98
K. B. Be´ c et al.
Diagonalization of the mass-weighted Hessian yields a matrix with 3N − N inv
columns consisting of orthonormal eigenvectors that describe the vibrational motion
of the system within the harmonic approximation, the so-called mass-weighted
normal modes. The 3N − N inv diagonal elements of the eigenvalue matrix h are
proportional to the square frequency of the associated normal mode.
h = U
T HU
(5.5)
The example of how the harmonic approximation simplifies the true behavior
of a vibrating molecule is demonstrated for the case of a water molecule, H1OH2
(Fig. 5.6), considering a two-dimensional example limited to the two OH stretching
vibrations. The corresponding two-dimensional potential energy surface V (r OH1 ,
r OH2 ) is described by the interatomic distances r OH1 and r OH2 (the corresponding
coordinates are depicted in Fig. 5.6a as black lines). In this example, the true potential
was determined with high accuracy using the CCSD(T)/aug-cc-pVTZ level of theory
employing a tight grid spacing.
As outlined above, the problem of the harmonic oscillator is only solvable at a
stationary point of the molecule’s PES. In the present example, this means that prior
to the evaluation of the Hessian, r OH1 and r OH2 need to be optimized to identify the
Fig. 5.6 Harmonic analysis at the example of the stretching vibrations of water v 1 (symmetric:
q sym ) and v 3 (antisymmetric, q sym ); a the true nature of normal modes on the potential energy
surface (red line: q sym ; blue line: q asym ); b the nature of the harmonic approximation applied to
these modes; c harmonic and anharmonic Morse-like potential curve of q sym ; the spacing between
subsequent energy levels is increasing; d harmonic and quartic anharmonic potential curve of q asym ;
in contrast to q sym , the spacing between the levels demonstrates a increase upon higher excitation
K. B. Be´ c et al.
Diagonalization of the mass-weighted Hessian yields a matrix with 3N − N inv
columns consisting of orthonormal eigenvectors that describe the vibrational motion
of the system within the harmonic approximation, the so-called mass-weighted
normal modes. The 3N − N inv diagonal elements of the eigenvalue matrix h are
proportional to the square frequency of the associated normal mode.
h = U
T HU
(5.5)
The example of how the harmonic approximation simplifies the true behavior
of a vibrating molecule is demonstrated for the case of a water molecule, H1OH2
(Fig. 5.6), considering a two-dimensional example limited to the two OH stretching
vibrations. The corresponding two-dimensional potential energy surface V (r OH1 ,
r OH2 ) is described by the interatomic distances r OH1 and r OH2 (the corresponding
coordinates are depicted in Fig. 5.6a as black lines). In this example, the true potential
was determined with high accuracy using the CCSD(T)/aug-cc-pVTZ level of theory
employing a tight grid spacing.
As outlined above, the problem of the harmonic oscillator is only solvable at a
stationary point of the molecule’s PES. In the present example, this means that prior
to the evaluation of the Hessian, r OH1 and r OH2 need to be optimized to identify the
Fig. 5.6 Harmonic analysis at the example of the stretching vibrations of water v 1 (symmetric:
q sym ) and v 3 (antisymmetric, q sym ); a the true nature of normal modes on the potential energy
surface (red line: q sym ; blue line: q asym ); b the nature of the harmonic approximation applied to
these modes; c harmonic and anharmonic Morse-like potential curve of q sym ; the spacing between
subsequent energy levels is increasing; d harmonic and quartic anharmonic potential curve of q asym ;
in contrast to q sym , the spacing between the levels demonstrates a increase upon higher excitation