A brief description of the computational methods used in this thesis and the discussion of some key methodological challenges tackled during its preparation are provided in Appendix 1. Subsequent appendices contain selected excerpts of the supplementary material for each publication appearing in Chapters 2 to 6. Due to space limitations, the full supporting information sections containing, in particular, Gaussian archive entries of optimised geometries for all studied species are not included in the printed material. These can be found either online or in the CD-ROM included inside the back cover of this thesis.
Appendix 1: Theoretical Procedures
Appendix 2: Additional Material for Chapter 2 Appendix 3: Additional Material for Chapter 3 Appendix 4: Additional Material for Chapter 4 Appendix 5: Additional Material for Chapter 5 Appendix 6: Additional Material for Chapter 6
the structures and properties of the chemical systems and the mechanisms of their reactions – a kind of knowledge that is often unobtainable directly and reliably from experiments. The ultimate aim of using quantum-chemical calculations in this thesis is to gain an in-depth understanding of the factors that define the stability and reactivity of the free radicals, and to apply it to the design of new and improved reagents and materials. This aim is achieved by using thoroughly benchmarked procedures at all stages of the ‘computational experiment’ including: the refinement of molecular geometry and associated conformational screening, the calculation of frequencies and the associated calculation of partition functions and gas-phase thermochemistry, the calculation of accurate single-point electronic energies, the calculation of accurate
Gibbs free energies of solvation, etc. This appendix provides a brief overview1 of the
corresponding methodology with a focus on specific techniques implemented in the studies, included in this thesis.
Ab Initio Methods. The term ab initio (Lat.) means ‘from the beginning’, and in the
context of quantum chemistry refers to a large family of methods originating in the
fundamental cornerstone concept of quantum mechanics – the Schrödinger equation2
(Eq. A1) – as applied to the chemical systems.
!!!!! (Eq. A1)
According to this time-independent Schrödinger equation (Eq. A1), for a stationary
state, the result of the Hamiltonian operator ! acting on the system’s wavefunction ! is
proportional to ! with a proportionality constant that is the energy of this state, !
(eigenvalue of the wavefunction). The molecular Hamiltonian has the general form of
! = {kinetic energy of electrons} + {electron-nucleus attraction} +
+ {electron-electron repulsion energy} + { kinetic energy of
nuclei} + {nucleus-nucleus repulsion energy} (Eq. A2)
Under the Born-Oppenheimer approximation,3 nucleic and electronic motions can be
decoupled due to the significant difference in their masses. This reduces Eq. A2 to an electronic problem that needs to be solved for a particular nuclei arrangement. However, even with this simplification, the equation can only be solved analytically for one- and two-particle systems, whereas certain assumptions and approximations are
required in the case of many-electron systems, thus giving rise to a range of ab initio
methods.1
Hartree-Fock theory (HF),4 also called the Self Consistent Field (SCF) method, is the
variational method that approximates an exact N-body wave function as a single-
reference Slater determinant of N orthonormal spin orbitals !!.5 The corresponding
Hartree-Fock equations have a form of
!!! !!!!! (Eq. A3)
where ! is the Fock operator and !! is the orbital energy. In an orthonormal basis set
(discussed below) these can be rewritten in a matrix form of Roothaan equations:
!" !!"# (Eq. A4)
where ! is the Fock matrix, ! is the matrix of molecular orbital (MO) coefficients and !
is the overlap matrix. These equations are solved iteratively starting with some guess for
! until convergence (i.e. until the eigenvalues are below the specified threshold).
There are several ‘flavours’ of the HF method depending on how the spin orbitals are
defined. Restricted Hartree-Fock method (RHF) is usually applied to the closed-shell
systems in which all electrons are paired, and thus " and! electrons of an electron pair
are assigned to the same spatial MO (energetically degenerate). Free radicals, on the
other hand, are open-shell systems, i.e. have uneven numbers of spin-up and spin-down
electrons. For such species, the Unrestricted Hartree-Fock method (UHF) can be used
instead, as it allows for separate sets of MOs for " and ! electrons. However, UHF
wave functions are often heavily spin-contaminated (i.e., have an excessive mixing of
excited states into the ground state) – a drawback that is overcome in a Restricted Open-
shell Hartree-Fock method (ROHF), which, in simple terms, first pairs all " and !
electrons similarly to RHF and only then allocates unpaired electron(s) in separate
MOs.6 Throughout this work, all of these methods were used and the degree of spin
contamination was monitored via the value of S2 operator.
While HF theory is extremely useful in optimising the initial orbital guess, it cannot
give accurate final energies due to a complete neglect of an electron correlation of the
opposite spin electrons. This contribution can be evaluated using one of the so-called
post-Hartree-Fock methods: perturbation theory (in particular, Moller-Plesset second-
order correction, or MP2),7coupled clusters theory (in particular, with single, double
8.1.1). Moreover, while truncated CI methods are variational (i.e., they yield energy that is always higher than the eigenvalue of a true wavefunction), they are not size-
consistent (i.e., the energy of a supramolecular complex of two species is not equal to
the sum of the energies of these species at an infinite separation). In contrast, MPn and
CC methods are size-consistent but non-variational.
Figure 8.1.1 Pople’s diagram showing the relationship between the accuracy of the computational method and its cost (FCI stands for full configuration interaction).
Basis Sets. A basis set is a set of functions (typically, atomic orbitals), the linear combination of which represents a molecular orbital:
!! ! !!!"!! (Eq. A5)
where !! is the molecular orbital (a spatial function) and !! is the atomic orbital (a
basis function). The two principal types of basis functions are Gaussian-type orbitals
(denoted ‘G’) that involve a !!!!!
term, and Slater-type orbitals (denoted ‘STO’) that
involve a !!!" term (here r is the distance from the nucleus and ! is called an
exponent). While the latter afford better accuracy with a relatively smaller number of them, the associated two-electron integrals for non-linear molecules are problematic. As a result, Gaussian-type basis functions are used more widely, however a larger number
of them is required. In a minimal basis set, one basis function is used for each orbital in
each occupied subshell of an atom in its ground state. An example of these is an STO- 3G basis set. However, more than one basis function per orbital is necessary for
accurate calculations – when each orbital is represented by two basis functions, such
basis is called a double-zeta basis set. Because the actual chemistry mostly involves the
valence electrons, it is rational to use only one basis function for the core, but more than
one – for the outer shell. Such basis sets are known as split-valence and follow the
notation x-yzG, where x is the number of primitive Gaussians comprising the core’s
basis function, and y and z are the numbers of primitive Gaussians comprising each of
the two basis functions (in this case for a double-zeta basis) of the valence shell. Such basis sets, used in most of this work, are often referred to as Pople’s basis sets.10 In
addition, a common practice is to add higher angular momentum functions (polarisation
basis functions), as well as basis functions with smaller exponents (diffuse basis
functions, essential for calculations on anions). Correlation-consistent basis sets,
developed by Dunning, are optimised at a post-HF level (e.g., CISD) and are designed
to systematically approach the complete basis set (CBS) limit.11 Similar rational is often
implemented in the composite high-level methods, which are discussed further below. Finally, to reduce the computational cost (and introduce relativistic effects) of the calculations involving heavy atoms, core electrons can be replaced with an effective
pseudopotential, or an effective core potential.
Density Functional Theory. Another approach to solving the time-independent Schrödinger equation (Eq. A1) involves Density Functional Theory (DFT). According
to the Hohenberg-Kohn theorem, the electronic energy of the ground state of a system
can be determined exclusively from its (exact) one-electron density, !!!!!. It therefore
follows that the exact ground-state energy can be calculated from the one-electron
density. Unlike ab initio calculations, which become increasingly demanding for larger
and larger number of electrons, the one-electron density depends on a constant number
of variables. Unfortunately the Hohenberg-Kohn theorem is an existence proof, rather
than a constructive proof, and the exact functional is not known. Correspondingly, DFT calculations are significantly less expensive computationally. The mathematical
apparatus behind the DFT is based upon the Kohn-Sham equations, i.e.:
! ! ! ! ! !!!"#$ ! !!!"# ! !!!" ! (Eq. A6)
The first three terms in Eq. A6 are the contribution from kinetic, electron-electron repulsion and electron-nuclei attraction energies, for which the functional forms are
defined. The only unknown in the Kohn-Sham equations is a so-called exchange-
correlation function, !!", defining which as accurately as possible being the greatest
a homogeneous electron gas (HEG). The exchange-correlation functional can be split into the exchange and correlation parts, and the exchange-energy density of HEG can be expressed analytically. Nowadays, many density functionals are hybrid in that they combine a certain percentage of various exchange and correlation functions, exact HF
exchange and/or diverse empirical corrections.12 As a result, performance of a given
functional is limited by its benchmarking test set and the fortuitous error-cancellation.13
In this thesis, several DFT methods, including a three-parameter exchange-correlation
hybrid functional, B3LYP,14 and a global-hybrid meta-GGA (generalized gradient
approximation) functional that incorporates dispersion implicitly, M06-2X,15 are
generally used in the geometry optimisations and frequency calculations, but not for accurate electronic energies except where their accuracy is supported by the benchmarking against experimental data and higher-level calculations.
Composite High-Level Methods. The accuracy of high-level ab initio methods can often be achieved at a fraction of a cost with so-called composite procedures. They
involve a combination (via additivity and/or extrapolation schemes) of different ab
initio and DFT methods of varying cost and accuracy in conjunction with basis sets of different sizes, as well as certain theoretical and empirical corrections. In general, a protocol of such a composite method includes optimisation of geometry and frequencies calculation (often using DFT), followed by a series of higher-level single-point energy calculations. Several examples of composite methods, used in this thesis, are given
below (Ee is the electronic energy, SO is spin-orbit correction, HLC is high-level
correction).
G3: Ee = MP4/6-31G(d) + [MP4/6-31+G(d) – MP4/6-31G(d)] +
[MP4/6-31G(2df,p) – MP4/6-31G(d)] + [QCISD(T)/6-31G(d) – – MP4/6-31G(d)] + [MP2(full)/G3Large – MP2/6-31G(2df,p) –
– MP2/6-31+G(d) + MP2/6-31G(d)] + SO + HLC (Eq. A7)
in conjunction with HF/6-31G(d) geometries and frequencies.16
G3(MP2)-RAD: Ee = U(R)CCSD(T)/6-31G(d) +
+ MP2/G3MP2Large – MP2/6-31G(d) + SO + HLC (Eq. A8)
G4: Ee= MP4/6-31G(d) + [MP4/6-31+G(d) – MP4/6-31G(d)] + + [MP4/6-31G(2df,p) – MP4/6-31G(d)] + [CCSD(T)/6-31G(d) – – MP4/6-31G(d)] + [MP2(full)/G3LargeXP – MP2/6-31G(2df,p) – – MP2/6-31+G(d) + MP2/6-31G(d)] + [ HF/limit –
– HF/G3LargeXP] + HLC (Eq. A9)
in conjunction with B3LYP/6-31G(2df,p) geometries and frequencies.18
CBS-RAD: Ee = HF/CBSB3 + [MP2/CBSB3 – HF/CBSB3] +
+ [CCSD(T)/6-31+G(d’) – MP4/6-31+G(d’)] + [MP4/CBSB4 –
– MP2/CBSB4] + %E(CBS) + %E(INT) + %E(EMP) + %E(SPIN) (Eq. A10)
in conjunction with B3LYP/6-31G(d) geometries and frequencies.19
W1: Ee has contributions from CCSD(T)/aug-cc-pVDZ+2df, CCSD(T)/aug-cc-
pVTZ+2df, CCSD/aug-cc-pVQZ+2df, CCSD(T,fc)/MTsmall and
CCSD(T,full)/MTsmall in conjunction with B3LYP/6-31G(d) geometries and
frequencies.20
All of these methods are generally capable of delivering accuracy in kcal or kJ range, at least for single-reference systems. However, these methods are still limited to a fairly small system size, e.g. 18-20 heavy atoms in the case of G3(MP2)-RAD. For larger
species, a multi-layer ONIOM-type approximation21 can be used instead. In this
approach, the remote substituent effects are treated at a lower-cost level of theory, such as MP2/6-311+G(3df,2p), while the actual chemistry is modelled using a more expensive composite method. In this way high-level calculation is performed only for
the small reactive core, defined as the reaction centre, its "-substituents and all
conjugated fragments, and hydrogen atoms in place of all remote substituents. The final
ONIOM energy of the full system is given by the following expression:13a,22
!!"#!$ ! !!"##!"#!!!
!"#$!!"!!!!"#$!"#! (Eq. A11)
Conformational Search. While it is clearly important to ensure that the level of theory used to obtain the electronic energy (and derivative properties) of the species is sufficiently accurate, this effort is essentially meaningless if the geometry, in particular the conformation of the investigated system is unrealistic. While in theory every molecular system exists as an ensemble of different conformers, in practice it is
low-lying conformers via one-dimensional hindered rotor partition functions.23 For
relatively small molecules it is generally feasible to perform a full conformational
search at the desired resolution (typically, 120°), i.e. geometry optimisation of each of
its 3N conformers (here N is the number of rotatable bonds), as shown in Figure 8.1.2a.
However, the potential energy landscape of the conformational space becomes increasingly complicated for larger species with many degrees of freedom, and it is extremely expensive computationally to explore it fully. One approach to this problem is to consider rotations independently of each other and identify the lowest energy
conformation of each rotation in sequence. This rational, known as the linear search
(Figure 8.1.2b), is much less expensive, however it is also not reliable, in particular for systems with weak intramolecular interactions, such as hydrogen bonding. Therefore, in
this thesis another protocol, developed by the Coote group, – the Energy-Directed Tree
Search24 (EDTS, Figure 8.1.2c) – has been employed instead. This method has been shown to yield reliable results at a fraction of a cost of a full search.
Figure 8.1.2 Various conformational searching algorithms (this image was prepared by me for Ref. 25).
It is also crucial to identify an appropriate level of theory to use in the conformational searching. In this case, ‘appropriate’ refers to a fine balance between the
relative accuracy of the geometry optimisation and resulting energy of the system, and the cost of such calculation. In this thesis, either a B3LYP or, for systems with significant dispersion interactions, an M06-2X method was used in conjunction with 6- 31G(d) or 6-31+G(d) (for calculations on anions) basis sets.
Finally, the conformational landscape can be greatly affected by solvation, e.g. intramolecular hydrogen bonding is often disrupted in highly polar solvents (water),
while intramolecular !-stacking disappears in toluene or styrene. Therefore, when such
effects are anticipated, it is rational to perform the conformer searching in solution to account for them.
Gas-Phase Thermochemistry. Once the lowest-energy conformer is identified, its geometry is improved (if necessary) and used to perform a so-called frequency calculation. Firstly, frequency calculation is necessary to ensure that the located geometry is a true minimum (a zero-order stationary point on PES with no imaginary frequencies) or a transition state (a first-order stationary point, or a saddle point with one imaginary frequency). Secondly, it yields information, subsequently used to evaluate various thermochemical quantities using statistical thermodynamics and
ultimately obtain realistic energetic measures – enthalpy !! and Gibbs free energy !!
at a given temperature !:
!! !!!!!!! ! !
!!!"#$!!"! !!!! (Eq. A12)
where !! is the entropy, !
! is the electronic energy, ZPVE is the zero-point vibrational
energy correction and !"! is the thermal correction.26ZPVE is a residual energy of the
system in its ground state, non-zero at 0 K due to the wave-like nature of all quantum mechanical systems under the Heisenberg uncertainty principle. The thermal correction (internal thermal energy) and entropy have contributions from translation, electronic motion, rotation and vibration, which can be evaluated using standard textbook formulae for statistical thermodynamics. Furthermore, it is important to choose an appropriate treatment of the low-frequency modes – either under the rigid-rotor
harmonic oscillator (HO) or the hindered rotor (HR) approximation.27 In most systems
studied in this thesis HO treatment produced sufficiently accurate results due to systematic error cancellation, however for modelling the energetics of RAFT polymerisation in Chapter 4 the HR treatment was necessary.
Geometry optimisations and frequency calculations are trivially performed using DFT methods, such as B3LYP and M06-2X. The accuracy of the geometry can thus be
Solvent Effects. Gas-phase energetics are valuable in their own right, e.g. to analyse the reactivity trends or interpret mass spectroscopy experiments, however it is often necessary to model chemical processes in condensed phase. In this thesis, solvation effects on reaction energies were evaluated using the following thermocycle:
where #Ggas is the reaction Gibbs free energy in the gas phase, #Gsoln is the reaction
Gibbs free energy in solution and ##Gsolv is the Gibbs free energy of solvation of each
species. This rational is only seemingly simple. Firstly, the thermocycle above assumes that the effect of the geometry change on reaction energy is negligible, which is not necessarily the case if the investigated system involves strong intramolecular interactions and dramatically changes the conformation upon solvation. Secondly, the thermodynamic quantities involved – gas-phase thermochemistry and solvation Gibbs free energy – are calculated assuming different standard-state concentrations, namely 1
atm vs. 1 mol L-1, and this needs to be corrected for each species by RTln(RT/P), where
P is the standard pressure of 1 atm. But the most complicated quantity here is the
solvation Gibbs free energy itself. Its very physical meaning is still debated in the
literature, but in simple terms ##Gsolv is a measure of how much energy is required to
transfer the species from the gas phase into the solution. This transfer includes the reorganisation and polarisation of both mediums (long-range effects), penetration through the phase interface, as well as the specific solute-solvent interactions (short- range effects).
Several approaches have been developed to describe the solvation and predict the
##Gsolv. They can be broadly divided into explicit, where the solvent is modelled as the
actual individual molecules surrounding the solute, and implicit, which model solvent as
a continuous medium with certain dielectric properties, and solute is placed in this field; the two approaches can also be combined. Only the implicit solvent models were used in this thesis and hence are discussed briefly below.
Agas + Bgas Cgas
Asoln + Bsoln Csoln !Ggas
!Gsoln
The Self-Consistent Reaction Field (SCRF) models solvent as a continuous universal
polarisable medium, characterised by the dielectric constant #. The solute molecule is
placed inside the cavity within the solvent reaction field (Figure 8.1.3), defined by the topology of this cavity and the relative permittivity of a solvent. Constructing the cavity is shaped by several parameters, including the surface type (solvent-accessible surface, Van der Waals surface, etc.), set of the atomic radii (Bondi, UAKS, etc.) and the scaling
factor " for these radii. Even the small changes in these parameters can greatly affect
the results, as is discussed in Chapter 5. The Gibbs free energy of solvation is expressed