Atoms and Molecules
161
H 2 , O 2 , N 2 , etc. The bonds resulting from the sharing of the valence electrons
are known as covalent or homopolar bonds.
Consider a particle moving in the presence of two similar, one-dimensional,
attractive potential V 1 and V 2 which are centred at positions x 1 and x 2 . If the
positions x 1 and x 2 are separated by a large distance d, the ground state energy
E 0 will be essentially degenerate, the degenerate eigenstates being ψ 1 and ψ 2
which are eigenstates with only V 1 or V 2 being present, respectively. As the
separation distance d decreases, one may consider as possible eigenstates,
ψ ± =
1
2
(ψ 1 ± ψ 2 )
(5.60)
where the overlap integral is ignored in the normalization. If it is also assumed
that ψ 1 is small at x 2 and ψ 2 is small at x 1 , the expectation value of the energy is
E ± ≈ E 0 ± ∫ ψ 1 * (x) V 2 ψ 2 (x) dx
(5.61)
Since the potential is attractive (V 1,2 may be taken to be negative), the integral
is expected to be negative. Therefore, the ground state splits into two states, the
symmetric state ψ + with energy E + and the antisymmetric state ψ
–
with energy
E –. For the symmetric state, which has a lower energy, it is seen that the
wavefunction is larger in between the centres of potentials than it is on the
outside.
Take the above model to be reasonable, it is seen that when two hydrogen
atoms are brought together, the two valence electrons prefer to be in between
the two atoms, and have a lower energy than when they are attached to the
atoms in isolation. The equilibrium situation is reached at some finite distance of
separation (the repulsion term in Eq. (5.58) becomes important at short distances),
in which the electrons are shared by the two atoms and the electrons prefer to
be in between the atoms. This gives rise to what are known as covalent or
homopolar bonds. The equilibrium distance in the case of the H 2 molecule is
about 0.74 Å and the corresponding binding energy is 4.75 eV. It is important to
note that in covalent bonds, since both the electrons prefer to be in between the
atoms, the spatial wave function is symmetric under the exchange of the two
electrons. It is, therefore, required by Pauli’s exclusion principle that the spins
must point in opposite directions and the total spin of the two electrons must be
zero. The state with parallel spins is antisymmetric in the spatial part of the
wave function, and will in general have a higher energy (see Fig. 5.12). The
antiparallel spin state is called the bonding state and the parallel spin state is
called the antibonding state. The situation is similar for other diatomic molecules.
161
H 2 , O 2 , N 2 , etc. The bonds resulting from the sharing of the valence electrons
are known as covalent or homopolar bonds.
Consider a particle moving in the presence of two similar, one-dimensional,
attractive potential V 1 and V 2 which are centred at positions x 1 and x 2 . If the
positions x 1 and x 2 are separated by a large distance d, the ground state energy
E 0 will be essentially degenerate, the degenerate eigenstates being ψ 1 and ψ 2
which are eigenstates with only V 1 or V 2 being present, respectively. As the
separation distance d decreases, one may consider as possible eigenstates,
ψ ± =
1
2
(ψ 1 ± ψ 2 )
(5.60)
where the overlap integral is ignored in the normalization. If it is also assumed
that ψ 1 is small at x 2 and ψ 2 is small at x 1 , the expectation value of the energy is
E ± ≈ E 0 ± ∫ ψ 1 * (x) V 2 ψ 2 (x) dx
(5.61)
Since the potential is attractive (V 1,2 may be taken to be negative), the integral
is expected to be negative. Therefore, the ground state splits into two states, the
symmetric state ψ + with energy E + and the antisymmetric state ψ
–
with energy
E –. For the symmetric state, which has a lower energy, it is seen that the
wavefunction is larger in between the centres of potentials than it is on the
outside.
Take the above model to be reasonable, it is seen that when two hydrogen
atoms are brought together, the two valence electrons prefer to be in between
the two atoms, and have a lower energy than when they are attached to the
atoms in isolation. The equilibrium situation is reached at some finite distance of
separation (the repulsion term in Eq. (5.58) becomes important at short distances),
in which the electrons are shared by the two atoms and the electrons prefer to
be in between the atoms. This gives rise to what are known as covalent or
homopolar bonds. The equilibrium distance in the case of the H 2 molecule is
about 0.74 Å and the corresponding binding energy is 4.75 eV. It is important to
note that in covalent bonds, since both the electrons prefer to be in between the
atoms, the spatial wave function is symmetric under the exchange of the two
electrons. It is, therefore, required by Pauli’s exclusion principle that the spins
must point in opposite directions and the total spin of the two electrons must be
zero. The state with parallel spins is antisymmetric in the spatial part of the
wave function, and will in general have a higher energy (see Fig. 5.12). The
antiparallel spin state is called the bonding state and the parallel spin state is
called the antibonding state. The situation is similar for other diatomic molecules.