Essentials of Computational Chemistry

336 9 CHARGE DISTRIBUTION AND SPECTROSCOPIC PROPERTIES
One way to simplify the problem is to recognize that most chemical systems of interest are
at sufficiently low temperature that only their lowest vibrational levels are significantly populated.
Thus, from a spectroscopic standpoint, only the transition from the zeroth vibrational
level to the first is observed under normal conditions, and so it is these transitions that we are
most interested in predicting accurately. Another way of thinking about this situation is that
we are primarily concerned only with regions of the PES relatively near to the minimum,
since these are the regions sampled by molecules in their lowest and first excited vibrational
states. Once we restrict ourselves to regions of the PES near minima, we may take advantage
of Taylor expansions to simplify our construction of V .
9.3.2.2 Harmonic oscillator approximation
Let us consider again our simple diatomic case. Using Eq. (2.2) for the potential energy from
a Taylor expansion truncated at second order, Eq. (9.37) transformed to internal coordinates
becomes
− 1
∂
2µ
2
1
2
+ k(r − req) (r) = E(r) (9.46)
∂r2 2
where µ is the reduced mass from Eq. (9.41), r is the bond length, and k is the bond force
constant, i.e., the second derivative of the energy with respect to r at req (see Eq. (2.1)). Eq.
(9.46) is the quantum mechanical harmonic oscillator equation, which is typically considered
at some length in elementary quantum mechanics courses. Its eigenfunctions are products of
Hermite polynomials and Gaussian functions, and its eigenvalues are
E = n + 1
hω (9.47)
2
where n is the vibrational quantum number and
ω = 1
k
2π µ
(9.48)
The selection rules for the QM harmonic oscillator permit transitions only for n =
±1 (see Section 14.5). As Eq. (9.47) indicates that the energy separation between any two
adjacent levels is always hω, the predicted frequency for the n = 0ton = 1 absorption
(or indeed any allowed absorption) is simply ν = ω. So, in order to predict the stretching
frequency within the harmonic oscillator equation, all that is needed is the second derivative
of the energy with respect to bond stretching computed at the equilibrium geometry, i.e.,
k. The importance of k has led to considerable effort to derive analytical expressions for
second derivatives, and they are now available for HF, MP2, DFT, QCISD, CCSD, MCSCF
and select other levels of theory, although they can be quite expensive at some of the more
highly correlated levels of theory.
Prior to proceeding, it is important to address the errors introduced by the harmonic
approximation. These errors are intrinsic to the truncation of the Taylor expansion, and will