Computational Quantum Chemistry for Free-Radical Polymerization

Vol. 9 COMPUTATIONAL CHEMISTRY 319

COMPUTATIONALQUANTUM CHEMISTRYFOR FREE-RADICALPOLYMERIZATION

Introduction

Chemistry is traditionally thought of as an experimental science, but recent rapidand continuing advances in computer power, together with the development of

Encyclopedia of Polymer Science and Technology. Copyright John Wiley & Sons, Inc. All rights reserved.

320 COMPUTATIONAL CHEMISTRY Vol. 9

efficient algorithms, have made it possible to study the mechanism and kineticsof chemical reactions via computer. In computational quantum chemistry, one cancalculate from first 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 affinities, radical stabilization energies, and singlet–triplet 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 Schrodinger 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 mechanism and 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 Schrodinger 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-anism and 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 (2–5). 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 (8–16).


Basic Principles of Quantum Chemistry

Ab 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 Schrodinger 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 = − h2

8π2

∑i

1mi

(∂2

∂x2i

+ ∂2

∂y2i

+ ∂2

∂z2i

)(3)

V =∑i<

∑j

(eiej

rij

)(4)

In these equations, the kinetic and potential terms are summed over allparticles in the system (nuclei and electrons), and each particle is characterizedby its Cartesian coordinates, its mass (m) and its electric charge (e). In addi-tion, h is Planck’s constant and rij is the distance separating particles i and j.It can thus be seen that the only empirical information required to solve theSchrodinger equation are the masses and charges of the nucleus and the electrons,and the values of some fundamental physical constants. For this reason, quantum-chemical calculations are referred to as “ab initio” (“without assumptions”)procedures.

In applying the Schrodinger equation to problems of chemical interest, twomajor simplifications can be made. Firstly, the Hamiltonian described by equa-tions 2–4 above is nonrelativistic, and it is valid provided that the velocities of theparticles do not approach the speed of light. This is generally reasonable exceptfor the inner shell electrons of heavy atoms, and in these cases corrections forrelativistic effects can be made (18). Secondly, the Hamiltonian is further simpli-fied by neglecting the kinetic energy contribution of the nuclei. This is known asthe Born–Oppenheimer approximation (19), and amounts to assuming that, sincenuclear motion is much slower than electronic motion, the electron distributiondepends only upon the positions of the nuclei and not on their motions. Thus, theSchrodinger equation can be rewritten as an electronic Schrodinger equation, as


follows:

Helec�elec(r, R) = Eeff(R)�elec(r, R) (5)

In this equation, �elec is the electronic wave function (which depends onthe electronic coordinates r, as well as the nuclear coordinates R) and Helec isthe electronic Hamiltonian in which the total kinetic energy is summed over allelectrons only. The Born–Oppenheimer approximation is valid provided that theratio of electron mass to nuclear mass is sufficiently small.

The objective of computational quantum chemistry is to solve this nonrel-ativistic, electronic Schrodinger equation for Eeff(R), that is, the energy corre-sponding to a given arrangement of nuclei. If the Schrodinger equation is solvedfor all possible arrangements of nuclei in a system, one obtains the potential en-ergy surface. (In fact, one yields the ground-state potential energy surface and anynumber of excited-state surfaces; however, for the remainder of this chapter wewill focus largely on the ground states.) This contains the information required forthe quantitative description of chemical structures and processes. For example,by identifying the nuclear coordinates corresponding to local minima in the po-tential energy surface, one can determine the equilibrium geometries of chemicalspecies. By comparing the energies of alternative local minima, one can determinethe relative energies of alternative conformers and isomers, and thus identify thepreferred (ie, global minimum energy) structure. Alternative local minima on apotential energy surface may correspond to the reactant(s) and product(s) in achemical reaction, and by comparing their total energies (including zero-point vi-brational energy) one can calculate the reaction enthalpy at 0 K. The transitionstructure for a chemical reaction can also be identified from the potential energysurface as a first-order saddle point, that is, a stationary point in which the energyis a local maximum in one dimension (corresponding to the reaction coordinate),and a local minimum in all other dimensions. By comparing the total energies atthe transition structure and the reactants, one can calculate the energy barrier forthe chemical reaction. Having identified the equilibrium geometry of a molecule,one can calculate the second derivative of the energy at that point in the potentialenergy surface, and this yields the vibrational frequencies of the molecule (ie, thepeak positions in its IR and Raman spectra) (see VIBRATIONAL SPECTROSCOPY). Thevibrational frequencies can be used to calculate the total zero-point vibrationalenergy, and also the vibrational contribution to the enthalpy and entropy of amolecule at any temperature. Finally, the potential energy surface provides theinformation required to calculate the rates of chemical reactions.

Solving the Schrodinger equation also yields the ground-state wave functionfor the given arrangement of nuclei. Since the square modulus of the wave func-tion is related to the electron probability density, quantum-chemical calculationsthus enable us to determine the distribution of electrons within a molecule, andhence the charge distribution. In fact, assigning charges to specific atoms within amolecule raises a number of philosophical problems, as under quantum mechanicselectrons do not “belong” to any specific nucleus but are distributed throughoutthe molecule in molecular orbitals. To determine the charge on a specific atomwithin a molecule, it is necessary to define a scheme for dividing the molecularvolume into atomic subspaces. A number of such schemes exist, and the main


ones include the Mulliken population analysis (8), the natural bond orbital analy-sis (20), and Atoms in Molecules (AIM) theory (21). In very broad terms, the formertwo schemes assign electrons to a nucleus if they are located in orbitals centeredon that nucleus, but differ in the manner in which those “orbitals” are defined.By contrast, AIM theory uses a spatial definition for the atom, and generally pro-vides the most rigorous and the most physically meaningful charges, though it isalso the most computationally expensive method. A detailed discussion of wavefunction analysis methods is provided in Reference (9).

In summary, by solving the Schrodinger equation, one has direct access toelectronic-structure information and can calculate from first principles the mech-anism, kinetics, and thermodynamics of chemical reactions. In principle, suchcalculations can be extremely accurate, relying only upon the validity of quan-tum mechanics, and values for the fundamental physical constants. However, inpractice, the multi-electron Schrodinger equation has no analytical solution andnumerical approximations must instead be made. These approximations are apotential source of error in the calculations, and it is thus important to under-stand their underlying assumptions. This article is primarily concerned with abinitio molecular orbital theory, which is one of the principal approaches to solvingthe electronic Schrodinger equation. In this section, the main principles of ab initiomolecular orbital theory are detailed. Other approaches to solving the Schrodingerequation include density functional theory and semiempirical methods, and theseare also briefly outlined. Finally, the additional theoretical calculations requiredin order to use the output of quantum-chemical calculations to obtain the ratecoefficients for chemical reactions are described. The method of molecular me-chanics (MM)—which is an empirical procedure that is not based on solving theSchrodinger equation—is described elsewhere (22–24).

Ab Initio Molecular Orbital Theory

In ab initio molecular orbital theory, the wave function is approximated using one-electron functions or “spin orbitals” (χ ). Each spin orbital is a product of a molecu-lar orbital, ψ(x, y, z), which depends on the Cartesian coordinates of the electron,and a spin function, α or β, which corresponds to the spin angular momentum ofthe electron being aligned along the positive or negative z-axes, respectively. Themolecular orbitals (ψ i) are represented mathematically as a linear combinationof a set of N one-electron functions called basis functions (φµ).

ψi =N∑

µ = 1

cµiφµ (6)

In this equation, the coefficients cµi are called the molecular orbital expan-sion coefficients, and are optimized during the computational procedure. In chem-ical terms, one can think of the basis functions as the sets of constituent atomicorbitals, which mix to form the molecular orbitals of the molecule. In order toapproach the exact solution to the Schrodinger equation, an infinite set of ba-sis functions would be required, as this would introduce sufficient mathematical


flexibility to allow for a complete description of the molecular orbitals. Of course,in practice a finite set of basis functions must be chosen, and this introduces apotential source of error to the calculations.

Having formed a set of N linearly independent molecular orbitals, theseorbitals must then be occupied to form the wave function. In ab initio molecu-lar orbital theory the wave function is formed as the determinant of a matrix,which for an n-electron closed-shell system [ie, n is even (If n were odd, and thesystem was a doublet species (having one unpaired electron), we could insteadform (n + 1)/2 orbitals, one of which would be singly occupied. The treatmentof open-shell systems is discussed in more detail below.)] might be written asfollows.

� = (n!)− 1/2

∣∣∣∣∣∣∣∣∣

ψ1(1)α(1) ψ1(1)β(1) ψ2(1)α(1) ψ2(1)β(1) · · · ψn/2(1)α(1) ψn/2(1)β(1)ψ1(2)α(2) ψ1(2)β(2) ψ2(2)α(2) ψ2(2)β(2) · · · ψn/2(2)α(2) ψn/2(2)β(2)

......

......

......

...ψ1(n)α(n) ψ1(n)β(n) ψ2(n)α(n) ψ2(n)β(n) · · · ψn/2(n)α(n) ψn/2(n)β(n)

∣∣∣∣∣∣∣∣∣(7)

This is usually abbreviated as follows.

� = |χ1(1)χ2(2). . .χn(n)| (8)

In this determinant, which is known as the Slater determinant (25), the firstrow corresponds to all possible assignments of electron 1 to all of the spin orbitals,the second to all possible assignments of electron 2, and so on. The factor of (n!)− 1/2

ensures that the total probability of finding an electron anywhere in space is 1.We have already seen that, from our set of N basis functions (where N should

approach infinity), we can form N linearly independent molecular orbitals. How-ever, in an n-electron system, we choose only n/2 of these orbitals (if n is even)to occupy in our Slater determinant. Clearly the most appropriate orbitals tooccupy should be the n/2 lowest energy orbitals, and this is the basis of thewell-known Hartree–Fock theory (26). While Hartree–Fock theory performs ad-equately in many cases, its use of a single determinant wave function can alsofrequently lead to considerable error. The problem is that this approach fails to ac-count for Coulombic electron correlation. That is, it is assumed that each electron“sees” the other electrons as an average electric field and thus no instantaneouselectron–electron interactions are included. To approach the exact solution to theSchrodinger equation, the wave function must instead be represented as a linearcombination of (an infinite number of) Slater determinants. In each of the addi-tional determinants, one or more of the lowest energy orbitals are substituted withthe (previously unoccupied) higher energy orbitals. In chemical terms, one mightthink of this as allowing for contributions to the wave function from the variouspossible excited configurations. This approach is known as configuration interac-tion (CI), and when all possible excited configurations are included the method isknown as Full CI.

Under ab initio molecular orbital theory, the exact solution to the Schrodingerequation could be obtained using the Full CI method in conjunction with an infinitebasis set. Since this is impractical, computational methods must place restrictions


Fig. 1. Pople diagram (8) illustrating the dependence of the accuracy of a computationalmethod on the basis set and the treatment of correlation.

on the number of basis functions included in the calculation, and (in general) sim-plify the method for treating correlation. The combination of method and basis setchosen for a calculation forms the “level of theory,” and by convention is written as“method/basis set” (using the standard abbreviations for the particular methodand basis set chosen). By increasing the size of the basis set and/or improving themethod, one can improve the accuracy of the calculation but also its computationalcost (see Fig. 1). Furthermore, the performance of a given level of theory can varyconsiderably depending upon the chemistry of the system, and the type of prop-erties being calculated. The choice of level of theory is therefore very important,and a qualitative understanding of the approximations made at the various lev-els of theory is thus helpful in choosing appropriate theoretical procedures. Forthe remainder of the present section, the main qualitative features of commonlyused methods and basis sets are described. The accuracy and applicability of thesemethods for studying the reactions of relevance to free-radical polymerization isdiscussed in a following section.

Basis Sets.LCAO Scheme. A basis set is a set of one-electron functions, which are

combined to form the molecular orbitals of the chemical species. This is knownas the Linear Combination of Atomic Orbitals (LCAO) scheme. To approach theexact solution to the Schrodinger equation, an infinite set of basis functions wouldbe required, as this would introduce sufficient mathematical flexibility to allowfor a complete description of the molecular orbitals. In practical calculations, wemust use a finite number of basis functions, and it is thus important to choose basisfunctions that allow for the most likely distribution of electrons within the system.This is achieved using basis functions that are based on the atomic orbitals of theconstituent atoms of the molecule. For example, if a chemical system containedan oxygen atom, the chosen basis set would include functions describing each ofthe 1s, 2s, and three 2p orbitals of an oxygen atom.


The basis functions that are typically used in practice are called Gaussian-type orbitals, and examples of their functional form for s-type and py-type orbitalsare shown below.

gs(α, r) =(

2α

π

)3/4

e− αr2(9)

gy(α, r) =(

128α5

π3

)1/4

y e− αr2(10)

The exponent α determines the size of the orbital, and standard values havebeen determined for the different orbitals on the different atoms. Gaussian-typeorbitals such as those shown in equations 9 and 10 are called primitive Gaussians.Contracted Gaussians are also used, and these consist of linear combinations ofprimitive Gaussians:

φµ =∑

pdµpgp (11)

The coefficients in this expression (dµp) are fixed for the basis set of a givenatom, and should not be confused with the molecular orbital expansion coefficients(Cµi) of equation 6, which are determined for a given system during the ab initiocalculation.

Basis sets thus contain sets of primitive or contracted Gaussians, whichcorrespond to the atomic orbitals of the constituent atoms. Minimal basis setstypically contain exactly the number of functions required to accommodate theelectrons in the system while maintaining the overall spherical symmetry. Forexample, a minimal basis set would contain a 1s function for each hydrogen atom,1s, 2s, and three 2p functions for each carbon atom, and so on. However, in caseswhere the low lying unoccupied orbitals are frequently involved in bonding (eg,the 2p orbitals of Li or Be), popular minimal basis sets (such as STO-3G) includethese additional functions as well.

The problem with minimal basis sets is that the size and shape of the atomicorbitals are fixed, and all that can be varied in the quantum chemical calculationis their contribution to the overall molecular orbitals. However, in reality, the sizeand shape of the atomic orbitals (and especially the valence orbitals) can dependheavily upon the molecular environment. For example, in polar environments, wemight expect a greater degree of asymmetry in the 2p orbitals of oxygen, com-pared with the same orbitals in nonpolar environments. One way of resolving thisproblem would be to create individual basis sets for each atom in each conceiv-able molecular environment, but this would not only be impractical, it would alsodefeat the purpose of performing ab initio calculations. Instead, a practical solu-tion to this problem is to include additional basis functions of varying sizes andshapes. By varying their individual contribution to the overall molecular orbitals,one effectively introduces flexibility to the size and shape of the constituent atomicorbitals. Some of the main extensions to a minimal basis set are outlined in thefollowing.


Fig. 2. Demonstration of how the mixing of two basis functions of different size is equiv-alent to introducing a single basis function with variable size. The basis functions arerepresented schematically as the boundary surface within which there is a 90% probabilityof finding the electron. In the first example, two s-type basis functions are mixed, while inthe second example two p-type functions are mixed.

(1) Double zeta, triple zeta, and quadruple zeta basis sets include additionalsets of each basis function (2, 3, or 4 sets respectively), with each additionalset having a different size. By varying the relative contribution of the al-ternative sizes to the overall molecular orbital, one effectively introduces asingle function with a variable size (see Fig. 2).

(2) Split-valence basis sets are a simplification to the double, triple, and quadru-ple zeta basis sets described above. Since the inner shell orbitals are not usu-ally involved in bonding, and their energies are reasonably independent oftheir molecular environment, it is usually only necessary to include the ex-tra basis functions for the valence orbitals. Basis sets that include differentnumbers of basis functions for the inner shell and valence electrons areknown as split-valence basis sets.

(3) Polarization functions are typically included in basis sets to allow for asym-metry in the electron distribution (see Fig. 3). They also allow for the par-ticipation of the low lying unfilled atomic orbitals in bonding. Polarizationfunctions typically consist of basis functions corresponding to the low ly-ing unfilled atomic orbitals. For example, the basis set for an oxygen atommight typically include a set of d-type orbitals, while those for hydrogenmight include a set of p-type orbitals. Larger basis sets often include ad-ditional polarization functions of higher angular momentum. For example,the 6-311 + G(3df,2pd) basis set includes a set of f -type functions and threesets of d-type functions for each first row atom, as well as two sets of p-typefunctions and a set of d-type functions for each hydrogen atom.

Fig. 3. Demonstration of the effect of polarization functions. In the first example an s-typeand a p-type function are mixed, while in the second case a p-type and a d-type function aremixed. The mixing of higher angular momentum basis functions allows for an asymmetricdistribution of electrons.


(4) Diffuse functions are basis functions that have very large amplitudes, farfrom the atomic nucleus. Sets of diffuse functions are typically added to thebasis set when describing species (such as anions) where the electrons arenot held very tightly to the nucleus.

(5) Effective core potentials (ECPs) are typically used to replace the basis func-tions of the inner shell electrons for atoms beyond the third row. This re-duces the computational cost of the calculations, and is possible because theinner shell electrons on such heavy atoms are relatively unaffected by themolecular environment. Effective core potentials also include correctionsfor relativistic effects, which are significant for the inner shell electrons ofheavy atoms. One of the most commonly used ECPs is called LANL2DZ.

Two commonly used families of extended basis sets are the Pople basis sets(8) and the Dunning basis sets (27). It is worth making a few brief comments ontheir notation.

(1) Pople basis sets have names such as “6-311 + G(3df,2p).” The “6-311” partrefers to the fact that it is a split valence set with one copy of each ba-sis function on the inner shell electrons, and three copies on the valenceelectrons. The “6” refers to the fact that the basis functions on the innershell electrons consist of a contracted Gaussian-type orbital formed from 6primitive Gaussians. The “311” part refers to the fact that one set of basisfunctions on the valence electrons are contracted Gaussians, each formedfrom 3 primitive Gaussians, while the other two sets of basis functions areprimitive Gaussians. The “+” part refers to the fact that one set of s-typeand p-type diffuse functions have been included for each heavy (ie, biggerthan hydrogen) atom. The 6-311 ++ G basis set includes an additional setof s-type diffuse functions on hydrogen atoms as well. The functions in thebracketed part of the expression (3df,2p) are the polarization functions. The“3df” part refers to the fact that 3 sets of d-type functions and one set of f -type functions are included for each heavy atom, and “2p” implies that twosets of p-type functions are included for each hydrogen atom. Finally, thenotations 6-31G∗ and 6-31G∗∗ are also frequently used. The first “∗” is anabbreviation for (d) and indicates that a set of d-type functions are includedfor each non-hydrogen atom, while “∗∗” stands for (d,p) and indicates that inaddition to the d-type functions for the non-hydrogen atoms, a set of p-typefunctions are included for each hydrogen atom.

(2) Dunning basis sets have names such as “cc-pVnZ.” This notation stands for“correlation-consistent polarized valence n-zeta.” For a double zeta basisset, n is replaced by a “D,” for a triple zeta basis set, n is replaced by a “T,”for a quadruple zeta basis set, n is replaced by a “Q,” for a quintuple basisset we use a “5,” and for a sextuple basis set we use a “6.” When diffusefunctions are included, an “aug” prefix is included in the name, as in “aug-cc-pVTZ.” The cc-pVTZ basis set generally has a performance similar to6-311G(2df,p). A special feature of the Dunning basis sets is that they havebeen designed so the series DZ, TZ, QZ, 5Z, 6Z. . . systematically convergeson the infinite basis set limit (27). This feature has been exploited in the


infinite basis set extrapolation procedure of Martin and Parthiban (28) (seebelow).

Plane Waves. While standard ab initio calculations use the LCAO schemealmost exclusively, density functional theory (DFT) calculations (discussed later)sometimes use plane wave basis sets. These are solutions of the Schrodinger equa-tion for a free particle and take the general form,

gPW = exp[i�k�r] (12)

where the vector �k is related to the momentum �p of the wave through �p = ��k. Oneof the advantages of plane wave basis sets is that they can handle calculations withperiodic boundary conditions, and they are thus used widely in solid-state physics.However, a disadvantage is that large basis sets are normally required to achieveacceptable accuracy and, as a result, plane waves are rarely used in calculationsof molecular systems. For a detailed discussion of plane wave basis sets, and theirapplication to Car–Parrinello (29) ab initio direct dynamics techniques, the readeris referred to a review by Blochl and co-workers (30).

Methods.Hartree–Fock (HF). The foundation of ab initio molecular orbital theory

is the Hartree–Fock (HF) method. As we saw above, it is based on a single-determinant wave function (eq. 7) in which the electrons are assigned to the lowestenergy orbitals. In fact the Slater determinant of equation 7 applies to closed-shellsystems, that is, the n electrons occupy the n/2 orbitals in pairs of opposite spin.This method is known more specifically as restricted Hartree–Fock (RHF). Foropen-shell systems (that is, those having one or more unpaired electrons), twoapproaches are possible. In restricted-open-shell Hartree–Fock (ROHF), the de-terminant is formed from a set of molecular orbitals, which are either doubly orsingly occupied, according to the multiplicity of the species. For example, for a rad-ical (doublet) species, the determinant would be formed from (n + 1)/2 orbitals,and one of these would be singly occupied. This ensures that there is exactly oneunpaired spin in the system, and the species is indeed a pure doublet. In unre-stricted Hartree–Fock (UHF), the α and β spin orbitals are defined and optimizedseparately. Thus there would be (n + 1)/2 occupied α spin orbitals, and (n −1)/2occupied β spin orbitals. The principal differences between the RHF, ROHF, andUHF theory are illustrated in Figure 4.

The UHF method offers some advantages over the ROHF method. In particu-lar, the additional freedom in the wave function (with the provision for nonintegernatural orbital occupation numbers) allows it to account in part for nondynamicelectron correlation, and leads to lower energies and a better qualitative descrip-tion of bond dissociation (9). Through its introduction of spin polarization effects,the UHF method can also provide a better (though by no means perfect) qualitativetreatment of hyperfine coupling constants (31). However, a disadvantage of UHFis that the independent optimization of the α and β spin orbitals can result innominally equivalent α and β orbitals having slightly different eigenvalues. Inother words, a doublet species could have effectively more than one unpairedelectron in the system. This physically unrealistic phenomenon is known as spin


Fig. 4. Electron configuration diagrams highlighting the differences between restrictedHartree–Fock theory (RHF), restricted open-shell Hartree–Fock theory (ROHF), and un-restricted Hartree–Fock theory (UHF).

contamination, and can be a particular problem for the transition structures in thepropagation steps of free-radical polymerization. Assessment studies (32,33) forsuch reactions have revealed that spin-contaminated UHF wave functions makepoor starting points for Moller–Plesset (MP) perturbation theory calculations (seebelow). By contrast, MP calculations based on ROHF wave functions (such asROMP2) show improved agreement with higher level values. Interestingly, it hasbeen found that for high level methods such as coupled cluster theory (see below),the choice of the starting wave function (ie, UHF vs ROHF) makes little differenceto the final calculated energies and geometries for radical reactions (34).

Having constructed a wave function, the remaining unknown parameters inthe Schrodinger equation are the molecular orbital expansion coefficients (cµi),as defined in equation 6. Determining the optimum values of these coefficients isthus the principal task of an ab initio calculation. It is beyond the scope of thischapter to outline the mathematical equations and numerical algorithms used toachieve this, but it is worth describing the general approach in qualitative terms.The basis of this procedure is the variational principle. It merely states that forany antisymmetric normalized function of the electronic coordinates (eg, a Slaterdeterminant), the energy of this function is always greater than the expectationvalue of the exact wave function for the ground state. In other words, the exactwave function serves as a lower bound to the energies calculated from our ap-proximate wave function, and the optimal coefficients cµi are merely those thatminimize the energy. This principle leads to the Roothaan–Hall equations (35),which are solved iteratively until the cµi coefficients converge. At convergencethe coefficients are “self-consistent,” and hence the HF theory is also known asself-consistent field (SCF) theory.

Configuration Interaction (CI). The description of the wave function usinga single determinant (as in HF theory) fails to take electron correlation into ac-count. To obtain the exact solution to the Schrodinger equation we instead need toconstruct the wave function as a linear combination of determinants, with each ad-ditional determinant corresponding to one of the various possible excited configu-rations obtained when electrons are promoted to the previously unoccupied higherenergy orbitals (see Fig. 5). The resulting wave function is written as follows.


Fig. 5. Electron configuration diagrams showing the configurations corresponding to theHF wave function, and the various possible excited configurations. For this 4-electron sys-tem, the combination of single, double, triple, and quadruple excitations constitutes fullCI. However, with an infinite basis set, there would be an infinite number of unoccupiedorbitals (instead of the two shown here) to promote the electrons into, and thus an infinitenumber of determinants would be required to obtain the exact solution to the Schrodingerequation.

� = a0�0 +∑s>0

as�s (13)

The �0 wave function is the HF wave function, while the various �s determi-nants correspond to the various excited configurations. The CI method introducesa further set of unknown parameters into the calculation, the coefficients (as).These coefficients 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 infinite basis set (and hence an infinite num-ber of virtual orbitals), or indeed with a finite 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 × 108 configurations. For this


Fig. 6. Electron configuration diagrams illustrating the lack of size consistency in trun-cated CI. In the first 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) infiniteseparation. 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 CID method.

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 infinite) 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 configuration 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 Schrodinger equation. However, these methods are still verycomputationally expensive.

Moller–Plesset (MP) Perturbation Theory. By convention, the correlationenergy is simply the difference between the Hartree–Fock energy and the exactsolution to the Schrodinger equation. Rather than approximate the exact solu-tion to the Schrodinger equation by attempting to build the exact wave functionthrough configuration interaction, an alternative (and considerably less expen-sive approach) is to estimate the correlation energy as a perturbation on the


Hartree–Fock energy. In other words, the exact wave function and energy areexpanded as a perturbation power series in a perturbation parameter λ asfollows.

Ψλ = � (0) + λ� (1) + λ2� (2) + λ3� (3) + · · · (14)

Eλ = E(0) + λE(1) + λ2E(2) + λ3E(3) + · · · (15)

Expressions relating terms of successively higher orders of perturbation areobtained by substituting equations 14 and 15 into the Schrodinger equation, andthen equating terms on either side of the equation. Having obtained these ex-pressions, it simply remains to evaluate the first terms in the series, and this isachieved by taking the �(0) term as the Hartree–Fock wave function. In practice,the MP series must be truncated at some finite order. Truncation at the first order(ie, E(1)) corresponds to the Hartree–Fock 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 finite order depends on the speed of convergence of the series, and thiswill vary considerably depending on how closely the Hartree–Fock energy approx-imates the exact energy. Indeed in some cases, the MP series can actually diverge,and the application of MP methods 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 finite 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 Schrodinger 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 of


Fig. 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 reflects the increasing possibility for cancelation of error. In the 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 final 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 tunnelingcoefficients (39–41). 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 first 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 maximum energystructure 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).


Fig. 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 identified 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 first 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 Set− Low Method/Small Basis Set

≈High Method/Large Basis Set

(16)

Procedures for extrapolating the energies obtained at a specific level of the-ory to the corresponding infinite basis set limit have also been devised. The twomain procedures are the extrapolation routine of Martin and 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.


Building on these strategies, several composite procedures for approximatingCCSD(T) or QCISD(T) energies with a large or infinite 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 affinities) and are generallyfound to be very accurate, typically showing mean absolute deviations fromexperiment of approximately 4 kJ · mol− 1.

(2) In the Wn methods, high level CCSD(T) calculations are extrapolated tothe infinite 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 from experiment of only 2.5 kJ · mol− 1 for the heats of formation of55 stable molecules. For the (more expensive) W2 theory, the correspondingdeviation was less than 1 kJ · mol− 1.

(3) In the CBS procedures, the complete basis extrapolation procedure of Pe-tersson and co-workers is used (42). This calculates the infinite 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 term for 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 big


as 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 definedas uncertainties of 4–8 kJ · mol− 1), 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 configuration starting wave function. At the impracti-cal limit of performing full CI (or summing all terms in the MP series) with aninfinite basis set, these methods will yield the exact solution to the nonrelativis-tic Schrodinger equation. However, when truncated to finite order, the use of asingle reference wave function can sometimes lead to significant 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-figurations, as follows.

� =∑

j

aj�j (17)

In this equation, the individual wave functions are formed from the lowest en-ergy configuration, and various excited configurations of the Slater determinants,and the aj coefficients are optimized variationally. While this method, which isknown as multireference self-consistent field (MCSCF), may seem analogous tothe single-reference CI methods discussed above, there is an important differencebetween them. In MCSCF, the molecular orbital coefficients (the cµi in eq. 6) are op-timized for all of the contributing configurations. In contrast, in single-referencemethods, the molecular orbital coefficients are optimized for the Hartree–Fockwave function, and are then held fixed at their HF values.

The optimization of both the orbital coefficients and the contribution of thevarious configurations to the overall wave function can be very computationallydemanding. As a result, MCSCF methods typically only consider a small numberof configurations, and one of the key problems is choosing which configurationsto include. In complete active space self-consistent field (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 possibleconfigurations that arise from distributing 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 chemical


Fig. 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 radical–radical reactions, a simplified 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 electron–electron 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 but


also more computationally demanding. For more information on multireferencemethods, the reader is referred to an excellent review by Schmidt and Gordon(46).

Semiempirical Methods

Semiempirical 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 Hartree–Fock 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 fitting 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 Theory

Density functional theory (DFT) is a different quantum chemical approach toobtaining electronic-structure information. The basis of DFT is the Hohenberg–Kohn 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 Hohenberg–Kohn 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 simplification 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, the


Hohenberg–Kohn theorem is 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 Schrodinger 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)

In this equation, ET is the kinetic energy term (arising from the motions of theelectrons), EV is the potential term (arising from the nuclear–electron attractionand the nuclear–nuclear repulsion), EJ is the electron–electron repulsion term, EX

is the exchange term (arising from the antisymmetry of the wave function), and EC

is the dynamical correlation energy of the individual electrons. The sum of the ET,EV, and EJ terms 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 functional–correlation functional, using standard abbreviations forthe various functionals.

Before discussing the functionals themselves, it is worth making a few com-ments on unrestricted Kohn–Sham theory (50). The effective potential of the Kohn–Sham equations contains no reference to the spin of 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 field.). Hence, if the exactfunctional were available, there would normally be no need to consider the α andβ 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 flexible if they explicitly depend on the α and β spin densities. In ananalogous manner to UHF, unrestricted Kohn–Sham theory allows the α and β

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 Kohn–Sham theory to reactions with bi-radical transition structures may be found in a paper by Goddard and Orlova (51).On balance, the unrestricted Kohn–Sham 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 specificchemical 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 may


be found in textbooks such as Refs. 10 and 11), but it is worth mentioning theirmain assumptions. Pure DFT methods may be loosely classified 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 (Vosko–Wilk–Nusair, 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 anoversimplification 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 · mol− 1 are 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 the“LYP” 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 obtainedin standard thermochemical tests of these methods are of the order of 25 kJ · mol− 1

(54). However, the GGA methods (as well as the LDA methods) perform poorly forweakly bound systems (such as those in which Van der Waal’s 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 Hartree–Fock 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, isdefined as follows (58).

EXCB3LYP = EX

LDA + c0(EX

HF − EXLDA

)+ cX(EX

B88 − EXLDA

)+ ECVWN + cC

(EC

LYP − ECVWN

)(19)

The coefficients in this expression, c0 = 0.20, cX = 0.72 and cC = 0.81, wereobtained by fitting the results of B3-LYP calculations to a test set of experimentalatomization energies, electron affinities, 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 fitted to


a 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 Data

Quantum and Classical Reaction Dynamics. In the quantum-chemical calculations described above, we solve the electronic Schrodinger equa-tion to determine the energy corresponding to a fixed 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 Born–Oppenheimer 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 specific 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 Schrodinger 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 Born–Oppenheimer 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 Schrodinger 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. Efficient 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 problem of solving the nuclearSchrodinger 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 Schrodinger 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 107 different points on the potential energy surface were required


in order to solve the nuclear Schrodinger equation, and this requirement scalesexponentially with the number of atoms in the system.

Classical reaction dynamics provides a strategy for calculating the rate co-efficients 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 first 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-fly ab-initio dynamics, this force is calculated (using quantum-chemical calculations) at each new position (63). The latter approach is simpler,but less computationally efficient, 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 field,” which is the setof potential functions that are used to govern relative motions of the constituentatoms. Force fields 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 fields for reacting systems, and prominent examples include ReaxFF(64) and MMVB (65). However, such force fields are nonetheless approximate,and only suitable for the types of reactions for which they were designed. Ac-curate force fields 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, in


the space represented by the coordinates and momenta of the reacting particles, itis possible to define 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. From these assumptions,the following simple equation relating the rate coefficient at a specific tempera-ture, k(T), to the properties of the reactant(s) and transition state can be derived(66).

k(T) = κ(c◦)1 − m kBTh

Q‡∏reactants

Qie− E0/RT (20)

In this equation κ is called the transmission coefficient 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 Boltzmann’s constant (1.380658 × 10− 23 J · molecule− 1 · K− 1),h is Planck’s constant (6.6260755 × 10− 34 J·s), 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 problem of 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 Gibb’s 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 these


species are also required. These serve as a bridge between the quantum mechan-ical states of a system and its thermodynamic properties, and are given by

Q =∑

i

gi exp(

− εi

kBT

)(21)

The values εi are the energy levels of a system, each having a number ofdegenerate states gi, and are obtained by solving the Schrodinger equation. Intheory, this equation should be solved for all active modes but in practice thecalculations can be greatly simplified by separating the partition function intothe product of the translational, rotational, vibrational, and electronic terms, asfollows.

Q = Qtrans × Qrot × Qvibr × Qelec (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(

2πMkBTh2

)3/2

= RTP

(2πMkBT

h2

)3/2

(23)

Qvib =∏

i

exp(

−12

hνi

kT

)×∏

i

1

1 − exp(−hνi

kT

) (24)

Qrot, linear = 1σr

(T�r

)where �r = h2

8π2IkB(25)

Qrot, nonlinear = π1/2

σr

(T3/2

(�r,x�r,y�r,z)1/2

)where �r,i = h2

8π2IikB(26)

Qelec = ω0 (27)

In equations 23–27 R is the universal gas constant (8.314 J · mol− 1 · K− 1);M is the molecular mass of the species; V is the reference volume, and T and Pthe corresponding reference temperature and pressure: νi 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; σ r is the symmetry number of the moleculewhich counts its number of symmetry equivalent forms (67); and ω0 is the elec-tronic spin multiplicity of the molecule (ie, ω0 = 1 for singlet species, 2 for doubletspecies, etc). The information required to evaluate these partition functions is rou-tinely accessible from quantum-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.


A number of additional comments need to be made concerning the use ofequations 23–27. 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 ratecoefficient, 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 first 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 first 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) = κkBT

h(c◦)1 − m e S‡/R e− H‡/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 (12–16). 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

∑i

hνi/kB (30)

H = R∑

i

hνi/kB

exp(hνi/kB/T) − 1+ 5

2RT + 3

2RT (31)

In equation 31, the first 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

(hνi/kBT

exp(hνi/kBT) − 1− ln(1 − exp(−hνi/kBT))

)(32)


St = R(

ln((

2πMkBTh2

)3/2 kBTP

)+ 1 + 3/2

)(33)

Sr, linear = R(

ln(

1σr

(T�r

))+ 1)

(34)

Sr, nonlinear = R(

ln(

π1/2

σr

(T3/2

(�r,x�r,y�r,z)1/2

))+ 3/2

)(35)

Se = R ln(ω0) (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 coefficients (15):

k(T) = Ae− Ea/RT (37)

The frequency factor (A) in this expression is related to the entropy of thesystem, as follows.

A = (c◦)1 − mem kBTh

exp(

S‡

R

)(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 main extensions to transition-state theory is variational transition-statetheory which, in its simplest form, locates the transition structure as that havingthe maximum Gibb’s 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 theory


equations 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 brieflydescribed 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 κ coefficient of equation 20, and are known astunneling coefficients.

There is an enormous variety of expressions available for calculating tunnel-ing coefficients. 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 Schrodingerequation describing motion of the molecules at every position along the reactioncoordinate. To calculate such tunneling coefficients, 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 Schrodinger 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 theSchrodinger equation, and hence an additional simplification is also often made.Instead of using the calculated energies along this path, some assumed functionalform for the potential energy is used instead. This is chosen so that the Schrodingerequation has an analytical solution, and thus a closed expression for the tunnelingcoefficient can be derived. The derivation of these simple tunneling coefficients isdescribed by Bell (74), and the main expressions used in practice are as follows.

The simplest tunneling coefficients are based on the assumption that thechange in energy along the minimum energy path can be described by a truncatedparabola. This functional form provides a good description of the energies nearthe transition structure (where tunneling is most significant), but a very poordescription elsewhere. The equation for the tunneling coefficient is given as thefollowing infinite series, which is frequently truncated after the first few terms.

κ =12 u‡

sin( 1

2 u‡) − u‡y− u‡/2π

{y

2π − u‡− y2

4π − u‡+ y3

6π − u‡− · · ·

}

where

u‡ = hv‡kT

and y = exp(

− 2πVu‡kT

) (40)


Equation 40 is called a Bell tunneling correction (74), and in this expressionV is the reaction barrier and ν‡ is the imaginary frequency (as obtained fromthe frequency calculation at the transition structure). By taking the first term ofequation 40, expanding it as an infinite series, and then truncating at an earlyorder, the (even simpler) Wigner (75) tunneling expression is obtained (74).

κ ≈12 u‡

sin( 1

2 u‡) ≈ 1 + u2

‡24

+ u4‡

5760+ · · · ≈ 1 + u2

‡24

(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 defined as the following functionsof the forward (V f) and reverse (Vr) reaction barriers (where the reaction is takenin the exothermic direction).

A = (√

Vf +√

Vr)2 and B = Vf − Vr (43)

The remaining parameter is chosen so as to give the most appropriatefit to the minimum energy path. If this fit is biased toward the points near thetransition structure (where tunneling is most important), it can be calculatedas the following function of the imaginary frequency ν‡ (where c is the speed oflight) (39,41):

= i2πcν‡

√18

(B2 − A2)2

A3(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) = 1 − cosh(α − β) + cosh(δ)cosh(α + β) + cosh(δ)

where

α = 4 π2

h

√2W, β = 4 π2

h

√2(W − B), δ = 4 π2

h

√2A − h2

16π2 2

(45)


The Eckart tunneling correction (κ) is then obtained by numerically inte-grating G(W) over a Boltzmann distribution of energies, via the formula (74):

κ = 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 few comments 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 coefficient expressionsshould always be used. However, in practice, the more convenient one-dimensionalexpressions are often adopted. Of these expressions, the Eckart tunneling coef-ficient is significantly more accurate and should be preferred. For example, thesmall curvature tunneling method gives a tunneling coefficient (κ) of approxi-mately 102 at 300 K, for the hydrogen abstraction reaction between •NH2 andC2H6 (77). At the same level of theory, the corresponding κ values for the Wigner,Bell, and Eckart corrections are approximately 5, 103, and 102 respectively, andhence only the Eckart method yields a tunneling coefficient 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 field 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 (ν >

200–300 cm− 1), 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 significant. However, for thelow frequency torsional modes, these errors can be significant and a more rigoroustreatment is often necessary, and this is especially the case for the reactions ofrelevance to free-radical polymerization (3–5,79).

The simplest method for treating the hindered internal rotations is to regardthem as one-dimensional rigid rotors. An appropriate rotation angle θ is identified,and then the potential V(θ ) 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 specific 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,


Fig. 10. Typical potentials associated with (a) a harmonic oscillator and (b) a hinderedinternal rotation.

one calculates two dimensionless quantities:

x = VkBT

and y = σinth√8π3ImkBT

(47)

In these equations, V is the barrier to rotation, σ 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,z

Amλ2mi

/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, and


λmi 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 befitted 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 Schrodinger equation.

− h2

8πIm

∂2�

∂θ2+ V(θ )� = ε� (49)

This yields the energy levels of the system (εi), which are then used to eval-uate the partition function via the following equation.

Qint rot = 1σint

∑i

exp(

− εi

kBT

)(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 significant, the use of a one-dimensional hinderedrotor model may actually introduce more error than the (fully decoupled) harmonicoscillator treatment. Hence, in these cases, the one-dimensional hindered rotormodel should be used cautiously.

Software

There 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 coeffi-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).


Accuracy and Applicability of Theoretical Procedures

By solving the Schrodinger 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 from the 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 influences 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,


the 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 addition barriers, owing to the cancelation of errors in the 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 coefficients, 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 reasonable“order-of-magnitude” estimates of rate coefficients.

Radical Addition to C S Bonds. Radical addition to C S bonds, andthe reverse β-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. In general, low levels of theory, such as 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.


Hydrogen 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 (59,94,97–99) 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, the“RAD” 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 (59,94,97–99),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 significant in hydrogen abstrac-

tion reactions, and hence accurate quantum-chemical studies of these systemsrequire the calculation of tunneling coefficients. The accuracy of tunneling coeffi-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 coefficients 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 coefficients 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 tunneling


coefficients, 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 coefficients ofseveral orders of magnitude (compared to high level CCSD(T)/6-311G(d,p) calcu-lations). The B3-LYP and MP2 methods performed significantly better, showingerrors of a factor of 2–3. 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 Models

Assuming high levels of theory are used, quantum-chemical calculations might beexpected to yield very accurate values of the rate coefficients for the specific 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 briefly 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 significant 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 specificinteraction 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 coefficients. Solvent effects such as these result in the experimentallymeasured rate coefficients for a free-radical polymerization varying according tothe solvent type.

Over and above these system-specific solvent effects, there is a more general“entropically based” difference between the rate coefficients 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-solvent“supermolecule” (101). Since the vibrational, translational, and rotational modesmake different quantitative contributions to the enthalpy and entropy of acti-vation, significant 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 translational


degrees 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 from the reaction rate.

The treatment of solvent effects varies according to their origin. The influ-ence of the dielectric constant on polar reactions can be dealt with routinely usingvarious continuum models (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 simplification, 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 simplified yet accuratemodels for dealing with solvent effects is an on-going field of research.

While strategies for calculating solution-phase rate coefficients 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 coefficients is required, a considerationof the entropically based gas-phase/solution-phase difference is necessary. This“entropic” solvent effect might be estimated by comparing corresponding exper-imental solution- and gas-phase rate coefficients 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 coefficient for a similar reaction, and comparing the calculatedresult with reliable experimental data. Fourthly, provided that specific interac-tions are not important, one might expect that solvent effects should largely cancelfrom relative rate coefficients, and hence the gas-phase values should generallybe suitable for studying substituent effects and solving mechanistic problems. Fi-nally, when specific interactions are important, simple gas-phase calculations arestill useful, as they can provide complementary information about the underlyinginfluences on the mechanism in the absence of the solvent.


Chain-Length Effects. The other simplification 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 γ -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 influence of penultimate unit effects must be taken into accountwhen interpreting the results of the calculations.

The entropic influence 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 coefficient 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 specifically, 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 sufficient 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 sufficientfor 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 firstpropagation step and the long chain limit, with the small radical additions havingfaster rate coefficients. 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 the


propagation rate coefficient for methyl acrylate at 298 K produced the value of 2.0× 104 L · mol− 1 · s− 1, which is in remarkable agreement with the corresponding ex-perimental value (also 2.0 × 104 L · mol− 1 · s− 1 at ambient pressure) (107), despitethe fact that both the medium and 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 coefficients.

Applications of Quantum Chemistry in Free-Radical Polymerization

Quantum chemistry 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 field, 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 Coefficients. The a priori pre-diction of accurate absolute rate coefficients 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 12–14 non-hydrogen atoms. This allows for the most common polymerizationsubstituents to be included, and often allows for penultimate unit effects to betaken into account. However, it does not allow for the accurate treatment of sol-vent or chain-length effects. Of course, with continuing rapid increases in com-puter power, these problems should soon be overcome. In addition, as was notedabove, for a bimolecular reaction, the chain-length effects and solvent effects actin opposite directions and may in fact largely cancel, leading to surprisingly accu-rate results. Certainly, with current available computing power, the prediction ofrate coefficients to within 1–2 orders of magnitude (or better) is possible for mostchemically controlled polymerization-related reactions, and further increases inaccuracy should be attainable in the near future.

The prediction of absolute rate coefficients via quantum chemistry is partic-ularly useful when direct experimental measurements are not possible. A goodexample of this is in the RAFT polymerization process (96), in which the experi-mentally observable quantities (such as the overall polymerization rate, the over-all molecular weight distribution, and the concentrations of the various species)are a complicated function of the rates of the various individual reactions. In orderto measure the rates of these individual steps, it is necessary to relate their rate co-efficients to the observable quantities via some kinetic-model-based assumptions.Depending upon the assumptions, enormous discrepancies arise in the estimated


rate coefficients. For example, alternative experimental measurements for thefragmentation rate in cumyl dithiobenzoate mediated polymerization of styreneat 60◦C differ by six orders of magnitude (109,110), and this difference arises, atleast in part, in the model-based assumptions of the alternative experimental tech-niques (For a discussion of problems measuring fragmentation rate coefficients,the reader is referred to Reference 111). Recent ab initio calculations of the frag-mentation rate coefficient for a model of this system were able to provide directevidence in support of one measurement (and hence one set of assumptions) overthe other, and in doing so provide an insight into the causes of rate retardation inthe RAFT process (6).

As accurate calculations become routine, there will be many other applica-tions for the a priori prediction of absolute rate coefficients. The calculation ofpropagation rate coefficients (which can be measured experimentally via pulsedlaser polymerization or PLP (112)) is currently used to benchmark theoretical pro-cedures, but accurate calculations may also be helpful as a stand-alone techniquefor toxic and hazardous monomers, and in other cases where PLP is difficult (dueto problems such as the monomer absorbing at the wavelength of the laser or reac-tions such as chain transfer broadening the molecular weight distribution). It hasalso been noted that ab initio calculations may provide the best means of study-ing long- and short-chain branching in free-radical polymerization (5). Quantumchemistry will also be helpful in extracting the rate coefficients of the individualreactions in other complicated reaction schemes, such as free-radical copolymer-ization, and the various types of controlled radical polymerization systems.

Prediction of Relative Rate Coefficients: Discriminating Models andMechanisms. High level ab initio calculations can already predict relative ratecoefficients with remarkable accuracy. This is because accurate relative valuesof quantities such as barriers and enthalpies can generally be obtained at lowerlevels of theory than corresponding absolute values, because of substantial cancel-lation of error. In addition, one might generally expect that chain-length effectsand solvent effects should be reasonably consistent for a series of similar reac-tions, and thus cancel from the comparative values. The prediction of relativerate coefficients is important for modeling and hence optimizing various aspectsof the polymerization process, and some of these applications are outlined below.The prediction of relative rate coefficients is also important for understanding theeffects of substituents on the various individual reactions, and these applicationsare discussed in the following section.

Copolymerization. In free-radical copolymerization (qv), the relative ratesof addition of the various types of radical to the alternative monomers are calledreactivity ratios, and are the key parameters governing the composition and se-quence distribution of the resulting copolymer. Experimentally, these parametersare “measured” for a given system by treating them as adjustable parameters ina least-squares fit of some assumed kinetic model to experimental values of thecomposition, sequence distribution, and/or propagation rate coefficients. However,there are two important problems with this approach (104). Firstly, it is not always(if ever) clear which copolymerization model (eg the terminal model, the implicitor explicit penultimate model, etc) is appropriate for a given system. If a phys-ically incorrect model is chosen, then the estimated parameters will lack theirassumed physical meaning, and will thus be unsuitable for mechanistic studies,


or for predicting other copolymerization properties. Secondly, even if the correctmodel is chosen, there are often more adjustable parameters than are numericallyneeded to fit the data, and this can result in large and highly correlated uncertain-ties in the estimated parameters. The problems in the experimental estimation ofreactivity ratios are discussed in more detail in the article on copolymerization(qv).

Quantum chemistry is able to address these problems, as it allows the re-activity ratios to be calculated directly from the rate coefficients for the vari-ous types of reaction, without making kinetic-model-based assumptions. Further-more, quantum chemistry can assist in determining which kinetic model is mostappropriate for a given system. For example, by measuring the effects of sub-stituents at the penultimate position of the propagating radical, quantum chem-istry can be used to determine whether a terminal or penultimate model is ap-propriate for a given system. In this respect, quantum chemistry has alreadymade an important contribution to this field by providing evidence for penulti-mate unit effects in the barriers (1,106) and frequency (2,106) factors of a varietyof different systems. Such calculations can thus simplify the model discrimina-tion process by ruling out either the terminal- or penultimate-based models for agiven system, and by providing estimates for the main model parameters (ie thereactivity ratios). In addition, quantum chemistry can provide a simple and cost-effective method for screening the reactivity ratios for a wide variety of monomerpairs (including hazardous and yet-to-be synthesized monomers). This may al-low the identification of monomer pairs with suitable reactivity ratios for a givenapplication.

Chain Transfer. The relative rate of abstraction to propagation for a spe-cific propagating radical is known as the chain-transfer constant, and is a keyparameter governing the molecular weight of the resulting polymer. High levelab initio calculations allow chain-transfer constants to be predicted for a givenpropagating radical and transfer agent, and thus provide an effective means ofscreening large numbers of transfer agents. This in turn may assist in the selec-tion of suitable transfer agents for a given polymerization system, and can alsoaid in our understanding of reactivity in the chain-transfer processes. Alreadyab initio calculations have been used to study intramolecular chain-transfer pro-cesses in free-radical polymerization (5,106), in order to provide estimates of theshort-chain branching ratios. Another study investigated the kinetics and ther-modynamics of the hydrogen abstraction by and from the monomer in ethylenepolymerization (113), and demonstrated that abstraction from the monomer wasthe kinetically (but not thermodynamically) preferred process. There is also anenormous body of literature concerning ab initio calculations of hydrogen abstrac-tion reactions in general, and in applications of relevance to other fields such asbiochemistry (114,119).

Controlled Radical Polymerization. In recent years, the field of free-radicalpolymerization has been revolutionalized by the development of techniques forcontrolling the molecular weight and architecture of the resulting polymer, in-cluding nitroxide-mediated polymerization (NMP) (120), atom-transfer polymer-ization (ATRP) (121), and reversible addition fragmentation chain transfer (RAFT)polymerization (96) (see LIVING RADICAL POLYMERIZATION). In order to controlpolymerization, these processes aim to minimize the influence of bimolecular


termination processes via a delicate balance of the relative rates of two or morecompeting reactions. Quantum chemistry can assist in optimizing these processesby providing reliable values for the rate coefficients of the competing reactions,across a wide range of systems. More generally, quantum chemistry can help toprovide a detailed understanding of the kinetics and mechanisms of the individualreactions, which can in turn allow for the design of improved control agents.

Quantum chemistry has already been used to study the RAFT process (6,7).As noted above, high level ab initio molecular orbital calculations have been usedto obtain direct measurements of the rates of addition and fragmentation in modeldithiobenozate-mediated systems, which in turn provided evidence that slow frag-mentation is responsible for rate retardation in these systems (6). More recently,ab initio calculations have revealed a new and unexpected side reaction (β-scissionof the alkoxy group) in certain xanthate-mediated systems, which may provide anexplanation for the experimentally observed inhibition in those systems (7). Theab initio calculations also indicated that fragmentation was considerably fasterin the xanthate-mediated systems (compared with other dithioester systems), be-cause of the stabilizing influence of the alkoxy group on the thiocarbonyl product(7). In addition to these ab initio calculations, semiempirical methods have beenused to survey the transfer enthalpies for a series of RAFT agents in styrene,methyl methacrylate, and butyl acrylate polymerization (122). Despite the lowlevel of theory used, and the limited accuracy of the quantitative predictions atthis level, the calculations were nonetheless shown to have some qualitative valuein determining which transfer agents would show the best molecular weight con-trol in RAFT polymerization.

Understanding Reactivity via the Curve-Crossing Model. We havealready seen that quantum chemistry can be used to calculate the absolute(and hence relative) rate coefficients for the individual reactions in free-radicalpolymerization, and is well suited to compiling systematic surveys of the ratecoefficients for homologous series of reactions. When such calculations arecombined with qualitative theoretical models, quantum chemistry can help toprovide an understanding of trends in reactivity. It is beyond the scope of thischapter to provide an account of the main qualitative models and conceptsin general theoretical chemistry. However, it is worth introducing one suchmodel—the curve-crossing model (123)—as this model has been used extensivelyto provide a qualitative rationalization of the effects of substituents in some of thekey radical reactions that occur in free-radical polymerization, including radicaladdition to alkenes (103) (ie the propagation step), and hydrogen abstraction(115,116,124) (ie, chain-transfer processes). The main qualitative features of themodel and its application to free-radical polymerization are described below; fora more detailed description, the reader is referred to the original references 123,and also the excellent book by Pross (125).

The curve-crossing model (also referred to as the valence-bond state corre-lation model, the configuration mixing model, or the state correlation diagram)was developed by Pross and Shaik (123) as a unifying theoretical framework forexplaining barrier formation in chemical reactions. It is largely based on valencebond (VB) theory (126,127), but also incorporates insights from qualitative molec-ular orbital theory (128). To understand the curve-crossing model, it is helpful tothink of a chemical reaction as being composed of a rearrangement of electrons,


accompanied by a rearrangement of nuclei (ie a geometric rearrangement). We canthen imagine holding the arrangement of electrons constant in its initial configu-ration (which we call the reactant VB configuration), and examining how the en-ergy changes as a function of the geometry. Likewise, we could hold the electronicconfiguration constant in its final form (the product VB configuration), and againexamine the variation in energy as a function of the geometry. If these two curves(energy vs geometry) are plotted, we form a “state correlation diagram.” The over-all energy profile for the reaction, which is also plotted, is formed by the resonanceinteraction between the reactant and product configurations (and any other lowlying configurations). State correlation diagrams allow for a qualitative explana-tion for how the overall energy profile of the reaction arises, and can then be usedto provide a graphical illustration of how variations in the relative energies of thealternative VB configurations affect the barrier height. This in turn allows us torationalize the effects of substituents on reaction barriers, and to predict whensimple qualitative rules (such as the Evans–Polanyi rule (129)) should break down.

This procedure is best illustrated by way of an example, such as the case ofradical addition to alkenes. For this type of reaction, the principal VB configura-tions that may contribute to the ground-state wave function are the four lowestdoublet configurations of the three-electron–three-center system formed by theinitially unpaired electron at the radical carbon (R) and the electron pair of theattacked π bond in the alkene (A) (103).

The first configuration (RA) corresponds to the arrangement of electrons inthe reactants, the second (RA3) to that in the products, and the others (R+A− andR− A+) to possible charge-transfer configurations. The state correlation diagramshowing (qualitatively) how the energies of these configurations vary as a functionof the reaction coordinate is provided in Figure 11 (103). Let us now examine howthis plot could be made for a specific system.

The anchor points on these diagrams are generally accessible from quantum-chemical calculations. For example, the energy difference between the RA config-uration at the reactant geometry, and the RA3 configuration at the product geom-etry, is simply the energy change of the reaction. The energy difference betweenthe RA and RA3 configurations at the reactant geometry is simply the verticalsinglet-triplet gap of the alkene. At the product geometry, the RA − RA3 energydifference is also an excitation energy, but this time between the ground state ofthe doublet product and an excited doublet state (In fact, studies of radical addi-tion to alkenes have ignored the influence of this excitation gap, without detrimentto the predictive value of the results (103). This is probably due to the fact thatthe transition structure is very early in these reactions. However, in studies ofhydrogen abstraction reactions, the RA and RA3 gaps at both the reactant andproduct geometries can be important. These are calculated as the singlet-tripletgap of the closed-shell substrate in each case.) The charge-transfer configurationscan be anchored at the reactant geometry, where they are given as the energy for


Fig. 11. State correlation diagram for radical addition to alkenes showing the variationin energy of the reactant (RA), the product (RA3), and the charge-transfer configurations(R+A− and R− A+) as a function of the reaction coordinate. The dashed line represents theoverall energy profile of the reaction.

complete charge transfer between the isolated reactants. For example, the energydifference between the R+A− and the RA configuration at the reactant geometrywould be given as the energy change of the reaction R + A → R+ + A− . It can beseen that the energy change of this reaction is simply the difference between theionization energy (R → R+ + e−) of the donor species and electron affinity (A− →A + e−) of the acceptor.

Although the anchor points in the diagram are obtained quantitatively, wegenerally interpolate the intervening points on the VB configuration curves qual-itatively, on the basis of spin pairing schemes and VB arguments (123). At thispoint it should be stressed that the overall energy profile for the reaction is ofcourse quantitatively accessible from our quantum-chemical calculations. The ob-jective of the curve-crossing model analysis is not to generate the overall reactionprofile but to understand how it arises—and a qualitative approach to generatingthe VB configuration curves is generally adequate for this purpose. If we considerfirst the product configuration, its energy is lowered during the course of the re-action because of bond formation between the radical and attacked carbon. At thesame time, the relative energy of the reactant configuration increases because theπ bond on the attacked alkene is stretched, and this is not compensated for bybond formation with the attacking radical. The energies of the charge-transfer


configurations are initially very high in energy, but are stabilized by Coloumbattraction as the reactants approach one another.

The overall energy profile for the reaction can be formed from the resonanceinteraction of these contributing configurations. In the early stages of the reac-tion, the reactant configuration is significantly lower in energy than the others anddominates the ground-state wave function. However, in the vicinity of the tran-sition structure, the reactant and product configurations have similar energies,and thus significant mixing is possible. This stabilizes the wave function, with thestrength of the stabilizing interaction increasing with the decreasing energy dif-ference between the alternate configurations. It is this mixing of the reactant andproduct configurations which leads to the avoided crossing, and accounts for bar-rier formation. Beyond the transition structure, the product configuration is lowerin energy and dominates the wave function. The charge-transfer configurationsgenerally lie significantly above the ground-state wave function for most of thereaction. However, in the vicinity of the transition structure, they can sometimesbe sufficiently low in energy to interact. In those cases, the transition structure isfurther stabilized, and (if one of the charge transfer configurations is lower thanthe other) the mixing is reflected in a degree of partial charge transfer betweenthe reactants. Since the charge distribution within the transition structure is ac-cessible from quantum-chemical calculations, this provides a testable predictionfor the model.

Using this state-correlation diagram, in conjunction with simple VB argu-ments, the curve-crossing model can be used to predict the influence of variousenergy parameters on the reaction barrier. For radical addition to alkenes (103),the barrier depends mainly on the reaction exothermicity (which measures theenergy difference between the reactant and product configurations at their opti-mal geometries), the singlet-triplet gap in the alkene (which measures the energydifference between the reactant and product configurations at the reactant ge-ometry), and the relative energies of the possible charge-transfer configurations.The effects of individual variations in these quantities are illustrated graphicallyin Figure 12. It can be seen that the barrier height is lowered by an increase inthe reaction exothermicity, a decrease in the singlet-triplet gap, or a decrease inthe relative energy of one or both of the charge-transfer configurations (providedthat these are sufficiently low in energy to contribute to the ground-state wavefunction).

A strategy for understanding the effects substituents in the barriers of rad-ical reactions, such as addition, is to calculate these key quantities [ie, the re-action exothermicity, the singlet-triplet excitation gap of the closed-shell sub-strate(s), and the energy for charge transfer between the reactants], and lookfor relationships between these quantities and the barrier heights. In this way,one could establish, for example, the extent of polar interactions in a particularclass of reactions. As noted at the beginning of this section, a number of suchstudies have already been performed for the key reactions in free-radical poly-merization. The curve-crossing analysis of radical addition reactions, which isreviewed in detail elsewhere (94,99), indicate that, in the absence of polar in-teractions, the barrier height depends on the reaction exothermicity, in accor-dance with the Evans–Polanyi rule (129). However, for combinations of electron-withdrawing and electron-donating reactants, polar interactions are significant,


Fig. 12. State correlation diagrams showing separately the qualitative effects of (a) in-creasing the reaction exothermicity, (b) decreasing the singlet–triplet gap, and (c) decreas-ing the energy of the charge-transfer configuration. For the sake of clarity the adiabaticminimum energy path showing the avoided crossing, as in Figure 9, is omitted from (a)and (b).

and cause substantial deviation from Evans–Polanyi behavior. More recently, thecurve-crossing model has been used to explain the relative reactivity of the C C,C O, and C S bonds (which is of relevance to RAFT polymerization) (130), andto examine why alkynes are less reactive to addition than alkenes (131). In thesecases the differing singlet-triplet gaps of the alternative substrates are also im-portant in governing their relative reactivities. Curve-crossing studies have alsobeen applied to various types of hydrogen abstraction reactions (115,116,124),and, depending upon the substituents, the singlet-triplet gaps (in this case of boththe reactant and product substrates), exothermicities, and polar interactions haveall been found to be important in governing reactivity in these reactions.

The Future

There are many more potential applications for quantum chemistry in free-radicalpolymerization. As computer power increases, one will be able to calculate ratecoefficients for yet larger systems, and at much greater accuracy. In this way,quantum chemistry has a role to play in modeling the kinetics and mechanism ofpolymerization processes, and predicting the properties of the resulting polymer(such as the molecular weight distribution, the copolymer composition, and thedegree of branching). Quantum chemistry may also help in designing improvedagents or processes for controlling polymerization, and in identifying their mech-anisms. Of course, quantum chemistry may also be used effectively to study othertypes of polymerization processes, such as ring-opening processes. Provided care


is taken to ensure the accuracy of the methods used, quantum-chemical methodsprovide a powerful tool for studying free-radical polymerization, and should beseen as a valuable complement to experimental approaches.

BIBLIOGRAPHY

1. M. L. Coote, T. P. Davis, and L. Radom, Theochem 461/462, 91–96 (1999); Macro-molecules 32, 2935–2940 (1999); Macromolecules 32, 5270–5276 (1999).

2. J. P. A. Heuts, R. G. Gilbert, and I. A. Maxwell, Macromolecules 30, 726 (1997).3. J. P. A. Heuts, R. G. Gilbert, and L. Radom, Macromolecules 28, 8771–8781

(1995).4. D. M. Huang, M. J. Monteiro, and R. G. Gilbert, Macromolecules 31, 5175 (1998).5. J. S.-S. Toh, D. M. Huang, P. A. Lovell, and R. G. Gilbert, Polymer 42, 1915–1920

(2001).6. M. L. Coote and L. Radom, J. Am. Chem. Soc. 125, 1490 (2003).7. M. L. Coote and L. Radom, Macromolecules 37, 590 (2004).8. W. J. Hehre, L. Radom, P. v. R. Schleyer, and J. A. Pople, Ab Initio Molecular Orbital

Theory, John Wiley & Sons, Inc., New York, 1986.9. F. Jensen, Introduction to Computational Chemistry, John Wiley & Sons, Inc., New

York, 1999.10. R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules, Oxford

University Press, New York, 1989.11. W. Koch and M. C. Holthausen, A Chemist’s Guide to Density Functional Theory,

Wiley-VCH, Weinheim, 2000.12. S. W. Benson, Thermochemical Kinetics, John Wiley & Sons, Inc., New York, 1976.13. D. A. McQuarrie, Statistical Mechanics, Harper & Row, New York, 1976.14. R. G. Gilbert and S. C Smith, Theory of Unimolecular and Recombination Reactions,

Blackwell Scientific, Oxford, 1990.15. J. I. Steinfeld, J. S. Francisco, and W. L. Hase, Chemical Kinetics and Dynamics, 2nd

ed., Prentice-Hall, Englewood Cliffs, N.J., 1999.16. P. W. Atkins, Physical Chemistry, 6th ed., W. H. Freeman and Co., San Francisco,

2000.17. E. Schrodinger, Ann. Physik 79, 361 (1926).18. P. Pyykko, Chem. Rev. 88, 563–594 (1988).19. M. Born and J. R. Oppenheimer, Ann. Physik 84, 457 (1927).20. A. E. Reed, L. A. Curtiss, and F. Weinhold, Chem Rev. 88, 899 (1988).21. R. F. W. Bader, Atoms in Molecules: A Quantum Theory, Clarendon Press, Oxford,

1990.22. C. J. Cramer, Essentials of Computational Chemistry: Theories and Models, John

Wiley and Sons, Inc., New York, 2002.23. E. G. Lewars, Computational Chemistry: Introduction to the Theory and Applications

of Molecular and Quantum Mechanics, Kluwer Academic Publishers, Dordrecht, theNetherlands, 2003.

24. A. K. Rappe and C. J. Casewit, Molecular Mechanics Across Chemistry, UniversityScience Books, Herndon, Va., 1997.

25. J. C. Slater, Phys. Rev. 34, 1293 (1929); J. C. Slater, Phys. Rev. 35, 509 (1930).26. D. R. Hartree, Proc. Cambridge Philos. Soc. 26, 376 (1928); V. Fock, Z. Phys. 61, 126

(1930).27. A. K. Wilson, T. van Mourik, and T. H. Dunning Jr., J. Mol. Struct. (Theochem) 388,

339–349 (1996), and references cited therein.


28. J. M. L. Martin and S. Parthiban, in J. Cioslowski, ed., Quantum Mechanical Predic-tion of Thermochemical Data, Kluwer Academic Publishers, Dordrecht, the Nether-lands, 2001, pp. 31–65, and references cited therein.

29. R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985).30. P. E. Blochl, P. Margl, and K. Schwarz, in B. B. Laird, R. B. Ross, and T. Ziegler,

eds., Chemical Applications of Density Functional Theory, American Chemical Society,Washington, D.C., 1996, pp. 54–69.

31. D. M. Chipman, Theor. Chim. Acta 82, 93–115 (1992).32. M. W. Wong and L. Radom, J. Phys. Chem. 99, 8582–8588 (1995).33. M. W. Wong and L. Radom, J. Phys. Chem. 102, 2237–2245 (1998).34. Y. Y. Chuang, E. L. Coitino, and D. G. Truhlar, J. Phys. Chem. A 104, 446 (2000).35. C. C. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951); G. G. Hall, Proc. R. Soc. (London)

A 205, 541 (1951).36. M. L. Coote, G. P. F. Wood, and L. Radom, J. Phys. Chem. A 106, 12124–12138 (2002).37. R. Gomez-Balderas, M. L. Coote, D. J. Henry, and L. Radom, J. Phys. Chem. A 108,

2874–2883 (2004).38. D. K. Malick, G. A. Petersson, and J. A. Montgomery, J. Chem. Phys. 108, 5704–5713

(1998).39. V. D. Knyazev and I. R. Slagle, J. Phys. Chem. 100, 16899–16911 (1996); V. D.

Knyazev, A. Bencsura, S. I Stoliarov, and I. R. Slagle, J. Phys. Chem. 100, 11346–11354 (1996).

40. M. Schwartz, P. Marshall, R. J. Berry, C. J. Ehlers, and G. A. Petersson, J. Phys. Chem.A 102, 10074–10081 (1998).

41. M. L. Coote, M. A. Collins, and L. Radom, Mol. Phys. 101, 1329–1338 (2003).42. J. A. Montgomery Jr., M. J. Frisch, J. W. Ochterski, and G. A. Petersson, J. Chem.

Phys. 112, 6532–6542 (2000), and references cited therein.43. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman,

J. A. Montgomery Jr., T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar,J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson,H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T.Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian,J. B. Cross, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J.Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth,P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C.Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J.V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A.Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham,C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M.W. Wong, C. Gonzalez, and J. A. Pople, Gaussian 03, Revision B.03, Gaussian, Inc.,Pittsburgh Pa., 2003. Available at http://www.gaussian.com/

44. L. A. Curtiss and K. Raghavachari, Theor. Chem. Acc. 108, 61–70 (2002), and refer-ences cited therein.

45. D. J. Henry, M. B. Sullivan, and L. Radom, J. Chem. Phys. 118, 4849–4860 (2003).46. M. W. Schmidt and M. S. Gordon, Annu. Rev. Phys. Chem. 49, 233–266 (1998).47. J. J. P Stewart, J. Compu. Aided Mol. Des. 4, 1–105 (1990).48. P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864 (1964).49. W. Kohn and L. J. Sham, Phys. Rev. A 140, 1133 (1965).50. J. P. Perdew, A. Savin, and K. Burke, Phys. Rev. A 51, 4531–4540 (1995).51. J. D. Goddard and G. Orlova, J. Chem. Phys. 111, 7705–7712 (1999).52. J. C. Slater, Quantum Theory of Molecular Solids, Vol. 4: The Self-Consistent Field for

Molecular Solids, McGraw-Hill, Inc., New York, 1974.53. S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, 1200 (1980).


54. W. Kohn, A. D. Becke, and R. G. Parr, J. Phys. Chem. 100, 12974–12980 (1996).55. A. D. Becke, Phys. Rev. A 38, 3098 (1988).56. C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, 785 (1988); B. Miehlich, A. Savin,

H. Stoll, and H. Preuss, Chem. Phys. Lett. 157, 200 (1989).57. J. P. Perdew, K. Burke, and Y. Wang, Phys. Rev. B 54, 16533 (1996), and references

cited therein.58. A. D. Becke, J. Chem. Phys. 98, 5648 (1993).59. B. J. Lynch, P. L. Fast, M. Harris, and D. G. Truhlar, J. Phys. Chem. A 104, 4811–4815

(2000); B. J. Lynch and D. G. Truhlar, J. Phys. Chem. A 105, 2936–2941 (2001);B. J. Lynch and D. G. Truhlar, J. Phys. Chem. A 106, 842–846 (2002).

60. M. A. Collins, Theor. Chem. Acc. 108, 313–324 (2002).61. D. H. Zhang, M. A. Collins, and S.-Y. Lee, Science 290, 961–963 (2000).62. K. Song and M. A. Collins, Chem. Phys. Lett. 335, 481 (2001).63. A. Sanchez-Galvez, P. Hunt, M. A. Robb, M. Olivucci, T. Vreven, and H. B. Schlegel,

J. Am. Chem. Soc. 122, 2911–2924 (2000); M. Sharma, Y. Wu, and R. Car, J. Chem.Phys. 95, 821–829 (2003).

64. A. C. T. van Duin, S. Dasgupta, F. Lorant, and W. A. Goddard III, J. Phys. Chem. A105, 9396–9409 (2001).

65. M. Garavelli, F. Ruggeri, F. Ogliaro, M. J. Bearpark, F. Bernardi, M. Olivucci, andM. A. Robb, J. Comput. Chem. 24, 1357–1363 (2003).

66. H. Eyring, J. Chem. Phys. 3, 107 (1935).67. For a more detailed definition of this term, see for example: A. J. Karas, R. G. Gilbert,

and M. A. Collins, Chem. Phys. Lett. 193, 181–184 (1992).68. D. G. Truhlar, B. C. Garrett, and S. J. Klippenstein, J. Phys. Chem. 100, 12771 (1996).69. R. T. Skodje, D. G. Truhlar, and B. C. Garrett, J. Phys. Chem. 85, 3019 (1981).70. B. C. Garrett, D. G. Truhlar, A. F. Wagner, and T. H. Dunning Jr., J. Chem. Phys. 78,

4400 (1983).71. Y. P. Liu, D. H. Lu, A. Gonzalez-Lafont, D. G. Truhlar, and B. C. Garrett, J. Am. Chem.

Soc. 115, 7806 (1993).72. J. C. Corchado, Y.-Y. Chuang, P. L. Fast, J. Villa, W.-P. Hu, Y.-P. Liu, G. C. Lynch,

K. A. Nguyen, C. F. Jackels, V. S. Melissas, B. J. Lynch, I. Rossi, E. L. Coitino,A. Fernandez-Ramos, J. Pu, T. V. Albu, R. Steckler, B. C Garrett, A. D. Isaacson, andD. G. Truhlar, POLYRATE 9.1, University of Minnesota, Minneapolis, Minn., 2002.Available at http://comp.chem.umn.edu/polyrate/

73. A. Kuppermann and D. G. Truhlar, J. Am. Chem. Soc. 93, 1840 (1971); B. C. Garrett,D. G. Truhlar, R. S. Grev, and A. W. Magnuson, J. Phys. Chem. 84, 1730 (1980).

74. R. P. Bell, The Tunnel Effect in Chemistry, Chapman and Hall, London, 1980.75. E. Wigner, J. Chem. Phys. 5, 720 (1937).76. C. Eckart, Phys. Rev. 35, 1303 (1930).77. Y.-X. Yu, S.-M. Li, Z.-F. Xu, Z.-S. Li, and C.-C. Sun, Chem. Phys. Lett. 302, 281 (1999).78. W. T. Duncan, R. L. Bell, and T. N. Truong, J. Comput. Chem. 19, 1039 (1998).79. V. Van Speybroeck, D. Van Neck, M. Waroquier, S. Wauters, M. Saeys, and G. B.

Martin, J. Phys. Chem. A 104 10939–10950 (2000).80. K. S. Pitzer and W. D. Gwinn, J. Chem. Phys. 10, 428–440 (1942); J. C. M. Li and

K. S. Pitzer, J. Phys. Chem. 60, 466–474 (1956).81. A. L. L. East and L. Radom, J. Chem. Phys. 106, 6655 (1997).82. V. Van Speybroeck, D. Van Neck, and M. Waroquier, J. Phys. Chem. A 106, 8945–8950

(2002).83. J. F. Stanton, J. Gauss, J. D. Watts, M. Nooijen, N. Oliphant, S. A. Perera, P. G. Szalay,

W. J. Lauderdale, S. A. Kucharski, S. R. Gwaltney, S. Beck, A. Balkova, D. E. Bern-holdt, K. K. Baeck, P. Rozyczko, H. Sekino, C. Hober, and R. J. Bartlett. Integral pack-ages included are VMOL (J. Almlof and P. R. Taylor), VPROPS (P. Taylor), ABACUS


(T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, J. Olsen, and P. R. Taylor); QuantumTheory Project, University of Florida, Available at http://www.qtp.ufl.edu/Aces2/

84. M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H.Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis,and J. A. Montgomery, J. Comput. Chem. 14, 1347–1363 (1993). Available athttp://www.msg.ameslab.gov/GAMESS/GAMESS.html

85. R. D. Amos, A. Bernhardsson, A. Berning, P. Celani, D. L. Cooper, M. J. O. Deegan,A. J. Dobbyn, F. Eckert, C. Hampel, G. Hetzer, P. J. Knowles, T. Korona, R. Lindh, A. W.Lloyd, S. J. McNicholas, F. R. Manby, W. Meyer, M. E. Mura, A. Nicklaβ, P. Palmieri,R. Pitzer, G. Rauhut, M. Schutz, U. Schumann, H. Stoll, A. J. Stone, R. Tarroni,T. Thorsteinsson, and H.-J. Werner, Available at http://www.molpro.net/

86. J. Kong, C. A. White, A. I. Krylov, C. D. Sherrill, R. D. Adamson, T. R. Furlani,M. S. Lee, A. M. Lee, S. R. Gwaltney, T. R. Adams, C. Ochsenfeld, A. T. B. Gilbert, G. S.Kedziora, V. A. Rassolov, D. R. Maurice, N. Nair, Y. Shao, N. A. Besley, P. E. Maslen,J. P. Dombroski, H. Daschel, W. Zhang, P. P. Korambath, J. Baker, E. F. C. Byrd,T. Van Voorhis, M. Oumi, S. Hirata, C.-P. Hsu, N. Ishikawa, J. Florian, A. Warshel,B. G. Johnson, P. M. W. Gill, M. Head-Gordon, and J. A. Pople, J. Comput. Chem. 21,1532–1548 (2000). Available at http://www.q-chem.com/

87. Wavefunction, Inc., Available at http://www.wavefun.com/88. G. Schaftenaar, Available at http://www.cmbi.kun.nl/∼schaft/molden/molden.html89. Cambridgesoft, Available at http://www.cambridgesoft.com/90. N. J. R. van Eikema Hommes, Available at http://www.ccc.uni-erlangen.de/molecule/91. Free Open Source Software, Available at http://jmol.sourceforge.net/92. B. M. Bode and M. S. Gordon, J. Mol. Graphics Mod. 16, 133–138 (1998). Available

at http://www.msg.ameslab.gov/GAMESS/GAMESS.html93. J. P. A. Heuts, R. G. Gilbert, and L. Radom, J. Phys. Chem. 100, 18997–19006 (1996).94. M. Saeys, M.-F. Reyniers, G. B. Marin, V. van Speybroeck, and M. Waroquier, J. Phys.

Chem. A 107, 9147–9159 (2003).95. A. P. Scott and L. Radom, J. Phys. Chem. 100, 16502–16513 (1996).96. J. Chiefari, Y. K. B. Chong, F. Ercole, J. Krstina, J. Jeffery, T. P. T. Le, R. T. A.

Mayadunne, G. F. Meijs, C. L. Moad, G. Moad, E. Rizzardo, and S. H. Thang, Macro-molecules 31, 5559 (1998).

97. J. K. Kang and C. B. Musgrave, J. Chem. Phys. 115, 11040 (2001).98. H. Basch and S. Hoz, J. Phys. Chem. A 101, 4416 (1997).99. M. L. Coote, J. Phys. Chem. A 108, 3865–3872 (2004).

100. M. L. Coote, T. P. Davis, B. Klumperman, and M. J. Monteiro, J. Macromol. Sci., Rev.Macromol. Chem. Phys. C 38, 567 (1998), and references cited therein.

101. M. Ben-Nun and R. D. Levine, Int. Rev. Phys. Chem. 14, 215–270 (1995); G. A. Vothand R. M. Hochstrasser, J. Phys. Chem. 100, 13034–13049 (1996).

102. J. R. Pliego Jr. and J. M. Riveros, J. Phys. Chem. A 105, 7241–7247 (2001).103. H. Fischer and L. Radom, Angew. Chem., Int. Ed. Engl. 40, 1340–1371 (2001).104. M. L. Coote and T. P. Davis, Prog. Polym. Sci. 24, 1217 (1999), and references cited

therein.105. M. Deady, A. W. H. Mau, G. Moad, and T. H. Spurling, Makromol. Chem. 194, 1691

(1993).106. J. Filley, T. McKinnon, D. T. Wu, and G. H. Ko, Macromolecules 35, 3731–3738 (2002).107. M. Buback, C. H. Kurz, and C. Schmaltz, Macromol. Chem. Phys. 199, 1721–1727

(1998).108. J. P. A. Heuts, in K. Matyjaszewski and T. P. Davis, eds., Handbook of Radical Poly-

merization, John Wiley & Sons, Inc., New York, 2002, pp. 1–76.109. C. Barner-Kowollik, J. F. Quinn, D. R. Morsley, and T. P. Davis, J. Polym. Sci., Part A:

Polym. Chem. 39, 1353 (2001).

Vol. 9 CONTROLLED RELEASE FORMULATION, AGRICULTURAL 371

110. Y. Kwak, A. Goto, Y. Tsuji, Y. Murata, K. Komatsu, and T. Fukuda, Macromolecules35, 3026 (2002).

111. C. Barner-Kowollik, M. L. Coote, T. P. Davis, L. Radom, and P. Vana, J. Polym. Sci.,Part A: Polym. Chem. 41, 2828–2832 (2003), and references cited therein.

112. M. L. Coote, M. D. Zammit, and T. P. Davis, Trends Polym. Sci. (Cambridge, U.K.) 4,189 (1996).

113. J. P. A. Heuts, A. Pross, and L. Radom, J. Phys. Chem. 100, 17087–17089 (1996).114. D. L. Reid, D. A. Armstrong, A. Rauk, and C. von Sonntag, Phys. Chem. Chem. Phys.

5, 3994–3999 (2003).115. L. Song, W. Wu, K. Dong, P. C. Hibberty, and S. Shaik, J. Phys. Chem. A 106, 11361–

11370 (2002).116. S. Shaik, W. Wu, K. Dong, L. Song, and P. C. Hiberty, J. Phys. Chem. A 105, 8226

(2001).117. S. L. Boyd and R. J. Boyd, J. Phys. Chem. A 105, 7096 (2001).118. R. Sumathi, H. H. Carstensen, and W. H. Green Jr., J. Phys. Chem. A 105, 8969–8984

(2001).119. R. Sumathi, H. H. Carstensen, and W. H. Green Jr., J. Phys. Chem. A 105, 6910–6925

(2001).120. C. J. Hawker, A. W. Bosman, and E. Harth, Chem. Rev. 101, 3661–88 (2001).121. J. S. Wang and K. Matyjaszewski, J. Am. Chem. Soc. 117, 5614–5615 (1995).122. S. C. Farmer and T. E. Patten, J. Polym. Sci., Part A: Polym. Chem. 40, 555 (2002).123. A. Pross and S. S. Shaik, Acc. Chem. Res. 16, 363 (1983); A. Pross, Adv. Phys. Org.

Chem. 21, 99–196 (1985); S. S. Shaik, Prog. Phys. Org. Chem. 15, 197–337 (1985).124. A. Pross, H. Yamataka, and S. Nagase, J. Phys. Org. Chem. 4, 135 (1991).125. A. Pross, Theoretical and Physical Principles of Organic Reactivity, John Wiley &

Sons, Inc., New York, 1995.126. W. Heitler and F. London, Z. Phys. 44, 455 (1927).127. L. Pauling, The Nature of the Chemical Bond, Cornell University Press, Ithaca, N.Y.,

1939.128. R. B. Woodward and R. Hoffman, Angew Chem. 81, 797 (1969); R. B. Woodward and

R. Hoffman, Angew Chem. Int. Ed. Engl. 8, 781 (1969).129. M. G. Evans, Disc. Faraday Soc. 2, 271–279 (1947); M. G. Evans, J. Gergely, and E. C.

Seaman,J. Polym. Sci. 3, 866–879 (1948).

130. D. J. Henry, M. L. Coote, R. Gomez-Balderas, and L. Radom, J. Am. Chem. Soc. 126,1732–1740 (2004).

131. R. Gomez-Balderas, M. L. Coote, D. J. Henry, H. Fischer, and L. Radom, J. Phys.Chem. A 107, 6082–6090 (2003).

MICHELLE L. COOTE

Australian National University

CONDUCTIVE POLYMER COMPOSITES. See Volume 5.

CONFORMATION AND CONFIGURATION. See Volume 2.

372 CONTROLLED RELEASE FORMULATION, AGRICULTURAL Vol. 9

CONTACT LENSES. See HYDROGELS.

CONTROLLED MICROSTRUCTURE. See SINGLE SITE CATALYSTS.

CONTROLLED RADICAL POLYMERIZATION.See LIVING RADICAL POLYMERIZATION.

Documents

Computational Quantum Chemistry for Free-Radical Polymerization