10. Appendix

Therefore the result of I ′ on „B is to change it to:
(1/ √ 2)[exp(i3apple/4))|S2〉 exp(iapple/4) exp(i3apple/4)|S1〉]
(1/ √ 2)[ exp(iapple/4))|S2〉 exp(apple)|S1〉]
(1/ √ 2)[exp(iapple/4))|S2〉 |S1〉]
Solution to Problem 2.20 599
„B.
Thus we find the lower energy bonding s-orbital to have odd parity under I ′ .
Again the energy separating the bonding and anti-bonding s-orbitals in Si is
only 2|Vss| 8.13 eV only. But the agreement with the full scale pseudopotential
calculation is better than the result in (a).
(c) One may ask then: what will the electron density look like for the odd
parity (bonding) orbital when k is at the L point? Will this electron density
have a maximum between the two atoms as we would expect for the bonding
state? The answer is yes. To understand why the bonding electronic wave
function has odd parity while the anti-bonding electronic wave function has
even parity we will examine the parity of the zone-center acoustic and optical
phonons in the diamond lattice. In Fig. 2.7 we have shown schematically
the displacement vectors for both phonon modes. We notice that the displacement
vectors point in the same direction for both atoms within the unit cell
in the case of the acoustic phonon. Clearly, if we extrapolate the atomic displacement
to the midpoint between the two atoms it will not be zero. In a
sense the acoustic phonon is the analogue of a bonding electronic state and
yet the acoustic phonon has odd parity under the operation I ′ . On the other
hand, the displacement vectors for the optical phonon point in opposite directions
at the two atoms inside the unit cell. When extrapolated to the midpoint
of the two atoms in the unit cell we expect the phonon wave function should
vanish as in an anti-bonding state but its parity is even under the operation
I ′ . The reason why the parity of these phonon modes seems to be opposite to
what one may expect intuitively lies in the translation by T of the lattice under
the operation I ′ . This translation causes the parity of the phonon modes
to change sign. Without this translation the acoustic phonon parity would indeed
be even while the parity of the optical phonon would be odd. The same
is true for the electron wave functions when k is at the L point. The translation
operation T introduces a phase factor: exp[ik · T] to the electronic wave
functions. When k (apple/a)(1, 1, 1) at the L point, this phase vector is equal to
exp(i3apple/4) and causes the parity of the electron wave function to reverse sign
as shown in (b). Thus the operation T causes both the phonon wave functions
at k 0 and the electronic wave functions for k (apple/a)(1, 1, 1) to reverse
the sign of their parity. This phase factor does not cause a problem when one
consider the “compatibility” between the zone-center and L point electronic
wave functions since this phase factor is zero when k (0, 0, 0). If we start at
k 0 we find that the parity of the bonding state (where the electron density
is non-zero at the mid-point between the two atoms in the unit cell) is even.
The parity of the electronic wave function changes sign as k approaches the L
point but the state remains a bonding state.