110
5 What has Quantum Chemistry Got to Do with It?
• Time symmetry. The reversal of all momenta and spins is nothing other than the
time-reversal operator we introduced earlier. As long as no external magnetic
fields are present, time is reversible.
• Elementary particles of the same kind are indistinguishable. The remaining symmetries do not refer to space or time but to the permutational symmetry of a set
of particles. All electrons are the same, and thus the Hamiltonian does not change
when we permute electron labels. This symmetry becomes apparent only when
multielectronic wavefunctions are considered, and these will be treated in the next
chapter. Likewise, identical nuclei can be permuted without changing the Hamiltonian. This is reflected in the potential energy terms, which consist of sums over
all pairwise interactions. Permutations of particle labels change only the order of
the terms in these summations.
As an example, in Fig. 5.1 we return to our favored ammonia molecule and list all
nuclear permutations, with and without the all-particle inversion operator, that leave
the full Hamiltonian invariant. Nuclear permutations are defined here in the same
way as in Sect. 3.3. A permutation such as (ABC) means that the letters A, B, and
C are replaced by B, C, and A, respectively. 1 The inversion operator, ˆ
E ∗ , inverts the
positions of all particles through a common inversion center, which can be conveniently chosen in the mass origin. In total, 12 combinations of such operations are
found, which together form a group that is isomorphic to D 3h . How is this related
to our previous C 3v point group? At this point it is very important to recall that the
state of a molecule is not only determined by its Hamiltonian but also, and to an
equal extent, by the boundary conditions. The eigenvalue equation is a differential
equation that has a very extensive set of mathematical solutions, but not all these
solutions are also acceptable states of the physical system. The role of the boundary
conditions is to define constraints that filter out physically unacceptable states of the
system. In most cases these constraints also lead to the quantization of the energies.
From our present perspective the invariance group should not only leave the
Hamiltonian invariant, but also it should not alter the boundary conditions.N o w ,
in most quantum-chemical applications a very stringent boundary condition is offered by the Born−Oppenheimer approximation. This simply states that the nuclei
are considered immobile. It seriously restricts the Longuet-Higgins list. Indeed, we
should retain only those operations that leave all nuclei in their rest positions. Applying this to ammonia, of the twelve operations we retain only those that do not
affect the starting structure, i.e., only those combinations of permutations, permutation inversion, and overall rotations that, as a net result, keep the nuclei fixed in
space. This immediately constrains the symmetry group to the familiar C 3v molecular point group. As an example, the permutation of nuclei B and C followed by the
inversion of all particles gives rise to the structure, marked (A)(BC) ∗ in Fig. 5.1,
in which the ammonia molecule has been turned upside down. The nuclei can be
moved back to their original positions by rotating the whole molecule by 180 ◦ about
1 See also [4, Chap. 1].
5 What has Quantum Chemistry Got to Do with It?
• Time symmetry. The reversal of all momenta and spins is nothing other than the
time-reversal operator we introduced earlier. As long as no external magnetic
fields are present, time is reversible.
• Elementary particles of the same kind are indistinguishable. The remaining symmetries do not refer to space or time but to the permutational symmetry of a set
of particles. All electrons are the same, and thus the Hamiltonian does not change
when we permute electron labels. This symmetry becomes apparent only when
multielectronic wavefunctions are considered, and these will be treated in the next
chapter. Likewise, identical nuclei can be permuted without changing the Hamiltonian. This is reflected in the potential energy terms, which consist of sums over
all pairwise interactions. Permutations of particle labels change only the order of
the terms in these summations.
As an example, in Fig. 5.1 we return to our favored ammonia molecule and list all
nuclear permutations, with and without the all-particle inversion operator, that leave
the full Hamiltonian invariant. Nuclear permutations are defined here in the same
way as in Sect. 3.3. A permutation such as (ABC) means that the letters A, B, and
C are replaced by B, C, and A, respectively. 1 The inversion operator, ˆ
E ∗ , inverts the
positions of all particles through a common inversion center, which can be conveniently chosen in the mass origin. In total, 12 combinations of such operations are
found, which together form a group that is isomorphic to D 3h . How is this related
to our previous C 3v point group? At this point it is very important to recall that the
state of a molecule is not only determined by its Hamiltonian but also, and to an
equal extent, by the boundary conditions. The eigenvalue equation is a differential
equation that has a very extensive set of mathematical solutions, but not all these
solutions are also acceptable states of the physical system. The role of the boundary
conditions is to define constraints that filter out physically unacceptable states of the
system. In most cases these constraints also lead to the quantization of the energies.
From our present perspective the invariance group should not only leave the
Hamiltonian invariant, but also it should not alter the boundary conditions.N o w ,
in most quantum-chemical applications a very stringent boundary condition is offered by the Born−Oppenheimer approximation. This simply states that the nuclei
are considered immobile. It seriously restricts the Longuet-Higgins list. Indeed, we
should retain only those operations that leave all nuclei in their rest positions. Applying this to ammonia, of the twelve operations we retain only those that do not
affect the starting structure, i.e., only those combinations of permutations, permutation inversion, and overall rotations that, as a net result, keep the nuclei fixed in
space. This immediately constrains the symmetry group to the familiar C 3v molecular point group. As an example, the permutation of nuclei B and C followed by the
inversion of all particles gives rise to the structure, marked (A)(BC) ∗ in Fig. 5.1,
in which the ammonia molecule has been turned upside down. The nuclei can be
moved back to their original positions by rotating the whole molecule by 180 ◦ about
1 See also [4, Chap. 1].