Solid State Physics. Philip Hofmann

alt="phi left-parenthesis bold r 1 comma bold r 2 right-parenthesis equals phi Subscript normal upper A Baseline left-parenthesis bold r 1 right-parenthesis phi Subscript normal upper B Baseline left-parenthesis bold r 2 right-parenthesis"/>, with and being the wave functions for atomic hydrogen.

However, this simple approach is physically incorrect because such a wave function does not obey the Pauli principle. Since the electrons are fermions, the total wave function must be antisymmetric with respect to particle exchange, and the simple product wave function does not meet this requirement. The total wave function consists of a spatial part and a spin part, and thus there are two possibilities for forming an antisymmetric wave function – we can either combine a symmetric spatial part with an antisymmetric spin part or vice versa. The spatial part of the wave function can be constructed in two ways,

(2.9)

(2.10)

      The plus sign in Eq. (2.9) returns a symmetric spatial wave function, which we can combine with an antisymmetric spin wave function with the total spin equal to zero (the so‐called singlet state); the minus in Eq. (2.10) results in an antisymmetric spatial wave function to be combined with a symmetric spin wave function with the total spin equal to 1 (the so‐called triplet state).

The antisymmetric wave function in Eq. (2.10) vanishes for – that is, the two electrons cannot be at the same location simultaneously. This leads to a depletion of the electron density between the nuclei and hence to an antibonding state. For the symmetric case, on the other hand, the electrons have opposite spins and can be at the same place, which leads to a charge accumulation between the nuclei and hence to a bonding state (see Figure 2.4).

Figure 2.4 The energy changes

and

for the formation of a hydrogen molecule. The dashed lines represent the approximation for long distances. The two insets show grayscale images of the corresponding electron probability density.

An approximate way to calculate the eigenvalues of Eq. (2.3) was suggested by W. Heitler and F. London in 1927. Their approach was to use the known single‐particle 1s wave functions for atomic hydrogen for and to form a two‐electron wave function , in the form of either Eq. (2.9) or Eq. (2.10). These wave functions might not be entirely correct because the atomic wave functions will certainly be modified by the presence of the other atom. However, even if they are only approximately correct, we can obtain the molecular energy levels as

      (2.11)

      According to the variational principle in quantum mechanics, the resulting energy will always be higher than the correct ground‐state energy, but it will approach it for a good choice of the trial wave functions.

The detailed calculation is quite lengthy and shall not be given here.¹ The resulting ground‐state energies for the singlet and triplet states can be written as

(2.12)

(2.13)

       is the ground‐state energy for one hydrogen atom; it is multiplied by two because we start with two atoms. The energies and are also shown in Figure 2.4. is always larger than zero and does not lead to any chemical bonding. , on the other hand, shows a minimum with negative energy at approximately . This is the bonding state.