Antibonding wave function

A molecular compound the antibonding wave function in an imino nitroxide free radical... 

Fig. 8.8. Illustration of bonding-antibonding wave function (po, which has 5 -symmetry and the non-bonding wave function 0i, which has d x — y ) symmetry. The amplitude of this wave function is zero along the [110] direction.

Now, the variational principle respecting the MO-coefficients of Eq. (1.214) upon the trial energy (1.215) yields that the bonding (+) and antibonding (-) wave-functions display the same forms as in the classical Heitler-London homopolar model (Heitler and London, 1927), while the corresponding energies are corrected by the bondonic BEC contribution... 

The optimum values of die oq and a coefficients are determined by the variational procedure. The HF wave function constrains both electrons to move in the same bonding orbital. By allowing the doubly excited state to enter the wave function, the electrons can better avoid each other, as the antibonding MO now is also available. The antibonding MO has a nodal plane (where opposite sides of this plane. This left-right correlation is a molecular equivalent of the atomic radial correlation discussed in Section 5.2. 

The wave function for the bonding and antibonding orbitals can be written as... 

Compared to an H atom, electrons with the function y/, are less energetic, and those with the function y/2 are more energetic. When the two available electrons occupy the molecular orbital y/(, this is energetically favorable y/, is the wave function of a bonding molecular orbital. y/2 belongs to an antibonding molecular orbital its occupation by electrons requires the input of energy. 

In order to calculate the orbitals for a methane molecule, the four Lv functions of the four hydrogen atoms and the functions 2s, 2px, 2py and 2pz of the carbon atom are combined to give eight wave functions, four of which are bonding and four of which are antibonding. The four bonding wave functions are ... 

The curve for the energy dependence as a function of k in Fig. 10.4 has a positive slope. This is not always so. When p orbitals are joined head-on to a chain, the situation is exactly the opposite. The wave function y/0 = Z.X, is then antibonding, whereas yKja is bonding (Fig. 10.5). 

In calculations and interaction diagrams, only the most simplistic MO models will be chosen to represent ground and excited states of reactants. An olefin then has a bond framework largely neglected in discussing the reactivity of the molecule. The bonding level will be characterized by a jr-electron wave function with no nodes between the two basis fi orbitals of the ir-bond. The first jr-antibonding level has one node in the wave function, and a first excited state has electron-occupancy of unity in each level. 

The two bonding 7r orbitals represented by these wave functions are degenerate. The wave functions for the antibonding states are identical in form except that negative signs are used in the combination of atomic wave functions and in the normalization constants. 

The root x = 21/2 leads to the wave function for the antibonding orbital,... 

A quantitative treatment of tt complex formation is, however, more complicated, since it is generally recognized that all three wave functions are necessary for an accurate description of the bond. For instance, it has been pointed out by Orgel (27) that n complex stability cannot solely be the result of n electron donation into empty metal d orbitals, since d and ions (Cu+, Ag+, Ni , Rh+, Pt , Pd++) form some of the strongest complexes with poor bases such as ethylene, tt Complex stability would thus appear to involve the significant back-donation of metal d electrons into vacant antibonding orbitals of the olefin. Because of the additional complication of back-donation plus the uncertainty of metal surface orbitals, it is only possible to give a qualitative treatment of this interaction at the present time. 

A theoretical approach in molecular orbital studies to formulate an expression for the wave function of a molecular orbital (both for bonding and antibonding orbitals) by linear combinations of the overlapping atomic orbitals with appropriate weighting factors. 

We take a very simple case of a pair of orbitals a and b that can bond. We assume the orbitals are at two different centers. The simplest LCAO approximation to the bonding orbital is cr = A a + b), and the antibonding coimterpart is o = B a — b). Here A = 1/V2(1 + S) and B = 1 /V2(l — S), where S is the overlap integral, are the normalization constants. Consider the simple three-electron doublet wave function... 

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