Solutions to Problems - Group Theory Applied to Chemistry - page 244

The normalized cyclic waves are thus given by

1
√ 2 π e ± ikφ |± k =

0.
2.3 The combination of transposition and complex conjugation is called the adjoint
operation, indicated by a dagger. A Hermitian matrix is thus self-adjoint. An
eigenfunction of this matrix, operating in a function space, may be expressed as
a linear combination

and these waves are orthogonal: − k | k =

| ψ m = c k | f k

k

We may arrange the expansion coefficients as a column vector c . This is called
the eigenvector. Its adjoint, c † , is then the complex-conjugate row vector. The
corresponding eigenvalue is denoted as E m . Now start by writing the eigenvalue
equation and multiply left and right with the adjoint eigenvector:

H c = E m c

E m c † c
Now take the adjoint and use the self-adjoint property of

c † H c =

H :

c †

† c

= E m c † c H

= E m c † c
A comparison of both results shows that the eigenvalue must be equal to its
complex conjugate and hence be real. If

c † H c

is skew-symmetric, a similar argu-
ment shows that the eigenvalue must be imaginary.

H

3.1 The table is a valid multiplication table of a group that is isomorphic to D 2 .The
element C is the unit element. There are six ways to assign the three twofold
axes to the letters A,B,D .
3.2 Any nonlinear triatomic molecule with three different atoms has only C s sym-
metry, e.g., a water molecule with one hydrogen replaced by deuterium. C 2
symmetry requires a nonplanar tetra-atomic molecule, such as H 2 O 2 .Inthe
free state the dihedral angle of this molecule is almost a right angle (see the
figure). To realize C i symmetry, one needs at least six atoms. Since three atoms
are always coplanar, the smallest molecule with no symmetry at all has at least
four atoms.