Vol. 9 COMPUTATIONAL CHEMISTRY 319COMPUTATIONALQUANTUM
CHEMISTRYFOR FREE-RADICALPOLYMERIZATIONIntroductionChemistry is
traditionally thought of as an experimental science, but recent
rapidand continuing advances in computer power, together with the
development ofEncyclopedia of Polymer Science and Technology.
Copyright John Wiley & Sons, Inc. All rights reserved.320
COMPUTATIONAL CHEMISTRY Vol. 9efcient algorithms, have made it
possible to study the mechanism and kineticsof chemical reactions
via computer. In computational quantum chemistry, one cancalculate
from rst principles the barriers, enthalpies, and rates of a given
chem-ical reaction, together with the geometries of the reactants,
products, and tran-sition structures. It also provides access to
useful related quantities such as theionization energies, electron
afnities, radical stabilization energies, and singlettriplet gaps
of the reactants, and the distribution of electrons within the
moleculeor transition structure. Quantum chemistry can provide a
window on the reac-tion mechanism, and assumes only the
nonrelativistic Schr odinger equation andvalues for the fundamental
physical constants.Quantum chemistry is particularly useful for
studying complex processessuch as free-radical polymerization (see
RADICAL POLYMERIZATION). In free-radicalpolymerization, a variety
of competing reactions occur and the observable quanti-ties that
are accessible by experiment (such as the overall reaction rate,
the overallmolecular weight distribution of the polymer, and the
overall monomer, polymer,and radical concentrations) are a
complicated function of the rates of these individ-ual steps. In
order to infer the rates of individual reactions from such
measurablequantities, one has to assume both a kinetic mechanismand
often some additionalempirical parameters. Not surprisingly then,
depending upon the assumptions,enormous discrepancies in the
so-called measured values can sometimes arise.Quantum chemistry is
able to address this problem by providing direct accessto the rates
and thermochemistry of the individual steps in the process,
withoutrecourse to such model-based assumptions.Of course, quantum
chemistry is not without limitations. Since the multi-electron Schr
odinger equation has no analytical solution, numerical
approxima-tions must instead be made. In principle, these
approximations can be extremelyaccurate, but in practice the most
accurate methods require inordinate amountsof computing power.
Furthermore, the amount of computer power required
scalesexponentially with the size of the system. The challenge for
quantum chemistsis thus to design small model reactions that are
able to capture the main chemi-cal features of the polymerization
systems. It is also necessary to perform carefulassessment studies,
in order to identify suitable procedures that offer a reason-able
compromise between accuracy and computational expense. Nonetheless,
withrecent advances in computational power, and the development of
improved algo-rithms, accurate studies using reasonable chemical
models of free-radical poly-merization are now feasible.Quantum
chemistry thus provides an invaluable tool for studying the
mech-anismand kinetics of free-radical polymerization, and should
be seen as an impor-tant complement to experimental procedures.
Already quantum chemical studieshave made major contributions to
our understanding of free-radical copolymer-ization kinetics, where
they have provided direct evidence for the importance ofpenultimate
unit effects (1,2). They have also helped in our understanding
ofsubstituent and chain-length effects on the frequency factors of
propagation andtransfer reactions (25). More recently, quantum
chemical calculations have beenused to provide an insight into the
kinetics of the reversible addition fragmen-tation chain transfer
(RAFT) polymerization process (6,7). For a more
detailedintroduction to quantum chemistry, the interested reader is
referred to severalexcellent textbooks (816).Vol. 9 COMPUTATIONAL
CHEMISTRY 321Basic Principles of Quantum ChemistryAb initio
molecular orbital theory is based on the laws of quantum
mechanics,under which the energy (E) and wave function () for some
arrangement of atomscan be obtained by solving the Schr odinger
equation 1 (17).H =E (1)This is an eigenvalue problem for which
multiple solutions or states arepossible, each state having its own
wave function and associated energy. The low-est energy solution is
known as the ground state, while the other higher energysolutions
are referred to as excited states. The wave function is an
eigenfunctionthat depends upon the spatial coordinates of all the
particles and also the spincoordinates. Its physical meaning is
best interpreted by noting that its squaremodulus is a measure of
the electron probability distribution. The term ( H) inequation 1
is called the Hamiltonian operator and corresponds to the total
kinetic(T) and potential energy ( V) of the system.H= T + V (2)T =
h282
i1mi_ 2x2i+ 2y2i+ 2z2i_(3)V =
i0as
s (13)The 0 wave function is the HF wave function, while the
various s determi-nants correspond to the various excited
congurations. The CI method introducesa further set of unknown
parameters into the calculation, the coefcients (as).These
coefcients are optimized as part of the ab initio calculation in
order tominimize the energy, in line with the variational
principle. CI methods can bebased on an RHF wave function (RCI), a
UHF wave function (UCI), or an ROHFwave function (URCI).Full CI is
impractical with an innite basis set (and hence an innite num-ber
of virtual orbitals), or indeed with a nite basis set and a
reasonably smallnumber of electrons. For example, even for water
with the small 6-31G(d) basisset, the full CI treatment would
involve nearly 5 108congurations. For this332 COMPUTATIONAL
CHEMISTRY Vol. 9Fig. 6. Electron conguration diagrams illustrating
the lack of size consistency in trun-cated CI. In the rst case, A
and B are treated separately by CID, and the treatment
thusconsiders the double excitations of electrons in each molecule.
In the second case, A andB are calculated as a supermolecule having
the A and B fragments at (effectively) inniteseparation. Now the
simultaneous excitation of two electrons from each of the A-type
andB-type orbitals constitutes a quadruple excitation, which is not
included in the CIDmethod.reason, methods based on a truncated CI
procedure are generally used in prac-tice. These methods consider a
limited number of excited determinants, such asall possible single
excitations (CIS) or all possible single and double
excitations(CISD). Restricting the CI procedure to single, double,
and possibly triple excita-tions is usually a reasonable
approximation, since excitations involving one, two,or three
electrons have a considerably higher probability of occurring, and
thuscontributing to the wave function, compared with excitations of
several electronssimultaneously.However, simple truncated CI
methods suffer from a lack of size consistency.That is, the error
incurred in calculating molecules A and B separately is differ-ent
from that incurred in calculating a single species, which contains
A and Bseparated by a large (effectively innite) distance. This can
be seen quite clearlyin the example shown in Figure 6. The lack of
size consistency can be a majorproblem as it introduces an
additional error to calculations of barriers and en-thalpies in
nonunimolecular reactions. This problem is addressed by
includingadditional terms in the wave function, and the methods
based on this approachinclude quadratic conguration interaction
(QCI) and coupled cluster theory (CC).These methods are typically
applied with single and double excitations (QCISDor CCSD), and the
triple excitations are often included perturbatively, leading
tomethods such as QCISD(T) and CCSD(T). When applied with an
appropriatelylarge basis set, these methods usually provide
excellent approximations to theexact solution to the Schr odinger
equation. However, these methods are still verycomputationally
expensive.M ollerPlesset (MP) Perturbation Theory. By convention,
the correlationenergy is simply the difference between the
HartreeFock energy and the exactsolution to the Schr odinger
equation. Rather than approximate the exact solu-tion to the Schr
odinger equation by attempting to build the exact wave
functionthrough conguration interaction, an alternative (and
considerably less expen-sive approach) is to estimate the
correlation energy as a perturbation on theVol. 9 COMPUTATIONAL
CHEMISTRY 333HartreeFock energy. In other words, the exact wave
function and energy areexpanded as a perturbation power series in a
perturbation parameter asfollows.=(0)+(1)+2
(2)+3
(3)+ (14)E=E(0)+E(1)+2E(2)+3E(3)+ (15)Expressions relating terms
of successively higher orders of perturbation areobtained by
substituting equations 14 and 15 into the Schr odinger equation,
andthen equating terms on either side of the equation. Having
obtained these ex-pressions, it simply remains to evaluate the rst
terms in the series, and this isachieved by taking the (0)term as
the HartreeFock wave function. In practice,the MP series must be
truncated at some nite order. Truncation at the rst order(ie, E(1))
corresponds to the HartreeFock energy, truncation at the second
orderis known as MP2 theory, truncation at the third order as MP3
theory, and so on.MP methods based on an RHF, UHF, or ROHF wave
function are referred to asRMP, UMP, or ROMP respectively.When
truncated at the second, third, or possibly fourth orders, the MP
meth-ods offer a very cost-effective method for estimating the
correlation energy. Theyare also size-consistent methods. However,
the validity of truncating the seriesat some nite order depends on
the speed of convergence of the series, and thiswill vary
considerably depending on howclosely the HartreeFock energy
approx-imates the exact energy. Indeed in some cases, the MP series
can actually diverge,and the application of MPmethods can in such
cases increase rather than decreasethe errors in the calculation.
As noted above, a relevant example of this problemoccurs in the
transition structures for radical addition to alkenes for which
UMP2calculations (based on the spin-contaminated UHF wave function)
are frequentlysubject to large errors (32,33). Furthermore, when
truncated at some nite order,the MP methods are not variational,
and may thus overestimate the correctionto the energy. Hence,
although MP procedures frequently provide excellent cost-effective
performance, they must be applied with caution.Composite
Procedures. The use of CCSD(T) or QCISD(T) methods with asuitably
large basis set generally provides excellent approximations to the
exactsolution of the Schr odinger equation. However, such methods
are computationallyexpensive, and in practical calculations smaller
basis sets and/or lower cost meth-ods must be adopted. A major
advance in recent years has been the developmentof high level
composite procedures, which approximate high level
calculationsthrough a series of lower level calculations. Some of
the main strategies that areused are described in the
following.Firstly, it has long been realized that geometry
optimizations and frequencycalculations are generally less
sensitive to the level of theory than are energy cal-culations. For
example, as will be discussed in a following section, detailed
assess-ment studies (36,37) have shown that even HF/6-31G(d) can
provide reasonableapproximations to the considerably more expensive
CCSD(T)/6-311 +G(d,p) levelof theory, for the geometries and
frequencies of the species in radical addition tomultiple bonds
(such as C C, C C, and C S). By contrast, very high levels of334
COMPUTATIONAL CHEMISTRY Vol. 9Fig. 7. Illustration of the relative
performance of the high and low levels of theory forgeometry
optimizations and energy calculations. The low level of theory
shows a very largeerror for the absolute energy of structure, a
smaller error for the Y X bond dissociationenergy (ie, the well
depth), and a very small error for the optimum geometry of the Y
Xbond. This reects the increasing possibility for cancelationof
error. Inthe bond dissociationenergy, errors in the absolute
energies of the isolated Y and X species are canceled to someextent
by errors in the Y X energies. In the geometry optimizations,
further cancelationis possible because the position of the minimum
energy structure depends on the relativeenergies of Y X compounds
having very similar Y X bond lengths.theory are required to
describe the barriers and enthalpies of these reactions.
Theimproved performance of low levels of theory in geometry
optimizations and fre-quency calculations can be understood in
terms of the increased opportunity forthe cancelation of error, as
such quantities depend only upon the relative energiesof very
similar structures (see Fig. 7). In contrast, reaction barriers and
enthalpiesdepend upon the relative energies of the reactants and
transition structures orproducts, and these can have quite
different structures, with different types ofchemical bonds. It is
thus possible to optimize the geometry of a compound at arelatively
low level of theory, and then improve the accuracy of its energy
usinga single higher level calculation (called a single point).
Since geometry opti-mizations and frequency calculations are more
computationally intensive thansingle-point energy calculations,
this approach leads to an enormous saving incomputational cost. By
convention, the nal composite level of theory is writtenas energy
method/energy basis set//geometry method/geometry basis
set.Secondly, an extension to the above strategy is known as the
IRCmax (in-trinsic reaction coordinate) procedure. It was developed
(38) for improving the ge-ometries of transition structures, though
techniques based on the same principlehave also been used to
calculate improved imaginary frequencies and tunnelingcoefcients
(3941). While low levels of theory are generally suitable for
optimiz-ing the geometries of stable species, the geometries of
transition structures aresometimes subject to greater error at
these low levels of theory. To address thisproblem, the minimum
energy path (MEP) for a reaction is rst calculated at alow level of
theory, and then improved via single-point energy calculations at
ahigher level of theory. Now, the transition structure is simply
the maximumenergystructure along the MEP for the reaction. By
identifying the transition structurefrom the high level MEP (rather
than the original low level MEP), one effectivelyoptimizes the
reaction coordinate at the high level of theory (see Fig. 8).Vol. 9
COMPUTATIONAL CHEMISTRY 335Fig. 8. Illustration of the IRCmax
procedure. The minimum energy path (MEP, alsoknown as the intrinsic
reaction coordinate or IRC) is optimized at a low level of
theory,and then improved using high level single-point energy
calculations. The improved tran-sition structure is then identied
as the maximum in the high level MEP. This effectivelyoptimizes the
reaction coordinate (often the most sensitive part of the geometry
optimiza-tion) at a high level of theory.Thirdly, one can improve
the single-point energy calculations themselvesusing additivity
and/or extrapolation procedures. In the former case, the energyis
rst calculated with a high level method (such as CCSD(T)) and a
small basisset. The effect of increasing to a large basis set is
then evaluated at a lower levelof theory (such as MP2). The
resulting basis set correction is then added to thehigh level
result, thereby approximating the high level method with a large
basisset. The calculation may be summarized as follows.High
Method/Small Basis Set+Low Method/Large Basis SetLow Method/Small
Basis SetHigh Method/Large Basis Set(16)Procedures for
extrapolating the energies obtained at a specic level of the-ory to
the corresponding innite basis set limit have also been devised.
The twomainprocedures are the extrapolationroutine of Martinand
Parthiban(18), whichtakes advantage of the systematic convergence
properties of the Dunning DZ, TZ,QZ, 5Z, . . . basis sets, and the
procedure of Petersson and co-workers (42), whichis based on the
asymptotic convergence of MP2 pair energies. For the mathemat-ical
details of these extrapolation routines, the reader is referred to
the origi-nal references. The Martin extrapolation procedure is
easily implemented on aspreadsheet, while the Petersson
extrapolation procedure has been coded into theGAUSSIAN (43)
computational chemistry software package.336 COMPUTATIONAL
CHEMISTRY Vol. 9Building onthese strategies, several composite
procedures for approximatingCCSD(T) or QCISD(T) energies with a
large or innite basis set have been devised.The main families of
procedures in current use are the G3 (44), Wn (28), and CBS(42)
families of methods. These are described in the following.(1) In
the G3 methods, the CCSD(T) or QCISD(T) calculations are
performedwith a relatively small basis set, such as 6-31G(d), and
these are then cor-rected to a large triple zeta basis set via
additivity corrections, carried outat the MP2 and/or MP3 or MP4
levels of theory (44). There are many vari-ants of the G3 methods,
depending upon the level of theory prescribed forthe geometry and
frequency calculations, the methods used for the basisset
correction, and depending on whether CCSD(T) or QCISD(T) is usedat
the high level of theory. Of particular note are the RAD variants
(45)of G3 (such as G3-RAD and G3(MP2)-RAD), which have been
designed forthe study of radical reactions. G3 methods include an
empirical correctionterm, which has been estimated against a large
test set of experimentaldata, and spin-orbit corrections (for
atoms). The G3 methods have been ex-tensively assessed against test
sets of experimental data (including heatsof formation, ionization
energies, and electron afnities) and are generallyfound to be very
accurate, typically showing mean absolute deviations fromexperiment
of approximately 4 kJ mol1.(2) In the Wn methods, high level
CCSD(T) calculations are extrapolated tothe innite basis set limit
using the extrapolation routine of Martin andParthiban (28).
Additional corrections are included for scalar relativisticeffects,
core-correlation, and spin-orbit coupling in atoms. No
additionalempirical corrections are included in this method. The Wn
methods arevery high level procedures, and have been demonstrated
to display chemicalaccuracy. For example, the W1 procedure was
found to have a mean absolutedeviation fromexperiment of only 2.5
kJ mol1for the heats of formation of55 stable molecules. For the
(more expensive) W2 theory, the correspondingdeviation was less
than 1 kJ mol1.(3) In the CBS procedures, the complete basis
extrapolation procedure of Pe-tersson and co-workers is used (42).
This calculates the innite basis setlimit at the MP2 level of
theory. This is then corrected to the CCSD(T)level of theory using
additivity procedures, as in the G3 methods. The CBSprocedures also
incorporate an empirical correction, and an additional
(em-pirically determined) correction for spin contamination. The
accuracy of thislatter termfor the transition structures of radical
addition reactions has re-cently been questioned (36,37).
Nonetheless, the CBS procedures also showsimilar (excellent)
performance to the G3 methods, when assessed againstthe same
experimental data for stable molecules (42).In summary, using
composite procedures, high level calculations can now beperformed
at a reasonable computational cost. With continuing rapid
increasesin computer power, details on the computational speeds of
the various methodswould be rapidly outdated. However, it is worth
noting that, at the time of writing,the most cost-effective G3
procedures can be routinely applied to molecules as bigVol. 9
COMPUTATIONAL CHEMISTRY 337as CH3SC(CH2Ph)SCH3, while the
state-of-the-art Wn methods are restricted tosmaller molecules,
such as CH3CH2CH(CH3). However, in the near future onecan look
forward to applying these methods to yet larger systems. In
general, thecomposite procedures described above offer chemical
accuracy (usually denedas uncertainties of 48 kJ mol1), with the
best methods offering accuracy inthe kJ range. However, careful
assessment studies are nonetheless recommendedwhen applying methods
to new chemical systems. A brief discussion of the per-formance of
computational methods for the reactions of relevance to
free-radicalpolymerization is provided in a following
section.Multireference Methods. The post-SCF methods discussed
above are allbased on a HF or single conguration starting wave
function. At the impracti-cal limit of performing full CI (or
summing all terms in the MP series) with aninnite basis set, these
methods will yield the exact solution to the nonrelativis-tic Schr
odinger equation. However, when truncated to nite order, the use of
asingle reference wave function can sometimes lead to signicant
errors. This isparticularly the case in the calculation of
diradical species (such as the transitionstructures for the
termination reactions in free-radical polymerization),
excitedstates, and unsaturated transition metals. In such
situations, the starting wavefunction itself should be represented
as a linear combination of two or more con-gurations, as follows.
=
jaj
j (17)Inthis equation, the individual wave functions are
formedfromthe lowest en-ergy conguration, and various excited
congurations of the Slater determinants,and the aj coefcients are
optimized variationally. While this method, which isknown as
multireference self-consistent eld (MCSCF), may seem analogous
tothe single-reference CI methods discussed above, there is an
important differencebetweenthem. InMCSCF, the molecular orbital
coefcients (the ci ineq. 6) are op-timized for all of the
contributing congurations. In contrast, in single-referencemethods,
the molecular orbital coefcients are optimized for the
HartreeFockwave function, and are then held xed at their HF
values.The optimization of both the orbital coefcients and the
contribution of thevarious congurations to the overall wave
function can be very computationallydemanding. As a result, MCSCF
methods typically only consider a small numberof congurations, and
one of the key problems is choosing which congurationsto include.
In complete active space self-consistent eld (CASSCF), the
molecularorbitals are divided into three groups: the inactive
space, the active space, andthe virtual space (see Fig. 9). The
wave function is then formed from all possiblecongurations that
arise fromdistributing the electrons among the active orbitals(ie,
full CI is performed within the active space). It then remains to
decide whichoccupied and virtual orbitals should be included in the
active space. Where pos-sible, it is advisable to include all
valence orbitals in the active space, togetherwith an equivalent
number of virtual orbitals. However, as with any full CI
calcu-lation, the computational cost rapidly increases with the
number of electrons andorbitals included, and CASSCF calculations
are currently limited to active spacesof approximately 16 electrons
in 16 molecular orbitals. Thus, for large chemical338 COMPUTATIONAL
CHEMISTRY Vol. 9Fig. 9. The partitioning of orbitals between the
inactive, active, and virtual spaces in aCASSCF
calculation.systems, full valence active spaces are not as yet
possible, and instead a restrictednumber of orbitals must be
chosen.In non-full valence CASSCF, the active space is typically
selected on thebasis of chemical intuition, and might include
orbitals that are directly involvedin the chemical reaction, or are
interacting strongly with the reacting orbitals.For example, in the
case of radicalradical reactions, a simplied multireferenceapproach
would be a CAS(2,2) method, in which the active space would
consistof the two singly occupied molecular orbitals. However, such
restricted methodsmust be used cautiously as they recover
correlation in the active space, but notin the inactive space or
between the active and inactive spaces. As a result, suchprocedures
can sometimes introduce a bias, which, for example, might lead toan
overestimation of the biradical character in systems with nearly
degeneratesinglet and triplet states (9).Multireference methods
primarily account for nondynamic electron corre-lation, which
arises from long-range interactions involving nearly
degeneratestates. It is still necessary to correct for dynamic
correlation, which arises fromshort-range electronelectron
interactions, and which is primarily addressed inthe
single-reference post-SCF methods, such as QCISD or MP2. For
example,in the case of CI-based methods, it would be necessary to
consider excitationsfrom the inactive (as well as active) space
orbitals, into all of the virtual orbitals.Multireference versions
of post-SCF methods have been derived, including mul-tireference CI
(eg, MR(SD)CI, which includes all single and double excitations)and
multireference perturbation theory (eg CASPT2, which is a
multireferenceanalogue of MP2). The former of these methods is
generally more accurate butVol. 9 COMPUTATIONAL CHEMISTRY 339also
more computationally demanding. For more information on
multireferencemethods, the reader is referred to an excellent
review by Schmidt and Gordon(46).Semiempirical MethodsSemiempirical
methods are often used to study large systems for which ab
initiocalculations are not feasible. A number of different
procedures are available, withthe main methods being CNDO, INDO,
MNDO, MINDO/3, AM1, and PM3. Thelatter two procedures are generally
the best performing of the current availablemethods, and are thus
the most popular in current use. Semiempirical methodsare based on
ab initio molecular orbital theory, but neglect several of the
computa-tionally intensive integrals that are required in
HartreeFock theory. Dependingon the procedure, certain interactions
between orbitals are either completely ne-glected or replaced by
parameters that are either derived from experimental datafor the
isolated atoms or obtained by tting the calculated properties of
moleculesto experimental data. This greatly reduces the
computational cost of the calcu-lations; however, it can also
introduce enormous errors. For more detailed infor-mation on the
principles and limitations of semiempirical methods, the reader
isreferred to an article by Stewart (47).In general, the
semiempirical methods perform reasonably well, providedthat the
species (and properties) being calculated are very similar to those
forwhich the method was parameterized. However, there are many
situations inwhich these methods fail dramatically, and hence such
methods should be ap-plied with caution and their accuracy should
always be checked against high levelcalculations for prototypical
reactions. In this context it should be noted thatsuch testing has
already been performed for the case of radical addition to C Cbonds
(32). In this work, semiempirical methods were shown to fail
dramatically,and hence (current) semiempirical methods are not
generally recommended forstudying the kinetics and thermodynamics
of the propagation step in free-radicalpolymerization.Density
Functional TheoryDensity functional theory (DFT) is a different
quantum chemical approach toobtaining electronic-structure
information. The basis of DFT is the HohenbergKohn theorem (48),
which demonstrates the existence of a unique functional
fordetermining the ground-state energy and electron density
exactly. In the ab initiomethods described above, we recall that
their objective was to determine the wavefunction (an
eigenfunction) of a system, which thereby enables the energy
(theeigenvalue) and electron density (the square modulus of the
wave function) tobe evaluated. The HohenbergKohn theorem implies
that the electronic energycan be calculated from the electron
density and there is thus no need to evaluatethe wave function.
This represents an enormous simplication to the calculationsince,
in an n-electron system, the wave function is a function of 3n
variables,whereas the electron density is a function of just 3
variables. Unfortunately, the340 COMPUTATIONAL CHEMISTRY Vol.
9HohenbergKohn theoremis merely an existence proof, rather than a
constructiveproof, and thus the exact functional for connecting the
energy and electron densityis not known. Hence, although in
principle DFT can provide the exact solution tothe Schr odinger
equation, in practice an approximate functional must be used,and
this introduces error to the calculations.The DFT methods used in
practice are based on the equations of Kohn andSham (49). They
partitioned the total electronic energy into the following
terms.E=ET+EV+EJ+EX+EC(18)Inthis equation, ETis the kinetic energy
term(arising fromthe motions of theelectrons), EVis the potential
term (arising from the nuclearelectron attractionand the
nuclearnuclear repulsion), EJis the electronelectron repulsion
term, EXis the exchange term(arising fromthe antisymmetry of the
wave function), and ECis the dynamical correlation energy of the
individual electrons. The sum of the ET,EV, and EJterms corresponds
to the classical energy of the charge distribution,while the
exchange and correlation terms account for the remaining
electronicenergy. The task of DFT methods is thus to provide
functionals for the exchangeand correlation terms. As a matter of
notation, DFT methods are typically namedas exchange
functionalcorrelation functional, using standard abbreviations
forthe various functionals.Before discussing the functionals
themselves, it is worth making a few com-ments
onunrestrictedKohnShamtheory (50). The effective potential of the
KohnShamequations contains no reference to the spinof the
electrons, and the energy issimply a functional of the total
electron density. (It will only become a functional ofthe
individual spin densities if the potential itself contains
spin-dependent parts,such as it would in the presence of an
external magnetic eld.). Hence, if the exactfunctional were
available, there would normally be no need to consider the and spin
densities individually, even for open-shell systems. However, in
practice wemust use approximate functionals, and (for open-shell
systems) these are gener-ally more exible if they explicitly depend
on the and spin densities. In ananalogous manner to UHF,
unrestricted KohnSham theory allows the and spin densities to
optimize independently, and this allows for a better
qualitativedescription of bond-breaking processes but leads to
physically unrealistic spindensities and symmetry breaking
problems. A more detailed discussion of the ad-vantages and
disadvantages of the unrestricted and spin-restricted theories
maybe found in the excellent textbook by Koch and Holthausen (11),
while an exampleof a practical application of unrestricted KohnSham
theory to reactions with bi-radical transition structures may be
found in a paper by Goddard and Orlova (51).On balance, the
unrestricted KohnSham theory is normally preferred for open-shell
systems; however, as always, careful assessment studies are
recommendedin order to establish the suitability of any
computational method for a specicchemical problem.Since the exact
functional relating the energy to the electron (or spin) densityis
unknown, it is necessary to design approximate functionals, and the
accuracyof a DFT method depends on the suitability of the
functionals employed. Manydifferent functionals for exchange and
correlation have been proposed, and it isbeyond the scope of this
article to outline their mathematical forms (these mayVol. 9
COMPUTATIONAL CHEMISTRY 341be found in textbooks such as Refs. 10
and 11), but it is worth mentioning theirmain assumptions. Pure DFT
methods may be loosely classied into local meth-ods and
gradient-corrected methods. The local DFT methods are based on
thelocal density approximation (LDA, also known as the local spin
density approxi-mation, LSDA), in which it is assumed that the
electron density may be treatedas that of a uniform electron gas.
From this assumption, functionals describingthe exchange (called
the Slater functional, S) (52) and correlation (VoskoWilkNusair,
VWN) (53) can be derived, and the resulting method is known as
S-VWN.The treatment of electron density as that of a uniform
electron gas is of course anoversimplication of the real situation,
and, while it can often provide reasonablemolecular structures and
frequencies, the LDA model fails to provide accuratepredictions of
thermochemical properties such as bond energies (for which errorsof
over 100 kJ mol1are typical) (54).Gradient-corrected DFT methods
(also sometimes referred to as nonlocal orsemilocal DFT) attempt to
deal with the shortcomings of the LDA model throughthe generalized
gradient approximation (GGA). This corrects the uniform gasmodel
through the introduction of the gradient of the electron density.
In intro-ducing the gradient, empirical parameters are often
incorporated. For example,the Becke-88 exchange functional (55) was
parameterized against the known ex-change energies of inert gas
atoms. This is commonly used in combination with theLYP
gradient-corrected correlation functional (56), to give the B-LYP
method.Another example of a gradient-corrected functional is the
Perdew-Wang 91 (PW91)functional, which has both an exchange and a
correlation component (57). TheGGA methods show improved
performance over the LDA model, especially withrespect to
thermochemical properties. In this regard, the errors generally
obtainedinstandard thermochemical tests of these methods are of the
order of 25 kJ mol1(54). However, the GGA methods (as well as the
LDA methods) perform poorly forweakly bound systems (such as those
in which Van der Waals interactions areimportant), and they also
perform poorly for reaction barriers (54).The DFT methods described
above are pure DFT methods. Another impor-tant class of methods is
called hybrid DFT. In these methods the exchange func-tional is
replaced by a linear combination of the HartreeFock exchange term
anda DFT exchange functional. In addition, the various exchange and
correlationfunctionals may themselves be constructed as linear
combinations of the vari-ous available methods. For example, the
popular hybrid DFT method, B3-LYP, isdened as follows
(58).EXCB3LYP=EXLDA+c0_EXHFEXLDA_+cX_EXB88EXLDA_+ECVWN+cC_ECLYPECVWN_(19)The
coefcients in this expression, c0 = 0.20, cX = 0.72 and cC = 0.81,
wereobtained by tting the results of B3-LYP calculations to a test
set of experimentalatomization energies, electron afnities, and
ionization potentials.Hybrid DFT methods frequently provide
excellent descriptions of the ge-ometries, frequencies, and even
reaction barriers and enthalpies for many chem-ical systems.
However, owing to their empirical parameters, such methods
areincreasingly becoming semiempirical in nature and as such can
frequently failwhen applied to systems other than those for which
they were parameterized. Agood example of this is the hybrid DFT
method MPW1K (59). This was tted to342 COMPUTATIONAL CHEMISTRY Vol.
9a test set of hydrogen abstraction barriers, and performs very
well for these re-actions. However, the same method has recently
been shown to have large errorswhen applied to the problem of
predicting the enthalpies for radical addition tomultiple bonds
(36,37). Nonetheless, hybrid DFT methods currently present themost
cost-effective option for studying larger chemical systems but, as
always, theperformance of such methods should be carefully assessed
for each new chemicalproblem.Calculation of Reaction Rates from
Quantum-Chemical DataQuantum and Classical Reaction Dynamics. In
the quantum-chemical calculations described above, we solve the
electronic Schr odinger equa-tion to determine the energy
corresponding to a xed arrangement of nuclei. Ifsuch calculations
are performed for all possible nuclear coordinates in a chemi-cal
system, this yields the potential energy surface. However, as we
saw above, inconstructing this potential energy surface we made the
BornOppenheimer ap-proximation, and thus ignored the contribution
of the motions of the nuclei to thetotal kinetic energy. This
approximation was appropriate for calculating the elec-tronic
energy at a specic geometry, but is clearly not very useful for
studying themotions of the atoms in chemical reactions. In order to
calculate reaction rates,we must construct a new Hamiltonian in
which the kinetic energy of the nuclei istaken into account. In
this Hamiltonian, the potential energy is simply the
totalelectronic energy, which we obtain from our quantum-chemical
calculations. Oncewe have formed our new Hamiltonian we can then
solve the Schr odinger equa-tion again, this time to follow the
motion of the nuclei. This procedure is knownas quantum dynamics,
and can in principle yield the exact reaction rates for achemical
system (within the BornOppenheimer approximation).However, there
are several practical limitations to quantum dynamics.Firstly, we
have already seen that, for all but the simplest chemical systems,
ob-taining accurate solutions to the electronic Schr odinger
equation for a single setof nuclear coordinates is very
computationally intensive. Secondly, to construct apotential energy
surface, these expensive calculations must be repeated for ev-ery
possible arrangement of nuclei. Efcient algorithms are available
for choos-ing only those geometries necessary for an adequate
description of the chemicalsystem (60). However, even using these
algorithms, large numbers of quantum-chemical calculations are
nonetheless required. For example, approximately
1000quantum-chemical calculations were required to construct a
reliable potential en-ergy surface for the OH+H2 system(61).
Furthermore, the number of data pointsrequired increases
substantially with the number of atoms in the system (due tothe
increasing dimensionality). Finally, we have the problemof solving
the nuclearSchr odinger equation. In practice, this is intractable
for all but the simplest sys-tems, as atoms (being heavier) require
even more basis functions than are neededto solve the electronic
Schr odinger equation. With current available computingpower,
quantum dynamics calculations are thus restricted to very small
systems,such as OH+H2 (61). In this (state-of-the-art) 4-atom
calculation, the energies atapproximately 107different points on
the potential energy surface were requiredVol. 9 COMPUTATIONAL
CHEMISTRY 343in order to solve the nuclear Schr odinger equation,
and this requirement scalesexponentially with the number of atoms
in the system.Classical reaction dynamics provides a strategy for
calculating the rate co-efcients of larger chemical systems. Having
used quantum-chemical techniquesto calculate the potential energy
surface, the motions of the nuclei are studied bysolving the laws
of classical or Newtonian dynamics. This is often a
reasonableapproximation, since the atoms (being heavier) are
considerably less subject toquantum effects than the electrons.
Nonetheless, standard classical reaction dy-namics calculations are
still limited by the need to calculate a full potential
energysurface (including the rst and second derivatives at each
point). As a result, stan-dard classical dynamics calculations
involving accurate ab initio potential energysurfaces are also
currently restricted to relatively small chemical systems, suchas
H3C3N3 (62).An alternative approach to constructing the entire
potential energy surfacefor a chemical system is provided by direct
dynamics. In both standard classicalreaction dynamics and direct
dynamics, the basic principle is the same. A startingarrangement of
atoms is adjusted (by a small amount) according to the forces
act-ing on them during a small step in time, using the laws of
classical mechanics.This time step is then repeated using the force
corresponding to the new geom-etry, and so on. The process is
repeated for many thousands of time steps, until acomplete
trajectory is mapped out. The process is then repeated for many
trajec-tories until the reaction probability (and other related
information) is establishedto within an acceptable level of
statistical error. As we saw above, in standardclassical reaction
dynamics, the force acting on the molecule as a function of
ge-ometry is obtained from the potential energy surface. In direct
dynamics, alsoknown as on-the-y ab-initio dynamics, this force is
calculated (using quantum-chemical calculations) at each new
position (63). The latter approach is simpler,but less
computationally efcient, and is still restricted to relatively
small systems(if accurate levels of theory are used to calculate
the forces).It should be noted that the classical reaction dynamics
of much larger sys-tems can be studied using approximate potential
energy surfaces, constructed us-ing empirical or semiempirical
procedures. In particular, the method of molecularmechanics (MM),
which is described elsewhere in this Encyclopedia, is commonlyused
to simulate the motion of polymers and proteins. However, the
accuracy ofMM simulations are limited by the accuracy of the force
eld, which is the setof potential functions that are used to govern
relative motions of the constituentatoms. Force elds are typically
derived on the basis of empirical and semiempir-ical information,
and are typically only accurate for the type of system for
whichthey were parameterized. Recently, much effort has been
directed at deriving ac-curate force elds for reacting systems, and
prominent examples include ReaxFF(64) and MMVB (65). However, such
force elds are nonetheless approximate,and only suitable for the
types of reactions for which they were designed. Ac-curate force
elds for studying the kinetics of free-radical polymerization do
notcurrently exist, and instead high level ab initio calculations
are necessary in orderto model these reactions
accurately.Transition-State Theory. To study reactions in larger
chemical systemsusing accurate ab initio calculations we need a
much simpler approach, and thisis provided by transition-state
theory (66). In its simplest form, it assumes that, in344
COMPUTATIONAL CHEMISTRY Vol. 9the space represented by the
coordinates and momenta of the reacting particles, itis possible to
dene a dividing surface such that all reactants crossing this
planego on to form products, and do not recross the dividing
surface. The minimumenergy structure on this dividing plane is
referred to as the transition structureof the reaction.
Transition-state theory also assumes there is an internal
statisti-cal equilibrium between the degrees of freedom of each
type of system (reactant,product, or transition structure), and
that the transition state is in statisticalequilibrium with the
reactants. In addition, it assumes that motion through
thetransition state can be treated as a classical translation.
Fromthese assumptions,the following simple equation relating the
rate coefcient at a specic tempera-ture, k(T), to the properties of
the reactant(s) and transition state can be derived(66).k(T)
=(c)1mkBThQ
reactantsQieE0/RT(20)In this equation is called the transmission
coefcient and is taken to beequal to unity in simple
transition-state theory calculations, but is greater thanunity when
tunneling is important (see below), c is the inverse of the
referencevolume assumed in calculating the translational partition
function (see below), mis the molecularity of the reaction (ie, m =
1 for unimolecular, 2 for bimolecular,and so on), kB is Boltzmanns
constant (1.380658 1023J molecule1 K1),h is Plancks constant
(6.6260755 1034Js), E0 (commonly referred to as thereaction
barrier) is the energy difference between the transition structure
andthe reactants (in their respective equilibrium geometries), Q is
the molecularpartition function of the transition state, and Qi is
the molecular partition functionof reactant i.Transition-state
theory thus reduces the problemof calculating the potentialenergy
surface for every possible geometric arrangement of nuclei, to the
con-sideration of a very small number of special geometries;
namely, the transitionstructure and the reactant(s). The transition
structure is the minimum energystructure on the dividing surface
between the reactants and products, and mustbe located so as to
make the no re-crossing assumption as valid as possible. Insimple
transition-state theory, the transition structure is located as the
maximumenergy structure, along the minimum energy path connecting
the reactants andproducts. This is generally a good approximation
for reactions having barriersthat are large compared to kBT.
However, for reactions with low or zero barriers,a more accurate
approach is required. To this end, in variational
transition-statetheory, the transition structure is located as the
structure (on the minimum en-ergy path) that yields the lowest
reaction rate. In thermodynamic terms, this maybe thought of as the
maximum in the Gibbs free energy of activation, rather thanthe
maximum internal energy of activation.In order to calculate
reaction rates via transition-state theory, one needsto identify
the equilibrium geometries of the reactants, and also the
transitionstructure, and calculate their energies. This information
is of course accessiblefrom quantum-chemical calculations. The
molecular partition functions for theseVol. 9 COMPUTATIONAL
CHEMISTRY 345species are also required. These serve as a bridge
between the quantum mechan-ical states of a system and its
thermodynamic properties, and are given byQ=
igi exp_ ikBT_(21)The values i are the energy levels of a
system, each having a number ofdegenerate states gi, and are
obtained by solving the Schr odinger equation. Intheory, this
equation should be solved for all active modes but in practice
thecalculations can be greatly simplied by separating the partition
function intothe product of the translational, rotational,
vibrational, and electronic terms, asfollows.Q=QtransQrotQvibrQelec
(22)This is generally a reasonable assumption, provided that the
reaction occurson a single electronic surface. Finally, if we
assume that reacting species are idealgas molecules, analytical
expressions for the partition functions are as
follows:Qtrans=V_2MkBTh2_3/2=RTP_2MkBTh2_3/2(23)Qvib=
iexp_12hikT_
i11exp_hikT_ (24)Qrot, linear= 1r_ T
r_where r= h282IkB(25)Qrot, nonlinear=1/2r_ T3/2(r,x
r,y
r,z)1/2_where r,i= h282IikB(26)Qelec=0 (27)In equations 2327 R
is the universal gas constant (8.314 J mol1 K1);M is the molecular
mass of the species; V is the reference volume, and T and Pthe
corresponding reference temperature and pressure: i are the
vibrational fre-quencies of the molecule; I is the principal moment
of inertia of a linear molecule,while for the nonlinear case Ix,
Iy, and Iz are the principal moments of inertiaabout axes x, y, and
z respectively; r is the symmetry number of the moleculewhich
counts its number of symmetry equivalent forms (67); and 0 is the
elec-tronic spin multiplicity of the molecule (ie, 0= 1 for singlet
species, 2 for doubletspecies, etc). The information required to
evaluate these partition functions is rou-tinely accessible
fromquantum-chemical calculations: the moments of inertia
andsymmetry numbers depend on the geometry of the molecule, while
the vibrationalfrequencies are obtained from the second derivative
of the energy with respect tothe geometry.346 COMPUTATIONAL
CHEMISTRY Vol. 9A number of additional comments need to be made
concerning the use ofequations 2327. Firstly, in the calculation of
the translational partition function(eq. 23), a reference volume
(or equivalently, a temperature and pressure) is as-sumed. This is
needed for the calculation of thermodynamic quantities such
asenthalpy and entropy, but the assumption has no bearing on the
calculated ratecoefcient, as the reference volume is removed from
equation 20 through the pa-rameter c(= 1/V). Secondly, the
vibrational partition function (eq. 24) has beenwritten as the
product of two terms. The rst of these corresponds to the
zero-pointvibrational energy of the molecule, while the latter
corresponds to its additionalvibrational energy at some nonzero
temperature T. The zero-point vibrationalenergy is often included
in the calculated reaction barrier E0. When this is thecase, this
rst term must be removed from equation 24, so as not to count this
en-ergy twice. Thirdly, the external rotational partition function
is calculated usingequation 25 if the molecule is linear, and
equation 26 if it is not.It is also worth noting that there is an
entirely equivalent thermodynamicformulation of transition-state
theory.k(T) =kBTh (c)1meS/ReH/RT(28)A derivation of this
expression, which is obtained by noting the relationshipbetween the
thermodynamic properties of a system (eg enthalpy, H, and
entropy,S) and the partition functions, can be found in textbooks
on statistical thermo-dynamics (1216). The enthalpy of activation
(H) for this expression can bewritten as the sum of the barrier
(Eo), the zero-point vibrational energy (ZPVE),and the temperature
correction (H).H=E0+ZPVE+H (29)The temperature correction (H) and
ZPVE for an individual species canbe calculated from the
vibrational frequencies as follows.ZPVE=R12
ihi/kB (30)H=R
ihi/kBexp(hi/kB/T) 1+52RT +32RT (31)In equation 31, the rst term
is the vibrational contribution to the enthalpy,the second term is
the translational contribution, and the third term is the
rota-tional contribution. The entropy of activation (S) is
calculated from the vibra-tional (Sv), translational (St),
rotational (Sr), and electronic (Se) contributions tothe entropies
of the individual species, in turn expressed as follows.Sv=R
i_ hi/kBTexp(hi/kBT) 1ln(1exp(hi/kBT))_(32)Vol. 9 COMPUTATIONAL
CHEMISTRY 347St=R_ln__2MkBTh2_3/2kBTP_+1+3/2_(33)Sr, linear=R_ln_
1r_ T
r__+1_(34)Sr, nonlinear=R_ln_1/2r_ T3/2(r,x
r,y
r,z)1/2__+3/2_(35)Se=R ln(0) (36)The parameters required to
evaluate these expressions are the same as thoseused in evaluating
the partition functions, as described above.Finally, by evaluating
the derivative of (28) with respect to temperature, itis possible
to derive a relationship between the above thermodynamic
quantitiesand the empirical Arrhenius expression for reaction rate
coefcients (15):k(T) =AeEa/RT(37)The frequency factor (A) in this
expression is related to the entropy of thesystem, as
follows.A=(c)1memkBTh exp_SR_(38)The Arrhenius activation energy is
related to the reaction barrier, as follows.Ea=E0+ZPVE+H+mRT
(39)From these expressions it can be seen that the so-called
temperature-independent parameters of the Arrhenius expression are
in fact functions of tem-perature, which is why the Arrhenius
expression is only valid over relativelysmall temperature ranges.
It should also be clear that the ZPVE-corrected barrier(E0 + ZPVE),
the enthalpy of activation (H), and the Arrhenius activation
en-ergy (Ea) are only equal to each other at 0 K. At nonzero
temperatures, thesequantities are nonequivalent and thus should not
be used interchangeably.Extensions to Transition-State Theory. Many
variants of transition-state theory have been derived, and a
comprehensive review of these recent de-velopments has been
provided by Truhlar and co-workers (68). As already noted,one of
the mainextensions to transition-state theory is variational
transition-statetheory which, in its simplest form, locates the
transition structure as that havingthe maximum Gibbs free energy
(rather than internal energy). Other variationsof transition-state
theory arise through making different assumptions as to
thestatistical distribution of the available energy throughout the
different molecularmodes, and through deriving expressions for the
partition functions for cases otherthan ideal gases. In addition,
two simple extensions to the transition-state theory348
COMPUTATIONAL CHEMISTRY Vol. 9equations are the inclusion of
corrections for quantum-mechanical tunneling, andthe improved
treatment of the low frequency torsional modes. Since these are
ofimportance in treating certain polymerization-related systems,
these are brieydescribed below.Tunneling Corrections. One of the
assumptions inherent in simpletransition-state theory is that
motion along the reaction coordinate can be con-sidered as a
classical translation. In general, this assumption is reasonably
validsince the reacting species, being atoms or molecules, are
relatively large and thustheir wavelengths are relatively small
compared to the barrier width. However,in the case of hydrogen (and
to a lesser extent deuterium) transfer reactions, themolecular mass
of the atom (or ion) being transferred is relatively small, and
thusquantum effects can be very important. Corrections for
quantum-mechanical tun-neling are incorporated into the coefcient
of equation 20, and are known astunneling coefcients.There is an
enormous variety of expressions available for calculating
tunnel-ing coefcients. The most accurate tunneling methods, such as
small curvaturetunneling (69), large curvature tunneling (70), and
microcanonical optimized mul-tidimensional tunneling (71), involve
solving the multidimensional Schr odingerequation describing motion
of the molecules at every position along the reactioncoordinate. To
calculate such tunneling coefcients, specialized software (such
asPOLYRATE (72)) is used, and additional quantum-chemical data
(such as thegeometries, energies, and frequencies along the entire
minimum energy path)are required. As a result, simpler (and hence
less accurate) expressions are of-ten adopted. These are derived by
treating motion along the reaction coordinateas a function of one
variable, the intrinsic reaction coordinate, and hence solv-ing a
one-dimensional Schr odinger equation. When this is done using the
calcu-lated energies along the reaction path, the procedure is
known as zero-curvaturetunneling (73). However, this procedure
still entails the numerical solution of theSchr odinger equation,
and hence an additional simplication is also often made.Instead of
using the calculated energies along this path, some assumed
functionalformfor the potential energy is usedinstead. This is
chosenso that the Schr odingerequation has an analytical solution,
and thus a closed expression for the tunnelingcoefcient can be
derived. The derivation of these simple tunneling coefcients
isdescribed by Bell (74), and the main expressions used in practice
are as follows.The simplest tunneling coefcients are based on the
assumption that thechange in energy along the minimumenergy path
can be described by a truncatedparabola. This functional form
provides a good description of the energies nearthe transition
structure (where tunneling is most signicant), but a very
poordescription elsewhere. The equation for the tunneling coefcient
is given as thefollowing innite series, which is frequently
truncated after the rst few terms. =12usin_12u_ uyu/2_ y2 u y24 u+
y36 u _whereu=hvkT and y =exp_2VukT_(40)Vol. 9 COMPUTATIONAL
CHEMISTRY 349Equation 40 is called a Bell tunneling correction
(74), and in this expressionV is the reaction barrier and is the
imaginary frequency (as obtained fromthe frequency calculation at
the transition structure). By taking the rst term ofequation 40,
expanding it as an innite series, and then truncating at an
earlyorder, the (even simpler) Wigner (75) tunneling expression is
obtained (74). 12usin_12u_ 1+u224+u45760+ 1+u224 (41)A slightly
more realistic description of the change in potential energy
alongthe minimum energy path is provided by the following Eckart
function (76):V(x) = Ay(1+y)2 + By(1+y) where y =ex/(42)To ensure
that the function passes through the reactants, products and
tran-sition structures, the parameters A and B are dened as the
following functionsof the forward (Vf) and reverse (Vr) reaction
barriers (where the reaction is takenin the exothermic
direction).A=(_Vf+_Vr)2and B=VfVr (43)The remaining parameter is
chosen so as to give the most appropriatet to the minimum energy
path. If this t is biased toward the points near thetransition
structure (where tunneling is most important), it can be
calculatedas the following function of the imaginary frequency
(where c is the speed oflight) (39,41):= i2c_18(B2A2)2A3 (44)The
value obtained from this expression is in mass-weighted
coordinates,which enables the reduced mass to be dropped from the
standard (76) Eckartformulae (41), resulting in the following
expression for the permeability of thereaction barrier G(W) as a
function of the energy W:G(W) =1cosh( ) +cosh()cosh( +)
+cosh()where =42h2W, =42h_2(WB), =42h_2A h2162
2(45)350 COMPUTATIONAL CHEMISTRY Vol. 9The Eckart tunneling
correction () is then obtained by numerically inte-grating G(W)
over a Boltzmann distribution of energies, via the formula (74): =
exp(VF/kBT)kBT_ 0G(W) exp(W/kBT) dW (46)Although this expression
requires numerical integration, it does not requiresophisticated
software and can be easily implemented on a spreadsheet.Finally, it
is worth making a fewcomments on the use of the tunneling
proce-dures. Firstly in very general terms, tunneling is important
for reactions involvingthe transfer of a hydrogen or deuterium atom
or ion. The importance of tunnelingcan also be established through
examination of the parameter u in equation 40, avalue of u < 1.5
usually indicating negligible tunneling effects (74). Secondly,
inprinciple, the more accurate multidimensional tunneling coefcient
expressionsshould always be used. However, in practice, the more
convenient one-dimensionalexpressions are often adopted. Of these
expressions, the Eckart tunneling coef-cient is signicantly more
accurate and should be preferred. For example, thesmall curvature
tunneling method gives a tunneling coefcient () of approxi-mately
102at 300 K, for the hydrogen abstraction reaction between NH2
andC2H6 (77). At the same level of theory, the corresponding values
for the Wigner,Bell, and Eckart corrections are approximately 5,
103, and 102respectively, andhence only the Eckart method yields a
tunneling coefcient of the right order ofmagnitude for this
(typical) reaction. The success of the Eckart tunneling methodhas
also been noted by Duncan and co-workers (78), who rationalized it
in termsof a favorable cancelation of errors.Treatment of Low
Frequency Torsional Modes. In the vibrational parti-tion function
(eq. 24), all modes are treated under the harmonic oscillator
approx-imation. That is, it is assumed that the potential eld
associated with their dis-tortion from the equilibrium geometry is
a parabolic well, as in a vibrating spring(see Fig. 10a). This is a
reasonable assumption for bond stretching motions, butnot for
hindered internal rotations (see Fig. 10b). For high frequency
modes ( >200300 cm1), the contribution of these motions to the
overall partition functionis negligible at room temperature (ie
Qvib,i 1) and thus the error incurred intreating these modes as
harmonic oscillators is not signicant. However, for thelowfrequency
torsional modes, these errors can be signicant and a more
rigoroustreatment is often necessary, and this is especially the
case for the reactions ofrelevance to free-radical polymerization
(35,79).The simplest method for treating the hindered internal
rotations is to regardthemas one-dimensional rigid rotors. An
appropriate rotation angle is identied,and then the potential V()
is calculated as a function of this angle (eg from 0 to360 in steps
of 10) via quantum chemistry. In general, it is recommended
thatthese potentials be obtained as relaxed scans; that is, in
calculating the energyat a specic angle, all geometric parameters
other than the rotational angle areoptimized (79). If the
rotational potential can be described as a simple cosinefunction,
the enthalpy and entropy associated with the mode can be
obtaineddirectly from the tables of Pitzer and co-workers (80). In
order to use these tables,Vol. 9 COMPUTATIONAL CHEMISTRY 351Fig.
10. Typical potentials associated with (a) a harmonic oscillator
and (b) a hinderedinternal rotation.one calculates two
dimensionless quantities:x = VkBT and y = inth_83ImkBT (47)In these
equations, V is the barrier to rotation, int is the symmetry
numberassociated with the rotation (which counts the number of
equivalent minima inthe potential energy curve), and Im is the
reduced moment of inertia associatedwith the rotation. This latter
parameter is given by the following formula:Im=Am_1
i =x,y,zAm2mi_Ii_(48)In this equation, Am is the moment of
inertia of the torsional coordinateitself, Ii is the principal
moment of inertia of the whole molecule about axis i, and352
COMPUTATIONAL CHEMISTRY Vol. 9mi is the direction cosine between
the axis of the top and the principal axis ofthe whole molecule.
More information on the calculation of reduced moments ofinertia
can be found in Reference (81). When the rotational potential
cannot betted with a simple cosine function (as in Fig. 10b), the
partition function (andhence the enthalpy and entropy) is obtained
instead by (numerically) solving theone-dimensional Schr odinger
equation. h28Im2
2 +V() = (49)This yields the energy levels of the system (i),
which are then used to eval-uate the partition function via the
following equation.Qint rot= 1int
iexp_ ikBT_(50)Having obtained the partition function (or
equivalently, the enthalpy andentropy) associated with a low
frequency torsional mode, this is used in place ofthe corresponding
harmonic oscillator contribution for that mode.The above treatment
of hindered rotors assumes that a given mode can beapproximated as
a one-dimensional rigid rotor, and studies for small systemshave
shown that this is generally a reasonable assumption in those cases
(82).However, for larger molecules, the various motions become
increasingly coupled,and a (considerably more complex)
multidimensional treatment may be needed inthose cases. When
coupling is signicant, the use of a one-dimensional hinderedrotor
model may actually introduce more error thanthe (fully decoupled)
harmonicoscillator treatment. Hence, in these cases, the
one-dimensional hindered rotormodel should be used
cautiously.SoftwareThere are a large number of software packages
available for performing compu-tational chemistry calculations.
Some of the programs available include ACES II(83), GAMESS (84),
GAUSSIAN (43), MOLPRO (85), and QCHEM (86). Otherprograms, such as
POLYRATE, (72), have been designed to use the output
ofquantum-chemistry programs to calculate reaction rates and
tunneling coef-cients. Whereas computational chemistry software has
traditionally been oper-ated on large supercomputers, versions for
desktop personal computers are in-creasingly becoming available. In
addition, programs for visualizing the outputof computational
chemistry calculations such as Spartan (87), Molden (88), CSChem3D
(89), Molecule (90), Jmol (91), Gauss View (43), and MacMolPlt (92)
arealso available. These programs allow one to visualize the
geometry and electronicstructure of the resulting molecule, and
animate its vibrational frequencies. Manyof these programs also
have built-in computation engines. Computational chem-istry is thus
increasingly becoming accessible to the nonspecialist user,
whichbrings with it its own problems (see also MOLECULAR
MODELING).Vol. 9 COMPUTATIONAL CHEMISTRY 353Accuracy and
Applicability of Theoretical ProceduresBy solving the Schr odinger
equation exactly, quantum chemistry can, in princi-ple, provide
accurate electronic-structure data. However, in practice,
approximatenumerical methods must be adopted, and these can
introduce error to the calcula-tions. As we have already seen, an
enormous number of approximate methods areavailable, and these
range fromthe accurate but computationally expensive to thecheap
but potentially nasty. Furthermore, the performance of a particular
methodvaries considerably depending upon the chemical system and
the property beingcalculated. It is therefore very important that
computational chemistry studiesare accompanied by rigorous
assessments of theoretical procedures. In such cali-bration
studies, prototypical systems are calculated at a range of levels
of theory,and the results are compared both internally (against the
highest level proce-dures) and externally (against reliable
experimental data) in order to identifythose methods which offer
the best compromise between accuracy and expense.In the present
section, the main conclusions from recent assessment stud-ies for
the reactions of importance to free-radical polymerization are
outlined.In presenting such studies, it must be acknowledged that,
with continuing rapidincreases in computer power, some of these
results will soon be outdated. In par-ticular, as computer power
increases, the need to rely upon lower levels of theoryfor large
polymer-related systems will diminish. Instead, the higher levels
of the-ory outlined below will be able to be used routinely.
Nonetheless, with increasingcomputer power, the temptation to apply
existing levels of theory to yet largersystems will no doubt ensure
that the main conclusions of these studies retainsome relevance
into the near future.Radical Addition to C C Bonds. Radical
addition to C C bonds are ofimportance for free-radical
polymerization as this reaction forms the propagationstep, and thus
inuences the reaction rate and molecular weight distribution inboth
conventional and controlled free-radical polymerization, and the
copolymercomposition and sequence distribution in free-radical
copolymerization. Numer-ous studies have examined the applicability
of high level theoretical methods forstudying radical addition to C
C bonds in small radical systems (32,33,37,93,94).The most recent
study (37) included W1 barriers and enthalpies, and geometriesand
frequencies at the CCSD(T)/6-311G(d,p) level of theory, and is the
highest levelstudy to date. The main conclusions from this study,
and (where still relevant) theprevious lower level studies, are
outlined below.Geometry optimizations are generally not very
sensitive to the level of the-ory, with even the low cost
HF/6-31G(d) and B3-LYP/6-31G(d) methods providingreasonable
approximations to the higher level calculations (37). In the latter
case,there is a small error arising from the tendency of B3-LYP to
overestimate theforming bond length in the transition structures,
and this can be reduced us-ing an IRCmax technique (94).
Alternatively, the error in the B3-LYP transitionstructures is also
reduced when the larger 6-311 +G(3df,2p) basis set is used(37). In
addition, the UMP2 method should generally be avoided for these
reac-tions, as they are subject to spin-contamination problems
(32,33,37). Frequencycalculations are also relatively insensitive
to the level of theory, especially whenthe frequencies are scaled
by their appropriate scale factors. (Scale-factors for themost
commonly used levels of theory may be found in Reference (95). In
particular,354 COMPUTATIONAL CHEMISTRY Vol. 9the B3-LYP/6-31G(d)
level of theory provides excellent performance for
frequencyfactors, temperature corrections, and zero-point
vibrational energy calculations,and would be a suitable low cost
method for studying larger systems (37).Barriers and enthalpies are
very sensitive to the level of theory. Where possi-ble, high level
composite procedures should be used for the prediction of
absolutereaction barriers and enthalpies, and of these methods the
RAD variants of G3provide the best approximations to the higher
level Wn methods (when the lat-ter cannot be afforded) (37). It
should also be noted that the (empirically based)spin-correction
term in the CBS-type methods appears to be introducing a
consid-erable error to the predicted reaction barriers for these
reactions and, until thisis revised, these methods should perhaps
be avoided for these reactions (37).When composite methods cannot
be afforded, the use of RMP2/6-311+G(3df,2p) single points provides
reasonable absolute values and excellentrelative values for the
barriers and enthalpies of these reactions (37). In contrast,the
hybrid DFT methods such as B3-LYP and MPW1K show considerable error
inthe reaction enthalpies, even when applied with large basis sets.
However, they doprovide reasonable additionbarriers, owing to the
cancelationof errors inthe earlytransition structures for these
reactions (37). Interestingly, for the closely relatedradical
addition to C C bonds, the situation is reversed and the B3-LYP
methodsperform well and the RMP2 methods perform poorly (37), and
this highlights theimportance of performing assessment studies
before tackling new chemical prob-lems. Finally, it should be
stressed that semiempirical methods do not provide anadequate
description of the barriers and enthalpies in these reactions
(32).For rate coefcients, the importance of treating the low
frequency torsionalmodes in radical addition reactions as hindered
internal rotations has been in-vestigated in a number of assessment
studies (37,79,93). For small systems suchas methyl addition to
ethylene and propylene, the errors are relatively minor(less than a
factor of 2) (37). However, for reactions of substituted radicals
(suchas n-alkyl radicals (93) and the ethyl benzyl radical (79)),
the errors are some-what larger (as much as a factor of 6), as
there are additional low frequencytorsional modes to consider.
Nonetheless, the errors are still relatively small, andthe harmonic
oscillator approximation might be expected to provide
reasonableorder-of-magnitude estimates of rate coefcients.Radical
Addition to C S Bonds. Radical addition to C S bonds, andthe
reverse -scission reaction, forms the key addition and
fragmentation steps ofthe RAFT polymerization process (96). Ab
initio calculations have a role to play inelucidating the effects
of substituents on this process, and in providing an under-standing
of the causes of rate retardation (6,7). A detailed assessment of
theoreti-cal procedures has been recently carried out for this
class of reactions (36), and themain conclusions are similar to
those for addition to C C bonds (37), as outlinedabove. Ingeneral,
lowlevels of theory, suchas B3-LYP/6-31G(d), are suitable for
ge-ometry optimizations and frequency calculations, provided an
IRCmax procedureis used to correct the transition structures.
However, high level composite methodsare required to obtain
reliable absolute barriers and enthalpies, though reason-able
relative quantities can be obtained at the RMP2/6-311 +G(3df,2p)
level. Asin the case of addition to C C bonds, the spin correction
term in the CBS-typemethods appears to require adjustment, and the
RAD variants of G3 should bepreferred when the higher level Wn
calculations are impractical.Vol. 9 COMPUTATIONAL CHEMISTRY
355Hydrogen Abstraction. Hydrogen abstraction reactions are
importantchain-transfer processes in free-radical polymerization.
In particular, hydrogenabstraction by the propagating polymer
radical from transfer agents (such asthiols), monomer, dead
polymer, or itself (ie intramolecular abstraction) can affectthe
molecular weight distribution, the chemical structure of the chain
ends, andthe degree of branching in the polymer. The accuracy of
computational proceduresfor studying hydrogen abstraction reactions
has received considerable attention,and the results of some of the
most recent and extensive studies (59,94,9799) aresummarized
below.Geometry optimizations are relatively insensitive to the
level of theory; how-ever, there are some important exceptions. In
particular, the HF and MP2 meth-ods should be avoided for
spin-contaminated systems (99). Moreover, the B3-LYPmethod does not
describe the transition structures very well for a number of
hy-drogen abstraction reactions (59,97). However, improved
performance is obtainedusing newer hybrid DFT methods such as MPW1K
(59) and KMLYP (97), andthese methods are suitable as low cost
methods, when high level procedures can-not be afforded.Barriers
and enthalpies are more sensitive to the level of the theory,
and,where possible, high level composite procedures should be used.
In particular, theRAD variants of G3 provide an excellent
approximation to the higher level Wnmethods, and would provide an
excellent benchmark level of theory when the lat-ter could not be
afforded (99). As in the case of the addition reactions, the
spincontamination correction term in the CBS-type methods appears
to be introduc-ing a systematic error to the predicted reaction
barriers and enthalpies and, un-til this is revised, this method
should perhaps be avoided for spin-contaminatedreactions (99). When
composite methods cannot be afforded, methods such asRMP2, MPW1K,
or KMLYP have been shown to provide good agreement with thehigh
level values (59,97,99), with a procedure such as
MPW1K/6-311+G(3df,2p)providing the best overall performance. By
contrast, the popular B3-LYP methodperforms particularly poorly for
reaction barriers and enthalpies (59,94,9799),and should thus be
generally avoided for abstraction reactions. Interestingly, ithas
been noted that the errors in B3-LYP increase with the increasing
polarity ofthe reactants (98), which suggests that assessment
studies based entirely on rela-tively nonpolar reactions (such as
CH3+CH4) may lead to the wrong conclusions.As noted in the previous
section, tunneling is signicant in hydrogen abstrac-tion reactions,
and hence accurate quantum-chemical studies of these systemsrequire
the calculation of tunneling coefcients. The accuracy of tunneling
coef-cients is profoundly affected by both the tunneling method and
the level of theoryat which it is applied. A systematic comparison
of the various tunneling methodsfor the hydrogen abstraction
reactions of relevance to free-radical polymerizationdoes not
appear to have been performed. However, in the example provided in
theprevious section, it was seen that the Eckart method was capable
of providingthe tunneling coefcients of the right order of
magnitude (when compared withthe more accurate multidimensional
methods), while the Wigner and Bell meth-ods respectively
underestimated and overestimated the tunneling coefcients byan
order of magnitude. Hence, when multidimensional tunneling methods
arenot convenient, the Eckart tunneling method should be preferred
as the bestlow cost method. An assessment of the effects of level
of theory on the tunneling356 COMPUTATIONAL CHEMISTRY Vol.
9coefcients, as calculated using this Eckart method, has recently
been published(41). It was found that errors in the imaginary
frequency at the HF level (witha range of basis sets) leads to
errors in the calculated tunneling coefcients ofseveral orders of
magnitude (compared to high level CCSD(T)/6-311G(d,p)
calcu-lations). The B3-LYP and MP2 methods performed signicantly
better, showingerrors of a factor of 23. However, even better
performance could be obtainedby correcting the HF values to the
CCSD(T)/6-311G(d,p) level via an IRCmaxprocedure.Applicability of
Chemical ModelsAssuming high levels of theory are used,
quantum-chemical calculations might beexpected to yield very
accurate values of the rate coefcients for the specic chem-ical
system being studied. With current available computing power, this
would inall likelihood consist of a small model reaction in the gas
phase. If, for exam-ple, this information is then to be used to
deduce something about solution-phasepolymerization kinetics, the
effects of the solvent and (in most cases) the effects ofchain
length need to be considered. Unfortunately, the treatment of these
effectsat a high level of theory is not generally feasible with
current available computingpower, and hence the neglect of these
effects (or their treatment at a crude levelof theory) remains a
potential source of error in quantum-chemical calculations.In this
section, these additional sources of error are briey
discussed.Solvent Effects. The presence of solvent molecules may
affect the poly-merization process in a variety of ways (100). For
example, if polar interactionsare signicant in the transition
structure of the reaction, the presence of a highdielectric
constant solvent may stabilize the transition structure and lower
the re-action barrier. Solvents may also affect the reactivity of
the reacting radicals, andeven the mechanism of the addition or
transfer reaction, through some specicinteraction such as hydrogen
bonding or complex formation. In addition, the pref-erential
sorption of the monomer or solvent around the reacting polymer
radicalmay lead to the effective concentrations available for
reaction being different tothose in the bulk solution, resulting in
a difference between the observed and pre-dicted rate coefcients.
Solvent effects such as these result in the experimentallymeasured
rate coefcients for a free-radical polymerization varying according
tothe solvent type.Over and above these system-specic solvent
effects, there is a more generalentropically based difference
between the rate coefcients for gas-phase andsolution-phase
systems. Whereas in the gas-phase an isolated molecule might
beexpected to have translational, rotational, and vibrational
degrees of freedom, inthe solvent phase the translational and
rotational degrees of freedom are effec-tively lost in collisions
with the solvent molecules. In their place, it is necessaryto
consider additional vibrational degrees of freedom involving a
solute-solventsupermolecule (101). Since the vibrational,
translational, and rotational modesmake different quantitative
contributions to the enthalpy and entropy of acti-vation, signicant
differences might be expected between the gas and solutionphases.
For bimolecular reactions this effect can be considerable, because
the maincontribution to the entropy of activation is the six
rotational and translationalVol. 9 COMPUTATIONAL CHEMISTRY
357degrees of freedom in the reactants, which are converted to
internal vibrationsin the transition structure. In contrast, for
unimolecular reactions, the entropi-cally based
gas-phase/solution-phase difference is generally much smaller, as
therotational and translational modes are similar for the
transition structure and re-actant molecule, and thus their
contribution largely cancels fromthe reaction rate.The treatment of
solvent effects varies according to their origin. The inu-ence of
the dielectric constant on polar reactions can be dealt with
routinely usingvarious continuummodels (102), implemented using
standard computational soft-ware, such as GAUSSIAN (43). However,
it should be stressed that these modelsdo not account for the
entropically based gas-phase/solvent-phase difference, nordo they
deal with direct solvent interactions in the transition structure.
Whendirect interactions involving the solvent are important, it is
necessary to includesolvent molecules in the quantum-chemical
calculation. In theory, one should in-clude many hundreds of
solvent molecules but in practice one includes a smallnumber of
molecules, and combines this with a continuum model (102).
However,even with this simplication, the additional solvent
molecules increase the com-putational cost of the calculations, and
it is not currently feasible to apply thesemethods (at any
reasonable level of theory) for polymer-related systems. Evenwhen
additional solvent molecules are included in the ab initio
calculations, vari-ous extensions to transition-state theory are
required in order to model the rates ofsolution-phase reactions
(68,101). Unfortunately, the existing models are compli-cated to
use and require additional parameters which are not readily
accessiblefor polymerization-related systems. The development of
simplied yet accuratemodels for dealing with solvent effects is an
on-going eld of research.While strategies for calculating
solution-phase rate coefcients exist, withthe current available
computing power these methods are not generally feasiblefor
polymer-related systems. Instead, the following practical
guidelines for dealingwith solvent effects are suggested. Firstly,
when the solvent participates directlyin the reaction, the
inclusion of the interacting solvent molecule in the
gas-phasecalculation is essential for gaining a mechanistic
understanding of the reaction.Secondly, when polar interactions are
expected to be important, the use of a con-tinuum model is
recommended, especially if the results are to be used to
interpretthe polymerization process in polar solvents. Thirdly, for
bimolecular reactions,if the a priori prediction of absolute rate
coefcients is required, a considerationof the entropically based
gas-phase/solution-phase difference is necessary. Thisentropic
solvent effect might be estimated by comparing corresponding
exper-imental solution- and gas-phase rate coefcients for that
class of reaction. Forexample, solution-phase experimental values
for radical addition reactions gen-erally exceed the corresponding
gas-phase values by approximately one order ofmagnitude (103). One
might also benchmark the gas-phase calculations by cal-culating the
rate coefcient for a similar reaction, and comparing the
calculatedresult with reliable experimental data. Fourthly,
provided that specic interac-tions are not important, one might
expect that solvent effects should largely cancelfrom relative rate
coefcients, and hence the gas-phase values should generallybe
suitable for studying substituent effects and solving mechanistic
problems. Fi-nally, when specic interactions are important, simple
gas-phase calculations arestill useful, as they can provide
complementary information about the underlyinginuences on the
mechanism in the absence of the solvent.358 COMPUTATIONAL CHEMISTRY
Vol. 9Chain-Length Effects. The other simplication that is
necessary in orderto use high level ab initio calculations on
polymer-related systems is to approxi-mate the propagating polymer
radical (which may be hundreds or thousands ofunits long) as a
short-chain alkyl radical. Provided that the reaction is
chemi-cally controlled, this is not an unreasonable assumption. In
chemical terms, theeffects of substituents decrease dramatically as
they are located at positions thatare increasingly remote from the
reaction center. For example, the terminal andpenultimate units of
a propagating polymer radical are known to affect its re-activity
and selectivity in the propagation reaction (104); however,
substituenteffects beyond the penultimate position are rarely
invoked in copolymerizationmodels. In order to include the most
important substituent effects, it is generallyrecommended that
propagating radicals be represented as -substituted propylradicals
(as a minimum chain length). For some systems this is not
currentlypossible without resorting to a low (and thus inaccurate)
level of theory. In thosecases, the possible inuence of penultimate
unit effects must be taken into accountwhen interpreting the
results of the calculations.The entropic inuence of the chain
length on the reaction rates extendsslightly beyond the penultimate
unit. For example, Deady and co-workers (105)showed experimentally
that there was a chain-length effect on the propagationrate
coefcient of styrene, which converged at the tetramer stage (ie an
octylradical). Heuts and co-workers (3) have explored this
chain-length dependencetheoretically, and suggested that it arises
predominantly in the translational androtational partition
functions. More specically, they suggest that there is a
smalleffect of mass that can be modeled by including an
unrealistically heavy isotopeof hydrogen at the remote chain end.
For example, in the propagation of ethylene,a model such as X
(CH2)n CH2 could be used, and in this model X is set as a hy-drogen
atom that happens to have a molecular mass of 9999 amu! They also
notedthat there is an effect of chain length on the rotational
entropy (and especially thehindered internal rotations), which
required the more subtle modeling strategyof using slightly longer
alkyl chains (ie n > 1). Nonetheless, using their heavyhydrogen
approach, their calculated frequency factors converged to within a
fac-tor of 2 of the long chain limit at even the propyl radical
stage (ie n = 2). Morerecently it was shown that the consideration
of the propagating radical as a sub-stituted hexyl radical (without
a heavy hydrogen at the remote chain end) wasalso sufcient to
reproduce experimental values for the frequency factors of
prop-agation reactions (106). For short-chain branching reactions,
it has been shownthat inclusion of just one methyl group beyond the
reaction center is sufcientfor modeling the long-chain reactions,
provided that the additional methyl groupis substituted with a
heavy (ie 9999 amu) hydrogen atom (5). Thus, in general,it appears
that small model alkyl radicals are capable of providing a
reasonabledescription of polymeric radicals in chemically
controlled reactions.Finally, it is worth noting that the
chain-length effects on the propagationsteps amount to
approximately an order of magnitude difference between the
rstpropagation step and the long chain limit, with the small
radical additions havingfaster rate coefcients. This chain-length
error is of the same magnitude and actsin the opposite direction to
the gas-phase/solvent-phase difference in bimolecu-lar reactions,
and hence substantial cancelation of error might be expected
inthese cases. Indeed an (unpublished) high level G3(MP2)-RAD
calculation of theVol. 9 COMPUTATIONAL CHEMISTRY 359propagation
rate coefcient for methyl acrylate at 298 K produced the value of
2.0104L mol1 s1, whichis inremarkable agreement withthe
corresponding ex-perimental value (also 2.0 104L mol1 s1at ambient
pressure) (107), despitethe fact that both the mediumand
chain-length effects were ignored in the calcula-tion. Hence,
careful efforts to correct for chain-length effects but not solvent
effects(or vice versa) may actually introduce greater errors to
calculated rate coefcients.Applications of Quantum Chemistry in
Free-Radical PolymerizationQuantumchemistry provides a powerful
tool for studying kinetic and mechanisticproblems in free-radical
polymerization. Provided a high level of theory is used,ab initio
calculations can provide direct access to accurate values of the
barriers,enthalpies, and rates of the individual reactions in the
process, and also provideuseful related information (such as
transition structures and radical stabilizationenergies) to help in
understanding the reaction mechanism. In the following, someof the
applications of quantum chemistry are outlined. This is not
intended tobe a review of the main contributions to this eld, nor
is it intended to providea theoretical account of reactivity in
free-radical polymerization (108). Instead,some of the types of
problems that quantum chemistry can tackle are described,with a
view to highlighting the potential of quantum-chemical calculations
as atool for studying free-radical polymerization (see RADICAL
POLYMERIZATION).A Priori Prediction of Absolute Rate Coefcients.
The a priori pre-diction of accurate absolute rate coefcients is
perhaps the most demanding taskin computational chemistry. For
example, high level ab initio calculations (at theG3(MP2)-RAD level
as a minimum) are required for the calculation of accuratereaction
barriers (and enthalpies) in radical addition reactions and, with
currentavailable computing power, these can be performed routinely
on systems of upto 1214 non-hydrogen atoms. This allows for the
most common po