Vol. 9 COMPUTATIONAL CHEMISTRY 319

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1 Vol. 9 COMPUTATIONAL CHEMISTRY 319 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.

2 320 COMPUTATIONAL CHEMISTRY Vol. 9 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 Schrödinger 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 Schrödinger 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).

3 Vol. 9 COMPUTATIONAL CHEMISTRY 321 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 Schrödinger equation 1 (17). Ĥ = E (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 modulus is a measure of the electron probability distribution. The term (Ĥ) in equation 1 is called the Hamiltonian operator and corresponds to the total kinetic ( ˆT) and potential energy ( ˆV) of the system. Ĥ = ˆT + ˆV (2) ˆT = h2 8π 2 i ( 1 m i ˆV = i< 2 x 2 i + 2 y 2 i ( ) ei e j j r ij ) + 2 z 2 i (3) (4) In these equations, the kinetic and potential terms are summed over all particles in the system (nuclei and electrons), and each particle is characterized by its Cartesian coordinates, its mass (m) and its electric charge (e). In addition, h is Planck s constant and r ij is the distance separating particles i and j. It can thus be seen that the only empirical information required to solve the Schrödinger equation are the masses and charges of the nucleus and the electrons, and the values of some fundamental physical constants. For this reason, quantumchemical calculations are referred to as ab initio ( without assumptions ) procedures. In applying the Schrödinger equation to problems of chemical interest, two major simplifications can be made. Firstly, the Hamiltonian described by equations 2 4 above is nonrelativistic, and it is valid provided that the velocities of the particles do not approach the speed of light. This is generally reasonable except for the inner shell electrons of heavy atoms, and in these cases corrections for relativistic effects can be made (18). Secondly, the Hamiltonian is further simplified by neglecting the kinetic energy contribution of the nuclei. This is known as the Born Oppenheimer approximation (19), and amounts to assuming that, since nuclear motion is much slower than electronic motion, the electron distribution depends only upon the positions of the nuclei and not on their motions. Thus, the Schrödinger equation can be rewritten as an electronic Schrödinger equation, as

4 322 COMPUTATIONAL CHEMISTRY Vol. 9 follows: Ĥ elec elec (r, R) = E eff (R) elec (r, R) (5) In this equation, elec is the electronic wave function (which depends on the electronic coordinates r, as well as the nuclear coordinates R) and Ĥelec is the electronic Hamiltonian in which the total kinetic energy is summed over all electrons only. The Born Oppenheimer approximation is valid provided that the ratio of electron mass to nuclear mass is sufficiently small. The objective of computational quantum chemistry is to solve this nonrelativistic, electronic Schrödinger equation for E eff (R), that is, the energy corresponding to a given arrangement of nuclei. If the Schrödinger equation is solved for all possible arrangements of nuclei in a system, one obtains the potential energy surface. (In fact, one yields the ground-state potential energy surface and any number of excited-state surfaces; however, for the remainder of this chapter we will focus largely on the ground states.) This contains the information required for the quantitative description of chemical structures and processes. For example, by identifying the nuclear coordinates corresponding to local minima in the potential energy surface, one can determine the equilibrium geometries of chemical species. By comparing the energies of alternative local minima, one can determine the relative energies of alternative conformers and isomers, and thus identify the preferred (ie, global minimum energy) structure. Alternative local minima on a potential energy surface may correspond to the reactant(s) and product(s) in a chemical reaction, and by comparing their total energies (including zero-point vibrational energy) one can calculate the reaction enthalpy at 0 K. The transition structure for a chemical reaction can also be identified from the potential energy surface as a first-order saddle point, that is, a stationary point in which the energy is a local maximum in one dimension (corresponding to the reaction coordinate), and a local minimum in all other dimensions. By comparing the total energies at the transition structure and the reactants, one can calculate the energy barrier for the chemical reaction. Having identified the equilibrium geometry of a molecule, one can calculate the second derivative of the energy at that point in the potential energy surface, and this yields the vibrational frequencies of the molecule (ie, the peak positions in its IR and Raman spectra) (see VIBRATIONAL SPECTROSCOPY). The vibrational frequencies can be used to calculate the total zero-point vibrational energy, and also the vibrational contribution to the enthalpy and entropy of a molecule at any temperature. Finally, the potential energy surface provides the information required to calculate the rates of chemical reactions. Solving the Schrödinger equation also yields the ground-state wave function for the given arrangement of nuclei. Since the square modulus of the wave function is related to the electron probability density, quantum-chemical calculations thus enable us to determine the distribution of electrons within a molecule, and hence the charge distribution. In fact, assigning charges to specific atoms within a molecule raises a number of philosophical problems, as under quantum mechanics electrons do not belong to any specific nucleus but are distributed throughout the molecule in molecular orbitals. To determine the charge on a specific atom within a molecule, it is necessary to define a scheme for dividing the molecular volume into atomic subspaces. A number of such schemes exist, and the main

5 Vol. 9 COMPUTATIONAL CHEMISTRY 323 ones include the Mulliken population analysis (8), the natural bond orbital analysis (20), and Atoms in Molecules (AIM) theory (21). In very broad terms, the former two schemes assign electrons to a nucleus if they are located in orbitals centered on that nucleus, but differ in the manner in which those orbitals are defined. By contrast, AIM theory uses a spatial definition for the atom, and generally provides the most rigorous and the most physically meaningful charges, though it is also the most computationally expensive method. A detailed discussion of wave function analysis methods is provided in Reference (9). In summary, by solving the Schrödinger equation, one has direct access to electronic-structure information and can calculate from first principles the mechanism, kinetics, and thermodynamics of chemical reactions. In principle, such calculations can be extremely accurate, relying only upon the validity of quantum mechanics, and values for the fundamental physical constants. However, in practice, the multi-electron Schrödinger equation has no analytical solution and numerical approximations must instead be made. These approximations are a potential source of error in the calculations, and it is thus important to understand their underlying assumptions. This article is primarily concerned with ab initio molecular orbital theory, which is one of the principal approaches to solving the electronic Schrödinger equation. In this section, the main principles of ab initio molecular orbital theory are detailed. Other approaches to solving the Schrödinger equation include density functional theory and semiempirical methods, and these are also briefly outlined. Finally, the additional theoretical calculations required in order to use the output of quantum-chemical calculations to obtain the rate coefficients for chemical reactions are described. The method of molecular mechanics (MM) which is an empirical procedure that is not based on solving the Schrödinger equation is described elsewhere (22 24). Ab Initio Molecular Orbital Theory In ab initio molecular orbital theory, the wave function is approximated using oneelectron functions or spin orbitals (χ). Each spin orbital is a product of a molecular orbital, ψ(x, y, z), which depends on the Cartesian coordinates of the electron, and a spin function, α or β, which corresponds to the spin angular momentum of the electron being aligned along the positive or negative z-axes, respectively. The molecular orbitals (ψ i ) are represented mathematically as a linear combination of a set of N one-electron functions called basis functions (φ µ ). ψ i = N c µi φ µ (6) µ = 1 In this equation, the coefficients c µi are called the molecular orbital expansion coefficients, and are optimized during the computational procedure. In chemical terms, one can think of the basis functions as the sets of constituent atomic orbitals, which mix to form the molecular orbitals of the molecule. In order to approach the exact solution to the Schrödinger equation, an infinite set of basis functions would be required, as this would introduce sufficient mathematical

6 324 COMPUTATIONAL CHEMISTRY Vol. 9 flexibility to allow for a complete description of the molecular orbitals. Of course, in practice a finite set of basis functions must be chosen, and this introduces a potential source of error to the calculations. Having formed a set of N linearly independent molecular orbitals, these orbitals must then be occupied to form the wave function. In ab initio molecular orbital theory the wave function is formed as the determinant of a matrix, which for an n-electron closed-shell system [ie, n is even (If n were odd, and the system was a doublet species (having one unpaired electron), we could instead form (n + 1)/2 orbitals, one of which would be singly occupied. The treatment of open-shell systems is discussed in more detail below.)] might be written as follows. ψ 1 (1)α(1) ψ 1 (1)β(1) ψ 2 (1)α(1) ψ 2 (1)β(1) ψ n/2 (1)α(1) ψ n/2 (1)β(1) ψ = (n!) 1/2 1 (2)α(2) ψ 1 (2)β(2) ψ 2 (2)α(2) ψ 2 (2)β(2) ψ n/2 (2)α(2) ψ n/2 (2)β(2) ψ 1 (n)α(n) ψ 1 (n)β(n) ψ 2 (n)α(n) ψ 2 (n)β(n) ψ n/2 (n)α(n) ψ n/2 (n)β(n) (7) This is usually abbreviated as follows. = χ 1 (1)χ 2 (2)...χ n (n) (8) In this determinant, which is known as the Slater determinant (25), the first row corresponds to all possible assignments of electron 1 to all of the spin orbitals, the second to all possible assignments of electron 2, and so on. The factor of (n!) 1/2 ensures that the total probability of finding an electron anywhere in space is 1. We have already seen that, from our set of N basis functions (where N should approach infinity), we can form N linearly independent molecular orbitals. However, in an n-electron system, we choose only n/2 of these orbitals (if n is even) to occupy in our Slater determinant. Clearly the most appropriate orbitals to occupy should be the n/2 lowest energy orbitals, and this is the basis of the well-known Hartree Fock theory (26). While Hartree Fock theory performs adequately in many cases, its use of a single determinant wave function can also frequently lead to considerable error. The problem is that this approach fails to account for Coulombic electron correlation. That is, it is assumed that each electron sees the other electrons as an average electric field and thus no instantaneous electron electron interactions are included. To approach the exact solution to the Schrödinger equation, the wave function must instead be represented as a linear combination of (an infinite number of) Slater determinants. In each of the additional determinants, one or more of the lowest energy orbitals are substituted with the (previously unoccupied) higher energy orbitals. In chemical terms, one might think of this as allowing for contributions to the wave function from the various possible excited configurations. This approach is known as configuration interaction (CI), and when all possible excited configurations are included the method is known as Full CI. Under ab initio molecular orbital theory, the exact solution to the Schrödinger equation could be obtained using the Full CI method in conjunction with an infinite basis set. Since this is impractical, computational methods must place restrictions

7 Vol. 9 COMPUTATIONAL CHEMISTRY 325 Fig. 1. Pople diagram (8) illustrating the dependence of the accuracy of a computational method on the basis set and the treatment of correlation. on the number of basis functions included in the calculation, and (in general) simplify the method for treating correlation. The combination of method and basis set chosen for a calculation forms the level of theory, and by convention is written as method/basis set (using the standard abbreviations for the particular method and basis set chosen). By increasing the size of the basis set and/or improving the method, one can improve the accuracy of the calculation but also its computational cost (see Fig. 1). Furthermore, the performance of a given level of theory can vary considerably depending upon the chemistry of the system, and the type of properties being calculated. The choice of level of theory is therefore very important, and a qualitative understanding of the approximations made at the various levels of theory is thus helpful in choosing appropriate theoretical procedures. For the remainder of the present section, the main qualitative features of commonly used methods and basis sets are described. The accuracy and applicability of these methods for studying the reactions of relevance to free-radical polymerization is discussed in a following section. Basis Sets. LCAO Scheme. A basis set is a set of one-electron functions, which are combined to form the molecular orbitals of the chemical species. This is known as the Linear Combination of Atomic Orbitals (LCAO) scheme. To approach the exact solution to the Schrödinger equation, an infinite set of basis functions would be required, as this would introduce sufficient mathematical flexibility to allow for a complete description of the molecular orbitals. In practical calculations, we must use a finite number of basis functions, and it is thus important to choose basis functions that allow for the most likely distribution of electrons within the system. This is achieved using basis functions that are based on the atomic orbitals of the constituent atoms of the molecule. For example, if a chemical system contained an oxygen atom, the chosen basis set would include functions describing each of the 1s, 2s, and three 2p orbitals of an oxygen atom.

8 326 COMPUTATIONAL CHEMISTRY Vol. 9 The basis functions that are typically used in practice are called Gaussiantype orbitals, and examples of their functional form for s-type and p y -type orbitals are shown below. g s (α, r) = ( ) 2α 3/4 e αr2 (9) π ( 128α 5 ) 1/4 g y (α, r) = y e αr2 (10) π 3 The exponent α determines the size of the orbital, and standard values have been determined for the different orbitals on the different atoms. Gaussian-type orbitals such as those shown in equations 9 and 10 are called primitive Gaussians. Contracted Gaussians are also used, and these consist of linear combinations of primitive Gaussians: φ µ = p d µp g p (11) The coefficients in this expression (d µp ) are fixed for the basis set of a given atom, and should not be confused with the molecular orbital expansion coefficients (C µi ) of equation 6, which are determined for a given system during the ab initio calculation. Basis sets thus contain sets of primitive or contracted Gaussians, which correspond to the atomic orbitals of the constituent atoms. Minimal basis sets typically contain exactly the number of functions required to accommodate the electrons in the system while maintaining the overall spherical symmetry. For example, a minimal basis set would contain a 1s function for each hydrogen atom, 1s, 2s, and three 2p functions for each carbon atom, and so on. However, in cases where the low lying unoccupied orbitals are frequently involved in bonding (eg, the 2p orbitals of Li or Be), popular minimal basis sets (such as STO-3G) include these additional functions as well. The problem with minimal basis sets is that the size and shape of the atomic orbitals are fixed, and all that can be varied in the quantum chemical calculation is their contribution to the overall molecular orbitals. However, in reality, the size and shape of the atomic orbitals (and especially the valence orbitals) can depend heavily upon the molecular environment. For example, in polar environments, we might expect a greater degree of asymmetry in the 2p orbitals of oxygen, compared with the same orbitals in nonpolar environments. One way of resolving this problem would be to create individual basis sets for each atom in each conceivable molecular environment, but this would not only be impractical, it would also defeat the purpose of performing ab initio calculations. Instead, a practical solution to this problem is to include additional basis functions of varying sizes and shapes. By varying their individual contribution to the overall molecular orbitals, one effectively introduces flexibility to the size and shape of the constituent atomic orbitals. Some of the main extensions to a minimal basis set are outlined in the following.

9 Vol. 9 COMPUTATIONAL CHEMISTRY 327 Fig. 2. Demonstration of how the mixing of two basis functions of different size is equivalent to introducing a single basis function with variable size. The basis functions are represented schematically as the boundary surface within which there is a 90% probability of finding the electron. In the first example, two s-type basis functions are mixed, while in the second example two p-type functions are mixed. (1) Double zeta, triple zeta, and quadruple zeta basis sets include additional sets of each basis function (2, 3, or 4 sets respectively), with each additional set having a different size. By varying the relative contribution of the alternative sizes to the overall molecular orbital, one effectively introduces a single function with a variable size (see Fig. 2). (2) Split-valence basis sets are a simplification to the double, triple, and quadruple zeta basis sets described above. Since the inner shell orbitals are not usually involved in bonding, and their energies are reasonably independent of their molecular environment, it is usually only necessary to include the extra basis functions for the valence orbitals. Basis sets that include different numbers of basis functions for the inner shell and valence electrons are known as split-valence basis sets. (3) Polarization functions are typically included in basis sets to allow for asymmetry in the electron distribution (see Fig. 3). They also allow for the participation of the low lying unfilled atomic orbitals in bonding. Polarization functions typically consist of basis functions corresponding to the low lying unfilled atomic orbitals. For example, the basis set for an oxygen atom might typically include a set of d-type orbitals, while those for hydrogen might include a set of p-type orbitals. Larger basis sets often include additional polarization functions of higher angular momentum. For example, the G(3df,2pd) basis set includes a set of f -type functions and three sets of d-type functions for each first row atom, as well as two sets of p-type functions and a set of d-type functions for each hydrogen atom. Fig. 3. Demonstration of the effect of polarization functions. In the first example an s-type and a p-type function are mixed, while in the second case a p-type and a d-type function are mixed. The mixing of higher angular momentum basis functions allows for an asymmetric distribution of electrons.

10 328 COMPUTATIONAL CHEMISTRY Vol. 9 (4) Diffuse functions are basis functions that have very large amplitudes, far from the atomic nucleus. Sets of diffuse functions are typically added to the basis set when describing species (such as anions) where the electrons are not held very tightly to the nucleus. (5) Effective core potentials (ECPs) are typically used to replace the basis functions of the inner shell electrons for atoms beyond the third row. This reduces the computational cost of the calculations, and is possible because the inner shell electrons on such heavy atoms are relatively unaffected by the molecular environment. Effective core potentials also include corrections for relativistic effects, which are significant for the inner shell electrons of heavy atoms. One of the most commonly used ECPs is called LANL2DZ. Two commonly used families of extended basis sets are the Pople basis sets (8) and the Dunning basis sets (27). It is worth making a few brief comments on their notation. (1) Pople basis sets have names such as G(3df,2p). The part refers to the fact that it is a split valence set with one copy of each basis function on the inner shell electrons, and three copies on the valence electrons. The 6 refers to the fact that the basis functions on the inner shell electrons consist of a contracted Gaussian-type orbital formed from 6 primitive Gaussians. The 311 part refers to the fact that one set of basis functions on the valence electrons are contracted Gaussians, each formed from 3 primitive Gaussians, while the other two sets of basis functions are primitive Gaussians. The + part refers to the fact that one set of s-type and p-type diffuse functions have been included for each heavy (ie, bigger than hydrogen) atom. The G basis set includes an additional set of s-type diffuse functions on hydrogen atoms as well. The functions in the bracketed part of the expression (3df,2p) are the polarization functions. The 3df part refers to the fact that 3 sets of d-type functions and one set of f - type functions are included for each heavy atom, and 2p implies that two sets of p-type functions are included for each hydrogen atom. Finally, the notations 6-31G and 6-31G are also frequently used. The first isan abbreviation for (d) and indicates that a set of d-type functions are included for each non-hydrogen atom, while stands for (d,p) and indicates that in addition to the d-type functions for the non-hydrogen atoms, a set of p-type functions are included for each hydrogen atom. (2) Dunning basis sets have names such as cc-pvnz. This notation stands for correlation-consistent polarized valence n-zeta. For a double zeta basis set, n is replaced by a D, for a triple zeta basis set, n is replaced by a T, for a quadruple zeta basis set, n is replaced by a Q, for a quintuple basis set we use a 5, and for a sextuple basis set we use a 6. When diffuse functions are included, an aug prefix is included in the name, as in augcc-pvtz. The cc-pvtz basis set generally has a performance similar to 6-311G(2df,p). A special feature of the Dunning basis sets is that they have been designed so the series DZ, TZ, QZ, 5Z, 6Z... systematically converges on the infinite basis set limit (27). This feature has been exploited in the

11 Vol. 9 COMPUTATIONAL CHEMISTRY 329 infinite basis set extrapolation procedure of Martin and Parthiban (28) (see below). Plane Waves. While standard ab initio calculations use the LCAO scheme almost exclusively, density functional theory (DFT) calculations (discussed later) sometimes use plane wave basis sets. These are solutions of the Schrödinger equation for a free particle and take the general form, g PW = exp[i k r] (12) where the vector k is related to the momentum p of the wave through p = k. One of the advantages of plane wave basis sets is that they can handle calculations with periodic boundary conditions, and they are thus used widely in solid-state physics. However, a disadvantage is that large basis sets are normally required to achieve acceptable accuracy and, as a result, plane waves are rarely used in calculations of molecular systems. For a detailed discussion of plane wave basis sets, and their application to Car Parrinello (29) ab initio direct dynamics techniques, the reader is referred to a review by Blöchl and co-workers (30). Methods. Hartree Fock (HF). The foundation of ab initio molecular orbital theory is the Hartree Fock (HF) method. As we saw above, it is based on a singledeterminant wave function (eq. 7) in which the electrons are assigned to the lowest energy orbitals. In fact the Slater determinant of equation 7 applies to closed-shell systems, that is, the n electrons occupy the n/2 orbitals in pairs of opposite spin. This method is known more specifically as restricted Hartree Fock (RHF). For open-shell systems (that is, those having one or more unpaired electrons), two approaches are possible. In restricted-open-shell Hartree Fock (ROHF), the determinant is formed from a set of molecular orbitals, which are either doubly or singly occupied, according to the multiplicity of the species. For example, for a radical (doublet) species, the determinant would be formed from (n + 1)/2 orbitals, and one of these would be singly occupied. This ensures that there is exactly one unpaired spin in the system, and the species is indeed a pure doublet. In unrestricted Hartree Fock (UHF), the α and β spin orbitals are defined and optimized separately. Thus there would be (n + 1)/2 occupied α spin orbitals, and (n 1)/2 occupied β spin orbitals. The principal differences between the RHF, ROHF, and UHF theory are illustrated in Figure 4. The UHF method offers some advantages over the ROHF method. In particular, the additional freedom in the wave function (with the provision for noninteger natural orbital occupation numbers) allows it to account in part for nondynamic electron correlation, and leads to lower energies and a better qualitative description of bond dissociation (9). Through its introduction of spin polarization effects, the UHF method can also provide a better (though by no means perfect) qualitative treatment of hyperfine coupling constants (31). However, a disadvantage of UHF is that the independent optimization of the α and β spin orbitals can result in nominally equivalent α and β orbitals having slightly different eigenvalues. In other words, a doublet species could have effectively more than one unpaired electron in the system. This physically unrealistic phenomenon is known as spin

12 330 COMPUTATIONAL CHEMISTRY Vol. 9 Fig. 4. Electron configuration diagrams highlighting the differences between restricted Hartree Fock theory (RHF), restricted open-shell Hartree Fock theory (ROHF), and unrestricted Hartree Fock theory (UHF). contamination, and can be a particular problem for the transition structures in the propagation steps of free-radical polymerization. Assessment studies (32,33) for such reactions have revealed that spin-contaminated UHF wave functions make poor starting points for Möller Plesset (MP) perturbation theory calculations (see below). By contrast, MP calculations based on ROHF wave functions (such as ROMP2) show improved agreement with higher level values. Interestingly, it has been found that for high level methods such as coupled cluster theory (see below), the choice of the starting wave function (ie, UHF vs ROHF) makes little difference to the final calculated energies and geometries for radical reactions (34). Having constructed a wave function, the remaining unknown parameters in the Schrödinger equation are the molecular orbital expansion coefficients (c µi ), as defined in equation 6. Determining the optimum values of these coefficients is thus the principal task of an ab initio calculation. It is beyond the scope of this chapter to outline the mathematical equations and numerical algorithms used to achieve this, but it is worth describing the general approach in qualitative terms. The basis of this procedure is the variational principle. It merely states that for any antisymmetric normalized function of the electronic coordinates (eg, a Slater determinant), the energy of this function is always greater than the expectation value of the exact wave function for the ground state. In other words, the exact wave function serves as a lower bound to the energies calculated from our approximate wave function, and the optimal coefficients c µi are merely those that minimize the energy. This principle leads to the Roothaan Hall equations (35), which are solved iteratively until the c µi coefficients converge. At convergence the coefficients are self-consistent, and hence the HF theory is also known as self-consistent field (SCF) theory. Configuration Interaction (CI). The description of the wave function using a single determinant (as in HF theory) fails to take electron correlation into account. To obtain the exact solution to the Schrödinger equation we instead need to construct the wave function as a linear combination of determinants, with each additional determinant corresponding to one of the various possible excited configurations obtained when electrons are promoted to the previously unoccupied higher energy orbitals (see Fig. 5). The resulting wave function is written as follows.

13 Vol. 9 COMPUTATIONAL CHEMISTRY 331 Fig. 5. Electron configuration diagrams showing the configurations corresponding to the HF wave function, and the various possible excited configurations. For this 4-electron system, the combination of single, double, triple, and quadruple excitations constitutes full CI. However, with an infinite basis set, there would be an infinite number of unoccupied orbitals (instead of the two shown here) to promote the electrons into, and thus an infinite number of determinants would be required to obtain the exact solution to the Schrödinger equation. = a s>0 a s s (13) 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 (a s ). 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 configurations. For this

14 332 COMPUTATIONAL CHEMISTRY Vol. 9 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 Schrödinger equation. However, these methods are still very computationally expensive. Möller Plesset (MP) Perturbation Theory. By convention, the correlation energy is simply the difference between the Hartree Fock energy and the exact solution to the Schrödinger equation. Rather than approximate the exact solution to the Schrödinger 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

15 Vol. 9 COMPUTATIONAL CHEMISTRY 333 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. Ψ λ = (0) + λ (1) + λ 2 (2) + λ 3 (3) + (14) E λ = E (0) + λe (1) + λ 2 E (2) + λ 3 E (3) + (15) Expressions relating terms of successively higher orders of perturbation are obtained by substituting equations 14 and 15 into the Schrödinger 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 Schrödinger 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)/ 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

16 334 COMPUTATIONAL CHEMISTRY Vol. 9 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).

17 Vol. 9 COMPUTATIONAL CHEMISTRY 335 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 High Method/Large Basis Set (16) 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.

18 336 COMPUTATIONAL CHEMISTRY Vol. 9 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

19 Vol. 9 COMPUTATIONAL CHEMISTRY 337 as CH 3 SC (CH 2 Ph)SCH 3, while the state-of-the-art Wn methods are restricted to smaller molecules, such as CH 3 CH 2 CH(CH 3 ). 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 Schrödinger 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. = j a j j (17) In this equation, the individual wave functions are formed from the lowest energy configuration, and various excited configurations of the Slater determinants, and the a j 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 c µi 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