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5…Chemical Bonding and Molecular Orbitals

5…Chemical Bonding and Molecular Orbitals

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P. Douglas et al.

electron density gives a bonding orbital of lower energy than either of the two

component atomic orbitals; where the orbitals overlap destructively between atoms

the decrease in electron density gives an antibonding orbital of higher energy than

either of the two component atomic orbitals. In addition, we may have orbitals that

are largely localised on one of the atoms such that there is no net interaction

between the two atoms, i.e. a non-bonding interaction. Thus, the combination of

any AOs on neighbouring atoms can be: bonding, antibonding, or non-bonding,

depending upon whether the overlap integral, the integral of the product of the two

atomic functions over space, is positive (bonding), negative (antibonding) or zero

(non-bonding). Consideration of the process shows that it is the symmetry properties of the orbitals that determine which of these interactions are possible. For any

combination of the signs of the orbitals which give a bonding interaction there is

also an opposite arrangement giving an antibonding interaction, so there is always a

pair of bonding and antibonding orbitals if the overlap integral is non-zero.

Although we have limited this discussion to molecular orbitals formed from the

combination of only two atomic orbitals, more orbitals can be involved, their

inclusion in the process being determined by their symmetry and energy. Figure

1.9 shows some molecular orbitals formed from combinations of s and p orbitals.

The important symmetry considerations for molecular orbitals are those concerning the internuclear axis. If the combination of the atomic orbitals is in the

direction of the axis joining the nuclei of the two atoms we term this a sigma (r)

orbital. Figure 1.9 shows how the combination of two s orbitals leads to a bonding

r orbital and an antibonding r* orbital (antibonding orbitals are indicated by the

superscript *). The r orbital has no nodal planes between the nuclei, while the r*

antibonding orbital has one nodal plane. A plot of the potential energy as a

function of the internuclear distance, the potential energy curve for the ground

state hydrogen molecule is given in Fig. 1.10. The minimum on the curve shows

both the molecular bond length (dH–H) and the bond dissociation energy (DH–H).

Sigma orbitals are also formed with p, d, and to a lesser extent, f, orbitals. With

f orbitals, as these are spatially closer to the nuclei and somewhat protected by the

outer s, p and d orbitals, the degree of bonding is more limited. This has important

consequences in photochemistry and spectroscopy, since these f orbitals have

characteristics more typical of atomic orbitals. This is particularly true for the

lanthanide (III) ions, where very sharp transitions involving the 4f orbitals are

observed in UV/Vis absorption and photoluminescence spectra. These have

applications in areas such as phosphors, lasers and photoluminescence probes (see

Chaps. 4 and 12). A second type of bonding involves combination of p or

d orbitals in which the internuclear axis is in a nodal plane. This is termed a pi (p)bond. When describing molecular orbitals we use Greek letters to represent similar

symmetry properties to those represented by Roman letters in atomic systems; thus

the s, p, and d orbitals in atoms become R; P and d orbitals in molecules. Similarly, for electronic states (which are the descriptions of the various multi-electron

combinations) the S, P, D states in atoms become R, p and D states in molecules.

There are the same number of antibonding p* orbitals as p orbitals.

1 Foundations of Photochemistry


Fig. 1.9 Formation of sigma bonding (r) and antibonding (r*) molecular orbitals (MOs) from

combination of s and p atomic orbitals (AOs) and of pi bonding (p) and antibonding (p*) MOs for

combination of p AOs


P. Douglas et al.

Fig. 1.10 Potential energy

curve for the combination of

two atoms to form a diatomic

molecule, such as H2

1.5.3 Orbital Hybridisation

The above ideas hold rigorously when only two atoms are involved (diatomic

molecules), but in polyatomic molecules, where three or more atoms are present, the

shapes of the molecules determined experimentally do not correspond to the orientation of the atomic orbitals. For example, water has two hydrogen and one

oxygen atom in a bent arrangement with an H–O–H bond angle of 104°. However,

in the simplest description, the central oxygen atom only has the 2p orbitals

available for bonding and these are at 90°. There is a convenient concept,

hybridisation (or mixing) of atomic orbitals, which, even though there are serious

theoretical criticisms of the concept [15], provides a useful way of visualising the

formation of molecules. The shapes of molecules are known from experiment, and

if we could solve the Schrödinger equation for molecules to the required level of

precision the result would be an energy minimum in keeping with the experimental

data. Hybridisation is a simple method to visualise the spatial result of a complete

Schrödinger analysis, using simple MOs without too much additional complexity.

Given the difficulty of the problem, the relatively small energy differences

involved, and the approximations used in MO theory, it is not surprising that a

model which gives a good representation of the spatial arrangement of atoms might

not necessarily give a good representation of other molecular properties [18]. With

the simplest molecular hydrides of oxygen (water, H2O), nitrogen (ammonia, NH3)

and carbon (methane, CH4), we assume that we mix the three 2p orbitals with the

one 2s orbital on the central atom to give four sp3 hybrid orbitals which are oriented

tetrahedrally. These can then combine with the hydrogen 1s orbitals to give two

(H2O), three (NH3) or four (CH4) sigma bonds. The other orbitals in water and

ammonia which are not involved in bonding will contain lone pairs of electrons and

be non-bonding. Hybridisation and bonding for methane is shown in Fig. 1.11 and

the redirected tetrahedral bond angle (109.288) is observed experimentally. The

bond angles in ammonia and water are slightly less due to the effect of the lone pair

1 Foundations of Photochemistry


Fig. 1.11 Results of hybridisation of s and p atomic orbitals. Hybridisation in methane is sp3

with four equivalent orbitals in a tetrahedral arrangement

non-bonding electrons which take up rather more space than a bond. Other forms of

hybridisation are possible. For example, with carbon we can use one 2s and two

2p orbitals to give sp2 hybridisation or one 2s and one 2p orbital to give sp

hybridisation. These hybrid orbitals can then form r bonds, while the remaining

2p orbitals can participate in p bonding with other atoms.

1.5.4 Electronic Occupation of Molecular Orbitals

Electron filling of MOs follows the same rules as described for AOs: lowest energy

orbitals first, with spins in parallel arrangements for degenerate orbitals, and

consideration of the spin-pairing energy for any decision about whether an electron

goes into a low energy orbital requiring spin-pairing or into a completely empty

higher energy orbital.


P. Douglas et al.

Fig. 1.12 Pi (p) molecular orbitals in ethene, butadiene and benzene

The number of bonds, the bond order, between atoms is given by:

Bond order ¼ ðnumber of bonding electrons À number of antibonding electronsÞ=2:


So, if when two atoms, X and Y, interact, the number of electrons in bonding

and antibonding orbitals is equal, then no bond is formed; if there is one more

bonding orbital filled than antibonding orbitals the bond order is 1, to give a single

1 Foundations of Photochemistry


bond, i.e. X–Y; when two more bonding than antibonding orbitals are filled, the

bond order is 2, i.e. a double bond X=Y; and when three, a triple bond is formed,


1.5.5 Conjugation and Extended Electron Orbitals

The overlap of AOs localised between neighbouring atoms gives rise to a localised

electron bond in which the electron wavefunction is determined by the electrostatic

attraction of only a few atoms, and the location of the electron is constrained to

within a relatively small volume around those atoms. Orbital overlap across a

number of atoms in a molecule leads to a different situation, one in which the MOs

extending across many atoms. The term for this behaviour is conjugation. Consider

the organic molecules ethene and butadiene (Fig. 1.12). In ethene, each carbon

atom forms three r bonds, one with the other carbon and one with each of the

hydrogen atoms. The remaining 2p orbitals on the two carbon atoms then form a p

bond. In butadiene, the terminal CH2 groups and two CH groups have sp2 hybrid

orbitals forming three r bonds. We now have 2p orbitals on each of the four carbon

atoms which can form four p bonds. As shown in Fig. 1.12, the combination of the

orbitals depends on the signs of the wavefunctions and will lead, in order of

increasing energy, to two bonding (p1, p2) and two antibonding (p*2, p*1) orbitals.

There are four electrons to go into these orbitals, and these will fill the two bonding

orbitals. Although the bond order between the middle two carbon atoms is less than

that for the end ones, the p electrons are now delocalised over the whole molecule.

Since the energy gaps between orbitals in conjugated structures lie in the UV

and visible spectral regions, and increasing conjugation decreases the energy

separation between the orbitals and, hence, increases the absorption wavelength,

conjugated structures make up many of the structures of interest to the photochemist. The orange colour of many plants and vegetables, such as carrots, is due

to the presence of the highly conjugated polyolefin b-carotene (see Chap. 4). A

particularly important case of conjugated systems is observed with certain planar

cyclic systems, such as benzene, which have 4n ? 2 electrons in p orbitals. This

provides increased delocalisation in the p orbitals and the systems are termed

aromatic (Fig. 1.12c).

1.5.6 Symmetry, Angular Momentum and Term Symbols

for Small Molecules

The symmetry labels and term symbols for small molecules are discussed in detail

in Ref. [14]. Here we illustrate the approach using H2 and O2 as examples.


P. Douglas et al.

Fig. 1.13 Electron correlation diagrams and ground state molecular orbital occupation for

a hydrogen (H2) and b oxygen (O2) molecules. For oxygen, the 1s orbitals are so low in energy,

i.e. so tightly held by their respective nuclei, that they do not contribute significantly to bonding

Figure 1.13 shows the filling with electrons of the molecular orbitals for the

ground state of the molecules H2 and O2. In H2, the two electrons enter the lower

energy r molecular orbital. To do this they must have opposite spins so the overall

spin is zero, the spin multiplicity, (2S ? 1), is one, and it is a singlet state. The

bond order is 1. The molecule has no node along the internuclear axis and, in

orbital terms, we say it has R symmetry. With centrosymmetric molecules we can

also indicate two other properties, the parity, discussed earlier, which is the effect

of inversion in a plane of symmetry. The sign of the wavefunction is unchanged by

this and we represent it with the symbol g. In addition, H2 also has planes of

symmetry. The symmetry operation associated with a plane is reflection, and if the

sign of the wavefunction is unchanged by this operation we give it the symbol +, if

it changes we represent it by a minus (-) sign. The electronic ground state of the

H2 molecule is represented 1R+g .

We will now consider the O2 molecule, which, in both ground and excited

states, is of major importance in photochemistry (e.g. see Chaps. 8, 9 and 15). The

electronic configuration of the oxygen atom is 1s2, 2s2, 2p4. The 1s2 electrons are

too tightly held to the nucleus to be involved in bonding and they remain as

essentially atomic orbitals. Since we have the same number of bonding and

antibonding electrons when molecular orbitals are formed from the 2s orbitals,

their net contribution to the bonding is zero. So, for bonding purposes, we only

need to consider the combinations of the 2p orbitals. The pair of p orbitals along

1 Foundations of Photochemistry


the internuclear axis (let us say pz) overlap strongly to give r and r * orbitals,

while each of the other pair of p orbitals, px and py overlap less strongly and give

two degenerate p, and p* orbitals. Introducing the electrons into the molecular

orbitals (Fig. 1.13), in the same way that we did with atoms, we arrive at the

important results that molecular oxygen has a bond order of 2, and that there are

two unpaired electrons in the p* orbital. With two unpaired electrons the spin

multiplicity of the lowest energy state, the ground state of oxygen is given by

(2S ? 1), i.e. 3, it is a triplet state. The orbital component has electron density

along the internuclear axis, and has R symmetry. O2 has both a centre and a plane

of symmetry. We showed in Fig. 1.6 the effect of inversion across the centre of

symmetry and reflection in the plane of symmetry on the signs of the wavefunctions. The sign is unchanged on inversion but changes on reflection. The ground

state is, thus, represented 3Rg . The lowest excited state has the two highest energy

electrons spin-paired in the same p* antibonding orbital, and therefore has S = 0.

This highly reactive species is termed singlet oxygen, and is represented 1Dg. There

is also a higher 1R+g excited state. The triplet character of ground state oxygen

means that it will be reactive with other species containing unpaired electrons,

most notably in photochemistry with other triplet states. The term oxidation historically is derived from reactions with oxygen, although its meaning now is much


1.5.7 Spectroscopic Nomenclature for More Complex


For polyatomic molecules, although we can always show the spin multiplicity, it is

no longer practical to give a complete orbital description. The degree of information we can present will depend on the symmetry of the molecule. For small,

symmetrical non-linear molecules, it is possible to describe the electronic states

using the effect of symmetry operations on the sign of the wavefunction [19]. The

electronic states are classified using the symmetry labels A, B, E and T, with

appropriate subscripts. These are obtained using Group Theory. We will not go

into detailed treatment of symmetry here. Appropriate treatments are given elsewhere [20]. However, we will consider the nomenclature for the electronic states

of the important case of the aromatic molecule benzene. This has a fairly high

symmetry and is said to belong to the point group D6h. A set of the possible

electronic states can be obtained from the so-called character tables given by

Group Theory for this symmetry. A number of electronic states are possible.

However, for reasons we will see shortly, only the ground-state and the lowest

electronic excited-states are of interest in photochemistry. The wavefunction in the

ground-state is fully symmetric in all operations and is termed A1. It has a spin

multiplicity of one (i.e. it is a singlet state), and has even parity. The full

description of this state is 1A1g. The lowest excited singlet state is classified 1B2u.


P. Douglas et al.

Benzene absorbs UV light around 256 nm due to an electronic transition between

these two states, which we indicate as a 1A1g ? 1B2u transition. A good experiment describing this is given elsewhere [21]. (Note: Photochemists tend to

describe the transition with the ground state first and the excited state second.

Spectroscopists often use the reverse order.)

With more complex molecules, it is not normally possible to indicate any

symmetry elements of the orbitals involved, and normally it is only possible to

describe these in terms of the total spin angular momentum. Typically, organic

molecules have a spin-paired singlet ground state, represented S0. They will have a

series of excited singlet (S1, S2, …, Sn) and triplet (T1, T2, …, Tn) excited states,

represented in order of increasing energy. However, for spectroscopic transitions it

may be possible to indicate the nature of the orbitals involved. For the above case

of benzene, the lowest energy transition is between a bonding p orbital and an

antibonding p* orbital, and can be represented as a p ? p* transition. The corresponding lowest excited state is described as a 1(p, p*) state.

1.5.8 HOMOs and LUMOs

The p molecular orbitals for butadiene and benzene are shown in Fig. 1.12. In both

cases, the orbitals are separated into p and p * orbitals of increasing energy. In the

molecular ground state, all the electrons are in the bonding p orbitals. For

chemistry and photochemistry, the two most important orbitals are the Highest

energy Occupied Molecular Orbital (HOMO; p2 for butadiene, p2 and p3 for

benzene) and the Lowest energy Unoccupied Molecular Orbital (LUMO, p*1 for

butadiene, p*1 and p*2 for benzene). Changes in the occupancy of these orbitals are

the lowest energy electronic processes possible: if an electron is added to a

molecule it goes into the LUMO, if an electron is removed it is removed from the

HOMO, the lowest electronic transition is from HOMO to LUMO, and even if

higher energy photons cause higher energy transitions, excited-state molecules

usually relax very quickly to the first excited-state, i.e. that corresponding to the

HOMO ? LUMO transition. For some molecules, HOMO-1 and LUMO ? 1

may also be important if higher excited-states play some role in the molecular


1.5.9 Molecular Excitons in Crystals and Large Systems

with Extended Conjugation

Two or more molecules close together can interact, leading to splitting of their

electronic states to produce delocalised electron systems. If these are in solid

crystals, the electronically excited states can be considered as chargeless quasi-

1 Foundations of Photochemistry


particles, termed excitons, which are considered to involve the separation of an

electron from a positively charged hole. These electron–hole pairs are capable of

migration over long distances [22]. In many inorganic systems, these are relatively

weakly bound, are delocalised over a large volume of the system, and are termed

Wannier (or Wannier-Mott) excitons. They are conventionally treated within a

semiconductor band model, as will be discussed in the next section. In other

systems, such as alkyl halide crystals or crystalline aromatic molecules, the

binding is much stronger, the exciton is more localised and is termed a Frenkel

exciton. This is particularly relevant for aromatic and other highly conjugated

systems and has important spectroscopic and photophysical consequences. There

is a third type, which is likely to be of particular importance for systems such as

conjugated polymers, where the electron and hole are separated by a distance

similar to the size of a molecule. These are termed charge-transfer excitons. The

theory of exciton interactions in organic molecular crystals was developed in

particular by Davydov [23], and applied by Kasha for molecular dimers and

polymers with various geometrical arrangements of chromophoric groups [24].

The concept of molecular excitons is important for treating electronic processes in

organic crystals, dye aggregates, conjugated polymers, etc. Electronic energy

migration in these systems can be treated in terms of exciton migration, either by

hopping from one site to another or rapid dissipation through delocalised bands.

1.5.10 Metallic Bonding

Wannier excitons correspond to a delocalisation of electrons (and holes) over a

solid structure. This corresponds to the model used in metals and semiconductors.

The picturesque view of metallic bonding is ‘‘a regular array of ions in a sea of

electrons’’. If many AOs are involved in creating MOs that extend across a great

many atoms, then a band structure is formed, in which there are two bands of

electron energy levels, one bonding, one antibonding, each made up of many

closely spaced energy levels, as shown in Fig. 1.14. Within a band there is a

pseudo-continuum of energy levels. If the band is partially filled or overlaps with

an empty band, electrons have access to many unoccupied levels, which are very

close together in energy (i.e. they have a high density of states), and can freely

move between these states. This accounts for the electrical conductivity of metals

and their metallic lustre. Metals are normally reflective across a wide range of

wavelengths because photons in the visible region penetrate deeply into the

strongly coupled band containing the conducting electrons and are typically

reflected. There are some important materials in which the electronic structure

allows transmission/reflection of certain frequencies and the metals are coloured,

such as copper and gold [25]. With certain conductors and semiconductors the

appropriate absorption/reflection is in the infrared and these transmit light in the

visible region. An important example is indium tin oxide (ITO) which is often used

as a transparent electrode material for a variety of optoelectronic devices including


P. Douglas et al.

Fig. 1.14 The formation of an energy band of orbitals as a result of adding N metal atoms to a

one-dimensional chain. The band contains a pseudo-continuum of energy levels, which exhibit

bonding character at the bottom of the band and antibonding character at the top of the band

dye sensitised solar cells (see Chap. 7). If the energy of a photon is high enough it

is possible to eject an electron completely from a metal. The energy required to do

this is termed the work function. This photoelectric effect can be used to detect

photons, and alloys of the alkali metals, which have relatively low work functions,

are used in the most important photodetector—the photomultiplier tube (PMT)

(see Chap. 14). Photomultipliers can be used to detect photons right across the

spectrum from the vacuum UV out to the NIR, but beyond this photons are of too

low an energy to overcome the work function of the alloy and different detectors

must be used.

1.5.11 Electronic Energy Band Structures

in Semiconductors

The ideas of electronic energy bands are also useful in describing electron

behaviour in semiconductors. In a semiconductor, energy levels are also band

structures, but in this case a low energy filled valence band is separated from a

higher energy, nominally empty, conduction band, by an energy gap termed the

semiconductor bandgap. If the bandgap is small enough, thermal excitation can

promote an electron from the valence band to the conduction band, where it is free

to move among the otherwise empty conduction band orbitals, thus giving the

structure some limited electrical conductivity. Excitation from the valence to

conduction bands can also be caused by absorption of a photon. This photoexcitation process, and the photochemistry resulting from formation of a mobile

electron (e-) and the mobile hole (h+), left in the valence band, are the dominant

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5…Chemical Bonding and Molecular Orbitals

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