Cramer C.J. Essentials of Computational Chemistry Theories and Models

of rotational and vibrational spectroscopy. The following subsections describe the relevant theory and detail the applicability of different methodologies for such computations.

9.3.1 Rotational

The simplest approach to modeling rotational spectroscopy is the so-called ‘rigid-rotor’ approximation. In this approximation, the geometry of the molecule is assumed to be constant at the equilibrium geometry qeq. In that case, V (qeq) in Eq. (9.37) becomes simply a multiplicative constant, so that we may write the rigid-rotor rotational Schrodinger¨ equation as

where E0, the eigenvalue for Eq. (9.38) corresponding to the lowest-energy rotational state, is taken to be the electronic energy for the equilibrium geometry.

Equation (9.38), if restricted to two particles, is identical in form to the radial component of the electronic Schrodinger¨ equation for the hydrogen atom expressed in polar coordinates about the system’s center of mass. In the case of the hydrogen atom, solution of the equation is facilitated by the simplicity of the two-particle system. In rotational spectroscopy of polyatomic molecules, the kinetic energy operator is considerably more complex in its construction. For purposes of discussion, we will confine ourselves to two examples that are relatively simple, presented without derivation, and then offer some generalizations therefrom. More advanced treatises on rotational spectroscopy are available to readers hungering for more.

The simplest possible case is a non-homonuclear diatomic (non-homonuclear because a dipole moment is required for a rotational spectrum to be observed). In that case, solution of Eq. (9.38) is entirely analogous to solution of the corresponding hydrogen atom problem, and

k

In the special case of a heteronuclear diatomic, rotation occurs exclusively about a single axis passing through the center of mass and perpendicular to the bond, and I is simply µreq2 , where the reduced mass µ is computed as

Because the wave functions are the spherical harmonics, each rotational level is (2J + 1)- fold degenerate (over the quantum number m); note that the lowest rotational level has a rotational energy of zero, consistent with the earlier statement that the total energy associated with this level is just the electronic energy of the equilibrium structure. Selection rules dictate that transitions occur only between adjacent levels, i.e., J = ±1 (see Section 14.5), in which case the energy change observed for transition from level J to level J + 1 is

When probed spectroscopically, the absorption frequency ν can be determined as

Non-linear molecules are more complicated than linear ones because they are characterized by three separate moments of inertia. In highly symmetric cases, however, relatively simple solutions of Eq. (9.38) continue to exist. For instance, in molecules possessing an axis of rotation that is three-fold or higher in symmetry, the two moments of inertia for rotation about the two axes perpendicular to the high-symmetry axis will be equal. For example, in fluoromethane, which is C3ν , there is one moment of inertia, IA, about the symmetry axis A, and there are two equal moments of inertia, IB and IC, about the axes perpendicular to axis A. In this particular case, the magnitude of the latter two moments is larger than that of the former moment because the heavy atoms have displacements of 0 from axis A but not from the other two, and such a molecule is called a prolate top. In the case of a prolate top, the rotational eigenvalues are given by

where K is the quantum number, running over −J , −J + 1, . . ., J −1, J , expressing the component of the angular momentum along the highest symmetry axis. The selection rules for a rotational transition in this case are J = ±1 and K = 0, and thus Eqs. (9.43) and (9.44) continue to be valid for absorption frequencies using I = IB.

Less symmetric molecules require a considerably more complicated treatment, but in the end their spectral transitions are functions of their three moments of inertia (see Section 10.3.5). From a computational standpoint, then, prediction of rotational spectral lines depends only on the moments of inertia, and hence only on the molecular geometry. Thus, any method which provides good geometries will permit an accurate prediction of rotational spectra within the regime where the rigid-rotor approximation is valid.

Since even very low levels of theory can give fairly accurate geometries, rotational spectra are quite simple to address computationally, at least over low rotational quantum numbers. For higher-energy rotational levels, molecular centrifugal distortion becomes an issue, and more sophisticated solutions of Eq. (9.37) are required.

9.3.2 Vibrational

When thinking about chemical thermodynamics and kinetics, it is a convenient formalism to picture a molecule as being a ball rolling on a potential energy surface. In this simple model, the exact position of the ball determines the molecular geometry and the potential energy, and its speed as it rolls in a frictionless way determines its kinetic energy. Of course, quantum mechanical particles are different than classical ones in many ways; one of the more important differences is that they are subject to the uncertainty principle. One consequence of the uncertainty principle is that polyatomic molecules, even at absolute zero, must vibrate – within the simple ball and surface picture, the ball must always be moving, with a sum of potential and kinetic energy that exceeds the energy of the nearest minimum by some non-zero amount. This energy is contained in molecular vibrations.

Transitions in molecular vibrational energy levels typically occur within the IR range of the frequency spectrum. Because vibrational motions tend to be highly localized within molecules, and the energy spacings associated with individual linkages tend to be reasonably similar irrespective of remote molecular functionality, IR spectroscopy has a long history of use in structure determination. Vibrational frequencies also have other important uses, for example in kinetics (Section 14.3) and computational geometry optimization (Section 2.4.1), so their accurate prediction has been a long-standing computational goal. We now examine different approaches towards that goal, and the utility of different levels of theory in application.

9.3.2.1 One-dimensional Schrodinger¨ equation

It is again useful to begin with the simplest possible case, the diatomic molecule. Equation (9.37), when restricted to the vibrational motion alone, is clearly a function of only a single variable, the interatomic distance r. Solutions of differential equations of only a single variable are typically reasonably straightforward. Our only challenge here is that we do not know exactly what the potential energy function V looks like as a function of r. Given a level of theory, however, we can compute V point by point to an arbitrary level of fineness (i.e., simply compute the electronic energy of the system for various fixed values of r). Those points may then be fit to any convenient analytic function – polynomial,

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

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

±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 = 0 to n = 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

remain even for an exact level of electronic structure theory. The most critical difference is that real bonds dissociate as they are stretched to longer and longer values. Thus, as indicated in Figure 9.7, the separation between vibrational levels narrows with increasing vibrational quantum number, and the total number of levels is finite. By contrast, the harmonic oscillator has an infinite number of levels, all equally spaced.

While the differences between the harmonic oscillator approximation and the true system are largest for higher vibrational levels, even at very short distances beyond the equilibrium bond length the true potential energy of the bond stretch curve is lower than that predicted by the parabolic potential of the harmonic approximation. Since the more shallow correct potential generates a lower vibrational frequency than that associated with the parabola, this means that an ‘exact’ harmonic frequency will always be greater than the true frequency. Over the few data available for diatomics that are sufficiently complete so that the PES can be constructed and harmonic frequencies inferred, the difference averages about 3%. Any level of theory that exceeds this accuracy using the harmonic approximation is presumably simply benefiting from a fortuitous cancellation of errors.

What about the polyatomic case? In that case, we must carry out a multi-dimensional Taylor expansion analogous to Eq. (2.26). This leads to the multi-dimensional analog of Eq. (9.46)

where x is the vector of atomic coordinates, xeq defines the equilibrium structure, and H is the Hessian matrix defined by Eq. (2.37).

While Eq. (9.49) has a well-defined potential energy function, it is quite difficult to solve in the indicated coordinates. However, by a clever transformation into a unique set of mass-dependent spatial coordinates q, it is possible to separate the 3N -dimensional Eq. (9.49) into 3N one-dimensional Schrodinger¨ equations. These equations are identical to Eq. (9.46) in form, but have force constants and reduced masses that are defined by the action of the transformation process on the original coordinates. Each component of q corresponding to a molecular vibration is referred to as a ‘normal mode’ for the system, and with each component there is an associated set of harmonic oscillator wave functions and eigenvalues that can be written entirely in terms of square roots of the force constants found in the Hessian matrix and the atomic masses.

Note that because Eq. (9.49) is over the full 3N coordinates, the transformed coordinate system q includes three translational and three rotational (two for linear molecules) ‘modes’. The eigenvalues associated with these modes are typically very close to zero, and indeed, the degree to which they are close to zero can be regarded as a diagnostic of how well optimized the structure is in terms of being at the local minimum geometry.

A few last technical points merit some discussion prior to an assessment of the relative utilities of different theoretical levels for prediction of IR spectra. First, note that the first derivatives in the Taylor expansion disappear only when the potential is expanded about a critical point on the PES (since then the gradients are all zero). Thus, the form of Eq. (9.49) is not valid if the level of theory used in the computation of the Hessian matrix differs from

that used for geometry optimization, since the two different levels of theory will almost inevitably have different minimum energy structures. Put more succinctly, there is little value in a frequency calculation for a particular geometry under the harmonic oscillator approximation unless the geometry in question was optimized at the same level of theory. (Note that mathematically one could certainly include the gradient term in the potential in Eq. (9.49), but the resulting differential equation is not worth working with.)

Another interesting point in this regard is that the form of Eq. (9.49) is valid for other stationary points that are not minima on the PES. However, in this instance there will be one or more normal mode force constants that will be negative, corresponding to motion along modes that lead to energy lowering. Insofar as the frequencies are computed from the square roots of the force constants, this leads to an imaginary frequency (one often sees these called negative frequencies in the literature, but this is simply sloppy). Frequency calculations thus are diagnostic as to the nature of stationary points. All positive frequencies implies a (local) minimum, one imaginary frequency implies a transition state structure, and two or more imaginary frequencies refers to stationary points characterized by additional negative force constants. Such structures are sometimes useful in searching for TS structures by following the various energy-lowering modes, but they have no chemical significance.

The utility of Eq. (9.49) depends on the ease with which the Hessian matrix may be constructed. Methods that allow for the analytic calculation of second derivatives are obviously the most efficient, but if analytic first derivatives are available, it may still be worth the time required to determine the second derivatives from finite differences in the first derivatives (where such a calculation requires that the first derivatives be evaluated at a number of perturbed geometries at least equal to the number of independent degrees of freedom for the molecule). If analytic first derivatives are not available, it is rarely practical to attempt to construct the Hessian matrix.

A technical point in this regard with respect to DFT is that when one refers to ‘analytic’ derivatives, what is actually meant is analytic derivatives to the quadrature schemes that are used to approximate the solution of the complicated integrals defining the exchange-correlation energy; analytic solutions to these integrals are not in general available, and hence neither are their derivatives. In practice, failure to converge the quadrature schemes has a considerably larger effect on second derivatives than it does on energies, and it is not uncommon to see potentially rather large changes in computed vibrational frequencies when switching from default to more dense quadrature-point densities (sometimes also called ‘grid’ densities) in standard electronic-structure packages. This effect can be particularly troubling with low frequencies, since the error can cause the frequencies to switch from real to imaginary and vice versa, and some care should be exercised where such issues are important.

With respect to absolute accuracy, Table 9.2 provides the mean unsigned errors in harmonic vibrational frequencies for a number of levels of theory over the 32 molecules in the reduced G2 test set. HF theory shows the poorest performance (AM1 and PM3 are in general somewhat worse than HF with a moderate basis set, however data are not available for this particular test set). MP2 shows significant improvement over HF, but substantial

a Test set includes 32 molecules containing only first-row atoms, see Johnson, Gill, and Pople (1993). Data from Adamo and Barone (1998).

error remains. CCSD(T) and some of the hybrid levels of density functional theory show the highest accuracies. In general, the BLYP combination seems to be more accurate than BPW91, whether pure or hybrid in formulation, but PWPW91 is nearly as accurate as BLYP, again whether pure or hybrid in formulation.

Of some interest in the error analysis is the degree to which the error is systematic. Although HF errors are large, they are very systematic. HF overemphasizes bonding, so all force constants are too large, and thus so are all frequencies. However, application of a constant scaling factor to the HF frequencies improves their accuracy enormously (Pople et al. 1993). Scott and Radom studied this issue in detail for eight different levels of theory using a database of 122 molecules and 1066 fundamentals (i.e., measured, anharmonic vibrational frequencies) and a summary of their results, together with a few other recommended scaling factors, is provided in Table 9.3 (Scott and Radom 1996; see also, Wong 1996). Note that even though the scale factor required for the HF/6-31G(d) level of theory is substantial, reducing every frequency by more than 10%, the final accuracy is quite high – better than the considerably more expensive MP2. Note also that the pure DFT functional BLYP requires essentially no scaling, i.e., its errors are random about the experimental values,

a Data from Scott and Radom (1996) unless otherwise indicated. b Number of frequencies still in error by more than 20% of the experimental value after scaling. c From analysis of 511 frequencies in 50 inorganic molecules (Bytheway and Wong 1998). d Pople et al. (1993). e From analysis of 900 frequencies for 111 molecules comprised of firstand second-row atoms and hydrogen (Halls, Velkovski, and Schlegel 2001). f From analysis of 110 frequencies for 31 small molecules having only first-row atoms and hydrogen (Jaramillo and Scuseria 1999). g Bauschlicher and Partridge (1995).

while the hybrid functionals require scale factors consistent with their inclusion of some HF character. Thus, including HF character results in proportionately too high predictions in vibrational frequencies, although the scaling procedure is very effective here as well. Finally, the errors in the semiempirical levels are quite high, and scaling is only modestly helpful. For those looking for the highest accuracy, the U.S. National Institute of Standards and Technology (NIST) maintains a web facility that permits users to select a focused set of molecules from NIST’s computational chemistry database (presumably based upon the user’s interest in a structurally related unknown) and then to compute least-squares best scaling

factors for specific levels of theory based only on those molecules (srdata.nist.gov/cccbdb/). One example of such an approach, albeit not using the NIST website, was provided by Yu, Srinivas, and Schwartz (2003) who optimized scale factors just for the C−O stretch of metal bound carbonyls.

Results from molecular mechanics can also be of reasonable accuracy, so long as the molecules addressed contain only functionality well represented in the force field training set. While extensive compilations of data are not available, Halgren has compared MM3 and MMFF94 over a test set of 157 frequencies from organic molecules and found RMS errors of 57 and 60 cm−1, respectively.

An interesting alternative to scaling the frequencies is instead to scale the force constants in the Hessian, which permits some sensitivity to different kinds of vibrations, e.g., stretches, bends, and torsions (see, for example, Grunenberg and Herges 1997; Baker, Jarzecki, and Pulay 1998; Arenas et al. 2000). Of course, as with any parameterization procedure, as the number of parameters increases so too does the requirement for additional data to ensure statistical reliability, and this approach has not yet seen wide application.

One final caveat with respect to comparing experimental IR spectra with theoretically predicted frequencies is that the latter do not account for such experimental complications as Fermi resonances (where two nearby fundamentals are shifted to higher and lower frequencies, respectively), overtones, etc. Such details require case-by-case evaluation.

In comparing complete theoretical spectra to complete experimental spectra in molecules of moderate to large size, there can be a large number of lines. To ensure proper correspondence of the normal modes, it is helpful to compare not only the absorption frequencies themselves but also the intensities of the absorptions. For a typical experimental spectrum, such intensities are usually reported simply as strong, medium, or weak, although in careful experiments absorption cross-sections can be measured accurately. From a computational standpoint, the prediction of IR intensities can be accomplished using the mixed second derivatives of the energy with respect to geometric motion and an external electric field (thereby permitting estimation of the changes in the dipole moment as a function of the vibrations, which is what IR intensities are proportional to). These mixed second derivatives are available analytically for all levels of theory for which analytic second derivatives with respect to the geometry are available, so it is a straightforward matter to compute IR intensities. The actual computed values tend to be no better than qualitative in the absence of using a very complete basis set and accounting for electron correlation, but insofar as most experimental intensities are essentially qualitative, this is not typically much of a drawback. Being able to line up strong absorptions in computed and experimental spectra is often quite helpful for assessing the validity of the comparison.

An alternative experiment that measures the same vibrational fundamentals subject to different selection rules is Raman spectroscopy. Raman intensities, however, are more difficult to compute than IR intensities, as a mixed third derivative is required to approximate the change in the molecular polarizability with respect to the vibration that is measured by the experiment. The sensitivity of Raman intensities to basis set and correlation is even larger than it is for IR intensities. However, Halls, Velkovski, and Schlegel (2001) have reported good results from use of the large polarized valence-triple-ζ basis set of Sadlej (1992) and