84
4 Representations
This gives an increase of frequency of 0.56 cm −1 , which is close to the experimental
value [6]o f0 . 6 0c m −1 . This confirms the dominant stretching character of the ν 3
mode.
What have we learned from this example? The Hessian matrix is block diagonal
over the irreps of the point group, and, as a result, the normal modes are characterized by symmetry labels. These labels are exact spectral assignments. In the long
run their relevance for the study of symmetry may be more important than the temporary gain in computational time for evaluation and diagonalization of the Hessian
matrix.
4.9 Application: Hückel Theory
The Hückel model for the chemist (or the analogous tight-binding model for the
condensed-matter physicist) is an extremely simplified molecular orbital model [7],
which nevertheless continues to play an important role in our understanding of electronic structures and properties. It emphasizes the molecule–graph analogy and uses
what is now regarded as spectral graph theory [8, 9] in order to obtain molecular orbitals. Its strength comes from the fact that, in spite of the approximations involved,
it incorporates the essential topological and symmetry aspects of electronic structures, and, as we keep repeating, these are simple but exact properties of complex
molecular quantum-systems. Hückel theory is preferentially applied to molecular
systems where each atom or node carries one atomic orbital, say |φ i . Molecular orbitals will be denoted as |Φ k . To find the molecular orbitals, one sets up the Hückel
Hamiltonian matrix, which in its most simplified form is proportional to the adjacency matrix, A, of the molecular graph. Elements of the adjacency matrix are zero,
unless row and column index refer to neighboring nodes, in which case the matrix
element is equal to one. The Hamiltonian matrix then is given by
φ i |H|φ j =αδ ij + βA ij
(4.110)
or, in operator form,
H =
i
α|φ i φ i |+
i =j
βA ij |φ i φ j |
(4.111)
Here, α is the so-called Coulomb integral, which corresponds to the on-site interaction element. It defines the zero-point of energy and thus has only a symbolic
significance in homogeneous systems. However, in hetero-atomic systems, it is important to differentiate the atoms. As an example, the Coulomb integral for nitrogen
will be more negative than the one for carbon because the heavier nitrogen nucleus
exerts a greater attraction on the electrons. The β parameter is the resonance or
inter-site hopping integral. It represents a bonding interaction and thus is negative.
The Hückel eigenvalues are thus of opposite sign as compared with the corresponding eigenvalues of the adjacency matrix. The molecular symmetry group is called in
4 Representations
This gives an increase of frequency of 0.56 cm −1 , which is close to the experimental
value [6]o f0 . 6 0c m −1 . This confirms the dominant stretching character of the ν 3
mode.
What have we learned from this example? The Hessian matrix is block diagonal
over the irreps of the point group, and, as a result, the normal modes are characterized by symmetry labels. These labels are exact spectral assignments. In the long
run their relevance for the study of symmetry may be more important than the temporary gain in computational time for evaluation and diagonalization of the Hessian
matrix.
4.9 Application: Hückel Theory
The Hückel model for the chemist (or the analogous tight-binding model for the
condensed-matter physicist) is an extremely simplified molecular orbital model [7],
which nevertheless continues to play an important role in our understanding of electronic structures and properties. It emphasizes the molecule–graph analogy and uses
what is now regarded as spectral graph theory [8, 9] in order to obtain molecular orbitals. Its strength comes from the fact that, in spite of the approximations involved,
it incorporates the essential topological and symmetry aspects of electronic structures, and, as we keep repeating, these are simple but exact properties of complex
molecular quantum-systems. Hückel theory is preferentially applied to molecular
systems where each atom or node carries one atomic orbital, say |φ i . Molecular orbitals will be denoted as |Φ k . To find the molecular orbitals, one sets up the Hückel
Hamiltonian matrix, which in its most simplified form is proportional to the adjacency matrix, A, of the molecular graph. Elements of the adjacency matrix are zero,
unless row and column index refer to neighboring nodes, in which case the matrix
element is equal to one. The Hamiltonian matrix then is given by
φ i |H|φ j =αδ ij + βA ij
(4.110)
or, in operator form,
H =
i
α|φ i φ i |+
i =j
βA ij |φ i φ j |
(4.111)
Here, α is the so-called Coulomb integral, which corresponds to the on-site interaction element. It defines the zero-point of energy and thus has only a symbolic
significance in homogeneous systems. However, in hetero-atomic systems, it is important to differentiate the atoms. As an example, the Coulomb integral for nitrogen
will be more negative than the one for carbon because the heavier nitrogen nucleus
exerts a greater attraction on the electrons. The β parameter is the resonance or
inter-site hopping integral. It represents a bonding interaction and thus is negative.
The Hückel eigenvalues are thus of opposite sign as compared with the corresponding eigenvalues of the adjacency matrix. The molecular symmetry group is called in