presented to provide the reader with enough insight to assess the limitations of the theoretical models from which the analytical tools emerge, without submerging him/her in mathematics. Though not always explicitly referred to, Refs. [26] and [71] often served as a guide in writing this chapter.
To put things in the right perspective, following remark should be made: DFT calculations are by all means superior to these older theoretical models when one wants to assess specific radical structures – as this doctoral thesis testifies. This is most pronounced when the radical geometry differs substantially from the pristine crystal structure. DFT calculations allow unambiguous identifications that otherwise would simply not be possible (or at least much more difficult) and often yield information that cannot be (directly) obtained experimentally. The principal use of the analytical tools lies in a priori reducing the number of plausible radical models and providing the necessary basic insight in the physics involved.
Some chemical vocabulary is important to avoid confusion and misinterpre- tations in this and the following chapters. Therefore, a few basic terms and concepts are first (re)defined.
4.2
Basic concepts
4.2.1 Nomenclature of radicals
Carbohydrates are compounds entirely consisting of hydrogen, carbon and oxygen. Hydrocarbons contain only hydrogen and carbon atoms, and are divided into two classes: aromatic compounds, which contain e.g. a benzene ring, and aliphatic compounds, which do not contain aromatic constituents. The term alkyl radical refers to a hydrocarbon radical with chemical formula CnH2n+1, e.g. C•H2-CH3. In the literature, however, the term alkyl radical
is regularly used for any carbon-centred radical C•(R1)(R2)(R3) with R1-R3 consisting of C, O and H, and we will adopt this convention. In a hydroxyalkyl radical at least one hydroxy group (i.e. an OH group) is bound to the radical centre, e.g. C•HOH-CH3. The allyl radical has the structure H2C=C-C•H2 ↔
H2C•-C=CH2. With alkoxy radical we refer to any oxygen-centred radical R1-
C(R2)-O•in which R1 and R2 may consist of C, O and H.
Figure 4.1 depicts a typical alkyl radical. The unpaired electron is formally localised in a 2pz orbital on a certain carbon. This carbon is denoted Cα and
is referred to as the α carbon. A proton attached to an α carbon is an α proton (Hα, one bond away from Cα). An adjacent carbon is referred to as a β carbon
(Cβ) and a proton bound to it as a β proton (Hβ, two bonds away from Cα).
4.2. Basic concepts
(four bonds away from Cα), . . . This nomenclature is also used
• for alkoxy radicals and for carbon-centred radicals containing one or more oxygen atoms. For instance, the hydrogen proton of a hydroxy group attached to a β carbon can be referred to as a γ proton (cf. Figure 4.1).
• in certain radicals where the spin density is delocalised over several atoms (e.g. on a ring oxygen or on a carbonyl group). The Cα carbon
is then the one carrying the largest part of the spin density.
Figure 4.1: Nomenclature of carbons and protons in a typical alkyl radical.
4.2.2 Electronic configuration of carbon and oxygen
The ground state valence electron configurations of carbon and oxygen atoms are shown in Figure 4.2. Carbon has only two 2p electrons that can engage in chemical bonding. The typical and well-known tetrahedral configuration of saturated carbon compounds is explained by hybridisation: the 2s orbital mixes with the three 2p orbitals, which yields four equivalent sp3hybrid orbitals that make angles of 109.5◦ with each other (sp3 hybridisation). Each sp3 orbital contains one electron and can engage in chemical bonding.
In an alkyl radical one of the carbon valence electrons is unpaired and does not participate in a chemical bond. Usually the carbon centre becomes approximately planar: the three bonds are in a plane and make angles of approximately 120◦. This can be explained in terms of sp2 hybridisation (Figure 4.2): the 2s orbital mixes with two 2p orbitals, yielding three sp2
4.2. Basic concepts
hybrid orbitals containing one electron each, while the remaining electron occupies one of the ’original’ 2p orbitals. Conventionally the unpaired electron is assigned to the 2pz orbital. We will adopt this convention throughout
this chapter. The hybridisation schemes for oxygen can be constructed in a completely analogous fashion (Figure 4.2).
The sp3 and sp2 orbitals can, in turn, mix with atomic orbitals of adjacent carbons to form a σ bond. (Figure 4.3). All single covalent bonds (e.g. C-H
Figure 4.2: The valence electronic configuration and orbitals of carbon and oxygen atoms in their (atomic) ground state (left), in sp3 hybridisation (middle) and in sp2 hybridisation (right). The sp3 orbitals are in a tetrahedral configuration, making angles of 109.5◦ with each other. In the sp2hybridisation, the hybridised orbitals are in a plane, making angles of 120◦with each other, and the 2pzorbital is perpendicular to this plane.
bonds) are σ bonds. The 2pz orbital of an sp2hybridised carbon can overlap
with the 2pz orbital of an adjacent oxygen or carbon and thus give rise to a π
bond (in addition to the σ bond). This is the case in, e.g., ethene (H2C=CH2,
Figure 4.3) and in a carbonyl group (C=O).2We note that also sp hybridisation can occur, where the two remaining 2p orbitals can both participate in π bonds. This is the case in, e.g., ethyn (HC≡CH).
A σ bond orbital results from head-on overlap of atomic orbitals and it has
4.2. Basic concepts
cylindrical symmetry around the bond axis with maximum electron density on this axis. A π orbital results from sideways overlap and is shaped like two ’arches’ with a nodal plane passing through the nuclei of both atoms. π bonds are weaker than σ bonds due to the lower degree of overlap of the constituting atomic orbitals in the former.
In the literature the symmetry of the lone-electron orbital (LEO) often de- termines the nomenclature, and not the distinction between atomic and molecular orbitals. For instance, atomic 2p orbitals can be referred to as π or 2pπ orbitals, even if they do not participate in a π bond. Similarly, ’π radical’
Figure 4.3: Electronic orbitals of the valence electrons in an ethene molecule. The carbons are sp2hybridised and have a σ as well as a π bond
or ’π-electron radical’ is often used in the literature for alkyl radicals (where the unpaired electron is mainly localised at a single atom) because the 2pz
orbital is approximately perpendicular to the hydrocarbon skeleton. This can be confusing but sometimes this habit is so deeply rooted in the literature that we have adopted it.
In an isolated atom the angular parts of the orbital wavefunctions are the spherical harmonics |L, MLi.3 These are eigenfunctions of the operators
ˆ
~L2 and ˆ~Lz with eigenvalues L(L+1) and ML respectively.4 Now, organic
molecules/crystals typically have a low local symmetry and the surrounding nuclei and electrons exert different (mainly electrostatic) forces in different directions. The result is that the (2L+1) orbital degeneracy present in the isolated atom is lifted. Since orbitally nondegenerate states have zero angular momentum (L = 0),5 this phenomenon is referred to as orbital quenching. Therefore it makes more sense, when dealing with the electronic configuration
3The spherical harmonics are often denoted Y
L MLor Y ML L . 4These operators are defined as ˆ~L=∑
i~Lˆiand ˆLz=∑iˆLz,i.
5This property of orbitally nondegenerate states can be proven mathematically [30]: essentially, nondegenerate states must be real and the only possible eigenvalue of the purely imaginary operator ˆ~Lioperating on a real eigenfunction is zero.