5 Introduction to Quantum Vibrational Spectroscopy
105
implementation of Numerov’s approach in the form of a matrix eigenvalue problem.
Accordingly, using the matrices A and B as well as the diagonal matrix V, containing
V (q) as elements, the solution can be written as
−
2
2m
A + BV
≈ BE
(5.24)
Rearrangement of Eq. 5.24 leads to the matrix representation of the timeindependent Schrödinger equation H = E, with
H = −
2
2m
B
−1 A + V
(5.25)
Eigen decomposition of H simultaneously yields all energy eigenvalues along the
diagonal of the energy matrix E, and the associated eigenvectors are collected in .
As the key advantage, these grid-based approaches do not require any assumption
or pre-defined building blocks (e.g., basis sets) to formulate the wavefunction .
These approaches are not limited to questions in vibrational spectroscopy and similar
methods have also been employed in the description of quantum tunnelling and the
electronic structure of atoms and small molecular systems.
The method can be extended to arbitrary orders in the numerical derivatives by
truncating the Taylor series of higher degree (e.g., A and B matrices with seven diagonal entries would require a Taylor series of eight degree). To predict IR intensities,
the transition moment integral μ mn , consisting of the respective wavefunctions of the
two involved states m and n , as well as the transition moment operator ˆ
μ(q), has
to be calculated.
μ mn =
∞
∫
−∞
m ˆ
μμ n dτ
(5.26)
In case of infrared spectroscopy, ˆ
μ(q) equals the dipole moment μ as a function of
the molecule’s normal coordinates q, making infrared measurements especially sensitive on polar function. The transition dipole moment is then employed to calculate
the associated oscillator strength f mn
f mn =
4π m e
3e 2
μ mn
2
ν mn
(5.27)
where m e denotes the electron mass, e the elementary charge, and v mn the transition
energy between the two states m and n. Typically, the oscillator strength is normalized
to that of the fundamental mode.
Within the 1D formalism, grid-based methods are capable of inherently taking
arbitrary anharmonicities into account. However, the main benefit lies in the generalization of the grid-based methods to higher dimensions, which enables the inclusion
105
implementation of Numerov’s approach in the form of a matrix eigenvalue problem.
Accordingly, using the matrices A and B as well as the diagonal matrix V, containing
V (q) as elements, the solution can be written as
−
2
2m
A + BV
≈ BE
(5.24)
Rearrangement of Eq. 5.24 leads to the matrix representation of the timeindependent Schrödinger equation H = E, with
H = −
2
2m
B
−1 A + V
(5.25)
Eigen decomposition of H simultaneously yields all energy eigenvalues along the
diagonal of the energy matrix E, and the associated eigenvectors are collected in .
As the key advantage, these grid-based approaches do not require any assumption
or pre-defined building blocks (e.g., basis sets) to formulate the wavefunction .
These approaches are not limited to questions in vibrational spectroscopy and similar
methods have also been employed in the description of quantum tunnelling and the
electronic structure of atoms and small molecular systems.
The method can be extended to arbitrary orders in the numerical derivatives by
truncating the Taylor series of higher degree (e.g., A and B matrices with seven diagonal entries would require a Taylor series of eight degree). To predict IR intensities,
the transition moment integral μ mn , consisting of the respective wavefunctions of the
two involved states m and n , as well as the transition moment operator ˆ
μ(q), has
to be calculated.
μ mn =
∞
∫
−∞
m ˆ
μμ n dτ
(5.26)
In case of infrared spectroscopy, ˆ
μ(q) equals the dipole moment μ as a function of
the molecule’s normal coordinates q, making infrared measurements especially sensitive on polar function. The transition dipole moment is then employed to calculate
the associated oscillator strength f mn
f mn =
4π m e
3e 2
μ mn
2
ν mn
(5.27)
where m e denotes the electron mass, e the elementary charge, and v mn the transition
energy between the two states m and n. Typically, the oscillator strength is normalized
to that of the fundamental mode.
Within the 1D formalism, grid-based methods are capable of inherently taking
arbitrary anharmonicities into account. However, the main benefit lies in the generalization of the grid-based methods to higher dimensions, which enables the inclusion