96
K. B. Be´ c et al.
wavefunction-based approaches. In combination with further computational techniques such as efficient pre-screening approaches, the use of sparse matrix algebra
routines, as well as convergence accelerators aimed at keeping the number of iterations small, DFT offers a remarkably favorable level of efficiency, an advantage
which is well-reflected by the popularity of its use in spectroscopic studies.
5.4 Harmonic Frequency Evaluation
5.4.1 Molecular Geometry Optimization Toward the Energy
Minimum
Geometry optimization, or energy minimization, is the procedure of determination of
the atomic (nuclear) coordinates of a molecule, which result in the lowest total potential energy of the system. A molecule’s potential energy V (Q) is a many-parameter
function of its atomic coordinates, represented as the vector Q = {q 1 , q 2 ,…, q 3N−Ninv }.
In principle, geometry optimization is a purely mathematical optimization problem
of finding Q that minimizes V (Q). In other words, it is a search for atomic coordinates of the molecule that minimize its potential energy. For a stationary point on
the potential energy surface (PES), the energy gradient (the derivative of the energy
with respect to all atomic coordinates, ∂V/∂q i ) is zero. A generic implementation
of the geometry optimization procedure is an iterative process of adjusting Q by
following the gradient toward zero. Note, the definition of the atomic coordinates is
not implicitly imposed. These may be, e.g., Cartesian coordinates, or internal coordinates describing bond lengths, bond angles, and dihedral angles. The quantum theory
model that provides V (Q) is also not imposed from the point of view of the optimization problem. As it will be demonstrated in the next section, geometry optimization
performed in order to bring the system to its local minimum on the potential energy
surface is a mandatory step prior to the execution of a harmonic frequency analysis.
5.4.2 Harmonic Approximation
Quantum chemical approaches to vibrational motion are in many points analogous to
the problem of electronic structure. Accordingly, the theory of the vibrational structure is based on the time-independent vibrational (nuclear) Schrödinger equation [2].
The Born–Oppenheimer approximation still applies, but in this case, the electronic
structure is reduced to the role of the source of an external potential upon which the
motion of nuclei depends. The vibrational Hamiltonian of a polyatomic oscillator
can be expressed as (Eq. 5.1)
K. B. Be´ c et al.
wavefunction-based approaches. In combination with further computational techniques such as efficient pre-screening approaches, the use of sparse matrix algebra
routines, as well as convergence accelerators aimed at keeping the number of iterations small, DFT offers a remarkably favorable level of efficiency, an advantage
which is well-reflected by the popularity of its use in spectroscopic studies.
5.4 Harmonic Frequency Evaluation
5.4.1 Molecular Geometry Optimization Toward the Energy
Minimum
Geometry optimization, or energy minimization, is the procedure of determination of
the atomic (nuclear) coordinates of a molecule, which result in the lowest total potential energy of the system. A molecule’s potential energy V (Q) is a many-parameter
function of its atomic coordinates, represented as the vector Q = {q 1 , q 2 ,…, q 3N−Ninv }.
In principle, geometry optimization is a purely mathematical optimization problem
of finding Q that minimizes V (Q). In other words, it is a search for atomic coordinates of the molecule that minimize its potential energy. For a stationary point on
the potential energy surface (PES), the energy gradient (the derivative of the energy
with respect to all atomic coordinates, ∂V/∂q i ) is zero. A generic implementation
of the geometry optimization procedure is an iterative process of adjusting Q by
following the gradient toward zero. Note, the definition of the atomic coordinates is
not implicitly imposed. These may be, e.g., Cartesian coordinates, or internal coordinates describing bond lengths, bond angles, and dihedral angles. The quantum theory
model that provides V (Q) is also not imposed from the point of view of the optimization problem. As it will be demonstrated in the next section, geometry optimization
performed in order to bring the system to its local minimum on the potential energy
surface is a mandatory step prior to the execution of a harmonic frequency analysis.
5.4.2 Harmonic Approximation
Quantum chemical approaches to vibrational motion are in many points analogous to
the problem of electronic structure. Accordingly, the theory of the vibrational structure is based on the time-independent vibrational (nuclear) Schrödinger equation [2].
The Born–Oppenheimer approximation still applies, but in this case, the electronic
structure is reduced to the role of the source of an external potential upon which the
motion of nuclei depends. The vibrational Hamiltonian of a polyatomic oscillator
can be expressed as (Eq. 5.1)