10580
J. Phys. Chem. 1996, 100, 10580-10594
Hybrid Models for Combined Quantum Mechanical and Molecular Mechanical Approaches
Dirk Bakowies† and Walter Thiel*
Organisch-Chemisches Institut, UniVersität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
ReceiVed: December 8, 1995; In Final Form: March 13, 1996X
A hierarchy of three models for combined quantum mechanical (QM) and molecular mechanical (MM)
approaches is presented. They simplify the QM description of large molecules by reducing it to the
electronically important fragment which interacts with the molecular mechanically treated remainder of the
molecule. In the simplest model A, the QM fragments are only mechanically embedded in their MM
environment. The more refined models B and C include a quantum mechanical treatment of electrostatic
interactions between the fragments and a semiclassical description of polarization. The implementation of
models A-C for MNDO type wavefunctions and the MM3 force field is outlined. Selected applications in
organic chemistry are discussed, addressing the ability of the proposed models to reproduce substituent effects
(MM) on chemical structure and reactivity (QM). These applications include protonations, deprotonations,
hydride transfer reactions, nucleophilic additions, and nucleophilic ring cleavage reactions.
1. Introduction
Quantum theory forms the physical basis for a large variety
of computational approaches to chemical structure and reactivity.
Due to the rapid development of computer hardware and
software, small to medium-sized molecules can be studied by
accurate ab initio methods.1 Density functional theory2 and
semiempirical methods3 extend the range of possible applications: For example, semiempirical calculations are routine for
systems up to about 100 atoms, and remain feasible for
molecules up to about 1000 atoms.4,5
Nevertheless, there are many problems of chemical interest
which cannot be handled by quantum mechanical treatments
now or in the near future. These include studies of chemical
reactivity in the presence of solvents and biochemical applications such as investigations of the catalytic effect of proteins in
enzymatic reactions. The explicit description of the solvent or
the protein leads to very large systems, and the use of quantum
mechanical potentials in Molecular Dynamics (MD) or Monte
Carlo (MC) simulations of such systems is clearly beyond the
scope of current computational ressources. Empirical force
fields6 have been employed successfully in many MD and MC
simulations, but they are not suitable for problems of chemical
reactivity since they do not allow an adequate description of
the formation or cleavage of chemical bonds.
This dilemma motivates the development of hybrid quantum
mechanical (QM) and molecular mechanical (MM) models
which combine the merits of both the quantum and the classical
approach. The first is more generally applicable and allows
the calculation of ground and excited state properties of
electronic properties such as the ionization potential or the
electron affinity and of chemical reactions. QM approaches
require parametrization not at all (ab initio) or only to a limited
extent (semiempirical), whereas the success of a classical force
field strongly depends on the careful calibration of a large
number of parameters against experimental reference data. This
ensures the reliability and accuracy of a given MM approach,
but also restricts its application to the classes of molecules it
has been designed for. Many force fields outperform simple
QM models in accuracy because specially tailored potential
† Current address: Department of Pharmaceutical Chemistry, University
of California, San Francisco, CA 94143-0446.
X Abstract published in AdVance ACS Abstracts, May 15, 1996.
S0022-3654(95)03651-3 CCC: $12.00
functions and flexible parametrizations allow a good reproduction of experimental data.6 A further advantage of MM force
fields is their computational efficiency which makes them
particularly attractive for MD or MC simulations. They are
much faster than QM approaches and show a more modest
scaling of the cpu requirements with the size of the system.
The development of hybrid QM/MM approaches is guided
by the general idea that large chemical systems may be
partitioned into an electronically important region which requires
a quantum mechanical treatment and a remainder which only
acts in a perturbative fashion and thus admits a classical
description. The perturbation may be mainly mechanical, e.g.
if the classical region forces the quantum region into a particular
geometry, but it may also include electronic effects such as
electrostatics and polarization. According to this concept a
chemical reaction is treated as a transformation involving only
the reactive center (QM region) which is influenced by its
environment (MM region, e.g. the solvent or a protein).
Obviously this concept is closely related to common chemical
argumentation.
Several hybrid QM/MM models have been developed in the
last two decades, combining semiempirical,7-12 density functional,13 valence bond,14,15 or ab initio Hartree Fock16 methodology with frequently used force fields. Most of the reported
applications are devoted to solvation phenomena,17-31 while
some deal with biochemical problems.32-36 Aiming for applications in organic chemistry (particularly organic reactions),
we have coupled semiempirical quantum mechanical MNDO
type methods37-39 with the molecular mechanical force field
MM3.40,41 Both approaches are well established in organic
chemistry. The wealth of successful semiempirical studies on
organic reactions42 and the almost experimental accuracy of
MM3 predictions for the geometries, relative energies, and heats
of formation for many organic molecules43 document their
relevance in this field.
A hierarchy of three coupling models is presented in this
paper. Model A is the simplest one and represents a mechanical
embedding of the quantum region. Model B includes electrostatic interactions between the QM and MM regions using
suitably defined classical point charges in the MM region which
enter the core hamiltonian. The relaxation of the density matrix
due to this external perturbation may be identified as the
polarization of the QM region. The most refined model C also
© 1996 American Chemical Society
Quantum Mechanical and Molecular Mechanical Hybrid Models
J. Phys. Chem., Vol. 100, No. 25, 1996 10581
includes the polarization of the MM region in the presence of
the electric field generated by the QM region. Conceptually
the coupling models focus on the theoretical description of the
QM/MM interactions, without introducing any changes in the
underlying QM and MM methods. This deliberate choice allows
a rather general formulation that may also be applied to other
QM/MM combinations.
This paper is organized as follows: Section 2 deals with the
general theory and includes a comparison of our models with
earlier treatments. Section 3 describes the implementation of
the models and briefly summarizes our semiempirical treatment
of electrostatic interactions in models B and C which has been
reported in detail elsewhere.44,45 Section 4 contains selected
numerical applications, with a discussion of the merits and
limitations of models A-C for particular problems. These
applications include the calculation of heats of formation, proton
affinities, and deprotonation enthalpies, as well as conformational analysis. Three types of organic reactions (hydride
transfer, nucleophilic addition to carbonyls, and nucleophilic
cleavage of oxiranes) serve as case studies which focus on the
effects of substituents on activation energies. In the concluding
section we report our experiences with the hybrid models and
provide some general guidelines for future applications.
total charge is not forced to be zero for neutral QM systems
which might pose additional problems in the calculation of
electrostatic QM/MM interactions.
A simpler solution to these problems is to satisfy the free
valencies with hydrogen atoms (“link atoms”, L) and to include
these in the SCF treatment of the QM fragment. The use of
link atoms corresponds to the chemically intuitive approximation
of the charge density of a fragment as part of a molecule. Many
ab initio or semiempirical studies take advantage of this concept
when replacing e.g. large side chains by hydrogen atoms to
define model systems which are feasible for QM calculations.
In a hybrid QM/MM model, additional effects due to these
omitted side chains are included in a perturbative fashion. The
link atom approach has been adopted in several earlier QM/
MM models8,16 and will also be used here. In contrast to the
earlier work, however, we start from a formula for the total
energy which implicitly includes a molecular mechanical
correction for the energetic contributions due to the link atoms.
At the moment we assume that the MM energy expressions for
X-Y and Y-L are well-defined quantities. In a QM/MM
scheme, the total energy of the molecule X-Y may then be
obtained from its MM energy by adding the QM energy for
Y-L and subtracting the MM energy for Y-L (The superscript
A in eq 2 refers to model A):
2. Theory
A
Etot
(X-Y) ) EMM(X-Y) - EMM(Y-L) + EQM(Y-L)
This section begins with a detailed discussion of the simplest
model A (“Mechanical Embedding”). Special attention is paid
to the treatment of QM/MM cuts within molecules where “link
atoms” are used to satisfy free valencies at the QM/MM
boundary. Models B (“Quantum Mechanical Treatment of
Electrostatics”) and C (“Classical Treatment of Polarization”)
are refined versions of the basic model A. They are only
characterized by discussing the modifications relative to model
A.
2.1 Model A: Mechanical Embedding. The combined QM/
MM description of a system composed of two separate
molecules X and Y does not invoke any severe problems. One
of the molecules (X) may be treated classically, and the other
(Y) quantum mechanically, so that the development of a
reasonable QM/MM model concentrates on a suitable representation of the intermolecular interaction energy E(X,Y). For
the total energy Etot(X,Y) we obtain in an obvious notation:
Etot(X,Y) ) EMM(X) + EQM(Y) + E(X,Y)
(1)
The present work, however, aims at a theoretical treatment
of large organic molecules which are decomposed into MM and
QM fragments. A suitable approach is much less evident in
this case since there is no unambiguous quantum chemical
treatment of a molecular fragment. None of the three obvious
substitutes Y- (anion), Y• (radical), Y+ (cation) adequately
represents the electron distribution of the fragment Y as part of
the molecule X-Y.
In the spirit of early work,7 Théry et al. have recently
proposed a fragment orbital approach that involves a transformation of the atomic orbitals of the “frontier” QM atom yielding
hybrid orbitals which are colinear with the bonds of this atom.10
Those hybrid orbitals which belong to bonds connecting QM
and MM atoms are excluded from the orbital basis, and the
associated electron densities act as external point charges on
the cationic QM fragment. Although physically appealing, this
approach suffers from the fact that the electron density of the
excluded orbital is not known unless a QM calculation for the
entire system is performed prior to the QM/MM treatment. The
(2)
EMM(X-Y) ) EMM(X) + EMM(X,Y) + EMM(Y)
(3)
EMM(Y-L) ) EMM(L) + EMM(Y,L) + EMM(Y)
(4)
Since the MM energy expression can unambiguously be
decomposed into contributions from each of the two parts of
the molecule and from their interaction (eqs 3, 4) one readily
obtains a simplified formula which avoids the double evaluation
of energy terms for Y
A
(X-Y) ) EMM(X) + EMM(X,Y) +
Etot
EQM(Y-L) + ELINK (5)
ELINK ) -EMM(L) - EMM(Y,L)
(6)
ELINK denotes the link atom correction which will be
discussed in more detail later. It should be kept in mind that
the derivation of eq 5 is only formally valid, since the basic
assumption of a well-defined expression EMM(X-Y) does not
generally hold. Assuming that the fragment Y undergoes a
chemical reaction, a molecular mechanical description is not
possible, and EMM(Y) is ill-defined. Due to cancelation,
however, this term does not appear in eq 5 which can therefore
be regarded as a useful master equation. Similar problems may
be encountered if the interaction terms EMM(X,Y) and EMM(Y,L) refer to atom types in Y for which no MM parameters
are available. This concerns bonded (bond stretching, angle
bending, torsion) interactions in the QM/MM overlap regions
and nonbonded (electrostatic, van der Waals) interactions for
the entire fragments X and Y. In practice, one either assumes
reasonable parameters for the few missing bonded terms (which
in many cases may be derived from similar bonding situations)
and accepts nonbonded parameters which are generally available
from either experimental data (van der Waals), other force fields
(van der Waals, electrostatics), or quantum chemical calculations
(mainly electrostatic), or one choses sufficiently large QM
regions and thus avoids the problems associated with missing
bonded parameters. Unlike most other force fields (e.g. UFF46
10582 J. Phys. Chem., Vol. 100, No. 25, 1996
Bakowies and Thiel
(Y-L). It can be readily verified that minima of EQM(Y-L)
with respect to link atom coordinates are found by minimizing
A
(X-Y) - ELINK. Consequently we define the QM/MM
Etot
potential energy by:
A
Epot
(X-Y) ) EMM(X) + EMM(X,Y) + EQM(Y-L) (7)
Figure 1. MM3 and MNDO C-H stretching potentials of methane.
Starting from the MNDO geometry of methane, one hydrogen atom is
moved along the C3 axis while all other atoms are fixed in their initial
position.
and those in AMBER,47-50 CHARMM51), MM3 includes many
explicit couplings between bonded interactions (e.g. stretchbend) as well as far-reaching implicit couplings in π systems
which are due to the readjustment of force field parameters using
geometry-dependent π-SCF (VESCF52-54) density matrix elements.55 These couplings pose additional problems to the
evaluation of EMM(X,Y), as discussed in detail elsewhere.44
Since they are expected to have only a marginal influence on
the total energy and are completely absent in other commonly
used force fields which are regarded as a future option for the
proposed QM/MM approaches, we decided to neglect any MM3
coupling of diagonal terms involving internal coordinates in the
QM region.44
With these specifications eq 5 provides a plausible and
convenient definition of the total energy of a molecule. It still
has to be examined, however, whether this energy formula is
suitable for geometry optimizations. This is not self-evident
since the formula implicitly includes the difference of the QM
and MM interaction energies involving the link atoms L. A
simple example demonstrates that this may cause problems.
Figure 1 shows the QM (MNDO) and MM (MM3) stretching
potentials for one of the CH bonds of the methane molecule,
employing the MNDO geometry for the others and preserving
C3V symmetry. While the MNDO and MM3 energy minima
correspond to nearly the same CH bond lengths (1.104 and 1.112
Å, respectively), the energy curves are very different far off
the minimum. The MM3 potential is much steeper left of the
minimum, causing large discrepancies between MM3 and
MNDO for shorter bond lengths. Since the corresponding
energy difference is part of the QM/MM total energy
A
(X-Y) for a system composed of a QM fragment CH3, a
Etot
link atom H, and some MM fragment X, a QM/MM geometry
optimization would continuously reduce the distance between
the C atom and the link atom until they finally collapse.
Geometry optimizations for several test cases show that this
artifact is not restricted to stretching potentials but also occurs
for other force field terms which refer to both QM and link
atoms. In all these cases, this behavior is caused by large
differences between the QM and MM potentials far away from
the equilibrium reference geometry. Therefore, eq 5 cannot be
used as a general definition of the QM/MM potential surface.
Since both MM3 and MNDO have been calibrated against
experimental data, the corresponding potential surfaces should
be most similar in the vicinity of energy minima. Hence we
may expect that ELINK (eq 6) is an appropriate correction to the
link atom interactions implicitly included in EQM(Y-L) only
for link atom positions which are close to the minimum of EQM-
Conceptually this is equivalent to the definition of the
potential energy and the total energy in the QM/MM model
proposed by Field et al.8 In our treatment, the link atom
correction ELINK is added to the potential energy EApot(X-Y),
once the geometry is optimized. Such a procedure is not
completely satisfactory from a theoretical point of view, since
one would like to have potential energies and total energies
which are the same. We have accepted this inconsistency in
our treatment because ELINK appears to be an integral part of
the total energy which corrects for some unphysical energy
contributions associated with link atoms (see section 4 for further
discussion).
An alternative quantum mechanical link atom correction can
easily be devised for MNDO type wavefunctions where the total
energy consists of only one-center and two-center contributions.44 A plausible definition of a QM link atom correction
would be the sum of all one-center and two-center contributions
referring to link atoms. The resulting QM energy is, however,
non-variational so that gradient evaluation becomes inefficient.
Therefore, this approach was discarded in favor of the MM link
atom correction discussed above.
Model A is defined by eqs 5-7. We refer to it as mechanical
embedding since all interactions between the QM and MM
regions are included in EMM(X,Y) and are thus calculated
molecular mechanically.
2.2 Model B: Quantum Mechanical Treatment of Electrostatics. MM3 (and its predecessor MM2) represent electrostatics by bond dipole interactions, contrary to many other
force fields which employ partial atomic charges. Weaknesses
of MM3 (and MM2) in the conformational analysis of molecules
containing strongly polar groups56-58 are at least partly due to
this approximation. A better representation of electrostatic QM/
MM interactions is an obvious goal for a refined hybrid model.
This may already be achieved by simply replacing the bond
dipoles with more appropriate potential-derived (PD)59 atomic
point charges. In addition, these charges can explicitly be
included in the hamiltonian for Y-L:
Y X q
J
Ĥel(Y-L;X) ) Ĥel(Y-L) - ∑∑
i J riJ
(8)
Ĥel(Y-L;X) and Ĥel(Y-L) are the electronic hamiltonians for
Y-L with and without the external field, respectively, which
is generated by the MM atomic point charges qJ. It should be
emphasized that Ĥel(Y-L;X) does not include any interactions
between link and MM atoms, consistent with our basic assumptions (eq 5).
In LCAO approximation, the inclusion of the MM charges
changes the elements hµV of the core hamiltonian to
X
J
h̃µV ) hµV - ∑qJVµV
(9)
J
The resulting Coulomb or electrostatic energy is given by:
Y Y X
Y X
J
Ecoul(X,Y) ) ∑∑∑PµVqJVµV
+ ∑∑ZAqJVAJ
µ
V
J
A
J
(10)
Quantum Mechanical and Molecular Mechanical Hybrid Models
Here, PµV and ZA are density matrix elements and nuclear
J
and VAJ denote nuclear attraction
charges, respectively. VµV
integrals and Coulomb repulsion terms which describe the
interaction between a unit charge at MM atom J and an electron
or core A in the QM region, respectively. Using the modified
core hamiltonian h̃ in the SCF procedure yields a perturbed
density matrix P̃ and, consequently, a different SCF energy ẼQM(Y-L). This process may be identified as polarization or
induction in the QM region and is accompanied by a net energy
gain of
Eind(Y) ) ẼQM(Y-L) - EQM(Y-L) - Ecoul(X,Y)
(11)
The refined model B includes the sum of the Coulomb and
induction energies from eqs 10 and 11 which replace the
electrostatic part of the QM/MM interactions from model A
vdW
(X,Y)) and bonded
while keeping the van der Waals (EMM
bonded
(EMM (X,Y)) parts. The potential energy and the total energy
are given by
B
(X-Y) ) EMM(X) + Ecoul(X,Y) + Eind(Y) +
Epot
(13)
The idea to include MM atomic point charges in the core
hamiltonian is not new.7-13 Apart from the link atom correction,
model B is in fact conceptually analogous to several previous
QM/MM approaches.8,9,11. We have adopted a different implementation, however, which aims at a realistic semiempirical
description of electrostatic potentials for both the QM and MM
regions of a molecule (see section 3 for a short summary and
ref 45 for further details).
2.3 Model C: Classical Treatment of Polarization. Model
B results in an unsatisfactory asymmetry for the description of
nonbonded QM/MM interactions. While the polarization of the
QM fragment is admitted, that of the MM fragment is not.
Obviously, a term Eind(X) is missing (see eq 12) which describes
the polarization of the MM fragment due to the presence of the
QM electric field.
A classical model frequently used in MD simulations of ionic
systems (see, e.g., ref 60) treats the induction energy as a sum
of atomic contributions:
1X
Eind(X) ) - ∑µJR⟨FJR⟩
2 J
(14)
and thus accounts for the nonuniform distribution of the electric
field.44 (The notation in section 2.3 as introduced in eq 14
implies the summation over repeated Greek indices.) In the
context of our QM/MM model the electric field may be
identified with the expectation value of the unit field operator.
In LCAO approximation, the corresponding R component for
location J reads as follows:
Y Y
⟨FJR⟩
) ∑∑
µ
V
Y
J
P̃µV∇RVµV
- ∑ZA∇RVAJ
ever, as was already recognized in 1917 by Silberstein.61 In
the spirit of this early work, Applequist proposed an induced
dipole interaction model62 which, however, suffered from
resonance problems for dipoles close in space (e.g. in a chemical
bond). The parametrization of his model consequently yielded
atomic polarizabilities which were much too small compared
to the available experimental and theoretical (ab initio) data and
showed the need to differentiate between identical atoms in
dissimilar chemical environments. Later, Thole proposed a
revised model which in an empirical fashion took into account
the physically well-known damping behavior of short-range
polarization.63 His model outperforms the older one in accuracy
although it only needs one parameter per element, the isotropic
atomic polarizability RJ. The calibration of his model against
observable molecular polarizabilities yields RJ values which are
in much closer agreement with experimentally available or
calculated (high level ab initio) polarizabilities of free atoms,
indicating the physically adequate description of short-range
damping behavior.44 According to Thole’s model the induced
dipole moment of atom I is given as:63
X
bonded
vdW
(X,Y) + EMM
(X,Y) (12)
EQM(Y-L) + EMM
B
B
(X-Y) ) Epot
(X-Y) + ELINK
Etot
J. Phys. Chem., Vol. 100, No. 25, 1996 10583
(15)
A
The atomic dipole moments µJR still need to be defined. The
simple assumption that they are independently induced by the
external field (µJR ) RJ⟨FJR⟩) would require strict additivity of
atomic polarizabilities (RX ) ∑JXRJ). This is not true, how-
IJ J
µIR ) RI(⟨FIR⟩ + ∑tRβ
µβ)
(16)
IJ
IJ
) ν4TRβ
+ 4(ν4 - ν3)rIJ-3δRβ
tRβ
(17)
J*I
IJ
TRβ
denotes the usual dipole-dipole interaction tensor
which may conveniently be defined as the second derivative
IJ
) ∇R∇βrIJ-1),
matrix of the inverse interatomic distance (TRβ
δRβ is the Kronecker delta, and ν is a scaling function which
mimics the damping behavior (ν ) rIJ/sIJ if rIJ < sIJ and ν ) 1
otherwise, where sIJ ) 1.662(RIRJ)1/6 63).
A physically reasonable and well-balanced model of induced
dipole interaction can thus reproduce molecular polarizabilities
from measurable atomic quantities. This observation provides
empirical evidence for the assumption that the interatomic
polarization is only a higher order correction to otherwise
additive polarizabilities. It has indeed only recently been shown
that the classical model of induced dipoles (or multipoles, in
general) which mutually polarize themselves until selfconsistency is achieved, corresponds to infinite-order intermolecular perturbation theory while the second-order expression
only includes the polarization due to the external field.64
Therefore it might be tempting to adopt the simple approximation of additive atomic polarizabilities. On the other hand, the
induced dipole interaction model is now well-established in
classical MD simulations although usually in the older, less
satisfactory implementation of Applequist. More importantly,
it represents the only way to describe molecular anisotropy using
simple scalar parameters, i.e. isotropic atomic polarizabilities.
Thole’s approach performs very satisfactory in this respect,
yielding polarizability tensor components which are usually
accurate to within 6%.63 From a theoretical point of view the
induced dipole interaction model is supported by the observation
that it treats the MM fragment in closer analogy to the quantum
mechanical description of the QM fragment. The SCF procedure obviously includes all orders of intermolecular perturbation
theory and thus accounts for the “internal polarization”.
However, a classical treatment of polarization formally equivalent to the QM (SCF) description would also include nonlinear
effects which are part of the third and higher order expressions
of intermolecular perturbation theory.64,65 A preliminary study
on the water dimer (one molecule treated by QM, the other by
MM) has indicated that the nonlinear contributions to the QM
induction energy are negligible at the semiempirical level.44
10584 J. Phys. Chem., Vol. 100, No. 25, 1996
Bakowies and Thiel
We adopt the induced dipole interaction model as specified
by eqs 14-17 to describe the polarization of MM fragments
by QM electric fields, and define model C by
charges and point polarizabilities without referring to a molecular mechanical treatment of the classical region.
3. Implementation
C
Epot
(X-Y)
)
B
Epot
(X-Y)
+ E (X)
ind
C
C
(X-Y) ) Epot
(X-Y) + ELINK
Etot
(18)
(19)
Test calculations using model C indicated an unreasonably
large polarization of MM fragments, due to the lack of a proper
damping mechanism for MM atoms participating in QM-MM
bonds. We remedied this artifact by simply treating all such
MM atoms at the QM/MM boundary as unpolarizable. This is
a very crude assumption, but one may argue that the major part
of the short range polarization is experienced by MM atoms
directly connected to the QM fragment and that these effects
are already implicitly included in the QM treatment of the link
atom which acts as a substitute. Nevertheless, future developments are likely to focus on a refinement of our treatment in
this respect.
The electric field ⟨FJR⟩ is evaluated using the perturbed
density matrix P̃ which refers to the modified core hamiltonian
h̃ (see eq 9). This procedure avoids the need for a second SCF
calculation using the initial core hamiltonian h. In this way,
induction energy terms referring to a third-order treatment of
the “interfragment” polarization are included.44 Contributions
from higher orders of intermolecular perturbation theory may
be included in a reaction field model which describes the mutual
polarization of QM and MM fragments. Two different types
of reaction field hamiltonians are known in the literature,
describing an average (ARF)66 and a direct (DRF)67 reaction
field, respectively, the latter referring to an instantaneous
response. There is some discussion about the physical implications of either approach,68 but the ARF hamiltonian seems to
model the reaction field more appropriately.69
We tested an implementation analogous to the ARF model
introduced by Tapia and Goscinski,66 with the difference that
the response is generated by a discrete reaction field (atomic
polarizabilities) rather than a continuum.44 Not too surprisingly,
the effects on the total induction energy were negligible, since
model C already includes all second-order and some third-order
contributions.44 In an independent study which appeared
recently,12 Thompson and Schenter presented a QM/MM model
conceptually almost identical with our reaction field implementation. Their main focus was the investigation of excited state
properties of the bacteriochlorophyll D dimer, and the inclusion
of MM polarization appeared to be necessary for a satisfactory
agreement with Stark effect experiments. Unfortunately, no
results are reported for a simpler polarization treatment analogous to our model C. According to our experiences, the
additional effort of a self-consistent reaction field is not justified
at the semiempirical level, since only marginal differences in
energy were observed in any of the test calculations. Hence
we did not further study the reaction field model but decided
to accept model C as our most refined QM/MM approach.
Apart from the recent work of Thompson and Schenter,12
there are two earlier QM/MM approaches which include
approximate treatments of polarization.7,16 The more recent
approaches, however, did not take up this feature. The
physically most appealing polarizability model of Thole63 has
so far only been considered in the DRF approach of van Duijnen
and co-workers67,70 which is, however, not a hybrid QM/MM
approach in the narrow sense, since it is mainly concerned with
a QM description of a molecule in the field of classical point
A computer program called MNDO/MM has been written
on the basis of MNDO9171 and MM3(89)72 which allows
practical applications of the proposed QM/MM models using
either MNDO37 or AM139 wavefunctions and the MM3 force
field.40,41 Details of the implementation are given elsewhere;44,45
here we only summarize the most important aspects which
include the semiempirical evaluation of electrostatic and induction interactions and the computation of energy gradients.
The adequate treatment of electrostatic QM/MM interactions
is the main focus for a proper implementation of models B and
C. Using eq 10 we may rewrite the Coulomb interaction
between QM and MM fragments as
X
Ecoul(X,Y) ) ∑qJΦJ
(20)
J
where ΦJ denotes the QM electrostatic potential at site J:
Y Y
Y
V
A
J
ΦJ ) -∑∑PµVVµV
+ ∑ZAVAJ
µ
(21)
The electrostatic potential is a well-defined quantity only in
ab initio quantum chemistry. In semiempirical methodology,
J
and VAJ. The
there are several options for the evaluation of VµV
73-76
which is characterized by
“quasi ab initio” approach
J
and VAJ employing deorthogoanalytic evaluation of both VµV
nalized semiempirical wavefunctions has become popular for
many applications including the calculation of potential derived
charges.77,78 It is, however, less satisfactory in the context of
QM/MM approaches, since the deorthogonalization procedure
is very time-consuming and affords additional three center
integrals (VµJ AVB) which make the approach even less efficient.
We adopted a parametrized approach which has first been
proposed by Ford and Wang79 and assumes an orthogonal basis
in line with the basic NDDO assumptions:45 Both quantities,
J
and VAJ, are represented by MNDO type parametric
VµV
functions which have been calibrated against ab initio RHF/631G* electrostatic potentials of a typical set of small organic
and inorganic molecules. Compared to the original parametrization of Ford and Wang79 which was mainly concerned with
minima of electrostatic potentials, we used a larger set of
reference points on several molecular surfaces with distances
ranging from 0.6 to 1.2 times the van der Waals radii and also
included electric field components in some of the parametrizations.45 This admitted a more balanced calibration of
electrostatic potentials and electric fields for regions close to
and far away from the molecules.
The MM atomic point charges qJ (eq 20) are derived from a
charge equilibration scheme (QEq)45,80 which is based on the
principle of electronegativity equalization. The original QEq
approach of Rappé and Goddard80 has been adapted to our
semiempirical QM/MM methodology, affording KlopmanOhno type two center integrals which converge to adjustable
atomic integrals in the one-center limit.45 This semiempirical
QEq approach involves two parameters per element which have
been calibrated to reproduce ab initio RHF/6-31G* potential
derived (PD) or Mulliken charges.45
In the present QM/MM study we employ the one-parametric
function f 1′ defined in eq 10a of ref 45 to calculate electrostatic
potentials and use the PD charge parametrization for the QEq
Quantum Mechanical and Molecular Mechanical Hybrid Models
charge equilibration scheme. According to previous experience,
an accuracy of about (20-25% and (0.1 e should be expected
for potentials and charges, respectively, when compared to ab
initio RHF/6-31G* reference data.45
Analytic energy gradients are readily available for model A
by simply combining the QM and MM gradients. In model B,
the inclusion of MM point charges in the core hamiltonian
matrix generates additional forces on QM (gxA) and MM (gxJ)
atoms:
X
∂VAJ
J
∂xA
gxA ) ∑ZAqJ
Y
∂VAJ
A
∂xJ
gxJ ) ∑ZAqJ
X A A
∂VµJ AVA
J µA VA
∂xA
- ∑∑∑P̃µAVAqJ
Y A A
∂VµJ AVA
A µA VA
∂xJ
- ∑∑∑P̃µAVAqJ
(22)
+ g′xJ (23)
J. Phys. Chem., Vol. 100, No. 25, 1996 10585
and electric field strengths (⟨F1x⟩, ⟨F1y⟩, ⟨F1z ⟩, ⟨F2x⟩, ..., ⟨Fnz ⟩),
respectively. The elements of the supermatrix A are given as:
JJ
ARβ
) (RJ)-1δRβ
(27)
IJ
IJ
) -TRβ
ARβ
(28)
Using eqs 26-28, the MM induction energy may be evaluated
as
1
Eind(X) ) - ⟨F+⟩A-1⟨F⟩
2
(29)
The matrix inversion technique is numerically stable but much
slower than the iterative procedure, typically by factors of 5-10.
Therefore, the latter has been preferred whenever possible.
4. Numerical Applications
The total QM energy expression (EQM(Y-L) + Ecoul(X,Y)
+ Eind(Y)) is stationary with respect to the perturbed density
matrix P̃ so that the QM gradient expressions do not involve
any density matrix derivatives.81 The QEq scheme yields
charges which depend on the geometry of the MM fragment.
This leads to additional forces g′xJ exerted on MM atoms:
Y
Y A A
∂qJ
∂qJ
g′xJ ) ∑ZA VAJ - ∑∑∑P̃µAVA VµJ AVA
∂xJ
∂xJ
A
A µA VA
(24)
The charge derivatives ∂qJ/∂xJ in eq 24 are readily available.45
No analytic energy gradients are provided for model C since
the QM energy expression is nonvariational in this case. A
proper implementation would require the solution of CPHF type
equations, a time-consuming procedure which seems inadequate
in the context of a simple QM/MM approach. Test calculations
have shown, however, that the additional consideration of
polarization in the MM fragment mainly affects the QM/MM
total energy, with little or no effect on the molecular geometry.44
Some examples are discussed in the next section which justify
the single point evaluation of model C energies at geometries
which have been optimized using model B. We report these
results in analogy to common notation as C//B (total energy//
geometry).
The evaluation of MM induction energies Eind(X) deserves a
final comment. The iterative procedure defined by eqs 14-17
may converge slowly or even diverge, especially when charged
QM fragments generate large electric fields. The convergence
behavior can be much improved by introducing a damping
mechanism
µJR(new): ) (1 + f)µJR(new) + fµJR(old)
(25)
Applying damping factors f in the range of 0.4 to 0.6 usually
leads to satisfactory convergence ((10-5 Debye) after 20
iterations.
A matrix inversion technique62,63 may be used if the iterative
procedure does not converge at all. For this purpose eq 16 may
be rewritten in matrix notation:
M ) A-1⟨F⟩
(26)
M and ⟨F⟩ denote column vectors which collect the compo)
, µind,1
, µind,2
, ..., µind,n
, µind,1
nents of dipole moments (µind,1
z
x
z
x
y
Selected numerical applications have been studied to establish
the accuracy of the proposed QM/MM models and their ability
to reproduce experimental trends reliably. In particular, we have
compared the performance of models A-C to assess the
appropriate levels of QM/MM embedding required for different
types of application. The QM/MM boundary has often deliberately been put close to the reactive center to emphasize
electrostatic and related effects, and different choices of QM/
MM boundaries have been investigated to derive guidelines for
future applications. Although the discussion in this section will
stress such methodological issues, most of the examples chosen
are also relevant from a chemical point of view and therefore
illustrate the potential of QM/MM applications.
4.1 Heats of Formation of Organic Molecules. MNDO
(AM1) and MM3 are designed to calculate heats of formation.
It is straightforward to convert QM/MM total energies to heats
of formation, using bond and structure heat increments for the
MM part of the energy expression, and atomic energies and
atomic heats of formation for the QM part.44 Figure 2 shows
the heat of formation of 1-octanol calculated for various QM/
MM divisions, treating the hydroxyl group either molecular
mechanically (top chart) or quantum mechanically (bottom
chart). The QM/MM results are of the same order as the MNDO
and MM3 values, with maximum deviations of ≈5 kcal/mol
(top) or ≈10 kcal/mol (bottom). These deviations are almost
entirely due to the differences between MNDO and MM3 for
the QM part of the molecule. For example, the deviation
between the MNDO and MM3 values for methanol to 1-heptanol decreases from -9.6 to -2.0 kcal/mol, in analogy to the
QM/MM results in the bottom chart. Apart from one single
exception, the heats of formation from model B are consistently
lower than those from model A, by 2 to 4 kcal/mol (see Figure
2), while the inclusion of MM polarization in model C has only
minor effects (less than 0.4 kcal/mol, not shown).
The above and other examples44 indicate that it is possible
to define a heat of formation in our QM/MM treatment and to
obtain reasonable results which are not overly dependent on
the choice for the QM/MM boundary. However, there are
several reasons to avoid the concept of a heat of formation in
our QM/MM models. MNDO (AM1) considers the heat of
formation as a conformational property which characterizes
distinct stationary points, while MM3 defines it as a molecular
property which in general requires the consideration of torsional
and population increments. Since the quality of MM3 heats of
formation largely depends on the careful calibration of heat
increments, meaningful QM/MM results would be restricted to
10586 J. Phys. Chem., Vol. 100, No. 25, 1996
Bakowies and Thiel
TABLE 1: Relative Energies of Various Ethylene Glycol
and Ethylenediamine Conformationsa,b
MNDO/MM3d
conformationc
MNDO
MM3
A//A
ab initioe
B//B
C//C
RHF
MP2
0.0
4.9
9.7
12.5
0.0
5.4
10.8
15.7
0.0
4.8
10.4
15.2
0.0
2.4
3.6
12.8
0.0
2.8
6.0
14.8
0.0
3.8
6.0
15.8
C2h (a)
C2h (s)
C2V (a)
C2V (s)
0.0
4.1
3.1
9.1
HOCH2CH2OH
0.0
0.0
0.0
4.7
4.3
4.9
7.9
8.0
9.7
12.9 12.4 12.7
C2h (a)
C2h (s)
C2V (a)
C2V (s)
0.0
0.2
3.1
5.5
H2NCH2CH2NH2
0.0
0.0
0.0
3.1
3.1
2.3
4.6
4.7
3.6
9.8 10.0 12.7
a All energies are in kcal/mol. b All methods predict a minimum and
a saddle point of first, second, and third order for the C2h (a), C2V (a),
C2h (s), and C2V (s) conformations of both molecules, respectively. c (a)
antiperiplanar, (s) synperiplanar conformation, with respect to the C-O
and C-N bonds, respectively. d QM(XCH2)/MM(CH2X). e 6-31G*
basis set.
Figure 2. MNDO/MM3 heats of formation of 1-octanol.
cases tractable by MM3 alone. Furthermore, the MM3 heat
increments have been derived in a manner consistent with the
MM3 treatment of electrostatics, so that they would not be
strictly applicable to models B and C with their own refined
treatment of QM/MM electrostatic interactions. In view of such
inconsistencies, we shall not refer to heats of formation in the
following subsections, but report QM/MM energies as defined
in eqs 5, 7, 12, 13, 18, and 19.
4.2 Conformational Analysis. Two molecules, ethylene
glycol and ethylenediamine, have been chosen to illustrate the
performance of the QM/MM models in conformational analysis.
Four symmetrical (C2V and C2h) conformations have been fully
optimized applying all three QM/MM models, MNDO, MM3,
as well as the RHF/6-31G*1,82 and MP2/6-31G*1,82 levels of
theory for reference. In the QM/MM calculations, the molecules
have been divided at the central CC bond, thereby lowering
the symmetry to Cs. Since MNDO/MM3 and AM1/MM3 yield
very similar results, only the first are shown and discussed. All
relevant data are collected in Table 1.
Apart from MNDO, all methods predict the same sequence
in energies. Not too surprisingly, MM3 and MNDO/MM3/A
(denoting a QM/MM calculation at the model A level, employing MNDO and MM3) yield very similar results. Both are much
more accurate than MNDO when compared to the ab initio
results as a reference. The quantum chemical treatment of
Coulomb interactions between the fragments improves the QM/
MM description for ethylene glycol, and for the least stable
conformation of ethylene diamine. The results from models B
and C are very similar, indicating only a small influence of
polarization effects on the relative energies.
The geometry optimizations using models B and C lead to
very small differences in bond lengths (e0.002 Å), bond angles
(e0.3 degrees), and dihedral angles (e0.3 degrees). C//B and
C//C energies differ by less than 0.01 kcal/mol. These and other
analogous results justify the use of geometries from model B
for energy evaluations using model C.
4.3 Protonations and Deprotonations. Two elementary
reactions with high relevance for organic chemistry and
biochemistry are protonations and deprotonations of Lewis bases
and acids, respectively. Due to large differences between the
charges of reactants and products, substituent effects of 10 kcal/
mol and more are common in these processes. We studied the
protonation of alcohols and pyridines as well as the deprotonation of alcohols and carboxylic acids.44 Here we report the
results for the alcohols and discuss the ability of the QM/MM
approaches to model the electrostatic and polarization effects
of alkyl substituents in these reactions.
The proton affinity PA of a base B and the deprotonation
enthalpy DPE of an acid HB are thermodynamically defined as
PA ) ∆Hf0(H+) + ∆Hf0(B) - ∆Hf0(HB+)
(30)
DPE ) ∆Hf0(H+) + ∆Hf0(B-) - ∆Hf0(HB)
(31)
Since the heat of formation ∆Hf0 of the proton is only poorly
reproduced by MNDO type methods, the experimental value is
normally used in semiempirical studies.83,84 We have proceeded
accordingly, with ∆Hf0(H+) ) 365.7 kcal/mol (“stationary
electron convention”).85 Several QM/MM divisions have been
tested, but the protonated or deprotonated part of the molecule
has always been treated quantum mechanically. Bonds which
are formed by protonation of anions or of neutral molecules
are treated as normal bonds in the molecular mechanical part
of the QM/MM coupling term. All quoted results refer to the
energetically most favorable conformations which may, however, be different for the various theoretical approaches (see
supporting information). The QM/MM results generally refer
to C//B calculations if not stated otherwise.
Proton Affinities of Alcohols. The proton affinities of
oxygen compounds as electron donors generally increase with
successive methyl substitution. In most experimental86-89 and
theoretical studies90 this phenomenon is attributed to inductive91
(charge transfer) and polarization effects. The relative significance of these and other factors such as electrostatics, however,
strongly depends on the type of correlation of experimental data
Quantum Mechanical and Molecular Mechanical Hybrid Models
J. Phys. Chem., Vol. 100, No. 25, 1996 10587
TABLE 2: Proton Affinities of Alcohols: Relative Valuesa
AM1(CH3OH)/MM3
molecule
RR
Rβ
Rγ
expb
AM1
C//B
R3COH
R3COH
R3COH
R3COH
R3COH
H
CH3
C2H5
CH3
CH3
H
H
H
CH3
CH3
H
H
H
H
CH3
0.0
6.4
8.9
9.3
11.8
0.0
6.2
7.3
10.8
15.9
0.0
7.7
8.3
13.0
17.4
AM1(H2O)/MM3
B//B
B
Epot
A//A
C//B
B//B
B
Epot
0.0
5.3
5.2
8.0
9.7
0.0
3.9
3.7
5.5
5.7
0.0
0.8
1.2
1.8
1.6
0.0
0.3
1.6
-0.9
-1.8
0.0
-0.7
-0.7
-2.3
-3.0
0.0
-0.9
-0.8
-2.5
-3.4
a Absolute values for methanol are: 181.9 (exp), 170.4 (AM1 and AM1(CH OH)/MM3), 176.2 (AM1(H O)/MM3/C//B). All entries are in
3
2
kcal/mol. b See: Lias, S. G.; Liebman, J. F.; Levin, R. D. J. Phys. Chem. Ref. Data 1984, 13, 695.
TABLE 3: Deprotonation Enthalpies of Alcohols: Relative Valuesa
AM1(CH3OH)/MM3
AM1(H2O)/MM3
molecule
RR
Rβ
Rγ
expb
AM1
C//B
B//B
B
Epot
C//B
B//B
B
Epot
R3COH
R3COH
R3COH
R3COH
R3COH
R3COH
R3COH
R3COH
R3COH
R3COH
H
CH3
C2H5
t-C4H9
CH3
CH3
C2H5
i-C3H7
t-C4H9
CH3
H
H
H
H
CH3
t-C4H9
t-C4H9
t-C4H9
t-C4H9
CH3
H
H
H
H
H
H
H
H
H
CH3
0.0
-3.1
-4.5
-7.1
-5.1
-8.5
-9.6
-10.7
-11.9
-5.9
0.0
0.2
-1.1
-3.3
0.3
-2.8
-3.8
-4.4
-5.3
-1.2
0.0
0.6
-0.3
-1.3
-0.9
-1.6
-2.3
-3.3
-3.6
-8.2
0.0
1.0
0.8
2.4
-1.6
0.1
0.7
1.0
1.8
-8.6
0.0
2.3
2.0
3.5
1.1
2.4
3.0
3.2
3.7
-4.0
0.0
-4.6
-6.4
-6.8
-9.2
-11.9
-12.9
-15.6
-16.6
-13.3
0.0
-2.0
-2.3
-0.5
-3.9
-2.6
-2.2
-2.5
-2.4
-5.7
0.0
-2.0
-2.3
-0.4
-3.9
-2.6
-2.2
-2.5
-2.5
-5.7
a
Absolute values for methanol are: 379.2 (exp), 384.2 (AM1 and AM1(CH3OH)/MM3), and 417.7 (AM1(H2O)/MM3/C//B). All entries are in
kcal/mol. b From ion cyclotron resonance experiments ((2 kcal/mol). See Bartmess, J. E.; Scott, J. A.; McIver, R. T., Jr. J. Am. Chem. Soc. 1979,
101, 6046.
or the particular energy partitioning scheme applied in theoretical
work.
Table 2 lists our results for some simple alcohols. It is
obvious that AM1 reproduces the trends in the experimental
proton affinities reasonably well. In the AM1(CH3OH)/MM3
partitioning, the mechanical embedding of model A yields only
very small changes of the proton affinities and thus fails
qualitatively which may be taken as indirect evidence for the
importance of electrostatic effects. The simple electrostatic
embedding of model B indeed accounts almost quantitatively
for the observed increase in the proton affinities upon adding
the first, second, and third methyl group to methanol (exp 6.4,
2.9, and 2.5 kcal/mol; calcd 5.3, 2.7, and 1.7 kcal/mol). The
dominant differential stabilization of the acid relative to the
conjugate base is provided by the electrostatic interactions in
model B (e.g. in the case of ethanol: 4.6 kcal/mol out of a
total of 5.3 kcal/mol). Model C includes the attractive induction
interaction in the MM part which favors the charged species
and increases with the size and the chain length of the
polarizable MM fragment. Consequently, the results from
model C further emphasize the increase of the proton affinities
with successive methyl substitution, and they give the correct
trend for extending the chain length (contrary to model B, see
ethanol vs 1-propanol).
Table 2 also shows the results for the AM1 (H2O)/MM3
partitioning which are clearly unsatisfactory. In this case, the
QM/MM boundary divides the polar CO bond directly adjacent
to the reactive center (i.e. the oxygen atom), and it is therefore
not too surprising that the electronic rearrangements during
protonation cannot be fully represented by the simple QM/MM
interactions included in our models. It is apparently more
appropriate to apply the QM/MM approach when the QM
reactive center and the MM fragment are separated by two
bonds, as e.g. in the AM1(CH3OH)/MM3 calculations.
Deprotonation Enthalpies of Alcohols. The deprotonation
enthalpies of alcohols decrease with successive alkyl substitution
and with extended chain lengths (see Table 3). AM1 severely
underestimates these substitution effects, but normally predicts
their sign correctly. In the AM1(CH3OH)/MM3 partitioning,
the inclusion of MM polarization in model C is necessary for
obtaining the correct sign, whereas model B fails in this regard
(see Table 3). The AM1(H2O)/MM3 calculations with model
C yield numerical results rather close to the experimental values
which must at least partly be due to fortuitous error compensation (considering that the QM/MM boundary cuts the CO bond,
see above). For both partitionings, however, model C provides
a significant improvement over model B indicating, in our
opinion, that model C captures an essential physical effect: The
indcuction energy increases with the polarizable volume of the
alkyl rest and preferentially stabilizes the anions whose rather
localized charges generate large electric fields in the alkyl
regions.
There are some interesting parallels between our computational results and empirical understanding.87,88 The first experiments have been conducted in solution phase and yielded the
acidity sequence CH3OH > C2H5OH > t-C4H9OH.92 Subsequent gas phase experiments, however, revealed a reversed
acidity sequence87,88 which could only be rationalized with
strong polarization favoring large anions.88,89,93 ”Thus, the
solution order is probably an artifact, and does not represent
any intrinsic property of the molecule.” 88 More recent gas phase
NMR studies on alcohols provided some evidence that electronic
factors affecting the neutral species might also be responsible
for the observed acidity sequence.94 The results did not admit,
however, any quantitative estimate of these effects.
Discussing their own QM/MM model, Field, Bash, and
Karplus attribute the large errors of AM1(H2O)/MM deprotonation enthalpies for alcohols to the poor AM1 results for small
anions (OH-).8 This is certainly not wrong, but the qualitatively
incorrect increase in deprotonation enthalpy from H2O to CH3OH which is characteristic of their (+10.5 kcal/mol)8 and our
(C//B: +6.9 kcal/mol, compare to AM1: -26.6 kcal/mol,
exp: -11.6 kcal/mol, see supporting information) result, is
merely caused by the exaggerated Coulomb repulsion between
the MM methyl fragment and the directly connected QM
10588 J. Phys. Chem., Vol. 100, No. 25, 1996
Bakowies and Thiel
Figure 4. QM/MM treatment of hydride transfer reactions. According
to option a, only the reactive center is treated quantum mechanically.
In option b, the classical region is very small and only consists of the
hydrocarbon chains connecting the bridgeheads of molecules 1-3.
TABLE 4: Activation Energies of Hydride Transfer
Reactionsa
AM1/MM3/A
Figure 3. Transition structures of hydride transfer reactions. Symmetry
point groups: Cs (1-4) and C2V (5).
reactive center. Again, this illustrates the problems of having
the QM/MM boundary too close to the reactive center.
Further Remarks. Tables 2 and 3 also report proton affinities
and deprotonation enthalpies obtained using AM1/MM3/B//B
potential energies (EBpot(X-Y)). Obviously, the link atom
correction is significant only when CH3OH is treated quantum
mechanically. In this case, however, it leads to a considerable
improvement of the results, especially for proton affinities.
Our discussion has been restricted to the AM1/MM3 results,
since MNDO/MM3 calculations do not show any qualitative
differences (see supporting information). All model C calculations have also been performed using fully optimized (C//C)
instead of model B geometries (see supporting information).
The total energies normally differed by no more than 0.1, 0.4,
and 1.0 kcal/mol for neutral, deprotonated, and protonated
molecules, respectively, except for t-C4H9OH (3 QM/MM
divisions; deviations: 0.2, 0.9, and 2.0 kcal/mol, respectively).
None of the qualitative conclusions is affected by the “accuracy”
of the applied geometries.
4.4 Hydride Transfer Reactions. The symmetric hydride
transfer of deprotonated hydroxy ketones 1-5 (see Figure 3)
has been studied kinetically by dynamic NMR spectroscopy.95-97
Watt and co-workers aimed at an experimental access to the
reaction path of Meerwein-Ponndorf-Verley-Oppenauer redox cycles by correlating characteristic geometrical parameters
of the sterically congested molecules with measured activation
barriers. Since the geometry of the reactive center is nearly
completely determined by the rigid hydrocarbon backbones, this
reaction provides an ideal example for mechanical embedding.
No significant electrostatic effects are expected to influence the
reaction, hence the simplest QM/MM model A should be able
to model the reaction satisfactorily.98
Here we summarize the main results of our AM1/MM399
study on these hydride transfer reactions and analyze the ability
of model A to reproduce trends in activation energies and
transition state geometries. The results are compared to recent
TSM100 (transition state modeling) calculations,101,102 and the
relation between TSM and QM/MM models is discussed. A
full report of our study is given elsewhere.44 It includes a
discussion of the reactant geometries and the effects of alkyl
substituents on activation energies, provides additional examples,
and also addresses correlations between geometries and activation energies.
Figure 4 illustrates the two different QM/MM divisions which
have been chosen for the calculations. According to option a,
only the reactive center (CH2O + H-+ CH2O) is treated
molecule
option
a
1
2
3
4
5
11.7
9.5
7.9
4.1
10.9
option
b
AM1
MP2/6-31+G*b
expc
18.0
15.6
14.0
d
d
18.1
16.8
15.7
7.3
12.5
16.5
13.0
11.4
2.9
6.6
21.7 (1.9)
19.0 (2.0)
17.3 (2.1)
13.7 (2.1)
19.4 (0.3)
a All entries in kcal/mol. b MP2/6-31+G*//AM1. c See refs 95-97.
The experimental errors are estimated to be (0.2 kcal/mol (3 to 5)
and (0.3 kcal/mol (2), respectively. The experimental value for 1 is
reported as a lower limit. Computed corrections to convert from
measured quantities to activation energies at 0 K are given in
parentheses (see text). d Option b is only defined for molecules 1-3.
quantum mechanically, while for the alternative option b, the
classical region comprises the hydrocarbon bridges (CH(CH2)nCH) present in molecules 1 to 3. The entire reaction path has
been modeled with just one predefined set of atom types and
bonds such that the number and type of force field terms entering
the MM part of the QM/MM coupling term remains the same
for reactants, transition states, and products. The symmetry of
the reaction requires that the hydride ion has no MM bonds to
any of the carbon atoms. Furthermore, the carbonyl and alkoxy
groups need to be represented by the same MM parameters.
The use of alkoxy parameters for both groups tends to
underestimate the rigidity of the reaction center. Employing
carbonyl parameters is not ideal either but prevents unreasonable
geometries. The best solution would be provided by a new
calibration, but we decided to use carbonyl parameters to avoid
any parameter proliferation at this early stage of testing the QM/
MM model.
To provide sufficient reference data for comparison, we
supplemented the available experimental results with AM1 and
ab initio calculations.82 Small model systems were used to
check the accuracy of AM1 and affordable ab initio levels
against MP2 calculations with large basis sets (up to triple ζ
quality, and augmented with diffuse and polarization functions).44 Surprisingly, AM1 turned out to predict geometries
more reliably than RHF/3-21G and RHF/3-21+G.44 For activation energies, the most adequate and still feasible level was MP2/
6-31+G*//AM1.44
Table 4 collects the AM1/MM3/A//A activation energies,
supplemented with theoretical (AM1, MP2/6-31+G*) and
experimental reference data. The latter are Gibbs free enthalpies
(∆Gqexp) and thus require conversion to activation energies
(∆Eq) for a meaningful comparison. Appropriate corrections
have been estimated from RHF/3-21G geometries and scaled
(0.9) vibrational frequencies using the harmonic oscillator and
rigid rotor approximation without explicit treatment of low-
Quantum Mechanical and Molecular Mechanical Hybrid Models
J. Phys. Chem., Vol. 100, No. 25, 1996 10589
TABLE 5: Relative Activation Energies Of Hydride
Transfer Reactionsa
AM1/MM3/A
molecule
1
2
3
4
5
option option
a
b
TSMb
AM1
MP2/6-31+G*c
expd
0.0
-2.2
-3.8
-7.6
-0.8
0.0
-1.3
-2.4
-10.8
-5.6
0.0
-3.5
-5.1
-13.6
-9.9
0.0
-2.6
-4.2
-7.8
-3.9
0.0
-2.4
-4.0
e
e
0.0
-2.0
-3.4
-6.2
2.3
a All entries in kcal/mol. b Transition state modeling results from
ref 101. c MP2/6-31+G*//AM1. d Corrected experimental values, see
Table 4. e Option b is only defined for molecules 1-3.
TABLE 6: Nonbonded Distances in the Transition
Structures (in Å)
AM1/MM3/A
distance
structure
option a
C‚‚‚Ca
1
2
3
4
5
1
2
3
4
5
2.495
2.465
2.443
2.528
2.749
1.448
1.434
1.425
1.393
1.444
C‚‚‚Hb
option b
AM1
2.412
2.387
2.369
c
c
1.458
1.443
1.432
c
c
2.411
2.390
2.373
2.425
2.676
1.448
1.436
1.428
1.373
1.418
a Distance between the C atoms of the CO groups. b Distance
between the migrating H atom and the C atom of a CO group. c Option
b is only defined for molecules 1-3.
frequency vibrations and neglecting electronic contributions.44
Applying these corrections, the experimental values appear to
be 7-13 kcal/mol higher than the best available ab initio data.
This discrepancy may partly be attributed to solvation effects.
Furthermore, the experimental values may not be accurate103
since the reaction rates strongly depend on substrate concentration and counterions.104 Apparently, the discrepancy is much
smaller for molecules 1-3 than for molecules 4 and 5.
Consequently, there is a good agreement between theory and
experiment (corrected values) for relatiVe activation energies
only within the groups (1, 2, 3) and (4, 5) (see Table 5). Facing
the uncertainties in the experimental values and admitting that
the theoretical level is far from being definitive, no decision
can be made as to which set of reference data is most reliable.
Higher level ab initio calculations are certainly desirable, but
appear to be prohibitive currently.
AM1/MM3/A (option b) predicts absolute activation energies
close to the AM1 values. This is not surprising at all, since
the QM region is very large in this case. The length of the
hydrocarbon chain (treated molecular mechanically) affects the
reactive center only indirectly through adjusting its geometry
at the bridgehead hinges. Interestingly, AM1/MM3/A (option
b) yields relatiVe activation energies in closer agreement than
AM1 with both experimental and ab initio reference data. If
just the reactive center is treated quantum mechanically (option
a), AM1/MM3/A predicts relatiVe activation energies in nearly
perfect agreement with the experimental values for 1-4 (see
Table 5), with differences of no more than 0.4 kcal/mol, a value
characteristic of the experimental error limits. Even the result
for 5 seems to be acceptable, given the discrepancies observed
for the other theoretical approaches. AM1/MM3/A (option a)
underestimates, however, the absolute activation energies.
Selected geometrical parameters characterizing the transition
structures are collected in Table 6. As expected, AM1/MM3/A
predicts nonbonded C‚‚‚C and C‚‚‚H distances at the reactive
Figure 5. Decomposition analysis of the AM1/MM3/A (option a)
activation energies of hydride transfer reactions. TOTAL, QM, and
LINK denote contributions from the QM/MM total energy
A
(X-Y), the QM energy for the QM subsystem ∆EQM(Y-L) and
∆Etot
the link atom correction ∆ELINK, respectively. REST denotes the
A
difference between ∆Etot
(X-Y) and the sum of ∆EQM(Y-L) and
∆ELINK.
center in close agreement with AM1 when only the hydrocarbon
chains are treated classically (option b). AM1/MM3/A thus
successfully models the mechanical compression of the reactive
center geometry due to extending MM hydrocarbon chains
which open the bridge heads.
Compared to AM1, the QM/MM model yields C‚‚‚C distances which are consistently larger (by 0.07-0.10 Å) when
only the reactive center is treated quantum mechanically (option
a). From a careful comparison of ab initio and AM1 data for
these molecules and smaller model systems,44 we expect that
AM1 overestimates the C‚‚‚C distances slightly and that the
AM1/MM3/A (option a) values are therefore probably too large
by about 0.1 Å. The AM1 sequence of increasing C‚‚‚C
distances (3 < 2 < 1 < 4 < 5) is, however, nicely reproduced
by the QM/MM model. Even the differences between absolute
values are very similar for AM1 and AM1/MM3/A (option a).
Both methods further agree quantitatively in the nonbonded
C‚‚‚H distances of 1-3. Compared to AM1, the QM/MM
model yields slightly larger C‚‚‚H distances for 4 and 5. This
reflects the larger discrepancies between the AM1 and AM1/
MM3/A relative activation energies calculated for these molecules (see Table 5).
Both the QM/MM model and AM1 show consistent sequences of activation energies and C‚‚‚H distances. A correlation between these parameters has indeed been established,44
supporting the assumption that the MM cages merely act as
geometrical anchors constraining the QM reactive center
geometry. A closer examination of the QM/MM results,
however, reveals a more complex situation (see Figure 5).
While the QM energy differences ∆EQM(Y-L) for the reactive
center follow the trend provided by the total activation energies
∆EAtot(X-Y), they account for only 40-50% quantitatively.
The link atom corrections ∆ELINK contribute significantly, both
by pronouncing the differences between the activation energies
and by increasing their absolute values. This example again
stresses the importance of ELINK as part of the total energy
expression.
The hydride transfer reactions of molecules 1-5 have also
been studied by TSM (transition state modeling).101,102 This
gives us the opportunity to comment on the relation between
QM/MM and TSM approaches. Both attempt a quantum
mechanical treatment of the reactive center, but in conceptually
different ways. TSM applies two separately calibrated force
fields for reactants and transition states, using QM geometries
of model compounds as penalty functions. The QM/MM
treatment, on the other hand, directly includes the QM model
10590 J. Phys. Chem., Vol. 100, No. 25, 1996
Bakowies and Thiel
Figure 6. Substituted 7-norbornanones chosen for the QM/MM study.
as integral part of the approach. Since the TSM studies and
our AM1/MM3/A (option a) calculations are based on the same
QM transition state model (CH2O + H- + CH2O) the results
may be compared directly: They are apparently of similar
overall quality, with a slight edge for the QM/MM results (see
Table 5). While TSM is certainly more economical in its
application, QM/MM approaches offer some conceptual advantages which make them more attractive for general purposes:
First, no specific parametrization is required. Second, all
stationary points are consistently treated in the same way. Since
TSM uses two different force fields for reactants and transition
states, it yields activation energies which are meaningless as
absolute numbers. Only relative values may be compared.
Third, QM/MM treats transition states in a physically correct
way as maxima of the reaction coordinate, contrary to TSM
which approximates them as minima of a constrained MM
potential surface. Fourth, QM/MM studies are not restricted
to specifically parametrized reaction channels. Finally, only
QM/MM approaches allow one to model the polarization of
the QM charge density due to MM point charges.
4.5 Nucleophilic Addition to Carbonyl Compounds. The
nucleophilic addition of metal hydrides or metal alkyls to
carbonyl compounds is similar in nature to the hydride transfer
reaction discussed above, but it affords the intermediate formation of at least one dipole complex. The reaction mechanism
has been studied theoretically in some detail,105-107 and much
work has been devoted to the origin of stereocontrol (see, e.g.,
refs 108-110). We examined the QM/MM models B and C
and tested their ability to reproduce substituent effects on
mechanistic details such as activation energies, transition
structures, stereochemical preferences, and the formation of
specific dipole complexes. Here we discuss energetic aspects
in some detail and give a short summary of the remaining
results. A full report may be found elsewhere.44
MNDO,111 MNDO/MM3,111,112 and ab initio calculations,82
(RHF/6-31G* and MP2/6-31G*//RHF/6-31G* 1) have been
carried out for the reactions of formaldehyde 6, acetone 7, and
substituted 7-norbornanones 8-10 with monomeric lithium
hydride113 (see Figure 6). In all QM/MM calculations for 7-10,
the subunit (CH2O + LiH) is treated quantum mechanically
while the remainder is described classically. The same bond
topology has been used for any of the stationary points,
employing carbonyl instead of alkoxy parameters even for the
product. The lithium parameter for MNDO electrostatic
potentials (RLi ) 1.11988 Å-1) has been obtained by the
parametrization procedure described in ref 45, using LiH, LiCH3,
and LiOH as reference molecules. All calculations refer to the
C//B level if not stated otherwise.
While molecules 6-9 consist of only one conformer each,
five conformations need to be considered for molecule 10,
resulting from different orientations of the two formyl groups.
Unlike both MNDO and MM3, the hybrid MNDO/MM3
approach succeeds in reproducing their energetical sequence as
predicted by ab initio methods, although the covered range of
energies is underestimated (4.2 kcal/mol vs 6.1 and 6.8 kcal/
mol at the RHF/6-31G* and MP2/6-31G* levels, see supporting
Figure 7. Nucleophilic addition of LiH to formaldehyde: Stationary
points (RHF/6-31G* geometries).
TABLE 7: Reaction Profile of the Nucleophilic Addition of
LiH to 6-10a
method
MNDO
MNDO/MM3/C//B
RHF/6-31G*
MP2/6-31G*//RHF/6-31G*
molecule
b
6
7
8
9
10
6b
7
8
9
10
6c
7d
8
9
10
6
7
8
9
10
RfD
DfT
TfP
-18.0
-17.8
-16.1
-16.0
-14.5
-18.0
-21.4
-21.0
-21.1
-18.7
-19.8
-23.9
-24.1
-24.5
-22.1
-19.9
-24.3
-24.1
-24.6
-22.3
4.3
12.0
17.5
18.9
15.7
4.3
5.5
8.2
7.9
3.3
3.6
10.0
10.2
11.4
5.3
3.5
4.9
3.6
4.4
-1.4
-35.5
-32.8
-38.0
-38.7
-39.5
-35.5
-48.4
-52.3
-51.6
-51.5
-36.8
-29.7
-34.2
-34.6
-35.3
-43.5
-33.9
-38.5
-38.5
-38.4
a
Difference in energy or heat of formation (kcal/mol) for the step
indicated. T and P always refer to the syn oriented attack. b The QM/
MM and QM treatments are identical by definition. c See also ref 106.
d
See also: Wu, Y.-D.; Houk, K. N. J. Am. Chem. Soc. 1987, 109,
908.
information for details). The energetically most favorable
conformer shows nearly orthogonally oriented formyl groups
and has no symmetry. The more favorable of the conformers
with Cs symmetry has nearly ecliptic CO and CC bonds. It
has been chosen for the present study since symmetry facilitates
reasonably accurate ab initio reference calculations.
The investigated reaction path has Cs symmetry throughout.
All theoretical approaches applied show 4 distinct stationary
points114 for any of the substrates 6-10: reactants R, dipole
complexes D with almost linear COLiH units, four center
transition states T, and products P (see Figure 7 and Table 7).
The formation of the dipole complex D is generally quite
exothermic. The RHF and MP2 levels of theory consistently
predict an energy gain of 20-25 kcal/mol which is probably
too large because of the basis set superposition error. The
intrinsic activation energy (D f T) is very low (4-11 kal/
mol) at the RHF level and is further reduced by electron
correlation. MP2 predicts even a negative barrier for the
reaction of 10. There is, however, an additional dipole complex
lower in energy (-2.1 kcal/mol) in this case which shows a
bent COLiH unit resembling the transition state geometry (see
supporting information). Large amounts of energy are released
in the product formation step (T f P, RHF: 30-37 kcal/mol,
MP2: 34-44 kcal/mol), resulting in a very exothermic overall
reaction. The transition state is generally formed with a net
gain of energy (R f T). Several differences in the reaction
Quantum Mechanical and Molecular Mechanical Hybrid Models
TABLE 8: Energy Contributions to the Reaction Profile:
MNDO/MM3/C//Ba
coul(X,
E
Eind(Y)
Eind(X)
ELINK
a
Y)
molecule
R
D
T
P
7
8
9
10
7
8
9
10
7
8
9
10
7
8
9
10
-10.0
-4.6
-4.7
-2.2
-6.9
-2.7
-2.6
-0.3
-4.4
-3.5
-3.7
-1.2
-1.4
-1.2
-1.1
-0.8
-12.4
-6.4
-6.4
-2.0
-6.9
-2.7
-2.6
-0.3
-6.0
-5.3
-5.7
-2.5
-0.8
-0.7
-0.7
-0.5
-10.5
-4.1
-5.1
-4.1
-6.9
-2.6
-2.6
-0.3
-3.9
-3.4
-3.5
-1.6
-5.5
-4.8
-4.8
-4.5
-6.7
-2.2
-2.6
-1.9
-6.7
-2.6
-2.6
-0.2
-2.5
-1.7
-1.8
-0.3
-33.1
-32.8
-32.6
-31.5
All entries are in kcal/mol. T and P refer to the syn oriented attack.
profiles of the substrates 6-10 are noticeable: (1) The complexation energy is lowest for formaldehyde 6, followed by the
diformyl substituted 7-norbornanone 10. (2) 10 has an exceptionally low intrinsic activation energy, the same is observed
for 6 at the RHF level of theory. (3) Acetone 7 shows the lowest
gain of energy for the product formation step.
MNDO/MM3 benefits from the satisfactory MNDO results
for 6 and reproduces most of the trends observed in the ab initio
calculations. The QM/MM model yields intrinsic activation
energies which are in between the RHF and MP2 results, and
predicts overall activation energies (referring to reactants rather
than to dipole complexes) which are close to the RHF values.
Most differences in the ab initio reaction profiles for 7-10 are
attributed to electrostatic and induction effects by the QM/MM
model (see Table 8). Compared to the unsubstituted 7-norbornanone 8, for example, the diformyl substituted molecule 10
shows less attractive Coulomb interactions for the reactant (2.4
kcal/mol) and for the dipole complex (4.4 kcal/mol). The
induction energies Eind(X) + Eind(Y) are generally lower for
10 than for 8, with only small variations along the reaction path
(4.5 ( 0.7 kcal/mol), resulting in both a lower complexation
energy and a lower intrinsic barrier for 10. The least exothermic
product formation step occuring for 7 may similarly be attributed
to electrostatic effects (see Table 8).
MNDO/MM3 consistently overestimates the stability of the
products by 10-15 kcal/mol. This is completely due to the
inappropriate representation of alkoxy groups by carbonyl
bonded
(X,Y).44 The
parameters in the MM coupling term EMM
large out of plane bending force constant associated with sp2
carbon atoms does not allow the required full pyramidalization
of the alkoxy unit. Unlike the MM carbon atom, its substitute
link atom experiences only a weak QM bending force and
therefore adopts a position consistent with a tetrahedral sp3
geometry. The link atom correction ELINK for the QM geometry
of Y-L is consequently too high and leads to excessive
stabilization of the product. The same artifact also occurred in
the QM/MM treatment of the hydride transfer reactions, but
was less obvious since the errors for reactants and products were
nearly compensated by smaller errors for two slightly bent
carbonyl groups in the transition state. The problem discussed
here should only appear for QM/MM divisions close to the
reactive center, and even then it can easily be removed by
appropriate parameter scaling.
It seems worth mentioning that MNDO/MM3 performs better
than MNDO in various aspects: For the first two steps (R f
D, D f T), the QM/MM model yields energy differences in
J. Phys. Chem., Vol. 100, No. 25, 1996 10591
much closer agreement with the ab initio values and correctly
estimates the transition states to be lower in energy than the
reactants. MNDO fails in three out of five cases. The QM/
MM model further predicts two additional dipole complexes,
with the COLiH unit being bent either within the CsC(dO)sC
plane (molecule 7) or perpendicular to it (molecule 10) (see
supporting information). Both dipole complexes have been
verified by RHF/6-31G* but cannot be located on the MNDO
potential surface. Unlike MNDO, the QM/MM model qualitatively reproduces the relative energies of products resulting
from syn and anti attack of LiH to 9 and 10 (see supporting
information). It fails, however, to reproduce relative transition
state energies and is thus unreliable in predicting the diastereoselectivities for this reaction (see supporting information).
This is surprising since electrostatic effects have been used to
explain syn or anti preferences,109 and the QM/MM approaches
have so far been successful in modeling these effects. A careful
examination of the ab initio and QM/MM results44 shows that
the Coulomb forces between the QM and MM fragments are
much less significant in directing the side of LiH attack than in
a previous model44,109 where LiH is represented by classical
point charges interacting with the ketone.
The ability of our QM/MM models to reproduce trends in
geometries has been thoroughly checked for this reaction.
Selected results are given in the supporting information. Apart
from a few exceptions, both the influence of MM substituents
on the reactive center geometry and changes in the substituent
geometries occurring along the reaction coordinate are reproduced qualitatively. The MNDO/MM3 geometries are not very
accurate in an absolute sense, however, since they suffer from
shortcomings in the MNDO parametrization115 of lithium.116
MNDO predicts LiH distances which are too short and LiO
distances which are too long, and these errors are present in
both the MNDO and the MNDO/MM3 results.
4.6 Nucleophilic Ring Cleavage of Oxiranes. The nucleophilic ring cleavage of unsymmetrically alkyl substituted
oxiranes is known to proceed regiospecifically. While the attack
at the less substituted carbon atom in basic media is commonly
explained by steric effects directing the pathway of a SN2
reaction,117 the more facile cleavage at the substituted carbon
atom in acidic media117,118 has caused a controversial
discussion.117-125 Kinetic studies favored a SN1 mechanism120
in which the regioselectivity is caused by electronic stabilization
(“+I effect”) of the intermediate carbenium ion. The observed
Walden inversion,119 however, could only be rationalized with
a SN2 mechanism, giving rise to the hypothesis of a “borderline
SN2” pathway117 which has, however, repeatedly been criticized.119,122
The reaction of 2,2-dimethyloxirane with OH- (basic, reaction
(a)) and of protonated 2,2-dimethyloxirane with H2O (acidic,
reaction (b)) has been studied by MNDO/MM3/C//B to examine
whether a simple QM/MM model can reproduce and explain
the experimental observations. Only MNDO calculations have
been carried out for reference, since we focus on qualitative
rather than quantitative aspects. The study has further been
restricted to the first part of the reaction, leading from reactants
R to dipole complexes D and transition states T (see Figure 8).
In the QM/MM treatment, the classical region comprises the
two methyl groups. None of the CO bonds is considered in
the MM part of the QM/MM coupling term to ensure a
consistent and unified description for all reaction paths. The
ring carbon atoms are represented by alkene parameters to
account for their (partial) sp2 character.122
The MNDO and MNDO/MM3/C//B results are collected in
Tables 9 and 10, respectively. Under base catalysis, the attack
10592 J. Phys. Chem., Vol. 100, No. 25, 1996
Bakowies and Thiel
experimentally observed regioselectivities. It is interesting to
note that the simpler QM/MM model A prefers attack at the
less substituted carbon atom for both reactions a and b. It further
yields much smaller differences in geometry for the two
transition states resulting from acid-catalyzed attack. These
results provide some insight in the origin of regioselectivity.
While minimization of steric repulsion appears to be the driving
force in basic media, Coulomb interactions between the QM
and MM fragments prevail in the acid-catalyzed reaction. They
are responsible for the large changes in geometry (supporting
the “borderline SN2” hypothesis) and facilitate the attack at the
sterically congested site. This explanation is closely related to
the empirical argumentation discussed above, except that
inductive (charge transfer) effects are replaced by electrostatic
effects which, however, show the same trend for obvious
reasons.
Figure 8. Ring cleavage of oxiranes: Reactants R, dipole complexes
D, and transition structures T; (a) base catalysis, (b) acid catalysis.
The activation energies and activation enthalpies are defined as
follows: ∆Eq1 ) E(T) - E(R), ∆Eq2 ) E(T) - E(D), ∆∆Hq1 )
∆Hf0(T) - ∆Hf0(R), ∆∆Hq2 - ∆Hf0(T) - ∆Hf0(D).
TABLE 9: Nucleophilic Ring Cleavage of Oxiranes: MNDO
Resultsa
reaction
R
attackb
∆∆Hq1
∆∆Hq2
rform
rbreak
a
a
a
bc
b
b
H, H
CH3, CH3
CH3, CH3
H, H
CH3, CH3
CH3, CH3
1
1
2
1
1
2
11.8
25.5
12.6
6.2
-1.5
12.9
17.4
31.2
18.3
12.5
4.0
18.4
1.942
2.000
1.897
3.021
3.606
3.077
1.571
1.674
1.533
1.943
1.798
2.034
a
Activation enthalpies ∆∆Hq in kcal/mol, transition structure
parameters in Å. See Figure 8 for reference. b See Figure 8 for
reference. c See also ref 123.
TABLE 10: Nucleophilic Ring Cleavage of Oxiranes:
MNDO/MM3/C//B Resultsa
reaction
R
attackb
∆Eq1
∆Eq2
rform
rbreak
a
a
b
b
CH3, CH3
CH3, CH3
CH3, CH3
CH3, CH3
1
2
1
2
21.8
13.7
-2.3
4.2
31.5
23.3
3.3
9.8
1.980
1.884
3.226
2.982
1.684
1.585
1.884
2.041
a MNDO/MM3(CH , CH ). Activation energies ∆Eq in kcal/mol,
3
3
transition structure parameters in Å. See Figure 8 for reference. b See
Figure 8 for reference.
at the unsubstituted carbon atom is strongly preferred (by 12.9
kcal/mol (MNDO) and 8.1 kcal/mol (MNDO/MM3), respectively). Both methods also agree that acid catalysis favors attack
at the substituted carbon atom (by 14.4 and 6.5 kcal/mol,
respectively). As expected, the lower activation energy corresponds to an earlier transition state as far as the breaking bond
is concerned. The forming bond, on the other hand, is always
longer for the attack at the substituted carbon atom, reflecting
the considerable steric repulsion between nucleophile and methyl
groups.
The QM/MM model reproduces the QM trends nicely, but
yields less pronounced differences, both in geometry and in
energy, for the alternative pathways. The geometries may not
be very accurate in either case, since MNDO is known to
overestimate nonbonded distances for the acid-catalyzed reaction
of unsubstituted oxirane44,123 while it underestimates them for
the base-catalyzed reaction.44,126 The considerably smaller
MNDO/MM3 values for reaction (a) are certainly more realistic
but probably still too large.
Nevertheless, the MNDO/MM3/C//B results may be regarded
as satisfactory since they are in qualitative agreement with the
5. Discussion and Conclusions
We have presented a hierarchy of three models to combine
quantum mechanical (QM) and molecular mechanical (MM)
approaches. They are designed to reduce the QM treatment of
large molecules to the electronically important fragment which
then interacts in a perturbative fashion with the molecular
mechanically described remainder. While model A represents
a simple mechanical embedding of the QM fragment, more
refined treatments of electrostatic and induction interactions are
included in models B and C. In contrast to earlier QM/MM
approaches, the proposed models include an explicit correction
for interactions involving the link atom which substitutes the
MM fragment in the QM treatment. This correction appears
to be an important part of the total energy expression, and
several test cases have shown that it can improve the results
considerably (see, e.g., sections 4.3 and 4.4). The link atom
correction behaves reasonably well for geometries close to the
optimized QM structures while it is inappropriate for geometries
far off the minimum. This prevents the use of the total energy
expression for geometry optimizations and makes the separate
definition of a potential energy inevitable. Inadequate MM
parameters for atoms at the QM/MM boundary may also lead
to ill-behaved link atom corrections. This causes problems
whenever the QM/MM treatment of a chemical reaction requires
a QM/MM boundary close to the reactive center with MM
parameters which are kept fixed for the entire reaction path (see,
e.g., section 4.5). Hence, future applications should consider
an appropriate scaling of the MM parameters in question. The
angle between the bond connecting the QM and MM fragments
and the bond of the frontier QM atom with the link atom
provides a useful check for the quality of these MM parameters.
It should not be larger than a few degrees, indicating a wellbalanced MM description of the boundary region.44
QM/MM approaches with an explicit link atom correction
have the formal disadvantage that they cannot be used in MD
simulations which require consistent total energies and gradients.
This problem may be circumvented by deliberately shifting the
QM/MM boundary as far away from the reactive center as is
necessary to achieve a sufficiently constant link atom correction
for the entire reaction path. In this case, it seems justified to
replace the total energy by the potential energy which leads to
consistent energies and gradients. The effects of this approximation need to be checked in any given application.
Model B includes a quantum chemical treatment of Coulomb
interactions between the QM and MM fragments. As a
refinement of earlier QM/MM approaches, we have adopted
parametrized models to calculate QM electrostatic potentials
and MM partial charges which have both been calibrated against
Quantum Mechanical and Molecular Mechanical Hybrid Models
RHF/6-31G* reference data.45 In the majority of cases, model
B successfully reproduces (ab initio or experimental) trends
arising from electrostatic effects which indicates a reasonable
description of Coulomb interactions. There are, however, some
deficiencies of model B: Electrostatic interactions involving
strongly charged MM atoms close to the QM/MM boundary
tend to be overestimated and to induce charge flux between
the QM fragment and the link atom which generally lowers the
QM/MM total energy.44 The link atom approach does not
provide any possibility to prevent this unphysical side effect.
Future developments should attempt to ameliorate this problem.
This might already be achieved by introducing secondary
conditions in the charge equilibration model which reduce the
MM charges at the QM/MM boundary. Such a refinement
would empirically account for the physically well-known
damping behavior of short-range interactions and simultaneously
reduce the undesirable charge flux.
In the present implementation, the calculation of relative
energies benefits from error cancellation.44 Problems connected
with either inappropriate MM parameters for the QM/MM
boundary or the poor representation of short range interactions
are likely to cause errors of similar magnitude for closely related
conformations or stationary points. The successful application
of the proposed QM/MM models to various problems of
chemical structure and reactivity has indeed shown that the more
interesting relative energies (or relative geometries) are barely
influenced by these shortcomings. As a general guideline, it is
obviously advantageous to put the QM/MM boundary as far
away from the reactive center as is computationally affordable
in order to minimize such errors and to allow for an optimum
error cancelation.
Model C includes the polarization of the MM fragment. It
adopts the physically justifiable treatment of Thole63 to describe
induced dipole interactions which is considered as a significant
improvement over the older treatment proposed by Applequist.62
Although some refinement is still possible for the description
of short range polarization at the QM/MM boundary, model C
currently provides one of the most advanced treatments of
polarization in semiempirical QM/MM approaches. The consideration of MM polarization appears to be crucial in applications involving charged QM fragments which generate large
electric fields. This has been exemplified for the calculation
of deprotonation enthalpies where considerable qualitative
improvements have been found compared with other QM/MM
treatments.
The proposed QM/MM approaches are generally not accurate
in a quantitative sense, due to deficiencies in the interaction
models and in the underlying QM and MM methods. In most
cases, however, they provide sufficiently reliable qualitative
results so that they can be used for an interpretation of
experimental data, especially with regard to trends in related
molecules.
Acknowledgment. This work has been supported in part
through the Alfried Krupp Förderpreis.
Supporting Information Available: Additional data are
provided for the protonation energies and deprotonation enthalpies of alcohols (Tables S1-S3) and for the nucleophilic
addition reaction of LiH with ketones (Tables S4-S9) (11
pages). Ordering information is given on any current masthead
page.
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