivatives, the PT time varies from 100 to
200 fs. These times correspond to the motion of a proton with a
great vibration amplitude and a frequency varying from 100 to
200 cm71 (see Ref. 350). A similar procedure has been used to
estimate the rate constant for the PT in a 1-pyrenol ± triethylamine
complex using a number of organic solvents.351 The characteristic
time was equal to 1 ps and almost did not depend on the solvent
polarity; the replacement of the bridging proton by deuterium also
had only a slight influence on the rate constant.351
The kinetics of PT in the lower triplet electronically excited
state has been studied in a number of publications.352 ± 355 The
keto ± enol tautomerism in the lower triplet state of 2-(20 -hydroxyphenyl)benzothiazole in hexane has been investigated.352 It was
found that the observed rate constant for PT almost does not
depend on temperature below 70 K. The H/D-isotope effect is 30
at 200 K and more than 100 at 100 K. Proton transfer was
concluded to follow a tunnelling mechanism.
Quantum-chemical calculations of the PES of a PT reaction in
the S1 state are usually carried out without explicit allowance for
the presence of a solvent. High-level ab initio calculations 278, 356
are used when considering model photochemical systems, for
instance, CIS calculations 357 ± 358 are used for real systems with
intramolecular PT (1-hydroxy-2-acetophenone 357) or double PT
(7-azaindole dimer 358, 359).
A number of kinetic data have been obtained for synchronous
transfer of several protons in hydrogen-bonded complexes in the
S0 state. Proton exchange in formic acid and methanol clusters in
perdeuterated tetrahydrofuran has been studied by NMR spectroscopy.360 The acid and methanol form 1 : 1 and 2 : 1 hydrogenbonded complexes in which transfer of two or three protons
(deuterons), respectively, takes place. The rate constant was
measured in the temperature range from 255 to 335 K. No
deviations from the linearity of the Arrhenius plots were found.
The activation energy varies from 6.5 to 10 kcal mol71.
The PT reaction in the porphin ion 361 and the double proton
transfer in the porphin molecule 362 in various aprotic solvents
have also been studied by NMR. In these systems, one or two
protons move in a four-well potential. The rate constants for
hydrogen, deuterium and tritium transfer were determined over a
broad temperature range. The Arrhenius dependence is clearly
non-linear. The kinetic H/D-isotope effect equals 93(34) at
T = 240 K and 16.5(11.4) at T = 298 K; the values for the porphin
molecule are given in parentheses.
The mechanism of double PT in the formic acid dimer in the S1
state in aprotic solvents has been studied in detail,363 although
there is still no consensus about its nature.
The PT structure and dynamics in the H-complexes of the
M7H_H7OR type, where M7H is the (PMe3)2(CO)2.
.(NO)W7H hydride ligand and H7OR is trifluoroacetic acid,
M V Basilevsky, M V Vener
have been vigorously studied in the last decade.364 Low-temperature NMR and IR spectroscopy were used to obtain information
on the PT kinetics and the activation barrier depending on the
properties of H_H complexes, determined by the nature of the
metal, the ligands and the medium (hexane, toluene and dichloroethane).365, 366
b. Proton transfer in protic solvents
Proton transfer in protic solvents (water, alcohols, etc.) requires
special consideration because the reaction complex forms additional H-bonds with solvent molecules.314 This complicates the PT
and in some cases, also changes the reaction mechanism. Usually,
four types of PT reactions in aqueous solutions are distinguished:367 neutralisation, protolysis, hydrolysis and PT in a
conjugate acid ± base system. The first three reactions are complex
processes, in which one cannot distinguish the individual characteristics of an elementary PT step. The last-mentioned reaction
can be described as follows
(AH) . nH2O + (B) . nH2O
(A7) . nH2O + (HB+) . nH2O,
(103)
where n is the number of water molecules that interact with
reacting molecules through the formation of H-bonds.
A large number of publications has been devoted to so-called
abnormally high proton mobility in aqueous solutions. To interpret the obtained data, it is necessary to know the structure of the
simplest stable proton hydrate. X-Ray diffraction and neutron
diffraction methods as well as NMR provide little information
due to the thermal motion of molecules in the liquid. Conclusions
about the structure of the simplest proton solvate in aqueous
solutions of acids are usually drawn by examining the vibrational
spectra. It has been suggested initially that the simplest proton
solvate is the hydroxonium ion, H3O+ (see Ref. 368). Thus, PT in
the aqueous medium can be represented as
H2O + H3O+
(104)
H3O+ + H2O.
17O
isotope has been
The PT process in water enriched in the
studied by NMR in the temperature range from 288 to
348 8C.369, 370 The rate constant was found from broadening of
the spectral lines. The results were described by the Arrhenius
equation in which the rate constant for the PT is
2:4
litre mol71 s71.
k = 6.0 6 1011 exp
RT
Thus, PT in water has a low activation energy. The lifetime of
the simplest proton hydrate is 1 ps. (Most of modern theoretical
studies rely on these experimental data; for example, see Ref. 371.)
The hydroxonium ion really exists in solid and liquid systems if
only one water molecule per acid proton is present.368 The
existence of this ion, whose vibrations are both IR- and Ramanactive, in dilute aqueous solutions of acids is not confirmed
experimentally.372, 373
The IR spectra of all systems containing proton hydrates
exhibit continuous intense absorption in the range from 3500 to
800 cm71 (see Ref. 374). Studies devoted to this phenomenon are
discussed in several reviews.45, 375 According to Zundel's
model,376 H9 O
4 is the most stable ion ± molecular species whose
vibrations can be observed in the IR spectrum of a solution. In this
model, a special role is played by the extremely high polarisability
of the hydrogen bond. This feature is postulated for the O7H_O
fragment. The origin of the continuous absorption can be
explained by assuming strong interaction of the bridging proton
with the medium. This model is widely used in many theoretical
and experimental works, despite the fact that continuous absorption is observed when the formation of the H9 O
4 ion is impossible
and the solvation effects are minimised (aprotic low-field solvent).377, 378 Thus, continuous absorption may be due not only to
the influence of the medium but also to intrinsic properties of ions
with a symmetrical hydrogen bond, the simplest of them being the
379
H5 O
2 ion.
Theoretical investigations of proton and hydrogen atom transfer in the condensed phase
A particular theory could, in principle, be validated by MD
simulation of the IR spectra. However, applicability of MD
techniques to the simulation of spectral properties of the proton
in aqueous solutions is quite limited. The trajectory length found
by so-called ab initio MD methods 312 is several picoseconds; this is
inadequate for calculating the IR spectrum of the solution.380
Molecular-dynamic approaches based on the valence bond
method either make use of a linear approximation of the system
dipole moment 381, 382 or imply calculation of the vibrational
spectrum by Fourier transform of the velocity autocorrelator.383
However, the linear approximation is inapplicable to the descrip379, 384 and the velocity
tion of the dipole moment of the H5 O
2 ion
autocorrelator was calculated for the Raman spectrum of this
ion.384
Experimental data on PT reactions in protic solvents can be
found in several reviews.385, 386 A widely used non-aqueous
solvent is methanol, which makes it possible to measure the rate
constant for the PT over a broad temperature range, from 797 to
64 8C.387 The double PT in the (CH3)3NH+ + CH3OH +
N(CH3)3 system in methanol has been studied by NMR spectroscopy.388 It was found that the activation energy decreases with an
increase in the temperature: from 3.8 kcal mol71 at 797.8 8C to
2.2 kcal mol71 at 25 8C; a tunnelling mechanism for double PT
was proposed.
The influence of the solvent on the PT can be taken into
account using the variational TS theory (see Section III.1.b),
numerical MD or Monte Carlo modelling (see Section III.2.b),
models in terms of the continual description of the medium (see
Section III.3), or various quantum-classical methods (see
Section III.4). Molecular dynamics methods have been successfully used to describe PT in acid ± base equilibria 199 and in model
systems 231 simulating ionisation of HCl in water 389 and ester
hydrolysis.227 Self-consistent reaction field methods were used to
simulate the PT dynamics in systems with strong hydrogen
bonds 223 and in amino acids.224 Combined (hybrid) approaches
were used to describe the solvation of the hydroxide ion,390 to
model PT in systems with strong hydrogen bonds 232 and to
calculate barriers to the PT reaction in water.391
c. Hydrogen atom transfer in liquids
A large number of reactions involving HAT take place in liquids.
In particular, HAT limits the chain termination during oxidation
of organic compounds in the liquid phase.392 Data on the rate
constants for HAT and the influence of the H/D isotope effect on
these constants are summarised and discussed in monographs and
reviews (see, for example, Refs 393 ± 395), although usually it is
impossible to draw an unambiguous conclusion on the magnitude
of the tunnelling effect on the basis of experimental data because
no reliable data on the temperature dependence of the kinetic
H/D-isotope effect in HAT are available.
The hydrogen atom transfer in the first step of oxidation of
4a,4b-dihydrophenanthrene with dioxygen in 2,2,4-trimethylpentane has been studied at T = 191 ± 263 K.396 In this case, the
Arrhenius plot is essentially non-linear. The experimental H/Disotope effect is 95 at 242 K.396 Arrhenius parameters were also
found for the bimolecular elimination of a hydrogen atom from
methanol in aqueous solutions in the temperature range from 283
to 359 K.397 The corresponding activation energy amounted to
30 kJ mol71 and the observed rate constant at 25 8C was
2.86106 litre mol71 s71. Comparison with the data for gasphase reactions 259 shows that the interaction with the solvent
leads to some acceleration of the reaction (*25%).
A lot of data have been accumulated for intramolecular
HAT.398 In most cases, the experimental values for the kinetic
isotope effect do not exceed 3.398 The intramolecular HAT
involved in the isomerisation of aryl radicals with bulky substituents has been characterised in sufficient detail.399, 400 Isomerisation of 2,4,6-tri-tert-butylphenyl into 3,5-di-tert-butylneophenyl in cyclopropane has been studied by EPR in the
temperature range from 113 to 247 K. It was found that due to
25
steric hindrance, the abstraction of hydrogen from the solvent
molecule is virtually impossible. Hydrogen atom transfer proceeds
via a five-membered cyclic TS. The Arrhenius plot for the transfer
of hydrogen and deuterium atoms is highly non-linear. The
experimental kinetic H/D-isotope effect varies from 80 at 250 K
to 13 000 at 120 K.
Various approaches are used for theoretical investigation of
HAT reactions in the liquid phase. The variational TS theory is
successfully used to calculate the temperature dependences of the
reaction rate constant, H/D-isotope effect, the addition of a
hydrogen atom to benzene 401 and elimination of a hydrogen
atom from methanol 402 in aqueous solutions. Empirical models
for the radical abstraction of hydrogen 403, 404 are used to classify
the results of analysis of the experimental rate constants and
activation energies for HAT reactions.
d. Proton transfer in ion ± molecular reactions in the liquid medium
A PT reaction results in highly reactive systems: carbanions,
carbocations, carbenes and ylides, etc. Therefore, ion ± molecular
reactions involving PT play an important role in organic chemistry.405, 406 The rate and equilibrium constants for such reactions
are determined by a variety of factors including the spatial and
electronic structures of the reactants, their concentrations, the
natures of the solvent and the catalyst and so on. Direct approach
to the problem of elucidating the structure ± reactivity relationship
based on quantum chemistry and statistical physics techniques is
seldom used for complex organic compounds. As a rule, methods
of physical organic chemistry are used.337, 405 ± 407 At this level, the
mechanisms of chemical reactions are considered in terms of
empirical and semiempirical rules. These rules can be formulated
in a quantitative form as linear free energy relationships.337
Interpretation of the obtained results can be simplified by considering so-called intrinsic barriers (i.e., barriers for symmetrical
reactions 26) and the intrinsic rate constants for the PT in ion ±
molecular reactions.408, 409 From the kinetic standpoint, this
situation occurs in the case where the rates of forward and back
reactions are equal, and from the thermodynamic standpoint,
where the reaction heat is zero irrespective of temperature and
pressure. The use of intrinsic rate constants allows one to separate
the thermodynamic effects from purely kinetic effects.
The dependences of the rate constant for PT on the nature of
the solvent, the substituents and the strength of the base are
studied most often.410 ± 413 In the case of PT in carboxylic acids,
this provides detailed characterisation for the relative contributions of the resonance and induction effects and the nature of the
solvent to the increase in the CH acidity.411, 413 For PT reactions
involving radical cations, negative activation energies and H/Disotope effect values smaller than unity are often obtained.412 This
may imply the existence of mechanisms alternative to the direct PT
from the radical cation to the base. The H/D-isotope effects in the
PT rate constant for the reaction of various methylarene radical
cations and pyridine bases in dichloroethane have been estimated.414 Depending on the structure of the reactants, they vary
from 31 to 47. In the studies cited, the rate constants were usually
determined at room temperature.
Numerous experimental works have been devoted to the PT
from nitroalkanes to anions of various bases.415 Usually these
reactions are studied by spectrophotometry because the anions
formed in the reaction exhibit strong absorption bands in the
visible region. The PT from 1-nitro-1-(4-nitrophenyl)ethane to the
hydroxyl ion has been studied. Aqueous acetonitrile was used as
the solvent. The reaction was found to occur in two steps, the PT
process taking place in the second step, which follows a unimolecular mechanism (irreversible decomposition of the `kinetically important' intermediate). The PT kinetics has been studied in
detail in the temperature range from 289 to 319 K. The rate
constant for the PT (289 K) is 11.5 s71 and the primary isotope
effect is 26. As the temperature increases to 319 K, the isotope
effect decreases to 16.9. The difference between the activation
energies and the ratio of the pre-exponential factors for proton
26
and deuteron transfer have been estimated.415 The researchers
believe that the obtained data point unambiguously to the
tunnelling mechanism of the PT reaction.
e. Coupled electron and proton transfer
Coupled electron and proton transfer can occur for the electronically excited states of organic molecules containing two functional
groups (for example, 6-hydroxyquinolinone) in aqueous solutions
of acids or bases;416 in donor ± acceptor pairs such as
417 in ruthenium comRu(bipyridine)2
3 ± 3,5-dinitrobenzene;
plexes on a polycrystalline gold electrode;418 in triads consisting
of carotenoid ± porphyrin ± quinone molecules;419 in proteins;420
in photoactive centres of bacteria.421
The reaction in question takes place in a complex system
which includes an electron and proton acceptor (Ac), several
H-bonds and heteroatoms with a free electron pair and a proton
donor (H-Dp). Its mechanism includes several successive steps:422
(i) electron transfer on Ac to give a radical anion having a high
proton affinity; (ii) the formation of a covalent bond between the
radical anion Ac7 and the proton; (iii) rearrangement of the
system of hydrogen bonds resulting in elimination of the proton
from the H-Dp donor to give the Dp7 anion; (iv) recovery of Dp7
to Dp; (v) the reverse rearrangement of hydrogen bonds resulting
in the formation of the initial Ac and H-Dp.
The central point of this scheme is a sharp enhancement of the
proton-acceptor properties of the molecule (functional group Ac)
and the proton-donor properties of the molecule (functional
group H-Dp) upon their oxidation. The rate-determining step of
the reaction is electron transfer;417 nevertheless, the rate constant
changes when the bridging H atoms are replaced by D in the
417
Ru(bipyridine)2
3 ± 3,5-dinitrobenzene donor ± acceptor pairs.
Experimental value for the H/D-isotope effect in the electron
transfer varies from 1.34 to 1.7.417
The theory of these reactions is considered in Section III.3.d;
the key references to theoretical studies are also given.
f. Proton and hydrogen atom transfer in biologically active
compounds
Proton, hydrogen atom, and hydride ion transfer underlie a large
number of processes which take place in biologically active
compounds.
First, proton transfer plays a key role in bioenergetics,423
which is concerned with the mechanisms of energy conversion in
redox processes in living organisms. The modern views in this field
are based on the `chemiosmotic theory,' which implies the
presence of an electrochemical potential difference (`proton
gradient') in protein complexes 424, 425 and in cytochrome c-oxidase.426, 427 Proton transfer through membranes takes place
against the proton gradient, the thickness of biological membranes being about 40 ± 60
A.425
Second, proton and hydride ion transfer are key steps in one of
the most versatile biological oxidation reactions, namely,
dehydrogenation of an alcohol to give a ketone or an aldehyde.428, 429 The process is catalysed by dehydrogenase enzymes.
These are dimers or tetramers composed of subunits with a
molecular mass of 20 000 ± 40 000. The removed hydrogen atoms
(or hydride ions, or protons) migrate to hydrogen-transporting
coenzymes, for example, nicotinamide adenine nucleotide, etc.
The mechanism of this process is not entirely clear.430
Third, a large number of studies have been devoted to the role
of strong (short) hydrogen bonds in enzyme-catalysed reactions.13
These hydrogen bonds are assumed to stabilise intermediates
and/or transition states in the reactions under interest.14 On the
PES of these H-bonds, two wells are separated by a very low
barrier (*2 kcal mol71).15
For the first two of the above-mentioned types of process,
detailed description of the thermodynamics and kinetics of PT and
hydride ion transfer is missing. Therefore, of considerable interest
are kinetic studies of HAT in enzymes.431 Enzyme-catalysed
C7H bond cleavage has been studied in detail.432 ± 433 A specific
M V Basilevsky, M V Vener
feature of this reaction is a very high kinetic H/D-isotope effect,
which varies from 17 (see Ref. 434) to 56 (see Ref. 432). Hydrogen
atom transfer in these systems is a rate-determining step of the
complex process of C7H bond oxidation,434 the rate constant for
HAT ranging from several to several hundred reciprocal seconds.
Study of the temperature dependence of the rate constants for H
and D made it possible to estimate the activation energy, which
can reach 24.5 kcal mol71 (see Ref. 433) and the pre-exponential
factor. These experimental data suggest that HAT follows a
tunnelling mechanism.
Extensive literature is devoted to the theoretical study of PT
and HAT reactions in enzymic systems. In order to give a general
idea of the methods and approaches used in this, rather specific
field, we will mention the results of some most recent studies.
Widely used are hybrid approaches 435 ± 437 which combine quantum-chemical calculations of the PES in the models of reaction
centres with calculation of the medium effects using MD or
molecular mechanics methods. Oxidation of the enzyme cytochrome P450eryF has been studied;435 proton tunnelling is
described as a one-dimensional process (see Section III.1.a). The
rate constants for HAT for three model enzymic reactions were
calculated in terms of the variational transition state theory;436 the
calculated kinetic H/D-isotope effect is in satisfactory agreement
with the experiment. The energy profiles of the PT involved in the
hydrolysis of peptide bonds by a model enzyme have been
obtained.437 Phenomenological models have been used to describe
the PT through biological membranes.438
3. Solids
The tunnelling mechanism of PT and HAT reactions in solids has
been proved experimentally for a relatively small number of
systems. These processes include both intermolecular and intramolecular reactions taking place in molecular crystals and in
glasses. The results obtained are analysed in several
reviews.99, 157, 439 ± 442 The conclusion concerning the tunnelling
mechanism of the reactions is based on three facts:
Ð the temperature dependence of the reaction rate constant k
found experimentally deviates from the Arrhenius dependence
over a broad temperature range, the Arrhenius dependence (i.e., a
linear dependence of log k on the reciprocal temperature) being
observed at relatively high temperatures;
Ð a large H/D-isotope effect for the PT and HAT rate
constants is found;
Ð the rate constants reach a plateau at relatively low temperatures.
a. Proton transfer reactions
The intramolecular PT in the molecular crystals of 9-hydroxyphenalenone derivatives has been studied by IR spectroscopy.443, 444 At 5 K, the IR spectra of 5-bromo- and 5-iodo-9hydroxyphenalenone exhibit intense bands at 83 and 68 cm71,
respectively. These bands disappear when the bridging proton is
replaced by deuterium. It was suggested 443, 444 that these bands
correspond to tunnel splitting of the ground vibrational state in
non-deuterated molecules.
The proton transfer in carboxylic acid dimers in the solid state
often differs sharply from similar processes in the gas phase or in
solutions. The reason is the crystal field effect, resulting in
asymmetry of the FES along the PT coordinate for the
O7H_O fragment, as opposed to the symmetric FES for the
gas phase.73, 74 In the case of a molecular crystal of the benzoic
acid dimer, the asymmetry was 35 cm71 (non-deuterated compound) and 108 cm71 (deuterated compound).236 By passing to
crystals doped with dye molecules, the crystal field effects can be
minimised.445 The double proton transfer in the crystals of
benzoic acid dimer was studied by optical spectroscopy,446
NMR 73, 74 and quasi-elastic neutron scattering 446, 447 over a
broad temperature range. A sharp increase in the rate of proton
transfer upon excitation of definite low-frequency vibrations has
been established experimentally.73 It was shown that the exper-
Theoretical investigations of proton and hydrogen atom transfer in the condensed phase
imental data (the NMR longitudinal relaxation time) can be
adequately interpreted only using at least two-dimensional FES.74
Some data on PT dynamics have been obtained by non-elastic
neutron scattering for various crystals with intra- and intermolecular hydrogen bonds.448 ± 454 Alternative models for the
N_H_O hydrogen bond, ionic 448, 449 and neutral
ones,451, 452, 454 have been proposed for PT in N-methylacetamide
crystals (this process is a prototype of PT in peptides). The reasons
for choosing these models are not analysed and remain obscure.
27
Langevin equation with a double-well potential corresponds to
the Kramers equation (13) and reduces to the latter if
b(t7 t) = g d(t 7t). This particular case is called the local or
Markovian limit. Equation (I.1) can be written in an alternative
form
t
mx x f t
x x GRF,
tx tdt m
(I.2)
0
with a renormalised force constant
b. Hydrogen atom transfer
Apparently, a tunnelling mechanism of the HAT reaction in a
solid was observed experimentally for the first time by Wang and
Williams,455 who studied the kinetics of hydrogen atom capture
by methyl radicals in crystalline methyl isocyanate using EPR in
the temperature range from 77 to 125 K. The Arrhenius plot for
the corresponding rate constant is non-linear. The observed
activation energy for the HAT reaction was 1.4 kcal mol71 at
77 K and increased to 4.5 kcal mol71 at 120 K. The isotope effect
at 110 K exceeded 1000. It was impossible to obtain a more precise
value, as the rate of abstraction of a deuterium atom was too low
in the given temperature range to be measured by EPR. Later, a
low-temperature limit of the HAT reaction rate has been found
experimentally.456 Detachment of a hydrogen atom from a
methanol molecule (in glasses) by a methyl radical was studied
by EPR in the temperature range from 15 to 89 K. At 40 K, a lowtemperature plateau was reached. Analogous results have been
obtained in a study of hydrogen atom elimination by a methyl
radical from acetonitrile,457 methanol 458 and other molecules.399, 400, 459
A large body of experimental data on the rate constant and
H/D isotope effect in the temperature range from 1.4 to 300 K and
for pressures from 1 to 30 atm has been obtained for HAT in the
fluorene molecular crystals doped with acridine molecules.153 In
the lower triplet electronically excited state of the acridine
admixture, the hydrogen atom is detached from the nearest
fluorene molecule and passes to the acridine nitrogen atom.
A typical feature of this reaction is the presence of several lowfrequency (intra- and intermolecular) modes whose excitation
accelerates HAT. This conclusion is based on investigation of
the Raman spectra of these crystals at different temperatures.154
For this reaction, calculations of the PES,460 in particular those
with explicit allowance for the crystal environment 216 were
published. (The interpretation of the temperature dependence of
the rate constant 152 is discussed in Section II.4.c.)
This review was written with financial support of the Russian
Foundation for Basic Research (Project Nos. 02-03-33049, 00-1597295 and 02-03-07029-ANO), and the RFBR ± INTAS Foundation (Project No. 97-03-71049).
V. Supplement I.
Generalised Langevin equation
For a stochastic variable x corresponding to an oscillator with the
mass mx , the frequency ox and the force constant mx = mxo2x , the
generalised Langevin equation (GLE) is written in the
form 55, 56, 132, 461
t
mx x b t
tx_ tdt mx x GRF,
(I.1)
0
where GRF is the Gaussian random force (time-dependent). Due
to the presence of the dissipative integral term, the dynamics of the
coordinate x(t) at time t is dictated by its preceding evolution
during the time period (0, t). Such equations are called equations
with memory or non-Markov equations. The function b(t) is
called the memory kernel. The regular force mxx is linearised; as a
result, Eqn (I.1) describes a harmonic oscillator in a medium. The
force linearisation is not obligatory; if the corresponding potential
U(x) is not quadratic, the force qU=qx is used. The generalised
x mx b t 0.
m
The dissipative force in Eqn I.2 depends on the coordinate
rather than on the rate. The memory kernels b(t) and f(t) are
related as follows:
?
b t
f t 0 dt 0 .
t
(I.3)
Important in the GLE theory is the Fourier ± Laplace transform 462, 463
?
f o
exp iotf tdt f1 o if2 o,
(I.4)
0
where f(o) is a complex function, its real [f1(o)] and imaginary
[f2(o)] components being even and odd functions of frequency,
respectively.
In the steady-state regime, the average x hxiT value is
constant. Here h:::iT means averaging over an equilibrated
ensemble with temperature T. The correlation function of the
coordinate x is the fluctuation thermal average
C t
x 0 x x t x
T
.
(I.5)
When t = 0, this gives a statistical distribution of the stochastic variable x with the variance D[x] = C(t = 0). The Fourier
transform of the function C(t)
?
C o
exp iotC tdt
(I.6)
?
is related to the imaginary part of the Fourier image of the
memory kernel by the relation
C o
hf2 octh
ho
.
2kB T
(I.7)
This relation is called fluctuation-dissipation theorem
(FDT).132, 461 ± 463 Relation (I.7) holds for quantum systems, its
range of applicability being wider than that for the classical GLE
(I.2). At the classical limit (
ho=kB T 5
5 1), equation (I.7) assumes
the form
C o
2kB T
f2 o.
o
(I.8)
Since the function C(t) can be calculated by molecular
dynamics methods, relations (I.7) and (I.8) open up the way to
microscopic calculation of the key GLE parameters. According to
the KGH theory, in the case of GLE with a double-well potential
U(x) having a barrier with the height U6, the following approximate expression for the reaction rate constant is
valid:55, 60, 464 ± 469
kKGH
O
2p
r
mx
U6
exp
.
kB T
m6
(I.9)
Here, O is the decay frequency, which is determined by the
characteristic equation
28
M V Basilevsky, M V Vener
mx O2 f O m6
f o 0 0;
6 2
6
m mx o ,
(I.10)
mx mx o2x .
The function f(o) is defined in Eqn (I.4); it is real for o = 0.
The values mx and m6 are force constants at the bottom of the
reactant well and on the top of the barrier. On the barrier, the
force constant is negative and the value m6 is found as its absolute
value. In a special case where the memory kernel b(t) is proportional to the delta-function (Markovian limit), expressions (I.9),
(I.10) lead to Eqns (11) and (14) for the rate constant. There exists
a multimode generalisation of the KGH theory.55, 60, 464 ± 469 Its
use is mentioned in Section III.3.a.
VI. Supplement II. Spectral density
In the linear approximation, the collective coordinate of the
medium which interacts with the coordinate of the reaction
subsystem is the linear combination
X
Q
Cn Qn ,
(II.1)
n
where Qn are modes of the medium corresponding to masses mn
and frequencies on. The spectral density 96, 131
J o
p X C2n
do
2 n mn on
on .
(II.2)
is associated with the coordinate Q.
This formula can be used to calculate the sums over the modes
of the medium containing the C2n coefficients. It is equivalent to the
sum rule:
X C2n
2
F on ,
J oF odo
mn on
p
n
(II.3)
where F(o) is an arbitrary (rather smooth) function. The combination appearing in the right-hand part is often found in applications [see relation (69)]. By using the sum rule, one can pass from
the discrete (simpler and more obvious) to a continuum (more
convenient for calculations and transformations) description of
the medium coordinates.
A different formulation is also useful. Let us consider a
`chemical' variable x which interacts with modes of the medium
according to a bilinear law. The interaction potential is
X7
hxO const xQ x
(II.4)
Cn Qn .
n
Here, const has the dimensionality of a force constant:
7
Cn = const Cn .
In this case, the spectral density J(o)
7
J(o)
7
p X C 2n
do
2 n mn on
on const2 J o,
(II.5)
131 here, the sum rule is derived from (II.3) by
is determined;96,7
replacing Cn by Cn and J by J.
The function J(o) is often specified parametrically. It should
disappear (J = 0) when o = 0 and should rapidly decay at
frequencies exceeding some critical value, o ? . The typical representation for non-polar media is
J o J0
o
o?
n
exp
o
.
o?
(II.6)
The power n is usually 1 for processes in disordered media
(liquids). In three-dimensional molecular crystals, relation (II.6)
corresponds to the Debye type phonon spectrum, n = 2. The
frequency o ? cutting the spectrum in the crystal corresponds to
the Debye frequency.462 In a similar representation for J(o),
the
constant factor J0 is modified.
The spectral density is closely related to the classical generalised Langevin equation, which describes, in the case of interaction
(II.4), the relaxation kinetics for the collective variable x. For the
GLE in form (I.2), the key result is that
7
J(o) = 7f2(o),
(II.7)
where f2(o) is the imaginary component of the memory kernel
[see Eqn (I.4)]. It allows one to calculate the spectral density for
microscopic models of the medium by MD methods because,
according to FDT (I.7), (I.8), the function f2(o) can be determined from the correlation function C(t) (I.5) available from an
MD computation.
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