5 Introduction to Quantum Vibrational Spectroscopy
99
minimum on the V (r OH1 , r OH2 ) surface. To obtain harmonic modes, V (r OH1 , r OH2 )
needs to be approximated via the potential of a 2D harmonic oscillator V
harm . The
key approximation in this case is that the harmonic potential is additive.
V
harm
= V (r OH1 ) + V (r OH2 )
(5.6)
Equation 5.6 requires that the potential does not depend simultaneously on r OH1
and r OH2, , i.e., there is no coupling potential. This implies that the vibrational wavefunction is a product of 1D wavefunctions, and the respective energy eigenvalues are
additive (same as the potential case shown above), as described by Eqs. 5.7 and 5.8.
|(r OH1 , r OH2 ) = | (r OH1 ) · | (r OH1 )
(5.7)
E
r OH1, r OH2,
= E(r OH1 ) + E(r OH2 )
(5.8)
In this case, the Hessian would be diagonal. To match the latter criterion, a reorientation of the coordinate frame is required, which corresponds mathematically to the
diagonalization of the mass-weighted Hessian described in Eq. 5.5. The frequency
of the harmonic vibration are obtained from the square root of the diagonal entries in
h, while the columns in the matrix U (i.e., the eigenvectors) provide the new coordinates highlighted in red and blue in Fig. 5.6. The data in the matrix U lead to Eqs. 5.9
and 5.10 describing how to recombine r OH1 and r OH2 to obtain the harmonic normal
modes, q 1 and q 3 .
q 1 = q sym =
1
√
2
r OH1 +
1
√
2
r OH2
(5.9)
q 3 = q asym =
1
√
2
r OH1 −
1
√
2
r OH2
(5.10)
This means for q 1 that if r OH1 increases, so does r OH2 . In contrast, for q 3 , if
r OH1 is elongated, r OH2 is shortened (and vice versa). Therefore, q 1 and q 3 refer
to symmetric and antisymmetric stretching normal modes, respectively. The representation of the potential in the harmonic approximation now corresponds to the
paraboloid depicted in yellow in Fig. 5.6b. Every position on this paraboloid (any
point on the harmonic potential surface) is given as the addition of the points lying
on the main axes of the re-oriented coordinate frame, i.e., the red and blue line shown
at the surface of the paraboloid. This surface dictates the stretching vibrations of the
water molecule; symmetric (v 1 ) and antisymmetric (v 3 ). Note that the additive character of the harmonic potential directly implies its paraboloid shape in a geometrical
sense.
In the following the principles of the harmonic approximation, a fundamental
simplification that has found extensive use in spectroscopy, are summarized. A
complex shape of the true vibrational potential is replaced by the corresponding
99
minimum on the V (r OH1 , r OH2 ) surface. To obtain harmonic modes, V (r OH1 , r OH2 )
needs to be approximated via the potential of a 2D harmonic oscillator V
harm . The
key approximation in this case is that the harmonic potential is additive.
V
harm
= V (r OH1 ) + V (r OH2 )
(5.6)
Equation 5.6 requires that the potential does not depend simultaneously on r OH1
and r OH2, , i.e., there is no coupling potential. This implies that the vibrational wavefunction is a product of 1D wavefunctions, and the respective energy eigenvalues are
additive (same as the potential case shown above), as described by Eqs. 5.7 and 5.8.
|(r OH1 , r OH2 ) = | (r OH1 ) · | (r OH1 )
(5.7)
E
r OH1, r OH2,
= E(r OH1 ) + E(r OH2 )
(5.8)
In this case, the Hessian would be diagonal. To match the latter criterion, a reorientation of the coordinate frame is required, which corresponds mathematically to the
diagonalization of the mass-weighted Hessian described in Eq. 5.5. The frequency
of the harmonic vibration are obtained from the square root of the diagonal entries in
h, while the columns in the matrix U (i.e., the eigenvectors) provide the new coordinates highlighted in red and blue in Fig. 5.6. The data in the matrix U lead to Eqs. 5.9
and 5.10 describing how to recombine r OH1 and r OH2 to obtain the harmonic normal
modes, q 1 and q 3 .
q 1 = q sym =
1
√
2
r OH1 +
1
√
2
r OH2
(5.9)
q 3 = q asym =
1
√
2
r OH1 −
1
√
2
r OH2
(5.10)
This means for q 1 that if r OH1 increases, so does r OH2 . In contrast, for q 3 , if
r OH1 is elongated, r OH2 is shortened (and vice versa). Therefore, q 1 and q 3 refer
to symmetric and antisymmetric stretching normal modes, respectively. The representation of the potential in the harmonic approximation now corresponds to the
paraboloid depicted in yellow in Fig. 5.6b. Every position on this paraboloid (any
point on the harmonic potential surface) is given as the addition of the points lying
on the main axes of the re-oriented coordinate frame, i.e., the red and blue line shown
at the surface of the paraboloid. This surface dictates the stretching vibrations of the
water molecule; symmetric (v 1 ) and antisymmetric (v 3 ). Note that the additive character of the harmonic potential directly implies its paraboloid shape in a geometrical
sense.
In the following the principles of the harmonic approximation, a fundamental
simplification that has found extensive use in spectroscopy, are summarized. A
complex shape of the true vibrational potential is replaced by the corresponding