Molecular orbitals in X2 molecules

We will see how a qualitative m.o. description can explain the differences found in Table 6.1. 

Molecule Bond energy/kJ mol Bond length/pm Magnetism 

This example will be discussed in some detail as it will serve as a paradigm for the cases to follow. It will largely resemble the discussion of excited 

In this expression, c ci in accordance with the dominant contribution of the low-lying 2s atomic orbitals. 

The energy difference between the anti-bonding and bonding tt m.o.s is smaller than for a m.o.s. This is partly explained on the basis of the expressions for the energies of tt and tt which are similar to those encountered on page 78 for the a m.o.s of H2 iV(a- -/3) and N (a — 0), respectively. The absolute value of the resonance integral (3 is small in view of the small 

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Molecular orbitals in X2 molecules overlap integral for 2p atomic orbitals ... 

And so the set of hydrogen Is orbitals in water transform as -i- All of the molecular orbitals of this molecule which contain hydros Is character win then be either of Aj or b2 symmetry. If of oi symmetry, then xi X2 w91 have the same sign and magnitude if of 62 symmetry, then x 1 and Xs wiU have equal weight but their orbital coefficients will be of opposite sign. 

Fig. 2.2 The approxiinate position of the energy levels of the molecular orbitals in the homonuclear molecule X2 formed from the atomic levels in X. Consider N2 as an example. Since every nitrogen contributes 5 outer shell electrons, the lowest 5 MOs are doubly occupied. It is the occupation of the

Wave functions for the orbitals of molecules are calculated by linear combinations of all wave functions of all atoms involved. The total number of orbitals remains unaltered, i.e. the total number of contributing atomic orbitals must be equal to the number of molecular orbitals. Furthermore, certain conditions have to be obeyed in the calculation these include linear independence of the molecular orbital functions and normalization. In the following we will designate wave functions of atoms by x and wave functions of molecules by i//. We obtain the wave functions of an H2 molecule by linear combination of the I5 functions Xi and X2 of the two hydrogen atoms ... 

In standard quantum-mechanical molecular structure calculations, we normally work with a set of nuclear-centred atomic orbitals Xv X2> - > Xn- CTOs are a good choice for the x s, if only because of the ease of integral evaluation. Procedures such as HF-LCAO then express the molecular electronic wavefunction in terms of these basis functions and at first sight the resulting HF-LCAO orbitals are delocalized over regions of molecules. It is often thought desirable to have a simple ab initio method that can correlate with chemical concepts such as bonds, lone pairs and inner shells. A theorem due to Fock (1930) enables one to transform the HF-LCAOs into localized orbitals that often have the desired spatial properties. 

One always begins with the monomer. What are its frontier levels The classical crystal field or molecular orbital picture of a square planar complex (Fig. 2) leads to a 4 below 1 splitting of the d block.11 For 16 electrons we have z2, xz, yz, and xy occupied and x2-y2 empty. Competing with the ligand field-destabilized x2-y2 orbital for being the lowest unoccupied molecular orbital (LUMO) of the molecule is the metal z. These two orbitals can be manipulated in understandable ways x acceptors push z down, x donors push it up. Better a donors push x2-y2 up. 

There are many other indications that the electrostatic effects of non-spherical features of the charge distribution, such as lone pairs and n electrons, can be important in determining molecular crystal structures. At the extreme of homonuclear diatomics (X2), the electrostatic potential outside the molecule arises from the non-spherical distribution of the valence electrons. Just as there are considerable variations in the bonding orbitals in the diatomics, there are also considerable variations in the lowest temperature ordered crystal structure. 

For a molecule with a total of m atomic basis functions (xi, X2 , Xm will be a total of m resultant molecular orbitals constructed from them. For most purposes these atomic orbitals can be assumed to be real functions and normaUzed (equation 1.7) such that the probability of finding an electron in Xju when integrated over all space is unity. Here Xpi is the complex conjugate of equation 1.8 we show an alternative, useful, way of writing such integrals. The molecular orbitals of... 

As we fill these molecular orbitals with electrons, (according to the Aufbau principle), the type and approximate strength of the bonding between the atoms in the various X2 molecules can be predicted. 

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