100
K. B. Be´ c et al.
harmonic potential; as described in Eq. 5.3, this step is determined by the diagonalization of the Hessian evaluated at the respective energy minimum. In the process,
a paraboloid approximating the shape of the true potential is derived. This process
may be interpreted as the rotation of the coordinate system until Eq. 5.3 is fulfilled.
A positive-definite Hessian (all-positive eigenvalues) corresponds to a positive
curvature of the potential along all directions from the reference point. On the other
hand, in case a negative curvature is present along a specific direction, an imaginary
frequency is obtained in the solution of the harmonic approximation, which is for
instance employed to evaluate the properties of a transition state or/and reaction
coordinates. Hence, the analysis of the Hessian at the stationary point (at which
g(Q) is equal to zero, i.e., no slope of the potential) enables the identification of the
local minima (positive curvatures), local maxima (negative curvatures), and transition
states (mixed occurrence of positive and negative curvature).
Since the harmonic potential depends on the Hessian, the efficiency of its determination by means of electronic structure theory is critical. The methods for which an
analytical solution to the Hessian is available (e.g., HF, DFT, MP2, CIS) are far more
efficient as the basis for a harmonic analysis than those for which the Hessian can only
be calculated numerically (e.g., CC). Regardless, the harmonic approximation leads
to a dramatic simplification of the vibrational problem in terms of complexity. The
diagonalization of the Hessian yields the full vibrational solution: harmonic frequencies and the associated normal modes. For small to intermediate-sized molecules,
this is a computationally inexpensive step (although it may become a bottleneck in
studies of large systems using FF approaches), which made the harmonic approximation particularly important for early advances in vibrational spectroscopy. However,
it is an extensive approximation of the real molecular oscillator. Firstly, the shape of
the potential is fixed as a quadratic function. This is well-reflected in Fig. 5.6, as seen
in three-dimensional space (Fig. 5.6a, b) as well as in one-dimensional projections
respective to each of the modes (Fig. 5.6c, d). In this example, the true potential
along the symmetric stretching mode of H 2 O is asymmetric with respect to the equilibrium position (i.e., anharmonic) and resembles a Morse-like curve (Fig. 5.6c).
This type of anharmonic potential is well-known, as it is often discussed in case
of diatomic molecular oscillators (Fig. 5.4). Unlike the harmonic solution, the true
vibrational levels are not equidistant. Morse-like anharmonicity (high contribution
from the cubic terms in Eq. 5.1) leads to a subsequent reduction of the energy gaps
between consecutive levels. However, the potential of the antisymmetric stretching
mode of H 2 O, although symmetric in shape, also deviates from the harmonic potential
(Fig. 5.6d). This is due to significant contribution in the quartic terms in Eq. 5.1. The
quartic anharmonicity leads to widened distances between consecutive vibrational
levels.
Nevertheless, with some exceptions, e.g., of X–H stretching modes, in many cases
the deviation between the harmonic approximation and the true molecular oscillator
is relatively moderate. Consequently, in case of fundamental transitions, harmonic
frequencies corresponding to those vibrations remain overestimated but not dramatically. This effect can be mitigated by an empirical correction, applied a posteriori
in the form of a scaling of harmonic frequencies. Hence, the calculation of harmonic
K. B. Be´ c et al.
harmonic potential; as described in Eq. 5.3, this step is determined by the diagonalization of the Hessian evaluated at the respective energy minimum. In the process,
a paraboloid approximating the shape of the true potential is derived. This process
may be interpreted as the rotation of the coordinate system until Eq. 5.3 is fulfilled.
A positive-definite Hessian (all-positive eigenvalues) corresponds to a positive
curvature of the potential along all directions from the reference point. On the other
hand, in case a negative curvature is present along a specific direction, an imaginary
frequency is obtained in the solution of the harmonic approximation, which is for
instance employed to evaluate the properties of a transition state or/and reaction
coordinates. Hence, the analysis of the Hessian at the stationary point (at which
g(Q) is equal to zero, i.e., no slope of the potential) enables the identification of the
local minima (positive curvatures), local maxima (negative curvatures), and transition
states (mixed occurrence of positive and negative curvature).
Since the harmonic potential depends on the Hessian, the efficiency of its determination by means of electronic structure theory is critical. The methods for which an
analytical solution to the Hessian is available (e.g., HF, DFT, MP2, CIS) are far more
efficient as the basis for a harmonic analysis than those for which the Hessian can only
be calculated numerically (e.g., CC). Regardless, the harmonic approximation leads
to a dramatic simplification of the vibrational problem in terms of complexity. The
diagonalization of the Hessian yields the full vibrational solution: harmonic frequencies and the associated normal modes. For small to intermediate-sized molecules,
this is a computationally inexpensive step (although it may become a bottleneck in
studies of large systems using FF approaches), which made the harmonic approximation particularly important for early advances in vibrational spectroscopy. However,
it is an extensive approximation of the real molecular oscillator. Firstly, the shape of
the potential is fixed as a quadratic function. This is well-reflected in Fig. 5.6, as seen
in three-dimensional space (Fig. 5.6a, b) as well as in one-dimensional projections
respective to each of the modes (Fig. 5.6c, d). In this example, the true potential
along the symmetric stretching mode of H 2 O is asymmetric with respect to the equilibrium position (i.e., anharmonic) and resembles a Morse-like curve (Fig. 5.6c).
This type of anharmonic potential is well-known, as it is often discussed in case
of diatomic molecular oscillators (Fig. 5.4). Unlike the harmonic solution, the true
vibrational levels are not equidistant. Morse-like anharmonicity (high contribution
from the cubic terms in Eq. 5.1) leads to a subsequent reduction of the energy gaps
between consecutive levels. However, the potential of the antisymmetric stretching
mode of H 2 O, although symmetric in shape, also deviates from the harmonic potential
(Fig. 5.6d). This is due to significant contribution in the quartic terms in Eq. 5.1. The
quartic anharmonicity leads to widened distances between consecutive vibrational
levels.
Nevertheless, with some exceptions, e.g., of X–H stretching modes, in many cases
the deviation between the harmonic approximation and the true molecular oscillator
is relatively moderate. Consequently, in case of fundamental transitions, harmonic
frequencies corresponding to those vibrations remain overestimated but not dramatically. This effect can be mitigated by an empirical correction, applied a posteriori
in the form of a scaling of harmonic frequencies. Hence, the calculation of harmonic