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COMPUTATIONAL QUANTUM CHEMISTRY FOR FREE-RADICAL POLYMERIZATION Introduction Chemistry is traditionally thought of as an experimental science, but recent rapid and continuing advances in computer power, together with the development of Encyclopedia of Polymer Science and Technology. Copyright John Wiley & Sons, Inc. All rights reserved.

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efficient algorithms, have made it possible to study the mechanism and kinetics of chemical reactions via computer. In computational quantum chemistry, one can calculate from first principles the barriers, enthalpies, and rates of a given chemical reaction, together with the geometries of the reactants, products, and transition structures. It also provides access to useful related quantities such as the ionization energies, electron affinities, radical stabilization energies, and singlet– triplet gaps of the reactants, and the distribution of electrons within the molecule or transition structure. Quantum chemistry can provide a “window” on the reaction mechanism, and assumes only the nonrelativistic Schr¨odinger equation and values for the fundamental physical constants. Quantum chemistry is particularly useful for studying complex processes such as free-radical polymerization (see RADICAL POLYMERIZATION). In free-radical polymerization, a variety of competing reactions occur and the observable quantities that are accessible by experiment (such as the overall reaction rate, the overall molecular weight distribution of the polymer, and the overall monomer, polymer, and radical concentrations) are a complicated function of the rates of these individual steps. In order to infer the rates of individual reactions from such measurable quantities, one has to assume both a kinetic mechanism and often some additional empirical 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 access to the rates and thermochemistry of the individual steps in the process, without recourse to such model-based assumptions. Of course, quantum chemistry is not without limitations. Since the multielectron Schr¨odinger equation has no analytical solution, numerical approximations must instead be made. In principle, these approximations can be extremely accurate, but in practice the most accurate methods require inordinate amounts of computing power. Furthermore, the amount of computer power required scales exponentially with the size of the system. The challenge for quantum chemists is thus to design small model reactions that are able to capture the main chemical features of the polymerization systems. It is also necessary to perform careful assessment studies, in order to identify suitable procedures that offer a reasonable compromise between accuracy and computational expense. Nonetheless, with recent advances in computational power, and the development of improved algorithms, accurate studies using reasonable chemical models of free-radical polymerization are now feasible. Quantum chemistry thus provides an invaluable tool for studying the mechanism and kinetics of free-radical polymerization, and should be seen as an important complement to experimental procedures. Already quantum chemical studies have made major contributions to our understanding of free-radical copolymerization kinetics, where they have provided direct evidence for the importance of penultimate unit effects (1,2). They have also helped in our understanding of substituent and chain-length effects on the frequency factors of propagation and transfer reactions (2–5). More recently, quantum chemical calculations have been used to provide an insight into the kinetics of the reversible addition fragmentation chain transfer (RAFT) polymerization process (6,7). For a more detailed introduction to quantum chemistry, the interested reader is referred to several excellent textbooks (8–16).

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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 atoms can be obtained by solving the Schr¨odinger equation 1 (17). ˆ = E H

(1)

This is an eigenvalue problem for which multiple solutions or “states” are possible, each state having its own wave function and associated energy. The lowest energy solution is known as the “ground state,” while the other higher energy solutions are referred to as “excited states.” The wave function is an eigenfunction that depends upon the spatial coordinates of all the particles and also the spin coordinates. Its physical meaning is best interpreted by noting that its square ˆ in modulus is a measure of the electron probability distribution. The term (H) equation 1 is called the Hamiltonian operator and corresponds to the total kinetic ˆ and potential energy (V) ˆ of the system. (T) ˆ = Tˆ + Vˆ H

(2)

 2  2 2 2 1 ∂ ∂ h ∂ + 2+ 2 Tˆ = − 2 8π i mi ∂x2i ∂yi ∂zi

(3)

Vˆ =

   ei ej i
0

The  0 wave function is the HF wave function, while the various  s determinants correspond to the various excited configurations. The CI method introduces a further set of unknown parameters into the calculation, the coefficients (as ). These coefficients are optimized as part of the ab initio calculation in order to minimize the energy, in line with the variational principle. CI methods can be based on an RHF wave function (RCI), a UHF wave function (UCI), or an ROHF wave function (URCI). Full CI is impractical with an infinite basis set (and hence an infinite number of virtual orbitals), or indeed with a finite basis set and a reasonably small number of electrons. For example, even for water with the small 6-31G(d) basis set, the full CI treatment would involve nearly 5 × 108 configurations. For this

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Fig. 6. Electron configuration diagrams illustrating the lack of size consistency in truncated CI. In the first case, A and B are treated separately by CID, and the treatment thus considers the double excitations of electrons in each molecule. In the second case, A and B are calculated as a supermolecule having the A and B fragments at (effectively) infinite separation. Now the simultaneous excitation of two electrons from each of the A-type and B-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 practice. These methods consider a limited number of excited determinants, such as all possible single excitations (CIS) or all possible single and double excitations (CISD). Restricting the CI procedure to single, double, and possibly triple excitations is usually a reasonable approximation, since excitations involving one, two, or three electrons have a considerably higher probability of occurring, and thus contributing to the wave function, compared with excitations of several electrons simultaneously. 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 different from that incurred in calculating a single species, which contains A and B separated by a large (effectively infinite) distance. This can be seen quite clearly in the example shown in Figure 6. The lack of size consistency can be a major problem as it introduces an additional error to calculations of barriers and enthalpies in nonunimolecular reactions. This problem is addressed by including additional terms in the wave function, and the methods based on this approach include quadratic configuration interaction (QCI) and coupled cluster theory (CC). These methods are typically applied with single and double excitations (QCISD or CCSD), and the triple excitations are often included perturbatively, leading to methods such as QCISD(T) and CCSD(T). When applied with an appropriately large basis set, these methods usually provide excellent approximations to the exact solution to the Schr¨odinger equation. However, these methods are still very computationally expensive. Moller–Plesset ¨ (MP) Perturbation Theory. By convention, the correlation energy is simply the difference between the Hartree–Fock energy and the exact solution to the Schr¨odinger equation. Rather than approximate the exact solution to the Schr¨odinger equation by attempting to build the exact wave function through configuration interaction, an alternative (and considerably less expensive approach) is to estimate the correlation energy as a perturbation on the

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Hartree–Fock energy. In other words, the exact wave function and energy are expanded as a perturbation power series in a perturbation parameter λ as follows. Ψλ =  (0) + λ (1) + λ2  (2) + λ3  (3) + · · ·

(14)

Eλ = E(0) + λE(1) + λ2 E(2) + λ3 E(3) + · · ·

(15)

Expressions relating terms of successively higher orders of perturbation are obtained by substituting equations 14 and 15 into the Schr¨odinger equation, and then equating terms on either side of the equation. Having obtained these expressions, it simply remains to evaluate the first terms in the series, and this is achieved 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 order is 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 as RMP, UMP, or ROMP respectively. When truncated at the second, third, or possibly fourth orders, the MP methods offer a very cost-effective method for estimating the correlation energy. They are also size-consistent methods. However, the validity of truncating the series at some finite order depends on the speed of convergence of the series, and this will vary considerably depending on how closely the Hartree–Fock energy approximates 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 decrease the errors in the calculation. As noted above, a relevant example of this problem occurs in the transition structures for radical addition to alkenes for which UMP2 calculations (based on the spin-contaminated UHF wave function) are frequently subject to large errors (32,33). Furthermore, when truncated at some finite order, the MP methods are not variational, and may thus overestimate the correction to the energy. Hence, although MP procedures frequently provide excellent costeffective performance, they must be applied with caution. Composite Procedures. The use of CCSD(T) or QCISD(T) methods with a suitably large basis set generally provides excellent approximations to the exact solution of the Schr¨odinger equation. However, such methods are computationally expensive, and in practical calculations smaller basis sets and/or lower cost methods must be adopted. A major advance in recent years has been the development of high level composite procedures, which approximate high level calculations through a series of lower level calculations. Some of the main strategies that are used are described in the following. Firstly, it has long been realized that geometry optimizations and frequency calculations are generally less sensitive to the level of theory than are energy calculations. For example, as will be discussed in a following section, detailed assessment studies (36,37) have shown that even HF/6-31G(d) can provide reasonable approximations to the considerably more expensive CCSD(T)/6-311 + G(d,p) level of theory, for the geometries and frequencies of the species in radical addition to multiple bonds (such as C C, C C, and C S). By contrast, very high levels of

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Fig. 7. Illustration of the relative performance of the high and low levels of theory for geometry optimizations and energy calculations. The low level of theory shows a very large error for the absolute energy of structure, a smaller error for the Y X bond dissociation energy (ie, the well depth), and a very small error for the optimum geometry of the Y X bond. This reflects the increasing possibility for cancelation of error. In the bond dissociation energy, errors in the absolute energies of the isolated Y• and X• species are canceled to some extent by errors in the Y X energies. In the geometry optimizations, further cancelation is possible because the position of the minimum energy structure depends on the relative energies of Y X compounds having very similar Y X bond lengths.

theory are required to describe the barriers and enthalpies of these reactions. The improved performance of low levels of theory in geometry optimizations and frequency calculations can be understood in terms of the increased opportunity for the cancelation of error, as such quantities depend only upon the relative energies of very similar structures (see Fig. 7). In contrast, reaction barriers and enthalpies depend upon the relative energies of the reactants and transition structures or products, and these can have quite different structures, with different types of chemical bonds. It is thus possible to optimize the geometry of a compound at a relatively low level of theory, and then improve the accuracy of its energy using a single higher level calculation (called a “single point”). Since geometry optimizations and frequency calculations are more computationally intensive than single-point energy calculations, this approach leads to an enormous saving in computational cost. By convention, the final composite level of theory is written as “energy method/energy basis set//geometry method/geometry basis set.” Secondly, an extension to the above strategy is known as the IRCmax (intrinsic reaction coordinate) procedure. It was developed (38) for improving the geometries of transition structures, though techniques based on the same principle have also been used to calculate improved imaginary frequencies and tunneling coefficients (39–41). While low levels of theory are generally suitable for optimizing the geometries of stable species, the geometries of transition structures are sometimes subject to greater error at these low levels of theory. To address this problem, the minimum energy path (MEP) for a reaction is first calculated at a low level of theory, and then improved via single-point energy calculations at a higher level of theory. Now, the transition structure is simply the maximum energy structure along the MEP for the reaction. By identifying the transition structure from the high level MEP (rather than the original low level MEP), one effectively optimizes the reaction coordinate at the high level of theory (see Fig. 8).

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Fig. 8. Illustration of the IRCmax procedure. The minimum energy path (MEP, also known 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 transition structure is then identified as the maximum in the high level MEP. This effectively optimizes the reaction coordinate (often the most sensitive part of the geometry optimization) at a high level of theory.

Thirdly, one can improve the single-point energy calculations themselves using additivity and/or extrapolation procedures. In the former case, the energy is first calculated with a high level method (such as CCSD(T)) and a small basis set. The effect of increasing to a large basis set is then evaluated at a lower level of theory (such as MP2). The resulting basis set correction is then added to the high level result, thereby approximating the high level method with a large basis set. The calculation may be summarized as follows. High Method/Small Basis Set +Low Method/Large Basis Set − Low Method/Small Basis Set

(16)

≈High Method/Large Basis Set Procedures for extrapolating the energies obtained at a specific level of theory to the corresponding infinite basis set limit have also been devised. The two main procedures are the extrapolation routine of Martin and Parthiban (18), which takes advantage of the systematic convergence properties of the Dunning DZ, TZ, QZ, 5Z, . . . basis sets, and the procedure of Petersson and co-workers (42), which is based on the asymptotic convergence of MP2 pair energies. For the mathematical details of these extrapolation routines, the reader is referred to the original references. The Martin extrapolation procedure is easily implemented on a spreadsheet, while the Petersson extrapolation procedure has been coded into the GAUSSIAN (43) computational chemistry software package.

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Building on these strategies, several composite procedures for approximating CCSD(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 performed with a relatively small basis set, such as 6-31G(d), and these are then corrected to a large triple zeta basis set via additivity corrections, carried out at the MP2 and/or MP3 or MP4 levels of theory (44). There are many variants of the G3 methods, depending upon the level of theory prescribed for the geometry and frequency calculations, the methods used for the basis set correction, and depending on whether CCSD(T) or QCISD(T) is used at 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 for the study of radical reactions. G3 methods include an empirical correction term, which has been estimated against a large test set of experimental data, and spin-orbit corrections (for atoms). The G3 methods have been extensively assessed against test sets of experimental data (including heats of formation, ionization energies, and electron affinities) and are generally found to be very accurate, typically showing mean absolute deviations from experiment of approximately 4 kJ · mol − 1 . (2) In the Wn methods, high level CCSD(T) calculations are extrapolated to the infinite basis set limit using the extrapolation routine of Martin and Parthiban (28). Additional corrections are included for scalar relativistic effects, core-correlation, and spin-orbit coupling in atoms. No additional empirical corrections are included in this method. The Wn methods are very high level procedures, and have been demonstrated to display chemical accuracy. For example, the W1 procedure was found to have a mean absolute deviation from experiment of only 2.5 kJ · mol − 1 for the heats of formation of 55 stable molecules. For the (more expensive) W2 theory, the corresponding deviation was less than 1 kJ · mol − 1 . (3) In the CBS procedures, the complete basis extrapolation procedure of Petersson and co-workers is used (42). This calculates the infinite basis set limit 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 CBS procedures also incorporate an empirical correction, and an additional (empirically determined) correction for spin contamination. The accuracy of this latter term for the transition structures of radical addition reactions has recently been questioned (36,37). Nonetheless, the CBS procedures also show similar (excellent) performance to the G3 methods, when assessed against the same experimental data for stable molecules (42). In summary, using composite procedures, high level calculations can now be performed at a reasonable computational cost. With continuing rapid increases in computer power, details on the computational speeds of the various methods would 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

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as CH3 SC• (CH2 Ph)SCH3 , while the state-of-the-art Wn methods are restricted to smaller molecules, such as CH3 CH2 CH(CH3 )• . However, in the near future one can look forward to applying these methods to yet larger systems. In general, the composite procedures described above offer “chemical accuracy” (usually defined as uncertainties of 4–8 kJ · mol − 1 ), with the best methods offering accuracy in the kJ range. However, careful assessment studies are nonetheless recommended when applying methods to new chemical systems. A brief discussion of the performance of computational methods for the reactions of relevance to free-radical polymerization is provided in a following section. Multireference Methods. The post-SCF methods discussed above are all based on a HF or single configuration starting wave function. At the impractical limit of performing full CI (or summing all terms in the MP series) with an infinite basis set, these methods will yield the exact solution to the nonrelativistic Schr¨odinger equation. However, when truncated to finite order, the use of a single reference wave function can sometimes lead to significant errors. This is particularly the case in the calculation of diradical species (such as the transition structures for the termination reactions in free-radical polymerization), excited states, and unsaturated transition metals. In such situations, the starting wave function itself should be represented as a linear combination of two or more configurations, as follows. =



aj j

(17)

j

In this equation, the individual wave functions are formed from the lowest energy configuration, and various excited configurations of the Slater determinants, and the aj coefficients are optimized variationally. While this method, which is known as multireference self-consistent field (MCSCF), may seem analogous to the single-reference CI methods discussed above, there is an important difference between them. In MCSCF, the molecular orbital coefficients (the cµi in eq. 6) are optimized for all of the contributing configurations. In contrast, in single-reference methods, the molecular orbital coefficients are optimized for the Hartree–Fock wave function, and are then held fixed at their HF values. The optimization of both the orbital coefficients and the contribution of the various configurations to the overall wave function can be very computationally demanding. As a result, MCSCF methods typically only consider a small number of configurations, and one of the key problems is choosing which configurations to include. In complete active space self-consistent field (CASSCF), the molecular orbitals are divided into three groups: the inactive space, the active space, and the virtual space (see Fig. 9). The wave function is then formed from all possible configurations that arise from distributing the electrons among the active orbitals (ie, full CI is performed within the active space). It then remains to decide which occupied and virtual orbitals should be included in the active space. Where possible, it is advisable to include all valence orbitals in the active space, together with an equivalent number of virtual orbitals. However, as with any full CI calculation, the computational cost rapidly increases with the number of electrons and orbitals included, and CASSCF calculations are currently limited to active spaces of approximately 16 electrons in 16 molecular orbitals. Thus, for large chemical

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Fig. 9. The partitioning of orbitals between the inactive, active, and virtual spaces in a CASSCF calculation.

systems, full valence active spaces are not as yet possible, and instead a restricted number of orbitals must be chosen. In non-full valence CASSCF, the active space is typically selected on the basis of “chemical intuition,” and might include orbitals that are directly involved in the chemical reaction, or are interacting strongly with the reacting orbitals. For example, in the case of radical–radical reactions, a simplified multireference approach would be a CAS(2,2) method, in which the active space would consist of the two singly occupied molecular orbitals. However, such restricted methods must be used cautiously as they recover correlation in the active space, but not in the inactive space or between the active and inactive spaces. As a result, such procedures can sometimes introduce a bias, which, for example, might lead to an overestimation of the biradical character in systems with nearly degenerate singlet and triplet states (9). Multireference methods primarily account for nondynamic electron correlation, which arises from long-range interactions involving nearly degenerate states. It is still necessary to correct for dynamic correlation, which arises from short-range electron–electron interactions, and which is primarily addressed in the 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 excitations from the inactive (as well as active) space orbitals, into all of the virtual orbitals. Multireference versions of post-SCF methods have been derived, including multireference CI (eg, MR(SD)CI, which includes all single and double excitations) and multireference perturbation theory (eg CASPT2, which is a multireference analogue of MP2). The former of these methods is generally more accurate but

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also more computationally demanding. For more information on multireference methods, 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 initio calculations are not feasible. A number of different procedures are available, with the main methods being CNDO, INDO, MNDO, MINDO/3, AM1, and PM3. The latter two procedures are generally the best performing of the current available methods, and are thus the most popular in current use. Semiempirical methods are based on ab initio molecular orbital theory, but neglect several of the computationally intensive integrals that are required in Hartree–Fock theory. Depending on the procedure, certain interactions between orbitals are either completely neglected or replaced by parameters that are either derived from experimental data for the isolated atoms or obtained by fitting the calculated properties of molecules to experimental data. This greatly reduces the computational cost of the calculations; however, it can also introduce enormous errors. For more detailed information on the principles and limitations of semiempirical methods, the reader is referred to an article by Stewart (47). In general, the semiempirical methods perform reasonably well, provided that the species (and properties) being calculated are very similar to those for which the method was parameterized. However, there are many situations in which these methods fail dramatically, and hence such methods should be applied with caution and their accuracy should always be checked against high level calculations for prototypical reactions. In this context it should be noted that such testing has already been performed for the case of radical addition to C C bonds (32). In this work, semiempirical methods were shown to fail dramatically, and hence (current) semiempirical methods are not generally recommended for studying the kinetics and thermodynamics of the propagation step in free-radical polymerization.

Density Functional Theory Density functional theory (DFT) is a different quantum chemical approach to obtaining electronic-structure information. The basis of DFT is the Hohenberg– Kohn theorem (48), which demonstrates the existence of a unique functional for determining the ground-state energy and electron density exactly. In the ab initio methods described above, we recall that their objective was to determine the wave function (an eigenfunction) of a system, which thereby enables the energy (the eigenvalue) and electron density (the square modulus of the wave function) to be evaluated. The Hohenberg–Kohn theorem implies that the electronic energy can be calculated from the electron density and there is thus no need to evaluate the wave function. This represents an enormous simplification to the calculation since, 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

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Hohenberg–Kohn theorem is merely an existence proof, rather than a constructive proof, and thus the exact functional for connecting the energy and electron density is not known. Hence, although in principle DFT can provide the exact solution to the Schr¨odinger equation, in practice an approximate functional must be used, and this introduces error to the calculations. The DFT methods used in practice are based on the equations of Kohn and Sham (49). They partitioned the total electronic energy into the following terms. E = E T + EV + EJ + EX + EC

(18)

In this equation, ET is the kinetic energy term (arising from the motions of the electrons), EV is the potential term (arising from the nuclear–electron attraction and 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 electronic energy. The task of DFT methods is thus to provide functionals for the exchange and correlation terms. As a matter of notation, DFT methods are typically named as exchange functional–correlation functional, using standard abbreviations for the various functionals. Before discussing the functionals themselves, it is worth making a few comments 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 is simply a functional of the total electron density. (It will only become a functional of the 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 exact functional were available, there would normally be no need to consider the α and β spin densities individually, even for open-shell systems. However, in practice we must use approximate functionals, and (for open-shell systems) these are generally more flexible if they explicitly depend on the α and β spin densities. In an analogous manner to UHF, unrestricted Kohn–Sham theory allows the α and β spin densities to optimize independently, and this allows for a better qualitative description of bond-breaking processes but leads to physically unrealistic spin densities and symmetry breaking problems. A more detailed discussion of the advantages and disadvantages of the unrestricted and spin-restricted theories may be found in the excellent textbook by Koch and Holthausen (11), while an example of a practical application of unrestricted Kohn–Sham theory to reactions with biradical transition structures may be found in a paper by Goddard and Orlova (51). On balance, the unrestricted Kohn–Sham theory is normally preferred for openshell systems; however, as always, careful assessment studies are recommended in order to establish the suitability of any computational method for a specific chemical problem. Since the exact functional relating the energy to the electron (or spin) density is unknown, it is necessary to design approximate functionals, and the accuracy of a DFT method depends on the suitability of the functionals employed. Many different functionals for exchange and correlation have been proposed, and it is beyond the scope of this article to outline their mathematical forms (these may

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be found in textbooks such as Refs. 10 and 11), but it is worth mentioning their main assumptions. Pure DFT methods may be loosely classified into “local” methods and “gradient-corrected” methods. The local DFT methods are based on the local density approximation (LDA, also known as the local spin density approximation, LSDA), in which it is assumed that the electron density may be treated as that of a “uniform electron gas.” From this assumption, functionals describing the 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 an oversimplification of the real situation, and, while it can often provide reasonable molecular structures and frequencies, the LDA model fails to provide accurate predictions of thermochemical properties such as bond energies (for which errors of over 100 kJ · mol − 1 are typical) (54). Gradient-corrected DFT methods (also sometimes referred to as nonlocal or semilocal DFT) attempt to deal with the shortcomings of the LDA model through the generalized gradient approximation (GGA). This corrects the uniform gas model through the introduction of the gradient of the electron density. In introducing the gradient, empirical parameters are often incorporated. For example, the Becke-88 exchange functional (55) was parameterized against the known exchange 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). The GGA methods show improved performance over the LDA model, especially with respect to thermochemical properties. In this regard, the errors generally obtained in 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 for weakly bound systems (such as those in which Van der Waal’s interactions are important), and they also perform poorly for reaction barriers (54). The DFT methods described above are pure DFT methods. Another important class of methods is called hybrid DFT. In these methods the exchange functional is replaced by a linear combination of the Hartree–Fock exchange term and a DFT exchange functional. In addition, the various exchange and correlation functionals may themselves be constructed as linear combinations of the various available methods. For example, the popular hybrid DFT method, B3-LYP, is defined as follows (58).  X   X   C  X X X C C EXC B3LYP = ELDA + c0 EHF − ELDA + cX EB88 − ELDA + EVWN + cC ELYP − EVWN (19) The coefficients in this expression, c0 = 0.20, cX = 0.72 and cC = 0.81, were obtained by fitting the results of B3-LYP calculations to a test set of experimental atomization energies, electron affinities, and ionization potentials. Hybrid DFT methods frequently provide excellent descriptions of the geometries, frequencies, and even reaction barriers and enthalpies for many chemical systems. However, owing to their empirical parameters, such methods are increasingly becoming semiempirical in nature and as such can frequently fail when applied to systems other than those for which they were parameterized. A good example of this is the hybrid DFT method MPW1K (59). This was fitted to

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a test set of hydrogen abstraction barriers, and performs very well for these reactions. However, the same method has recently been shown to have large errors when applied to the problem of predicting the enthalpies for radical addition to multiple bonds (36,37). Nonetheless, hybrid DFT methods currently present the most cost-effective option for studying larger chemical systems but, as always, the performance of such methods should be carefully assessed for each new chemical problem.

Calculation of Reaction Rates from Quantum-Chemical Data Quantum and Classical Reaction Dynamics. In the quantumchemical calculations described above, we solve the electronic Schr¨odinger equation to determine the energy corresponding to a fixed arrangement of nuclei. If such calculations are performed for all possible nuclear coordinates in a chemical system, this yields the potential energy surface. However, as we saw above, in constructing this potential energy surface we made the Born–Oppenheimer approximation, and thus ignored the contribution of the motions of the nuclei to the total kinetic energy. This approximation was appropriate for calculating the electronic energy at a specific geometry, but is clearly not very useful for studying the motions 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 is taken into account. In this Hamiltonian, the potential energy is simply the total electronic energy, which we obtain from our quantum-chemical calculations. Once we have formed our new Hamiltonian we can then solve the Schr¨odinger equation again, this time to follow the motion of the nuclei. This procedure is known as quantum dynamics, and can in principle yield the exact reaction rates for a chemical 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, obtaining accurate solutions to the electronic Schr¨odinger equation for a single set of nuclear coordinates is very computationally intensive. Secondly, to construct a potential energy surface, these expensive calculations must be repeated for “every” possible arrangement of nuclei. Efficient algorithms are available for choosing only those geometries necessary for an adequate description of the chemical system (60). However, even using these algorithms, large numbers of quantumchemical calculations are nonetheless required. For example, approximately 1000 quantum-chemical calculations were required to construct a reliable potential energy surface for the OH + H2 system (61). Furthermore, the number of data points required increases substantially with the number of atoms in the system (due to the increasing dimensionality). Finally, we have the problem of solving the nuclear Schr¨odinger equation. In practice, this is intractable for all but the simplest systems, as atoms (being heavier) require even more basis functions than are needed to solve the electronic Schr¨odinger equation. With current available computing power, 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 at approximately 107 different points on the potential energy surface were required

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in order to solve the nuclear Schr¨odinger equation, and this requirement scales exponentially with the number of atoms in the system. Classical reaction dynamics provides a strategy for calculating the rate coefficients of larger chemical systems. Having used quantum-chemical techniques to calculate the potential energy surface, the motions of the nuclei are studied by solving the laws of classical or Newtonian dynamics. This is often a reasonable approximation, since the atoms (being heavier) are considerably less subject to quantum effects than the electrons. Nonetheless, standard classical reaction dynamics calculations are still limited by the need to calculate a full potential energy surface (including the first and second derivatives at each point). As a result, standard classical dynamics calculations involving accurate ab initio potential energy surfaces are also currently restricted to relatively small chemical systems, such as H3 C3 N3 (62). An alternative approach to constructing the entire potential energy surface for a chemical system is provided by direct dynamics. In both standard classical reaction dynamics and direct dynamics, the basic principle is the same. A starting arrangement of atoms is adjusted (by a small amount) according to the forces acting 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 geometry, and so on. The process is repeated for many thousands of time steps, until a complete trajectory is mapped out. The process is then repeated for many trajectories until the reaction probability (and other related information) is established to within an acceptable level of statistical error. As we saw above, in standard classical reaction dynamics, the force acting on the molecule as a function of geometry is obtained from the potential energy surface. In direct dynamics, also known as on-the-fly ab-initio dynamics, this force is calculated (using quantumchemical 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 systems can be studied using approximate potential energy surfaces, constructed using empirical or semiempirical procedures. In particular, the method of molecular mechanics (MM), which is described elsewhere in this Encyclopedia, is commonly used to simulate the motion of polymers and proteins. However, the accuracy of MM simulations are limited by the accuracy of the “force field,” which is the set of potential functions that are used to govern relative motions of the constituent atoms. Force fields are typically derived on the basis of empirical and semiempirical information, and are typically only accurate for the type of system for which they were parameterized. Recently, much effort has been directed at deriving accurate 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. Accurate force fields for studying the kinetics of free-radical polymerization do not currently exist, and instead high level ab initio calculations are necessary in order to model these reactions accurately. Transition-State Theory. To study reactions in larger chemical systems using accurate ab initio calculations we need a much simpler approach, and this is provided by transition-state theory (66). In its simplest form, it assumes that, in

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the space represented by the coordinates and momenta of the reacting particles, it is possible to define a dividing surface such that all reactants crossing this plane go on to form products, and do not recross the dividing surface. The minimum energy structure on this dividing plane is referred to as the transition structure of the reaction. Transition-state theory also assumes there is an internal statistical equilibrium between the degrees of freedom of each type of system (reactant, product, or transition structure), and that the transition state is in statistical equilibrium with the reactants. In addition, it assumes that motion through the transition state can be treated as a classical translation. From these assumptions, the following simple equation relating the rate coefficient at a specific temperature, k(T), to the properties of the reactant(s) and transition state can be derived (66).

k(T) = κ(c◦ )1 − m

Q‡ kB T  h

Q reactants i

e − E0 /RT

(20)

In this equation κ is called the transmission coefficient and is taken to be equal to unity in simple transition-state theory calculations, but is greater than unity when tunneling is important (see below), c◦ is the inverse of the reference volume assumed in calculating the translational partition function (see below), m is 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 the reaction barrier) is the energy difference between the transition structure and the reactants (in their respective equilibrium geometries), Q‡ is the molecular partition function of the transition state, and Qi is the molecular partition function of reactant i. Transition-state theory thus reduces the problem of calculating the potential energy surface for “every” possible geometric arrangement of nuclei, to the consideration of a very small number of “special” geometries; namely, the transition structure and the reactant(s). The transition structure is the minimum energy structure on the dividing surface between the reactants and products, and must be located so as to make the “no re-crossing” assumption as valid as possible. In simple transition-state theory, the transition structure is located as the maximum energy structure, along the minimum energy path connecting the reactants and products. This is generally a good approximation for reactions having barriers that are large compared to kB T. However, for reactions with low or zero barriers, a more accurate approach is required. To this end, in variational transition-state theory, the transition structure is located as the structure (on the minimum energy path) that yields the lowest reaction rate. In thermodynamic terms, this may be thought of as the maximum in the Gibb’s free energy of activation, rather than the maximum internal energy of activation. In order to calculate reaction rates via transition-state theory, one needs to identify the equilibrium geometries of the reactants, and also the transition structure, and calculate their energies. This information is of course accessible from quantum-chemical calculations. The molecular partition functions for these

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species are also required. These serve as a bridge between the quantum mechanical states of a system and its thermodynamic properties, and are given by Q=

 i

 εi gi exp − kB T

(21)

The values εi are the energy levels of a system, each having a number of degenerate states gi , and are obtained by solving the Schr¨odinger equation. In theory, this equation should be solved for all active modes but in practice the calculations can be greatly simplified by separating the partition function into the product of the translational, rotational, vibrational, and electronic terms, as follows. Q = Qtrans × Qrot × Qvibr × Qelec

(22)

This is generally a reasonable assumption, provided that the reaction occurs on a single electronic surface. Finally, if we assume that reacting species are ideal gas molecules, analytical expressions for the partition functions are as follows:  Qtrans = V

2πMkB T h2

3/2 =

 RT 2πMkB T 3/2 P h2

  1 1 hνi Qvib = × exp −  hνi  2 kT i i 1 − exp − kT  1 T h2 Qrot, linear = where r = σr r 8π 2 IkB 

Qrot, nonlinear =

 T 3/2 π 1/2 σr (r,x r,y r,z )1/2 Qelec = ω0

where

r,i =

h2 8π 2 Ii kB

(23)

(24)

(25)

(26)

(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 P the corresponding reference temperature and pressure: ν i are the vibrational frequencies 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 inertia about axes x, y, and z respectively; σ r is the symmetry number of the molecule which counts its number of symmetry equivalent forms (67); and ω0 is the electronic spin multiplicity of the molecule (ie, ω0 = 1 for singlet species, 2 for doublet species, etc). The information required to evaluate these partition functions is routinely accessible from quantum-chemical calculations: the moments of inertia and symmetry numbers depend on the geometry of the molecule, while the vibrational frequencies are obtained from the second derivative of the energy with respect to the geometry.

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A number of additional comments need to be made concerning the use of equations 23–27. Firstly, in the calculation of the translational partition function (eq. 23), a reference volume (or equivalently, a temperature and pressure) is assumed. This is needed for the calculation of thermodynamic quantities such as enthalpy and entropy, but the assumption has no bearing on the calculated rate coefficient, as the reference volume is removed from equation 20 through the parameter c◦ (= 1/V). Secondly, the vibrational partition function (eq. 24) has been written as the product of two terms. The first of these corresponds to the zero-point vibrational energy of the molecule, while the latter corresponds to its additional vibrational energy at some nonzero temperature T. The zero-point vibrational energy is often included in the calculated reaction barrier E0 . When this is the case, this first term must be removed from equation 24, so as not to count this energy twice. Thirdly, the external rotational partition function is calculated using equation 25 if the molecule is linear, and equation 26 if it is not. It is also worth noting that there is an entirely equivalent thermodynamic formulation of transition-state theory. k(T) = κ

kB T ◦ 1 − m S‡ /R − H‡ /RT (c ) e e h

(28)

A derivation of this expression, which is obtained by noting the relationship between the thermodynamic properties of a system (eg enthalpy, H, and entropy, S) and the partition functions, can be found in textbooks on statistical thermodynamics (12–16). The enthalpy of activation ( H ‡ ) for this expression can be written 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 can be calculated from the vibrational frequencies as follows. ZPVE = R

H = R

 i

1 hνi /kB 2 i

(30)

5 3 hνi /kB + RT + RT exp(hνi /kB /T) − 1 2 2

(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 rotational contribution. The entropy of activation ( S‡ ) is calculated from the vibrational (Sv ), translational (St ), rotational (Sr ), and electronic (Se ) contributions to the entropies of the individual species, in turn expressed as follows. Sv = R

 i



hνi /kB T − ln(1 − exp(−hνi /kB T)) exp(hνi /kB T) − 1

(32)

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  2π MkB T 3/2 kB T + 1 + 3/2 St = R ln P h2

(33)

   1 T Sr, linear = R ln +1 σr r

(34)

  1/2  π T 3/2 + 3/2 Sr, nonlinear = R ln σr (r,x r,y r,z )1/2

(35)

Se = R ln(ω0 )

(36)

The parameters required to evaluate these expressions are the same as those used in evaluating the partition functions, as described above. Finally, by evaluating the derivative of (28) with respect to temperature, it is possible to derive a relationship between the above thermodynamic quantities and 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 the system, as follows. ◦ 1 − m m kB T

A = (c )

e

h



S‡ exp 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 temperatureindependent parameters of the Arrhenius expression are in fact functions of temperature, which is why the Arrhenius expression is only valid over relatively small temperature ranges. It should also be clear that the ZPVE-corrected barrier (E0 + ZPVE), the enthalpy of activation ( H ‡ ), and the Arrhenius activation energy (Ea ) are only equal to each other at 0 K. At nonzero temperatures, these quantities are nonequivalent and thus should not be used interchangeably. Extensions to Transition-State Theory. Many variants of transitionstate theory have been derived, and a comprehensive review of these recent developments has been provided by Truhlar and co-workers (68). As already noted, one of the main extensions to transition-state theory is variational transition-state theory which, in its simplest form, locates the transition structure as that having the maximum Gibb’s free energy (rather than internal energy). Other variations of transition-state theory arise through making different assumptions as to the statistical distribution of the available energy throughout the different molecular modes, and through deriving expressions for the partition functions for cases other than ideal gases. In addition, two simple extensions to the transition-state theory

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equations are the inclusion of corrections for quantum-mechanical tunneling, and the improved treatment of the low frequency torsional modes. Since these are of importance in treating certain polymerization-related systems, these are briefly described below. Tunneling Corrections. One of the assumptions inherent in simple transition-state theory is that motion along the reaction coordinate can be considered as a classical translation. In general, this assumption is reasonably valid since the reacting species, being atoms or molecules, are relatively large and thus their wavelengths are relatively small compared to the barrier width. However, in the case of hydrogen (and to a lesser extent deuterium) transfer reactions, the molecular mass of the atom (or ion) being transferred is relatively small, and thus quantum effects can be very important. Corrections for quantum-mechanical tunneling are incorporated into the κ coefficient of equation 20, and are known as tunneling coefficients. There is an enormous variety of expressions available for calculating tunneling coefficients. The most accurate tunneling methods, such as small curvature tunneling (69), large curvature tunneling (70), and microcanonical optimized multidimensional tunneling (71), involve solving the multidimensional Schr¨odinger equation describing motion of the molecules at every position along the reaction coordinate. To calculate such tunneling coefficients, specialized software (such as POLYRATE (72)) is used, and additional quantum-chemical data (such as the geometries, energies, and frequencies along the entire minimum energy path) are required. As a result, simpler (and hence less accurate) expressions are often adopted. These are derived by treating motion along the reaction coordinate as a function of one variable, the intrinsic reaction coordinate, and hence solving a one-dimensional Schr¨odinger equation. When this is done using the calculated energies along the reaction path, the procedure is known as zero-curvature tunneling (73). However, this procedure still entails the numerical solution of the Schr¨odinger equation, and hence an additional simplification is also often made. Instead of using the calculated energies along this path, some assumed functional form for the potential energy is used instead. This is chosen so that the Schr¨odinger equation has an analytical solution, and thus a closed expression for the tunneling coefficient can be derived. The derivation of these simple tunneling coefficients is described by Bell (74), and the main expressions used in practice are as follows. The simplest tunneling coefficients are based on the assumption that the change in energy along the minimum energy path can be described by a truncated parabola. This functional form provides a good description of the energies near the transition structure (where tunneling is most significant), but a very poor description elsewhere. The equation for the tunneling coefficient is given as the following infinite series, which is frequently truncated after the first few terms.

κ=

1 u 2 ‡ 1  sin 2 u‡

− u‡ y − u‡ /2π



 y y2 y3 − + −··· 2π − u‡ 4π − u‡ 6π − u‡

where u‡ =

hv‡ kT

and

 2π V y = exp − u‡ kT

(40)

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

κ≈

1 u 2 ‡ 1  sin 2 u‡

≈1+

u2‡ 24

+

u4‡ 5760

+···≈1+

u2‡ 24

(41)

A slightly more realistic description of the change in potential energy along the 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 transition structures, the parameters A and B are defined as the following functions of the forward (V f ) and reverse (V r ) reaction barriers (where the reaction is taken in the exothermic direction).   A = ( Vf + Vr )2

and

B = Vf − Vr

(43)

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

= 2π cν‡

1 (B2 − A2 )2 8 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) Eckart formulae (41), resulting in the following expression for the permeability of the reaction barrier G(W) as a function of the energy W: G(W) = 1 − where α=

cosh(α − β) + cosh(δ) cosh(α + β) + cosh(δ)

4 π 2 √ 4 π 2  4 π 2 h2 2W, β = 2(W − B), δ = 2A − h h h 16π 2 2

(45)

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

κ=

exp(VF /kB T) kB T



∞

G(W) exp(−W/kB T) dW

(46)

0

Although this expression requires numerical integration, it does not require sophisticated software and can be easily implemented on a spreadsheet. Finally, it is worth making a few comments on the use of the tunneling procedures. Firstly in very general terms, tunneling is important for reactions involving the transfer of a hydrogen or deuterium atom or ion. The importance of tunneling can also be established through examination of the parameter u‡ in equation 40, a value of u‡ < 1.5 usually indicating negligible tunneling effects (74). Secondly, in principle, the more accurate multidimensional tunneling coefficient expressions should always be used. However, in practice, the more convenient one-dimensional expressions are often adopted. Of these expressions, the Eckart tunneling coefficient is significantly more accurate and should be preferred. For example, the small curvature tunneling method gives a tunneling coefficient (κ) of approximately 102 at 300 K, for the hydrogen abstraction reaction between • NH2 and C2 H6 (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, and hence only the Eckart method yields a tunneling coefficient of the right order of magnitude for this (typical) reaction. The success of the Eckart tunneling method has also been noted by Duncan and co-workers (78), who rationalized it in terms of a favorable cancelation of errors. Treatment of Low Frequency Torsional Modes. In the vibrational partition function (eq. 24), all modes are treated under the harmonic oscillator approximation. That is, it is assumed that the potential field associated with their distortion 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, but not 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 function is negligible at room temperature (ie Qvib,i ≈ 1) and thus the error incurred in treating these modes as harmonic oscillators is not significant. However, for the low frequency torsional modes, these errors can be significant and a more rigorous treatment is often necessary, and this is especially the case for the reactions of relevance to free-radical polymerization (3–5,79). The simplest method for treating the hindered internal rotations is to regard them 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 to 360◦ in steps of 10◦ ) via quantum chemistry. In general, it is recommended that these potentials be obtained as relaxed scans; that is, in calculating the energy at a specific angle, all geometric parameters other than the rotational angle are optimized (79). If the rotational potential can be described as a simple cosine function, the enthalpy and entropy associated with the mode can be obtained directly from the tables of Pitzer and co-workers (80). In order to use these tables,

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

one calculates two dimensionless quantities: x=

V kB T

and

y= 

σint h 8π 3 Im kB T

(47)

In these equations, V is the barrier to rotation, σ int is the symmetry number associated with the rotation (which counts the number of equivalent minima in the potential energy curve), and Im is the reduced moment of inertia associated with the rotation. This latter parameter is given by the following formula:  Im = A m 1 −



Am λ2mi





Ii

(48)

i = x,y,z

In this equation, Am is the moment of inertia of the torsional coordinate itself, Ii is the principal moment of inertia of the whole molecule about axis i, and

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λmi is the direction cosine between the axis of the top and the principal axis of the whole molecule. More information on the calculation of reduced moments of inertia can be found in Reference (81). When the rotational potential cannot be fitted with a simple cosine function (as in Fig. 10b), the partition function (and hence the enthalpy and entropy) is obtained instead by (numerically) solving the one-dimensional Schr¨odinger equation. −

h2 ∂ 2  + V(θ ) = ε 8π Im ∂θ 2

(49)

This yields the energy levels of the system (ε i ), which are then used to evaluate the partition function via the following equation.  1  εi exp − Qint rot = σint i kB T

(50)

Having obtained the partition function (or equivalently, the enthalpy and entropy) associated with a low frequency torsional mode, this is used in place of the corresponding harmonic oscillator contribution for that mode. The above treatment of hindered rotors assumes that a given mode can be approximated as a one-dimensional rigid rotor, and studies for small systems have 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 in those cases. When coupling is significant, the use of a one-dimensional hindered rotor model may actually introduce more error than the (fully decoupled) harmonic oscillator treatment. Hence, in these cases, the one-dimensional hindered rotor model should be used cautiously.

Software There are a large number of software packages available for performing computational chemistry calculations. Some of the programs available include ACES II (83), GAMESS (84), GAUSSIAN (43), MOLPRO (85), and QCHEM (86). Other programs, such as POLYRATE, (72), have been designed to use the output of quantum-chemistry programs to calculate reaction rates and tunneling coefficients. Whereas computational chemistry software has traditionally been operated on large supercomputers, versions for desktop personal computers are increasingly becoming available. In addition, programs for visualizing the output of computational chemistry calculations such as Spartan (87), Molden (88), CS Chem3D (89), Molecule (90), Jmol (91), Gauss View (43), and MacMolPlt (92) are also available. These programs allow one to visualize the geometry and electronic structure of the resulting molecule, and animate its vibrational frequencies. Many of these programs also have built-in computation engines. Computational chemistry is thus increasingly becoming accessible to the nonspecialist user, which brings with it its own problems (see also MOLECULAR MODELING).

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Accuracy and Applicability of Theoretical Procedures By solving the Schr¨odinger equation exactly, quantum chemistry can, in principle, provide accurate electronic-structure data. However, in practice, approximate numerical methods must be adopted, and these can introduce error to the calculations. As we have already seen, an enormous number of approximate methods are available, and these range from the accurate but computationally expensive to the cheap but potentially nasty. Furthermore, the performance of a particular method varies considerably depending upon the chemical system and the property being calculated. It is therefore very important that computational chemistry studies are accompanied by rigorous assessments of theoretical procedures. In such “calibration” studies, prototypical systems are calculated at a range of levels of theory, and the results are compared both internally (against the highest level procedures) and externally (against reliable experimental data) in order to identify those methods which offer the best compromise between accuracy and expense. In the present section, the main conclusions from recent assessment studies for the reactions of importance to free-radical polymerization are outlined. In presenting such studies, it must be acknowledged that, with continuing rapid increases in computer power, some of these results will soon be outdated. In particular, as computer power increases, the need to rely upon lower levels of theory for large polymer-related systems will diminish. Instead, the higher levels of theory outlined below will be able to be used routinely. Nonetheless, with increasing computer power, the temptation to apply existing levels of theory to yet larger systems will no doubt ensure that the main conclusions of these studies retain some relevance into the near future. Radical Addition to C C Bonds. Radical addition to C C bonds are of importance for free-radical polymerization as this reaction forms the propagation step, and thus influences the reaction rate and molecular weight distribution in both conventional and controlled free-radical polymerization, and the copolymer composition and sequence distribution in free-radical copolymerization. Numerous studies have examined the applicability of high level theoretical methods for studying 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 geometries and frequencies at the CCSD(T)/6-311G(d,p) level of theory, and is the highest level study to date. The main conclusions from this study, and (where still relevant) the previous lower level studies, are outlined below. Geometry optimizations are generally not very sensitive to the level of theory, with even the low cost HF/6-31G(d) and B3-LYP/6-31G(d) methods providing reasonable 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 the forming bond length in the transition structures, and this can be reduced using an IRCmax technique (94). Alternatively, the error in the B3-LYP transition structures 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 reactions, as they are subject to spin-contamination problems (32,33,37). Frequency calculations are also relatively insensitive to the level of theory, especially when the frequencies are scaled by their appropriate scale factors. (Scale-factors for the most commonly used levels of theory may be found in Reference (95). In particular,

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the B3-LYP/6-31G(d) level of theory provides excellent performance for frequency factors, 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 possible, high level composite procedures should be used for the prediction of absolute reaction barriers and enthalpies, and of these methods the “RAD” variants of G3 provide the best approximations to the higher level Wn methods (when the latter 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 considerable error to the predicted reaction barriers for these reactions and, until this is revised, these methods should perhaps be avoided for these reactions (37). When composite methods cannot be afforded, the use of RMP2/6311 + G(3df,2p) single points provides reasonable absolute values and excellent relative 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 in the reaction enthalpies, even when applied with large basis sets. However, they do provide reasonable addition barriers, owing to the cancelation of errors in the early transition structures for these reactions (37). Interestingly, for the closely related radical addition to C C bonds, the situation is reversed and the B3-LYP methods perform well and the RMP2 methods perform poorly (37), and this highlights the importance of performing assessment studies before tackling new chemical problems. Finally, it should be stressed that semiempirical methods do not provide an adequate description of the barriers and enthalpies in these reactions (32). For rate coefficients, the importance of treating the low frequency torsional modes in radical addition reactions as hindered internal rotations has been investigated in a number of assessment studies (37,79,93). For small systems such as methyl addition to ethylene and propylene, the errors are relatively minor (less than a factor of 2) (37). However, for reactions of substituted radicals (such as n-alkyl radicals (93) and the ethyl benzyl radical (79)), the errors are somewhat larger (as much as a factor of 6), as there are additional low frequency torsional modes to consider. Nonetheless, the errors are still relatively small, and the 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, and the reverse β-scission reaction, forms the key addition and fragmentation steps of the RAFT polymerization process (96). Ab initio calculations have a role to play in elucidating the effects of substituents on this process, and in providing an understanding of the causes of rate retardation (6,7). A detailed assessment of theoretical procedures has been recently carried out for this class of reactions (36), and the main conclusions are similar to those for addition to C C bonds (37), as outlined above. In general, low levels of theory, such as B3-LYP/6-31G(d), are suitable for geometry optimizations and frequency calculations, provided an IRCmax procedure is used to correct the transition structures. However, high level composite methods are required to obtain reliable absolute barriers and enthalpies, though reasonable relative quantities can be obtained at the RMP2/6-311 + G(3df,2p) level. As in the case of addition to C C bonds, the spin correction term in the CBS-type methods appears to require adjustment, and the RAD variants of G3 should be preferred when the higher level Wn calculations are impractical.

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Hydrogen Abstraction. Hydrogen abstraction reactions are important chain-transfer processes in free-radical polymerization. In particular, hydrogen abstraction by the propagating polymer radical from transfer agents (such as thiols), monomer, dead polymer, or itself (ie intramolecular abstraction) can affect the molecular weight distribution, the chemical structure of the chain ends, and the degree of branching in the polymer. The accuracy of computational procedures for studying hydrogen abstraction reactions has received considerable attention, and the results of some of the most recent and extensive studies (59,94,97–99) are summarized below. Geometry optimizations are relatively insensitive to the level of theory; however, there are some important exceptions. In particular, the HF and MP2 methods should be avoided for spin-contaminated systems (99). Moreover, the B3-LYP method does not describe the transition structures very well for a number of hydrogen abstraction reactions (59,97). However, improved performance is obtained using newer hybrid DFT methods such as MPW1K (59) and KMLYP (97), and these methods are suitable as low cost methods, when high level procedures cannot 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 Wn methods, and would provide an excellent benchmark level of theory when the latter could not be afforded (99). As in the case of the addition reactions, the spin contamination correction term in the CBS-type methods appears to be introducing a systematic error to the predicted reaction barriers and enthalpies and, until this is revised, this method should perhaps be avoided for spin-contaminated reactions (99). When composite methods cannot be afforded, methods such as RMP2, MPW1K, or KMLYP have been shown to provide good agreement with the high 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 method performs particularly poorly for reaction barriers and enthalpies (59,94,97–99), and should thus be generally avoided for abstraction reactions. Interestingly, it has been noted that the errors in B3-LYP increase with the increasing polarity of the reactants (98), which suggests that assessment studies based entirely on relatively nonpolar reactions (such as CH3 • + CH4 ) may lead to the wrong conclusions. As noted in the previous section, tunneling is significant in hydrogen abstraction reactions, and hence accurate quantum-chemical studies of these systems require the calculation of tunneling coefficients. The accuracy of tunneling coefficients is profoundly affected by both the tunneling method and the level of theory at which it is applied. A systematic comparison of the various tunneling methods for the hydrogen abstraction reactions of relevance to free-radical polymerization does not appear to have been performed. However, in the example provided in the previous section, it was seen that the Eckart method was capable of providing the tunneling coefficients of the right order of magnitude (when compared with the more accurate multidimensional methods), while the Wigner and Bell methods respectively underestimated and overestimated the tunneling coefficients by an order of magnitude. Hence, when multidimensional tunneling methods are not convenient, the Eckart tunneling method should be preferred as the best low cost method. An assessment of the effects of level of theory on the tunneling

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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 (with a range of basis sets) leads to errors in the calculated tunneling coefficients of several orders of magnitude (compared to high level CCSD(T)/6-311G(d,p) calculations). The B3-LYP and MP2 methods performed significantly better, showing errors of a factor of 2–3. However, even better performance could be obtained by correcting the HF values to the CCSD(T)/6-311G(d,p) level via an IRCmax procedure.

Applicability of Chemical Models Assuming high levels of theory are used, quantum-chemical calculations might be expected to yield very accurate values of the rate coefficients for the specific chemical system being studied. With current available computing power, this would in all likelihood consist of a small model reaction in the gas phase. If, for example, this information is then to be used to deduce something about solution-phase polymerization kinetics, the effects of the solvent and (in most cases) the effects of chain length need to be considered. Unfortunately, the treatment of these effects at a high level of theory is not generally feasible with current available computing power, and hence the neglect of these effects (or their treatment at a crude level of 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 polymerization process in a variety of ways (100). For example, if polar interactions are significant in the transition structure of the reaction, the presence of a high dielectric constant solvent may stabilize the transition structure and lower the reaction barrier. Solvents may also affect the reactivity of the reacting radicals, and even the mechanism of the addition or transfer reaction, through some specific interaction such as hydrogen bonding or complex formation. In addition, the preferential sorption of the monomer or solvent around the reacting polymer radical may lead to the effective concentrations available for reaction being different to those in the bulk solution, resulting in a difference between the observed and predicted rate coefficients. Solvent effects such as these result in the experimentally measured rate coefficients for a free-radical polymerization varying according to the 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 and solution-phase systems. Whereas in the gas-phase an isolated molecule might be expected to have translational, rotational, and vibrational degrees of freedom, in the solvent phase the translational and rotational degrees of freedom are effectively “lost” in collisions with the solvent molecules. In their place, it is necessary to consider additional vibrational degrees of freedom involving a solute-solvent “supermolecule” (101). Since the vibrational, translational, and rotational modes make different quantitative contributions to the enthalpy and entropy of activation, significant differences might be expected between the gas and solution phases. For bimolecular reactions this effect can be considerable, because the main contribution to the entropy of activation is the six rotational and translational

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degrees of freedom in the reactants, which are converted to internal vibrations in the transition structure. In contrast, for unimolecular reactions, the entropically based gas-phase/solution-phase difference is generally much smaller, as the rotational and translational modes are similar for the transition structure and reactant molecule, and thus their contribution largely cancels from the reaction rate. The treatment of solvent effects varies according to their origin. The influence of the dielectric constant on polar reactions can be dealt with routinely using various continuum models (102), implemented using standard computational software, such as GAUSSIAN (43). However, it should be stressed that these models do not account for the entropically based gas-phase/solvent-phase difference, nor do they deal with direct solvent interactions in the transition structure. When direct interactions involving the solvent are important, it is necessary to include solvent molecules in the quantum-chemical calculation. In theory, one should include many hundreds of solvent molecules but in practice one includes a small number of molecules, and combines this with a continuum model (102). However, even with this simplification, the additional solvent molecules increase the computational cost of the calculations, and it is not currently feasible to apply these methods (at any reasonable level of theory) for polymer-related systems. Even when additional solvent molecules are included in the ab initio calculations, various extensions to transition-state theory are required in order to model the rates of solution-phase reactions (68,101). Unfortunately, the existing models are complicated to use and require additional parameters which are not readily accessible for polymerization-related systems. The development of simplified yet accurate models for dealing with solvent effects is an on-going field of research. While strategies for calculating solution-phase rate coefficients exist, with the current available computing power these methods are not generally feasible for polymer-related systems. Instead, the following practical guidelines for dealing with solvent effects are suggested. Firstly, when the solvent participates directly in the reaction, the inclusion of the interacting solvent molecule in the gas-phase calculation is essential for gaining a mechanistic understanding of the reaction. Secondly, when polar interactions are expected to be important, the use of a continuum model is recommended, especially if the results are to be used to interpret the polymerization process in polar solvents. Thirdly, for bimolecular reactions, if the a priori prediction of absolute rate coefficients is required, a consideration of the entropically based gas-phase/solution-phase difference is necessary. This “entropic” solvent effect might be estimated by comparing corresponding experimental solution- and gas-phase rate coefficients for that class of reaction. For example, solution-phase experimental values for radical addition reactions generally exceed the corresponding gas-phase values by approximately one order of magnitude (103). One might also benchmark the gas-phase calculations by calculating the rate coefficient for a similar reaction, and comparing the calculated result with reliable experimental data. Fourthly, provided that specific interactions are not important, one might expect that solvent effects should largely cancel from relative rate coefficients, and hence the gas-phase values should generally be suitable for studying substituent effects and solving mechanistic problems. Finally, when specific interactions are important, simple gas-phase calculations are still useful, as they can provide complementary information about the underlying influences on the mechanism in the absence of the solvent.

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Chain-Length Effects. The other simplification that is necessary in order to use high level ab initio calculations on polymer-related systems is to approximate the propagating polymer radical (which may be hundreds or thousands of units long) as a short-chain alkyl radical. Provided that the reaction is chemically controlled, this is not an unreasonable assumption. In chemical terms, the effects of substituents decrease dramatically as they are located at positions that are increasingly remote from the reaction center. For example, the terminal and penultimate units of a propagating polymer radical are known to affect its reactivity and selectivity in the propagation reaction (104); however, substituent effects beyond the penultimate position are rarely invoked in copolymerization models. In order to include the most important substituent effects, it is generally recommended that propagating radicals be represented as γ -substituted propyl radicals (as a minimum chain length). For some systems this is not currently possible without resorting to a low (and thus inaccurate) level of theory. In those cases, the possible influence of penultimate unit effects must be taken into account when interpreting the results of the calculations. The entropic influence of the chain length on the reaction rates extends slightly beyond the penultimate unit. For example, Deady and co-workers (105) showed experimentally that there was a chain-length effect on the propagation rate coefficient of styrene, which converged at the tetramer stage (ie an octyl radical). Heuts and co-workers (3) have explored this chain-length dependence theoretically, and suggested that it arises predominantly in the translational and rotational partition functions. More specifically, they suggest that there is a small effect of mass that can be modeled by including an unrealistically heavy isotope of 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 hydrogen atom that happens to have a molecular mass of 9999 amu! They also noted that there is an effect of chain length on the rotational entropy (and especially the hindered internal rotations), which required the more subtle modeling strategy of using slightly longer alkyl chains (ie n > 1). Nonetheless, using their “heavy hydrogen” approach, their calculated frequency factors converged to within a factor of 2 of the long chain limit at even the propyl radical stage (ie n = 2). More recently it was shown that the consideration of the propagating radical as a substituted hexyl radical (without a heavy hydrogen at the remote chain end) was also sufficient to reproduce experimental values for the frequency factors of propagation reactions (106). For short-chain branching reactions, it has been shown that inclusion of just one methyl group beyond the reaction center is sufficient for modeling the long-chain reactions, provided that the additional methyl group is 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 reasonable description of polymeric radicals in chemically controlled reactions. Finally, it is worth noting that the chain-length effects on the propagation steps amount to approximately an order of magnitude difference between the first propagation step and the long chain limit, with the small radical additions having faster rate coefficients. This chain-length error is of the same magnitude and acts in the opposite direction to the gas-phase/solvent-phase difference in bimolecular reactions, and hence substantial cancelation of error might be expected in these cases. Indeed an (unpublished) high level G3(MP2)-RAD calculation of the

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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 experimental value (also 2.0 × 104 L · mol − 1 · s − 1 at ambient pressure) (107), despite the fact that both the medium and chain-length effects were ignored in the calculation. 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 mechanistic problems 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 provide useful related information (such as transition structures and radical stabilization energies) to help in understanding the reaction mechanism. In the following, some of the applications of quantum chemistry are outlined. This is not intended to be a review of the main contributions to this field, nor is it intended to provide a 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 a tool for studying free-radical polymerization (see RADICAL POLYMERIZATION). A Priori Prediction of Absolute Rate Coefficients. The a priori prediction of accurate absolute rate coefficients is perhaps the most demanding task in computational chemistry. For example, high level ab initio calculations (at the G3(MP2)-RAD level as a minimum) are required for the calculation of accurate reaction barriers (and enthalpies) in radical addition reactions and, with current available computing power, these can be performed routinely on systems of up to 12–14 non-hydrogen atoms. This allows for the most common polymerization substituents to be included, and often allows for penultimate unit effects to be taken into account. However, it does not allow for the accurate treatment of solvent or chain-length effects. Of course, with continuing rapid increases in computer power, these problems should soon be overcome. In addition, as was noted above, for a bimolecular reaction, the chain-length effects and solvent effects act in opposite directions and may in fact largely cancel, leading to surprisingly accurate results. Certainly, with current available computing power, the prediction of rate coefficients to within 1–2 orders of magnitude (or better) is possible for most chemically controlled polymerization-related reactions, and further increases in accuracy should be attainable in the near future. The prediction of absolute rate coefficients via quantum chemistry is particularly useful when direct experimental measurements are not possible. A good example of this is in the RAFT polymerization process (96), in which the experimentally observable quantities (such as the overall polymerization rate, the overall molecular weight distribution, and the concentrations of the various species) are a complicated function of the rates of the various individual reactions. In order to measure the rates of these individual steps, it is necessary to relate their rate coefficients to the observable quantities via some kinetic-model-based assumptions. Depending upon the assumptions, enormous discrepancies arise in the estimated

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rate coefficients. For example, alternative experimental measurements for the fragmentation rate in cumyl dithiobenzoate mediated polymerization of styrene at 60◦ C differ by six orders of magnitude (109,110), and this difference arises, at least in part, in the model-based assumptions of the alternative experimental techniques (For a discussion of problems measuring fragmentation rate coefficients, the reader is referred to Reference 111). Recent ab initio calculations of the fragmentation rate coefficient for a model of this system were able to provide direct evidence in support of one measurement (and hence one set of assumptions) over the other, and in doing so provide an insight into the causes of rate retardation in the RAFT process (6). As accurate calculations become routine, there will be many other applications for the a priori prediction of absolute rate coefficients. The calculation of propagation rate coefficients (which can be measured experimentally via pulsed laser polymerization or PLP (112)) is currently used to benchmark theoretical procedures, but accurate calculations may also be helpful as a stand-alone technique for toxic and hazardous monomers, and in other cases where PLP is difficult (due to problems such as the monomer absorbing at the wavelength of the laser or reactions such as chain transfer broadening the molecular weight distribution). It has also been noted that ab initio calculations may provide the best means of studying long- and short-chain branching in free-radical polymerization (5). Quantum chemistry will also be helpful in extracting the rate coefficients of the individual reactions in other complicated reaction schemes, such as free-radical copolymerization, and the various types of controlled radical polymerization systems.

Prediction of Relative Rate Coefficients: Discriminating Models and Mechanisms. High level ab initio calculations can already predict relative rate coefficients with remarkable accuracy. This is because accurate relative values of quantities such as barriers and enthalpies can generally be obtained at lower levels of theory than corresponding absolute values, because of substantial cancellation of error. In addition, one might generally expect that chain-length effects and solvent effects should be reasonably consistent for a series of similar reactions, and thus cancel from the comparative values. The prediction of relative rate coefficients is important for modeling and hence optimizing various aspects of the polymerization process, and some of these applications are outlined below. The prediction of relative rate coefficients is also important for understanding the effects of substituents on the various individual reactions, and these applications are discussed in the following section. Copolymerization. In free-radical copolymerization (qv), the relative rates of addition of the various types of radical to the alternative monomers are called reactivity ratios, and are the key parameters governing the composition and sequence distribution of the resulting copolymer. Experimentally, these parameters are “measured” for a given system by treating them as adjustable parameters in a least-squares fit of some assumed kinetic model to experimental values of the composition, 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 implicit or explicit penultimate model, etc) is appropriate for a given system. If a physically incorrect model is chosen, then the estimated parameters will lack their assumed physical meaning, and will thus be unsuitable for mechanistic studies,

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or for predicting other copolymerization properties. Secondly, even if the correct model is chosen, there are often more adjustable parameters than are numerically needed to fit the data, and this can result in large and highly correlated uncertainties in the estimated parameters. The problems in the experimental estimation of reactivity ratios are discussed in more detail in the article on copolymerization (qv). Quantum chemistry is able to address these problems, as it allows the reactivity ratios to be calculated directly from the rate coefficients for the various types of reaction, without making kinetic-model-based assumptions. Furthermore, quantum chemistry can assist in determining which kinetic model is most appropriate for a given system. For example, by measuring the effects of substituents at the penultimate position of the propagating radical, quantum chemistry can be used to determine whether a terminal or penultimate model is appropriate for a given system. In this respect, quantum chemistry has already made an important contribution to this field by providing evidence for penultimate unit effects in the barriers (1,106) and frequency (2,106) factors of a variety of different systems. Such calculations can thus simplify the model discrimination process by ruling out either the terminal- or penultimate-based models for a given system, and by providing estimates for the main model parameters (ie the reactivity ratios). In addition, quantum chemistry can provide a simple and costeffective method for screening the reactivity ratios for a wide variety of monomer pairs (including hazardous and yet-to-be synthesized monomers). This may allow the identification of monomer pairs with suitable reactivity ratios for a given application. Chain Transfer. The relative rate of abstraction to propagation for a specific propagating radical is known as the chain-transfer constant, and is a key parameter governing the molecular weight of the resulting polymer. High level ab initio calculations allow chain-transfer constants to be predicted for a given propagating radical and transfer agent, and thus provide an effective means of screening large numbers of transfer agents. This in turn may assist in the selection of suitable transfer agents for a given polymerization system, and can also aid in our understanding of reactivity in the chain-transfer processes. Already ab initio calculations have been used to study intramolecular chain-transfer processes in free-radical polymerization (5,106), in order to provide estimates of the short-chain branching ratios. Another study investigated the kinetics and thermodynamics of the hydrogen abstraction by and from the monomer in ethylene polymerization (113), and demonstrated that abstraction from the monomer was the kinetically (but not thermodynamically) preferred process. There is also an enormous body of literature concerning ab initio calculations of hydrogen abstraction reactions in general, and in applications of relevance to other fields such as biochemistry (114,119). Controlled Radical Polymerization. In recent years, the field of free-radical polymerization has been revolutionalized by the development of techniques for controlling the molecular weight and architecture of the resulting polymer, including nitroxide-mediated polymerization (NMP) (120), atom-transfer polymerization (ATRP) (121), and reversible addition fragmentation chain transfer (RAFT) polymerization (96) (see LIVING RADICAL POLYMERIZATION). In order to control polymerization, these processes aim to minimize the influence of bimolecular

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termination processes via a delicate balance of the relative rates of two or more competing reactions. Quantum chemistry can assist in optimizing these processes by providing reliable values for the rate coefficients of the competing reactions, across a wide range of systems. More generally, quantum chemistry can help to provide a detailed understanding of the kinetics and mechanisms of the individual reactions, 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 used to obtain direct measurements of the rates of addition and fragmentation in model dithiobenozate-mediated systems, which in turn provided evidence that slow fragmentation is responsible for rate retardation in these systems (6). More recently, ab initio calculations have revealed a new and unexpected side reaction (β-scission of the alkoxy group) in certain xanthate-mediated systems, which may provide an explanation for the experimentally observed inhibition in those systems (7). The ab initio calculations also indicated that fragmentation was considerably faster in the xanthate-mediated systems (compared with other dithioester systems), because of the stabilizing influence of the alkoxy group on the thiocarbonyl product (7). In addition to these ab initio calculations, semiempirical methods have been used to survey the transfer enthalpies for a series of RAFT agents in styrene, methyl methacrylate, and butyl acrylate polymerization (122). Despite the low level of theory used, and the limited accuracy of the quantitative predictions at this level, the calculations were nonetheless shown to have some qualitative value in determining which transfer agents would show the best molecular weight control in RAFT polymerization. Understanding Reactivity via the Curve-Crossing Model. We have already seen that quantum chemistry can be used to calculate the absolute (and hence relative) rate coefficients for the individual reactions in free-radical polymerization, and is well suited to compiling systematic surveys of the rate coefficients for homologous series of reactions. When such calculations are combined with qualitative theoretical models, quantum chemistry can help to provide an understanding of trends in reactivity. It is beyond the scope of this chapter to provide an account of the main qualitative models and concepts in general theoretical chemistry. However, it is worth introducing one such model—the curve-crossing model (123)—as this model has been used extensively to provide a qualitative rationalization of the effects of substituents in some of the key radical reactions that occur in free-radical polymerization, including radical addition to alkenes (103) (ie the propagation step), and hydrogen abstraction (115,116,124) (ie, chain-transfer processes). The main qualitative features of the model and its application to free-radical polymerization are described below; for a 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 correlation model, the configuration mixing model, or the state correlation diagram) was developed by Pross and Shaik (123) as a unifying theoretical framework for explaining barrier formation in chemical reactions. It is largely based on valence bond (VB) theory (126,127), but also incorporates insights from qualitative molecular orbital theory (128). To understand the curve-crossing model, it is helpful to think of a chemical reaction as being composed of a rearrangement of electrons,

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accompanied by a rearrangement of nuclei (ie a geometric rearrangement). We can then imagine holding the arrangement of electrons constant in its initial configuration (which we call the reactant VB configuration), and examining how the energy changes as a function of the geometry. Likewise, we could hold the electronic configuration constant in its final form (the product VB configuration), and again examine 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 overall energy profile for the reaction, which is also plotted, is formed by the resonance interaction between the reactant and product configurations (and any other low lying configurations). State correlation diagrams allow for a qualitative explanation for how the overall energy profile of the reaction arises, and can then be used to provide a graphical illustration of how variations in the relative energies of the alternative VB configurations affect the barrier height. This in turn allows us to rationalize the effects of substituents on reaction barriers, and to predict when simple 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 of radical addition to alkenes. For this type of reaction, the principal VB configurations that may contribute to the ground-state wave function are the four lowest doublet configurations of the three-electron–three-center system formed by the initially unpaired electron at the radical carbon (R) and the electron pair of the attacked π bond in the alkene (A) (103).

The first configuration (RA) corresponds to the arrangement of electrons in the reactants, the second (RA3 ) to that in the products, and the others (R+ A − and R − A+ ) to possible charge-transfer configurations. The state correlation diagram showing (qualitatively) how the energies of these configurations vary as a function of the reaction coordinate is provided in Figure 11 (103). Let us now examine how this plot could be made for a specific system. The anchor points on these diagrams are generally accessible from quantumchemical calculations. For example, the energy difference between the RA configuration at the reactant geometry, and the RA3 configuration at the product geometry, is simply the energy change of the reaction. The energy difference between the RA and RA3 configurations at the reactant geometry is simply the vertical singlet-triplet gap of the alkene. At the product geometry, the RA − RA3 energy difference is also an excitation energy, but this time between the ground state of the doublet product and an excited doublet state (In fact, studies of radical addition to alkenes have ignored the influence of this excitation gap, without detriment to the predictive value of the results (103). This is probably due to the fact that the transition structure is very early in these reactions. However, in studies of hydrogen abstraction reactions, the RA and RA3 gaps at both the reactant and product geometries can be important. These are calculated as the singlet-triplet gap of the closed-shell substrate in each case.) The charge-transfer configurations can be anchored at the reactant geometry, where they are given as the energy for

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Fig. 11. State correlation diagram for radical addition to alkenes showing the variation in 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 the overall energy profile of the reaction.

complete charge transfer between the isolated reactants. For example, the energy difference between the R+ A − and the RA configuration at the reactant geometry would be given as the energy change of the reaction R + A → R+ + A − . It can be seen that the energy change of this reaction is simply the difference between the ionization 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, we generally interpolate the intervening points on the VB configuration curves qualitatively, on the basis of spin pairing schemes and VB arguments (123). At this point it should be stressed that the overall energy profile for the reaction is of course quantitatively accessible from our quantum-chemical calculations. The objective of the curve-crossing model analysis is not to generate the overall reaction profile but to understand how it arises—and a qualitative approach to generating the VB configuration curves is generally adequate for this purpose. If we consider first the product configuration, its energy is lowered during the course of the reaction because of bond formation between the radical and attacked carbon. At the same time, the relative energy of the reactant configuration increases because the π bond on the attacked alkene is stretched, and this is not compensated for by bond formation with the attacking radical. The energies of the charge-transfer

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configurations are initially very high in energy, but are stabilized by Coloumb attraction as the reactants approach one another. The overall energy profile for the reaction can be formed from the resonance interaction of these contributing configurations. In the early stages of the reaction, the reactant configuration is significantly lower in energy than the others and dominates the ground-state wave function. However, in the vicinity of the transition structure, the reactant and product configurations have similar energies, and thus significant mixing is possible. This stabilizes the wave function, with the strength of the stabilizing interaction increasing with the decreasing energy difference between the alternate configurations. It is this mixing of the reactant and product configurations which leads to the avoided crossing, and accounts for barrier formation. Beyond the transition structure, the product configuration is lower in energy and dominates the wave function. The charge-transfer configurations generally lie significantly above the ground-state wave function for most of the reaction. However, in the vicinity of the transition structure, they can sometimes be sufficiently low in energy to interact. In those cases, the transition structure is further stabilized, and (if one of the charge transfer configurations is lower than the other) the mixing is reflected in a degree of partial charge transfer between the reactants. Since the charge distribution within the transition structure is accessible from quantum-chemical calculations, this provides a testable prediction for the model. Using this state-correlation diagram, in conjunction with simple VB arguments, the curve-crossing model can be used to predict the influence of various energy parameters on the reaction barrier. For radical addition to alkenes (103), the barrier depends mainly on the reaction exothermicity (which measures the energy difference between the reactant and product configurations at their optimal geometries), the singlet-triplet gap in the alkene (which measures the energy difference between the reactant and product configurations at the reactant geometry), and the relative energies of the possible charge-transfer configurations. The effects of individual variations in these quantities are illustrated graphically in Figure 12. It can be seen that the barrier height is lowered by an increase in the reaction exothermicity, a decrease in the singlet-triplet gap, or a decrease in the relative energy of one or both of the charge-transfer configurations (provided that these are sufficiently low in energy to contribute to the ground-state wave function). A strategy for understanding the effects substituents in the barriers of radical reactions, such as addition, is to calculate these key quantities [ie, the reaction exothermicity, the singlet-triplet excitation gap of the closed-shell substrate(s), and the energy for charge transfer between the reactants], and look for relationships between these quantities and the barrier heights. In this way, one could establish, for example, the extent of polar interactions in a particular class of reactions. As noted at the beginning of this section, a number of such studies have already been performed for the key reactions in free-radical polymerization. The curve-crossing analysis of radical addition reactions, which is reviewed in detail elsewhere (94,99), indicate that, in the absence of polar interactions, the barrier height depends on the reaction exothermicity, in accordance with the Evans–Polanyi rule (129). However, for combinations of electronwithdrawing and electron-donating reactants, polar interactions are significant,

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Fig. 12. State correlation diagrams showing separately the qualitative effects of (a) increasing the reaction exothermicity, (b) decreasing the singlet–triplet gap, and (c) decreasing the energy of the charge-transfer configuration. For the sake of clarity the adiabatic minimum 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, the curve-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), and to examine why alkynes are less reactive to addition than alkenes (131). In these cases the differing singlet-triplet gaps of the alternative substrates are also important in governing their relative reactivities. Curve-crossing studies have also been applied to various types of hydrogen abstraction reactions (115,116,124), and, depending upon the substituents, the singlet-triplet gaps (in this case of both the reactant and product substrates), exothermicities, and polar interactions have all been found to be important in governing reactivity in these reactions.

The Future There are many more potential applications for quantum chemistry in free-radical polymerization. As computer power increases, one will be able to calculate rate coefficients 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 of polymerization processes, and predicting the properties of the resulting polymer (such as the molecular weight distribution, the copolymer composition, and the degree of branching). Quantum chemistry may also help in designing improved agents or processes for controlling polymerization, and in identifying their mechanisms. Of course, quantum chemistry may also be used effectively to study other types of polymerization processes, such as ring-opening processes. Provided care

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

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