1

The Born-Oppenheimer approximation supposes that the nuclei, being so muchheavier than the electron, move relatively slow and may be treated as stationary as theelectron moves around them. We can therefore choose the nuclei to have a definite separationR and solve the Schrödinger equation for the electron alone, which is far easier than trying tosolve the complete Schrödinger equation by treating all three particles on an equal footing.The approximation is quite good for ground-state molecules, for calculations suggest that thenuclei in H2

+ move through only 1 pm while the electron speeds about 1000 pm, and so theerror of assuming that the nuclei are stationary is small. In molecules other than H2

+ thenuclei are even heavier, and the approximation is generally (but not always) better.Exceptions to its validity include certain excited states of polyatomic molecules and theground states of cations; both types of species are important when considering photoelectronspectroscopy and mass spectroscopy.

The Born-Oppenheimer approximation reduces the full problem to a single-particleSchrödinger equation for an electron in the field of two stationary protons at a separation R.

The potential energy of the electron is V =−e2

4πε01rA

+1rB

Introduction to Molecular StructureThe Born-Oppenheimer approximation

rA and rB are the electron’s distances from the nuclei A and B. This expression can be

used in the one-particle Schrödinger equation

−

h2

2me

∇2ψ +Vψ = Eψ

and exact solution can be obtained.

2

The total energy of the molecule at the selected separation R is then thesum of the eigenvalue E and the nucleus-nucleus repulsion

Vnuc−nuc =+e2

4πε0×1R

The Born-Oppenheimer approximation allows us to decide on aparticular internuclear separation, and to solve the Schrödinger equationfor the electron distribution. Then we can choose a different separation

and repeat the calculation, and so on. In this way we can calculate how the energy of themolecule – with the nuclear kinetic energy ignored – varies with bond length (and, in morecomplex systems, with bond angles and dihedral angles too), and obtain the molecularpotential energy curve (surface). We can then identify the equilibrium bond length of themolecule with the lowest point on this curve.

The valence bond theoryVB theory was the first quantum mechanical theory of bonding to be developed. It has

undergone much less computational development than molecular orbital theory, but thelanguage it introduced (spin pairing, σ and π bonds, is widely used.

3

The hydrogen molecule H2 – the simplest molecule with an electron pair. If H atoms are far part, we can write the wavefunction as

€

ψ = ψH1sA r1( )ψH1sB r2( ) Electron 1 is on atom A and electron 2 is on atom B. For simplicity we can write

€

ψ = A 1( )B 2( ) .When the atoms are close, it is impossible to know to what atom the electrons belong. Therefore, an equally valid description is

€

ψ = A 2( )B 1( ) . When two outcomes are equally probable, we should describe the true state of the system as a superposition of the wavefunctions for each possibility:

€

ψ = A 1( )B 2( ) ± A 2( )B 1( ) For H2, the lower in energy combination appears to be

€

ψ = A 1( )B 2( ) + A 2( )B 1( ) The formation of the bond in H2 – due to the high probability that the two electrons will be found between two nuclei and hence will bind them together. The wave pattern represented by A(1)B(2) interferes constructively with the wave pattern represented by A(2)B(1) and there is an enhancement in the value of the wavefunction in the interrnuclear region. Such electron distribution is called a σ bond. A σ bond has cylindrical symmetry around the internuclear axis. When viewed along the internuclear axis, it resembles a pair of electrons in an s orbital. The electrons in a σ bond have zero orbital

angular momentum about the internuclear axis. The molecular potential energy curve is calculated by changing the H-H distance R and evaluating the expectation value for energy at each distance.

4

The energy falls below that of two separated H atoms as the two atoms approach each other within bonding distance and each electron is free to migrate to the other atom. However, the energy reduction is opposed by an increase in energy from the Coulombic repulsion of the two positively charged nuclei. The positive contribution to energy becomes large as R becomes small – the total potential energy curve passes through a minimum and then climbs to a strongly positive value at small R.

To write the total wavefunction properly we have to include spin wavefunction and, to follow the Pauli principle, the only allowed wavefunction is:

€

ψ = A 1( )B 2( ) + A 2( )B 1( ){ }σ− 1,2( ) where

€

σ− 1,2( ) = 1/21/2( ) α 1( )β 2( )−α 2( )β 1( ){ } The state of lower energy (and hence the formation of a chemical bond) is achieved if the electron spins are paired.

Homonuclear diatomic molecules The essential features of VB theory – the pairing of electrons and the accumulation of electron density in the internuclear region because of this pairing. To construct the VB description of N2, consider the valence electron configuration of each atom: 2s22px

12py12pz

1. z-axis is conventionally the internuclear axis. We can imagine each atom as having a 2pz orbital pointing towards a 2pz orbital on the other atom. A σ bond is the formed by spin pairing between the two electrons in the

two 2pz orbitals: in the spatial wavefunction A and B now stand for the two 2pz orbitals.

5

The remaining p orbitals cannot merge to give σ bonds – they do not have cylindrical symmetry. Instead, they merge to form π bonds. A π bond arises from the spin pairing of electrons in two p orbitals. Viewed along the internuclear axis, a π bond resembles a pair of electrons in a p orbital. An electron in a π bond has one unit of orbital angular momentum about the internuclear axis; the wavefunction has one angular node. There are two π bonds in N2,

one formed by spin pairing in two neighboring 2px orbitals and the other by spin pairing in two 2py orbitals. The overall bonding pattern in N2 includes a σ bond plus two π bonds – consistent with the Lewis structure :N≡N: for nitrogen. Polyatomic molecules Each σ bond in a polyatomic molecule is formed by spin pairing of electrons in atomic orbitals with cylindrical symmetry about the relevant internuclear axis; π bonds are formed by pairing electrons that occupy atomic orbitals of the appropriate symmetry (normally, p or d).

H2O molecule: the valence electron configuration of O is 2s22px22py

12pz1.

The two unpaired electrons in the O2p orbitals can each pair with an electron in an H1s orbital to give two cylindrically symmetric σ bonds. Because the 2py and 2pz orbitals lie at 90°, the two σ bonds also lie at 90° to

each other - H2O should be an angular molecule. However, the actual bond angle is 104.5°.

6

NH3 molecule: the valence electron configuration of N is 2s22px12py

12pz1 – three H atoms can

form bonds by spin pairing with the electrons in the three half-filled 2p orbitals, perpendicular to each other. We predict a trigonal pyramidal molecule with a bond angle 90°. Experimental studies confirm the predicted shape, but the actual bond angle is 107°. Promotion A deficiency of VB theory – inability to account for carbon’s tetravalence. The ground state configuration of C is 2s22px

12py1, which suggests that a carbon atom should be capable

of forming only two bonds, not four. This deficiency is overcome by allowing for promotion, the excitation of an electron to an orbital of higher energy. Although electron promotion requires an investment of energy, that investment is worthwhile if the energy can be more than recovered from the greater strength or number of bonds that it allows to be formed. In carbon, the promotion of a 2s electron can lead to the configuration 2s12px

12py12pz

1, with four unpaired electrons in separate orbitals. These electrons may pair with four electrons in orbitals provided by four other atoms (like in CH4) and hence form four σ bonds. Promotion, and the formation of four bonds, is a characteristic feature of carbon because the promotion energy is quite small: the promoted electron leaves a doubly occupied 2s orbital and enters a vacant 2p orbital.

Hybridization

The description of the bonding in CH4 is still incomplete because it implies the presence of three σ bonds of one type (formed from H1s and C2p orbitals) and a fourth σ bond of another type (formed from H1s and C2s orbitals).

7

This problem can be solved by introducing hybrid orbitals formed by interference between C2s and C2p orbitals. The waves (wavefunctions) intefere destructively and constructively in different regions and give rise to four new shapes. Linear combinations specifying four equivalent hybrid orbitals:

€

h1 = s+ px + py + pz

€

h2 = s− px − py + pz

€

h3 = s− px + py − pz

€

h4 = s+ px − py − pz Each hybrid orbital consists of a large lobe pointing in the direction of one corner of a regular tetrahedron. The angle between the axes of the hybrid orbitals is the tetrahedral angle, 109.47°. These are sp3 hybrid orbitals. In CH4, each hybrid orbital of the promoted C atom contains a single unpaired electron; an H1s electron can pair with each one, giving rise to a σ bond pointing in a tetrahedral direction:

€

ψ = h1 1( )A 2( ) +h1 2( )A 1( ) Because each sp3 hybrid orbital has the same composition, all four σ bonds are identical apart from their orientation in space. A hybrid orbital has a pronounced directional character – it has an enhanced amplitude in the internuclear region. This is due the constructive interference between the s orbital and the positive lobes of the p orbitals. As a result of the enhanced amplitude in the internuclear region, the bond strength is greater than for a bond formed from an s or p orbital alone.

8

Ethene molecule: H2C=CH2 – planar, with HCC and HCH bond angles close to 120°. To reproduce the σ bonding structure, we promote each C atom to a 2s12p3 configuration. Then, we form sp2 hybrid orbitals by superposition of an s orbital and 2 p orbitals:

€

h1 = s+21/2 py

€

h2,3 = s± 3/2( )1/2 px − 1/2( )1/2 py The three hybrid orbitals lie in a plane and point toward the corners of an equilaterial triangle. The third 2p orbital (2pz) is not included; its axis is perpendicular to the plane in which the hybrids lie. The proportion of each orbital in the mixture is given by the square of the corresponding coefficient: in the first hybrid the ratio of s to p contributions is 1:2. The total p contribution in each of h2 and h3 is 3/2 + 1/2 = 2, so the ratio for these orbitals is also 1:2. The different signs of the coefficients ensure that constructive interference takes place in different regions of space.

In H2C=CH2, the sp2-hybridized C atoms each form three σ bonds by spin pairing with H1s orbitals. The σ framework consists of C-H and C-C σ bonds at 120° to

each other. When the two CH2 groups lie in the same plane, the two electrons in the unhybridized p orbitals can pair and form a π bond. The formation of this π bond locks the framework into the planar arrangement: any rotation of one CH2 group leads to a weakening of the π bond (an increase in energy of the molecule).

9

Ethyne, HC≡CH – a linear molecule. The C atoms are sp hybridized, and the σ bonds are formed using hybrid atomic orbitals:

€

h1 = s+ pz

€

h2 = s− pz These two orbital lie along the internuclear axis. The electrons in them pair either with an electron in the corresponding hybrid orbital on the other C atom or on in one of the H1s orbitals. Electrons in the two remaining p orbitals on each atom, which are perpendicular to the molecular axis, pair to form to perpendicular π bonds.

Other hybridization schemes may involve d orbitals and are useful to predict molecular geometries qualitatively. The hybridization of N atomic orbitals always gives N hybrids, which may form bonds or contain lone pairs of electrons. Example: sp3d2 hybridization – six hybrid orbitals - octahedral molecules like SF6. The molecular orbital approximation The one-electron wavefunctions obtained by solving the Schrödinger equation are called molecular orbitals. A molecular orbital ψ gives, through the value of ψ2, the distribution of the electron in the molecule. A molecular orbital is like an atomic orbital, but spreads throughout the molecule. Exact, analytical molecular orbitals may be obtained for H2

+ (within the Born-Oppenheimer approximation), but they are very complicated functions and do not give much insight into the form of the orbitals and contributions to the energy. Therefore, we consider a simpler procedure that, while more approximate, gives more insight.

10

Linear combinations of atomic orbitals The approximation we use is based on the fact that when the electron is very close to nucleus A, the term 1/rA in V is very much bigger than 1/rB. Then, the potential energy reduces to

V ≈−e2

4πε 0×1rA

The Schrödinger equation for the electron in the molecule is then the same as that for an isolated H atom, and its lowest energy solution is a 1s orbital on A, ψ1s(A). Thus, close to A, the molecular orbital resembles an atomic 1s orbital. Likewise, close to B the molecular orbital resembles a 1s orbital on B, ψ1s(B). This discussion suggests that we can approximate the overall wavefunction ψ as a sum of the two atomic orbitals: ψ = N ψ 1s A( ) +ψ 1s B( ){ } where N is a normalization factor. In accord with a preceding discussion, when the electron is close to A its distance from B is large, ψ1s(B) is small, and therefore the wavefunction is almost pure ψ1s(A). Similarly, ψ is almost pure ψ1s(B) close to B. We can call the sum as a linear combination of atomic orbitals (LCAO), and we shall use that name from now on. An approximate molecular orbital formed from a linear combination of atomic orbitals is called an LCAO-MO.

11

A molecular orbital that has cylindrical symmetry around the internuclear axis, such as the one we are discussing, is called a σ orbital (because it resembles an s orbital when viewed along the axis). Since the σ orbital in the case we consider now is formed from 1s orbitals, it is more fully described as a 1sσ orbital. We should keep in mind the approximations we have made so far. The Born-Oppenheimer approximation separates the electronic and nuclear motions and allows us to talk in terms of the molecular orbitals of the electron in the field of stationary nuclei. The LCAO approximation goes one step further and approximates a molecular orbital as a sum (linear combination) of atomic orbitals. It allows us to use atomic orbitals to discuss the distribution of electrons in molecules. σ orbitals According to the Born interpretation, the probability density of the electron in H2

+ is proportional to the square of its wavefunction. The probability density of the LCAO-MO 1sσ is

ψ 2 = N 2 ψ 1s A( ) +ψ 1s B( ){ }2

= N 2 ψ 1s A( )2 +ψ 1s B( )2 + 2ψ 1s A( )ψ 1s B( ){ }

12

The calculation of the normalization factor gives N2 =0.31 and since ψ1s(A) is the function

ψ1s A( ) =1πa0

3

1/ 2

e− rA /a0

with rA the distance of the electron from A, and similarly for ψ1s(B), it is easy to evaluate ψ and the probability density at any point. An important feature of the equation for ψ2 becomes apparent when we examine the probability density in the internuclear region, where both atomic orbitals have similar amplitudes. The total probability density is proportional to the sum of (a) ψ1s(A)2, the probability density if the electron were confined to the orbital on A. (b) ψ1s(B)2, the probability density if the electron were confined to the orbital on B.

(c) 2ψ1s(A)ψ1s(B), an extra contribution to the density.

13

The last contribution, the overlap density, is crucial, since it represents an enhancement of the probability of finding the electron in the internuclear region above what it would be if it

were confined to one of the two atoms. That is, because the electron is free to move from one nucleus to the other, the electron density in the internuclear region is increased. The enhancement can be traced to the constructive interference of the two atomic orbitals: each has a positive amplitude in the internuclear region, and so the total amplitude is greater there than if the electron were confined to a single atomic orbital. Thus, electrons accumulate in regions where atomic orbitals overlap and interfere constructively, The accumulation of electron density between the nuclei puts the electron in a position where it interacts strongly with both nuclei. Hence the energy of the molecule is lower than that of the separate atoms, where each electron can interact strongly with only one nucleus. Bonding orbitals The 1sσ orbital is an example of a bonding orbital, an orbital which, if occupied, contributes to a lowering of the energy of a molecule. An electron that occupies a σ orbital is called a σ electron, and if that is the only electron present in the molecule (as in the ground state of H2

+), we report the configuration of the molecule as 1sσ1.

14

The energy of the 1sσ orbital decreases as R decreases from large values because electron density accumulates in the internuclear regions and the two atomic orbitals increasingly overlap. However, at small separations, there is too little space between the nuclei for significant accumulation of electron density there. In addition, the nucleus-nucleus repulsion Vnuc-nuc becomes large. As a result, the total energy rises at short distances, and there is a minimum in the potential energy curve. The internuclear separation at the minimum of the curve is called the equilibrium bond length Re and the depth of the minimum is called the bond dissociation energy De. Calculations on H2

+ give Re = 130 pm and De = 1.77 eV (171 kJ mol-1); the experimental values are 106 pm and 2.6 eV, and so this simple LCAO-MO description of the molecule, while inaccurate, is not absurdly wrong.

Antibonding orbitals The argument that led to the expression of ψ as the sum of two atomic orbitals is equally well satisfied by writing a molecular orbital as the difference

′ ψ = N ψ 1s A( ) −ψ 1s B( ){ }

15

In this case too, the molecular orbital resembles one or other of the atomic orbitals close to the two nuclei. However, this linear combination corresponds to a higher energy than the orbital written as a sum, and is in fact a good approximation to the next-higher exact solution of the Schrödinger equation for H2

+. Since ψ’ is cylindrically symmetrical around the internuclear axis it is also a σ orbital, and since it is formed from 1s orbitals it too is a 1sσ orbital. We distinguish it from the bonding 1sσ orbital by denoting it 1sσ*. We can see that the 1sσ* orbital has a node where ψ1s(A) and ψ1s(B) cancel. Consequently there is zero probability of finding the electron half-way between the nuclei if it occupies this orbital. We express this reduction in amplitude by saying that because the atomic orbitals are combined with opposite signs, they interfere destructively where they overlap. In terms of the probability density, ψ 2 = N 2 ψ 1s A( ) −ψ 1s B( ){ }2

= N 2 ψ 1s A( )2 +ψ 1s B( )2 − 2ψ 1s A( )ψ 1s B( ){ } and the third term reduces the probability of finding the electron between the nuclei relative to its value if the electron were confined to one of the atomic orbitals.

16

The 1sσ* orbital is an example of an antibonding orbital, an orbital that, if occupied, raises the energy of the molecule relative to the separated atoms. Antibonding orbitals are labeled with a *. An antibonding electron destabilizes the molecule relative to the separated atoms. This is partly because, since it is excluded from the internuclear region, the electron is distributed largely outside the bonding region. In effect, the electron pulls the nuclei apart from outside. The combined effect of the electron distribution and the internuclear repulsion results in an antibonding orbital being more strongly antibonding than the corresponding bonding orbital is bonding. The structures of diatomic molecules For atoms, we used the hydrogenic atomic orbitals and the building-up principle to deduce the ground state electronic configurations of many-electron atoms. We can do the same for many-electron diatomic molecules (such as H2 with two electrons and Br2 with 70), but using H2

+ molecular orbitals instead. The general procedure is to construct molecular orbitals by combining the atomic orbitals supplied by the atoms. The electrons supplied by the atoms are then accommodated in the orbitals so as to achieve the lowest overall energy subject to the constraint of the Pauli exclusion principle that no more than two electrons may occupy a single orbital (and they must be paired). As in the case of atoms, if several degenerate orbitals are available, we add the electrons to each individual orbital before doubly occupying any one orbital (because that reduces electron-electron repulsions). We also take note of Hund’s rule, that if electrons do occupy different degenerate orbitals, they

do so with parallel spins. We shall illustrate the general procedure by considering H2, the simplest diatomic many-electron molecule.

17

The hydrogen and helium molecules First, we need to build the molecular orbitals. Since each H atom of H2 contributes a 1s orbital (as in H2

+), we can form the 1sσ and 1sσ* orbitals from them. At the experimental internuclear separation we can draw energy levels for these orbitals, which is called a molecular orbital energy level diagram. From two atomic orbitals we can build two molecular orbitals, and this a special case of the general rule that from N atomic orbitals we can build N molecular orbitals.

Electronic configurations There are two electrons to accommodate, and both can enter 1sσ by pairing their spins. The ground state configuration is therefore 1sσ2 and the atoms are joined by a bond consisting of an electron pair in a bonding σ orbital. This approach shows that an electron pair represents the maximum number of electrons that can enter a bonding molecular orbital.

The same argument shows why He does not form diatomic molecules. Each He atom contributes a 1s orbital, and so the same two molecular orbitals can be constructed. There are four electrons to accommodate. Two can enter the 1sσ orbital, but then it is full, and the next two must enter the 1sσ* orbital. The ground state electronic configuration of He2 is therefore 1sσ21sσ*2. We see that there is one bond and one antibond. Since an

antibond is slightly more antibonding than a bond is bonding, the He2 molecule has a higher energy than the separated atoms, and so does not form.

18

Bond order A measure of the net bonding in a diatomic molecule is its bond order b, defined as b =1/ 2(n − n*) n is the number of electrons in bonding orbitals and n* the number in antibonding orbitals. Thus each electron pair in a bonding orbital increases the bond order by 1 and each pair in an antibonding orbital decreases it by 1. For H2, b = 1, corresponding to a single bond, H-H, between the two atoms. In He2, b = 0, and there is no bond. The bond order is a useful parameter for discussing the characteristics of bonds, since it correlates with bond length, and the greater the bond order between atoms of a given pair of elements the shorter the bond. It also correlates with bond strength, and the greater the bond order the greater the strength. Period-2 diatomic molecules We now see how the concepts we have introduced apply to homonuclear diatomic molecules in general, which are diatomic molecules formed from identical atoms, such as N2 and Cl2. In line with the building-up procedure, we first consider the molecular orbitals that may be formed. The atomic orbitals available are the core orbitals, those of the inner, closed shells, the valence orbitals, those of the valence shell, and the virtual orbitals, those of the atom that are unoccupied in its ground state. In elementary treatments (but not in the sophisticated treatments using computers), the core orbitals are ignored as being too compact to have significant overlap with orbitals on other atoms. The virtual orbitals are ignored on

the grounds they are too high in energy to participate in bonding. Therefore we form molecular orbitals using only the valence orbitals.

19

In Period-2, the valence orbitals are 2s and 2p. The 2s orbitals on the two atoms overlap to give a bonding 2sσ orbital and antibonding 2sσ* orbital, in exactly the same way as we have already seen for 1s orbitals. The new feature of these molecules is that p orbitals are also available for bonding. The two pz orbitals directed along the internuclear axis overlap strongly, and may do so either constructively or destructively, to give a bonding or antibonding 2pσ orbital, respectively. π orbitals Now consider the 2px and 2py orbitals of each atom, which are perpendicular to the internuclear axis and may overlap broadside-on. This overlap may be constructive or destructive, and results in a bonding or an antibonding π orbital. The notation π is the analog of p in atoms, for when viewed along the axis of the molecule, a π orbital looks like a p orbital. The two 2px orbitals overlap to give a bonding and antibonding 2pxπ orbital, and the two 2py orbitals overlap to give two 2pyπ orbitals. The 2pxπ and 2pyπ orbitals have the same energies, as do their antibonding partners. In some cases, 2pπ orbitals are less strongly bonding than 2pσ orbitals because their maximum overlap occurs off-axis, away from the optimum bonding region. However, there is no guarantee that this order of energies

should prevail, and it is found experimentally (by spectroscopy) and by detailed calculation that the order varies along Period 2.

20

The relative order is controlled by the separation of the 2s and 2p orbitals in the atoms, which increases across the period. The switch in order occurs at N2.

21

s,p overlap

The origin of the variation in relative energy levels lies in the ability of both 2s and 2p orbitals to contribute to the same molecular orbitals. So far, we have considered s,s overlap and p,p overlap as occurring distinctly, and have constructed distinct 2sσ and 2pσ orbitals. One reason for this is that strong bonds arise from the overlap of atomic orbitals having similar energies. In homonuclear diatomic molecules, the energies of the 2s orbitals of the two atoms are identical, as are those of the 2p orbitals, and so the principal contributions to molecular orbitals will come from their respective overlaps. However, we should really write a σ orbital as a linear combination of all possible atomic orbitals that have the correct symmetry to contribute, and express it as

ψ = c2s A( )ψ 2 s A( ) + c2 pz A( )ψ 2 pz

A( ) +

+c2 s B( )ψ 2s B( ) + c2 pz B( )ψ 2 pzB( )

A 2s and a 2pz orbital can both contribute to a σ orbital because they both have cylindrical symmetry around the internuclear axis. In other words, s and pz orbitals on different atoms have nonzero overlap and can participate in orbital formation with each other. On the other hand, 2s,2px overlap

makes no contribution to bonding because the effect of the constructive overlap in one region is exactly cancelled by the effect of the destructive overlap in another, and so there is no net overlap and no net contribution to bonding.

22

The structures of homonuclear diatomic molecules The general layout of the valence-shell atomic orbitals of Period-2 atoms is shown in Figures in p. 20. The lines in the middle are an indication of the energies of the molecular orbitals that can be formed by overlap of atomic orbitals: from the eight valence shell orbitals (four from each atom), we can form eight molecular orbitals. With the orbitals established, we can deduce the ground state configurations of the molecules by adding the appropriate number of electrons to the orbitals and following the building-up rules. Charged species (such as the peroxide ion, O2

2-) need either more or fewer electrons than the neutral molecules. We shall illustrate the procedure with N2, which has ten valence electrons. Two electrons pair, enter, and fill the 2sσ orbital; the next two enter and fill the 2sσ* orbital. Six electrons remain. There are two 2pπ orbitals, and so four electrons can be accommodated in them. The last two enter 2pσ orbital. The ground state configuration of N2 is therefore N2 2sσ22sσ*22pπ42pσ2 and the bond order is b = 1/2(8-2) = 3 This bond order accords with Lewis structure of the molecule (N≡N) and is consistent with its high dissociation energy (944 kJ mol-1). For O2, the electron configuration is then O2 2sσ22sσ*22pσ22pπ42pπ*2 and bond order is b = 2. According to the building-up principle, the two 2pπ* electrons in O2 will occupy different orbitals: one will enter 2pxπ* and the other will enter 2pyπ*. Since they are in different orbitals, they will have parallel spins. Therefore, we can predict that an O2 molecule will

have a net spin angular momentum S = 1 and be in a triplet state. Since electron spin is the source of a magnetic moment, we can go on to predict that oxygen should be paramagnetic, which is in fact the case.

23

An F2 molecule has two more electrons than an O2 molecule and the configuration and bond order are F2 2sσ22sσ*22pσ22pπ42pπ*4, b = 1 We conclude that F2 is a singly-bonded molecule, in agreement with its Lewis structure F-F. The low bond order is consistent with its low dissociation energy (154 kJ mol-1). Ne2 has two further electrons: Ne2 2sσ22sσ*22pσ22pπ42pπ*42pσ*2 b = 0 The zero bond order agrees with the monatomic structure of Ne. Heteronuclear diatomic molecules A heteronuclear diatomic molecule is a diatomic molecule formed from atoms of two different elements, such as CO and HCl. The electron distribution in the covalent bond between the atoms is not symmetrical because it is energetically favorable for the electron pair to be found closer to one atom than the other. The imbalance results in a polar bond, which is a covalent bond in which the electron pair is shared unequally by the two atoms. The bond in HF, for instance, is polar, with the electron pair closer to the F atom. The accumulation of the electron pair near the F atom results in that atom having a net negative charge, which is called a partial negative charge and denoted δ-. There is a compensating partial positive charge δ+ on the H atom.

24

Polar covalent bond A polar covalent bond consists of two electrons in an orbital on the form ψ = cAψ A( ) + cBψ B( ) with unequal coefficients. The proportion of the atomic orbital ψ(A) in the bond is cA

2 and that of ψ(B) is cB

2. A nonpolar bond has cA2 = cB

2, and a pure ionic bond has one coefficient zero (so that A+B- would have cA = 0 and cB = 1). The atomic orbital with the lower energy makes the larger contribution to the bonding molecular orbital. The opposite is true of the antibonding orbital, for which the dominant component comes from the atomic orbital with higher energy.

These points can be illustrated by considering HF, and judging the energies of the orbitals that contribute to the molecular orbital by noting the ionization energies of the atoms. The general form of the molecular orbitals is ψ = cHψ H( ) + cFψ F( ) where ψ(H) is an H1s orbital and ψ(F) is an F2p orbital. The H1s orbital lies at 13.6 eV below the zero energy (the separated proton and electron and the F2p orbital lies at 18.6 eV below zero energy. Hence, the bonding σ orbital in HF is mainly F2p and the antibonding σ orbital is mainly H1s orbital in character. The two electrons in the bonding orbital are most likely to be

found in the F2p orbital, so there is a partial negative charge on the F atom

and a partial positive charge on the H atom.

25

The variation principle A systematic way of finding the coefficients in the linear combinations used to build molecular orbitals is provided by the variation principle: If an arbitrary wavefunction is used to calculate the energy, than the value calculated is never less than the true energy. The arbitrary wavefunction is called the trial wavefunction. The principle implies that if we vary coefficients in the trial wavefunction until we achieve the lowest energy, then those coefficients will be the best. We might get a lower energy if we use a more complicated wavefunction (for example, by taking a linear combination of several atomic orbitals on each atom), but we shall have the optimum molecular orbital that can be built form the given set of atomic orbitals. The method can be illustrated by the trial wavefunction

ψ = cAψ A( ) + cBψ B( ) This function is real but not normalized (because the coefficients can take arbitrary values),

so in the following we cannot assume that ψ 2∫ dτ =1.

The energy of the orbital is the expectation value of the energy operator (the Hamiltonian, H):

E = ψHψdτ∫ / ψ 2dτ∫

26

We must search for the values of the coefficients in the trial function that minimize the value of E. This is a standard problem in calculus, and is solved by finding the coefficients for which ∂E /∂cA = 0 and ∂E /∂cB = 0 The secular equations The first step is to express the two integrals in terms of the coefficients. The denominator is

ψ 2dτ∫ = cAψ A( ) + cBψ B( ){ }∫2dτ =

= cA2 ψ A( )∫

2dτ + cB

2 ψ B( )∫2dτ + 2cAcB ψ A( )ψ B( )∫

= cA2 + cB

2 + 2cAcBS since the individual atomic orbitals are normalized and the third integral is the overlap

integral S. The numerator is ψHψ =∫

cAψ A( ) + cBψ B( ){ }H∫ cAψ A( ) + cBψ B( ){ }dτ

= cA2 ψ A( )∫ Hψ A( )dτ + cB

2 ψ B( )∫ Hψ B( )dτ +2cAcB ψ A( )∫ Hψ B( )dτ

27

There are some complicated integrals in this expression, but we can denote them by the

constants α A = ψ A( )∫ Hψ A( )dτ

α B = ψ B( )∫ Hψ B( )dτ , β = ψ A( )∫ Hψ B( )dτ

Then, ψHψ = cA2α A +∫ cB

2α B + 2cAcB

α is called a Coulomb integral. It is negative, and can be interpreted as the energy of the electron when it occupies ψ(A) (for αA) and ψ(B) (for αB). In a homonuclear diatomic molecule, αA = αB. β is called a resonance integral. It vanishes when the orbitals do not overlap, and at equilibrium bond lengths it is normally negative. The complete expression for E is

E = cA2α A + cB

2α B + 2cAcBβcA2 + cB

2 + 2cAcBS

Its minimum is found by differentiation with respect to the two coefficients. The result is the two secular equations

α A − E( )cA + β − ES( )cB = 0

β − ES( )cA + α B − E( )cB = 0

28

As for any set of simultaneous equations, these two have a solution if the secular determinant, the determinant of coefficients, is zero; that is, if

α A − E β − ESβ − ES α B − E

= 0

This determinant expands to a quadratic equation in E. Its two roots give the energies of the bonding and antibonding molecular orbitals formed from the atomic orbitals and, according to the variation principle, these are the best energies for the given set of atomic orbitals. The values of the coefficients in the linear combination are obtained by solving the secular equations using the two energies: the lower energy gives the coefficients for the bonding molecular orbital, the upper energy the coefficients for the antibonding molecular orbital. The secular equations give expressions for the ratio of the coefficients in each case, and so we need a further equation in order to find their individual values. This is obtained by demanding that the best wavefunction should be normalized, which means that, at the final stage, we also must ensure that

ψ 2dτ = cA2 +∫ cB

2 + 2cAcBS =1

Let us, for example, find the energies E of the bonding and antibonding orbitals of a homonuclear diatomic molecule by solving the secular equations.

29

α − E β − ESβ − ES α − E

= α − E( )2 − β − ES( )2 = 0

The solutions of this equation are E+ =α + β1+ S

E− =α − β1− S

For the bonding orbital cA = cB =1

2 1+ S( )

1/ 2

and for the antibonding orbital cA = −cB =1

2 1− S( )

1/ 2

In this case, the best bonding function has the form

ψ + =1

2 1+ S( )

1/ 2

ψ A( ) +ψ B( ){ }

and the corresponding antibonding function is

ψ− =1

2 1− S( )

1/ 2

ψ A( ) −ψ B( ){ }

30

Let’s consider the second simple case: a heteronuclear diatomic molecule but with S = 0 (for simplicity). The secular determinant is then

€

αA − E β

β αB − E= αA − E( ) αB − E( )− β2 = 0

The solutions can be expressed in terms of the parameter ζ:

€

ζ =12arctan

2βαB −αA

€

E− = αB − β tanζ

€

ψ− = −Asinζ +Bcosζ

€

E+ = αA +β tanζ

€

ψ+ = Acosζ +Bsinζ An important feature: as the difference

€

αA −αB increases, the value of ζ decreases. When the energy difference is large the energies of the molecular orbitals differ only slightly from those of the atomic orbitals – the bonding and antibonding effects are small. The strongest bonding and antibonding effects are obtained when the two contributing orbitals have closely similar energies. The difference in energy between core and valence orbitals is the justification for neglecting the contribution of core orbitals to bonding. On the other hand, the core orbitals of one atom have a similar energy to the core orbitals of the other atom; but core-core interaction is largely negligible because the overlap between them (and hence the value of β) is very small.

31

Calculate the wavefunctions and energies of the σ orbitals in the HF molecules taking β = -1.0 eV and the following ionization energies: H1s: 13.6 eV, F2s: 40.2 eV, F2p: 18.6 eV. Because the F2p and H1s orbitals are much closer in energy than the F2s and H1s orbitals, to a first approximation we neglect the contribution of the F2s orbital. We need to know the values of the Coulomb integrals αH and αF. Because these integrals represent the energies of the H1s and F2p electrons, they are approximately equal to the negative of the ionization energies of the atoms: αH = -13.6 eV αF = -18.6 eV Then, 2tanζ = 0.40 ζ = 10.9° E- = -13.4 eV ψ- = 0.98ψH – 0.19ψF E+ = -18.8 eV ψ+ = 0.19ψH + 0.98ψF The lower energy orbital (-18.8 eV) has a composition that is more F2p orbital than H1s, and the opposite is true of the higher energy, antibonding orbital. Molecular orbitals for polyatomic systems The molecular orbitals of polyatomic molecules are built in the same way as in diatomic molecules, but we use more atomic orbitals to construct them:

€

ψ = ciψii∑

ψI – an atomic orbital and the sum extends over all the valence orbitals of all atoms in the molecule. To find the coefficient, we set up the secular equations and the secular determinant, solve the latter for the energies, and then use these energies in the secular equations to find the coefficients for each molecular orbital. The shape of a polyatomic

Example. Calculating the molecular orbitals of HF

molecule (its bond lengths and bond angles) can be predicted by calculating the total energy of the molecule for a variety of nuclear positions, and then identifying the structure that corresponds the lowest energy.

32

The LCAO basis set approach We may imagine constructing wave functions in any fashion and we may judge the quality of our wave functions by evaluation of the energy eigenvalues associated with each. The one with the lowest energy will be the most accurate and presumably the best one to use for computing other properties by the application of other operators. How can an arbitrary function be represented by a combination of more convenient functions? The convenient functions are called a ‘basis set’. In our QM system, we have temporarily restricted ourselves to systems of one electron. If our system has only one nucleus, the Schrödinger equation can be solved exactly – the eigenfunctions are hydrogenic orbitals: 1s, 2s, 2p, 3s, 3p, 3d, … These orbitals, as functions, may be useful in the construction of more complicated molecular orbitals. We will construct a guess wave function φ as a linear combination of

atomic wave functions ϕ:

€

φ = aiϕii=1

N

∑

This construction is known as the linear combination of atomic orbitals (MO-LCAO) approach. LCAO are normally centered on the atoms of the molecule. The summation in the equation has an upper limit N – we cannot work with an infinite basis set in any convenient way. However, the more atomic orbitals we allow into our basis set, the closer our basis will come to covering the true molecular orbital space. There may be very many ‘true’ one-electron MOs that are very high in energy. Accurately describing these MOs may require

some unusual basis functions, e.g. very diffuse functions to describe weakly bound electrons – in Rydberg states. In general, we have to distinguish between chemically meaningful AOs and basis functions as a mathematical tool.

33

The Secular Equation

€

E =

aiϕii∑

∫ H a jϕ j

j∑

dr

aiϕii∑

∫ a jϕ j

j∑

dr

=

aia j ϕiHϕ jdr∫ij∑

aia j ϕiϕ jdr∫ij∑

=

aia jHijij∑

aia jSijij∑

€

Hij = ϕiHϕ jdr∫ - a resonance integral

€

Sij = ϕiϕ jdr∫ - an overlap integral – characterizes the extent to which any two basis functions overlap in a phase-matched fashion in space. Hii – corresponds to the energy of a single electron occupying basis function i – essentially equivalent to the ionization potential of AO. According to the variational principle, as we get closer and closer to the ‘true’ one-electron ground-state wave functions, we will obtain lower and lower energies from our guess. Thus, once we have selected a basis set, we would like to choose the coefficients ai so as to minimize the energy for all possible combinations of our basis functions. A necessary condition for a function to be at its minimum is that its derivatices with respect to all of its

free variables are zero: for all k

€

∂E∂ak

= 0

After partial differentiation, we obtain the following equations:

34

€

H11 − ES11 H12 − ES12 H1N − ES1NH21 − ES21 H22 − ES22 H2N − ES2N

HN1 − ESN1 HN2 − ESN 2 HNN − ESNN

= 0 Secular Equation

In general, there will be N roots E which satisfy the secular equation – N energies Ej, each of them give rise to a different set of coefficients, aij, which can be found by solving the set of linear equations using Ej, and these coefficients will define an optimal wave function ϕj

within the given basis set:

€

ϕ j = aijϕii

N

∑

In a one-electron system, the lowest energy molecular orbital defines the ground state of the system and the higher energy orbitals are excited states. These are different MOs and they have different basis function coefficients. The variational principle holds for the excited states as well: the calculated energy of a guess wave function for an excited state will be bounded from below by the true excited state energy. To find the optimal one-electron wave function for a molecular system:

1. Select a set of N basis functions. 2. For that set of basis functions, determine all N2 values of both Hij and Sij. 3. Form the secular determinant, and determine the N roots Ej of the secular equation.

For all k

€

ai Hki − ESki( )i=1

N

∑ = 0

4. For each of the N values of Ej, solve the set of linear equations in order to determine the basis set coefficients aij for that MO.

35

Hückel Theory Hückel theory – was developed in the 1930s to explain some of the unique properties of unsaturated and aromatic hydrocarbons:

(a) The basis set is formed entirely from parallel carbon 2p orbitals, one per atom (π system).

(b) The overlap matrix is defined by Sij = δij (c) Matrix elements Hii are set equal to the negative of the ionization potential of the

methyl radical – the orbital energy of the singly occupied 2p orbital in the prototypical system defining sp2 carbon hybridization (written as α).

(d) Matrix elements Hij between neighbors are also derived from experimental information: 90° rotation in ethylene

Ep = α Eπ = 2α + 2β ΔE = 2Ep – Eπ ΔE/2 = β

So, the matrix elements between neighbors are set equal to β. Matrix elements Hij between carbon 2p orbitals more distant than nearest neighbors are set equal to zero. Then the secular determinant has the following structure:

E = Eπ E = 2Ep

36

1. All diagonal elements: α - E 2. Off-diagonal elements between neighboring atoms: β 3. All other elements: 0.

For ethylene, we express π orbitals as LCAO of the C2p orbitals perpendicular to the molecular plane:

€

ψ = cAA+ cBB A, B – C2p orbitals on atoms A and B, respectively. Then the secular determinant is

€

α− E β − ESβ − ES α− E

= 0

Using Hückel approximations:

€

α− E β

β α − E= α− E( )2 − β2 = 0

The roots of the equation are

€

E± = α ±β The + sign corresponds to the bonding combination (β is negative) and the – sign corresponds to the antibonding combination. The configuration is 1π2. We can also estimate the π*←π excitation energy as 2|β|. The constant β is often used as an adjustable parameter; an approximate value for (C2p,C2p)-overlap π bonds is about –75 kJ mol-1 (-0.8 eV).

37

The highest occupied molecular orbital in ethylene (HOMO) is the 1π orbital; the lowest unfilled molecular orbital (LUMO) is the 2π* orbitals. These two orbitals are the frontier orbitals of the molecule. The frontier orbitals are important because they are largely responsible for chemical and spectroscopic properties of the molecule. In matrix form, solving the secular determinant is equivalent to matrix diagonalization:

€

C−1HC = E To find the eigenvalues Ei, we have to find a transformation of matrix H that makes it diagonal. The diagonal elements then correspond to the eigenvalues Ei and the columns of the matrix C that brings about this diagonalization are the coefficients of the members of the basis set and hence give us the composition of the molecular orbitals. Example: the π orbitals of butadiene

€

H =

H11 H12 H13 H14

H21 H22 H23 H24

H31 H32 H33 H34

H41 H42 H43 H44

=

α β 0 0β α β 00 β α β

0 0 β α

38

Diagonalization of this matrix gives

€

E =

α +1.62β 0 0 00 α +0.62β 0 00 0 α− 0.62β 00 0 0 α−1.62β

The matrix that achieves the diagonalization is

€

C =

0.372 0.602 0.602 −0.3720.602 0.372 −0.372 0.6020.602 −0.372 −0.372 −0.6020.372 −0.602 0.602 0.372

€

E = α +1.62β

€

ψ = 0.372ψ1 +0.602ψ2 +0.602ψ3 +0.372ψ4

€

E = α +0.62β

€

ψ = 0.602ψ1 +0.372ψ2 − 0.372ψ3 − 0.602ψ4

€

E = α− 0.62β

€

ψ = 0.602ψ1 − 0.372ψ2 − 0.372ψ3 +0.602ψ4

€

E = α−1.62β

€

ψ = −0.372ψ1 +0.602ψ2 − 0.602ψ3 +0.372ψ4

39

The ground state configuration is 1π22π2 The total π-electron binding energy, Eπ: Ethylene - Eπ = 2(α + β) = 2α + 2β Butadiene - Eπ = 2(α + 1.62β) + 2(α + 0.62β)= 4α + 4.48β The energy of the butadiene molecule lies lower by 0.48β (-36 kJ mol-1) than the sum of two individual π bonds. Extra stabilization of a conjugated system – delocalization energy. The π-bond formation energy – the energy released when a π bond is formed: Ebf = Eπ - Nα Ebf = 4.48β for butadiene Benzene and aromatic stability The six C2p orbitals perpendicular to the molecular plane overlap to give six π orbitals spreading around.

€

H =

α

β

000β

β

α

β

000

0β

α

β

00

00β

α

β

0

000β

α

β

β

000β

α

E = α + 2β, α + β, α + β

40

The ground state configuration is a2u2e1g

4 The π-electron energy of benzene: Eπ = 2(α + 2β) + 4(α + β)= 6α + 8β If we ignored delocalization, the π-electron energy would be only 3(2α + 2β) = 6α + 6β. The delocalization energy is 2β (-150 kJ mol-1). The π-bond formation energy is 8β. Semi-empirical and ab initio methods We can now see how the LCAO coefficients are found: the secular equations are solved for the energies, and those energies are used to obtain the optimum coefficients. There is still the problem of knowing the values of the Coulomb and resonance integrals. One approach has been to estimate them from spectroscopic information (for example, ionization energies). This combination of empirical data and quantum mechanical calculation has given rise to the semi-empirical methods of molecular structure calculation. The modern tendency is to calculate the integrals from the first principles. These ab initio methods demand extensive numerical computation.

41

Application of Hückel theory to the allyl system

€

α− 2β

€

α+ 2β

CC

CH

H

H

H

H

21 3

pC

E

φ1 =

φ2 =

φ3 =

α

42

Secular equation:

€

α− E β 0β α − E β

0 β α − E= 0

(α - E)3 + (β2×0) – [0×(α - E)×0] - β2(α - E) - (α - E)β2 = 0

€

E = α + 2β , α,

€

α− 2β The lowest (ground) state solution is

€

α + 2β (α, β < 0)

€

a1 α− α + 2β( ) ⋅1[ ]+a2 β − α + 2β( ) ⋅0[ ]+a3 0− α + 2β( ) ⋅0[ ] = 0

€

a1 β − α + 2β( ) ⋅0[ ]+a2 α− α + 2β( ) ⋅1[ ]+a3 β − α + 2β( ) ⋅0[ ] = 0

€

a1 0− α + 2β( ) ⋅0[ ]+a2 β − α + 2β( ) ⋅0[ ]+a3 α− α + 2β( ) ⋅1[ ] = 0

€

a2 = 2a1

€

a3 = a1

From normalization:

€

a11 =12

,

€

a21 =22

,

€

a31 =12

43

€

φ1 =12p1 +

22p2 +

12p3

By choosing the higher energy roots, we may solve two other sets of linear equations and determine the coefficients required to construct φ2 (from E = α) and φ3 (from

€

E = α− 2β):

€

a12 =22

,

€

a22 = 0,

€

a32 = −22

€

a13 =12

,

€

a23 = −22

,

€

a32 =12