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