: Improvements on the approximation : An example of the : The trial wavefunction: a

## Bonding and anti-bonding orbitals

We look, first, at the form of the orbitals that correspond to the energies , respectively. These can be obtained by solving for the variational coefficients, and . These will be given by the matrix equation:

For example, using , the following equations for the coefficients are obtained:

which are not independent but are satisfied if . Similarly, for , we obtain the two equations:

which are satisfied if . Thus, the two states corresponding to are

The overall constants are determined by requiring that both be normalized. For , for example, we find

which requires that

Similarly, it can be shown that

Thus, the two states become

Notice that these are orthogonal:

Projecting onto a coordinate basis, we have

The state , which corresponds to the energy admits a chemical bond and is, therefore, called a bonding state. The state , which corresponds to the energy does not admit a chemical bond and is, therefore, called an anti-bonding state. and are examples of what are called molecular orbitals. In this case, they are constructed from linear combinations of atomic orbitals.

The functional form of the two molecular orbitals for H within the current approximation scheme is

The contours of these functions are sketched below (the top plot shows the two individual two atomic orbitals, while the middle and bottom show the linear combinations and , respectively):

Since and have the same value at the origin and very similarly values near the origin, it is clear that the electron probability density in this region will intensify for which is the sum of and . This clearly corresponds to a chemical bonding situation. In contrast, for , which is the difference between and , there will be a deficit of electron density in the region between the two protons, which is indicative of a non-bonding situation.

: Improvements on the approximation : An example of the : The trial wavefunction: a
Mark Tuckerman 平成17年2月2日