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

have determined frequency scaling factors for its use in conjunction with several levels of theory (see Table 9.3).

9.3.2.3 Vibrationally averaged expectation values

With fairly few exceptions, all discussion of computed molecular properties up to this point has proceeded under the assumption that the value computed for the stationary equilibrium structure is relevant in comparison to experiment. However, the experimental population is in constant vibrational motion, even at 0 K, so the experimental measurement actually samples structures having a distribution dictated by the molecular vibrational wave function. Thus, for some property A, the measured value is the expectation value given by

A = (q)A(q) (q)dq (9.50)

where is the wave function for nuclear motion and q is the coordinate system.

Some general analysis of Eq. (9.50) is warranted. Note that the zeroth vibrational level for a normal mode within the harmonic approximation is characterized by a Gaussian wave function. The total molecular vibrational wave function is a product of the wave functions of all of the individual modes, so if every vibration is in its ground state, is an even function if the origin is taken to be qeq (‘even’ meaning that the function has the same value for equal displacements in the positive and negative directions along any axis). It is now helpful to consider not the expectation value of A, but the expectation value for the deviation of A from the value at the equilibrium position, i.e.,

If the deviation of A from A(qeq) is coupled with only a single component of q, and if its dependence on displacement from the equilibrium structure is linear, then A is an odd function of q (‘odd’ meaning now that the function takes on positive and negative values of equal magnitude when displaced an equal distance along any axis). From elementary calculus, we know that the product of two even functions and an odd function is an odd function, and that the integral over all space of an odd function is zero, so under the conditions outlined above the expectation value defined by Eq. (9.51) is zero and the value of A must be A(qeq). An example of such a situation would be a harmonic oscillator having a dipole moment. The change in dipole moment is linear in the displacement from the equilibrium bond length, so the expectation value of the dipole moment over the first vibrational wave function (indeed, over any of the vibrational wave functions in this case, since the system is harmonic) is exactly equal to the dipole moment at the equilibrium bond length.

Of course, in a real system with many atoms, the coupling of the property to the individual degrees of freedom is more complicated, and there is no guarantee that A will be an odd function. Nevertheless, the assumption that Eq. (9.51) is equal to zero is often sufficiently accurate for everyday computational predictions.

Considering all five h.f.s. values, the agreement with experiment improves in every case when vibrational averaging is taken into account, and the RMS error drops from 11.8 G to 4.8 G.

9.4 NMR Spectral Properties

Nuclear magnetic resonance (NMR) is probably the most widely applied spectroscopic technique in modern chemical research. Its high sensitivity and the mild conditions required for its application render it peerless for structure determination and kinetics measurements in many instances. As an experimental technique, its use is extraordinarily widespread.

Until quite recently, however, theoretical prediction of NMR spectral properties significantly lagged experimental work. The ultimate factor slowing theoretical work has been simply that it is more difficult to model the interactions of a wave function with a magnetic field than it is to model interactions with an electric field. Nevertheless, great progress has been made over the last decade, particularly with respect to DFT, and calculation of chemical shifts is becoming much more routine than had previously been true.

This section begins with a very brief summary of some of the technical issues associated with NMR spectral calculations. Subsequent subsections address the various utilities of modern methods for predicting chemical shifts and nuclear coupling constants.

9.4.1 Technical Issues

NMR measurements assess the energy difference between a system in the presence and absence of an external magnetic field. For a chemical shift measurement on a given nucleus, there are two magnetic fields of interest: the external field of the instrument and the internal field of the nucleus. The chemical shift is proportional to the second derivative of the energy with respect to these two fields, and it can be computed using second-derivative analogs of Eqs. (9.33) or (9.34). However, the integrals in question are more complex because, unlike the electric field, which perturbs the potential energy term of the Hamiltonian, the magnetic field perturbs the kinetic energy term (it is the motion of the electrons that generates electronic magnetic moments). The nature of the perturbed kinetic energy operator is such that an origin must be specified defining a coordinate system for the calculation. This origin is called the ‘gauge origin’.

346 9 CHARGE DISTRIBUTION AND SPECTROSCOPIC PROPERTIES

absolute values, but instead as values relative to some standard compound, e.g., tetramethylsilane, which is often used for 1H and 13C. Comparison between computed and experimental numbers requires some care to ensure the same convention is being used for reporting data. To compute a relative chemical shift, obviously one must carry out a separate calculation for the reference compound.

For molecules composed of only first-row atoms, heavy-atom chemical shifts can be computed with a fair degree of accuracy, as indicated in Table 9.5. Happily, even HF theory gives acceptable accuracy in most instances, although some improvements are available in favorable instances from DFT (provided it is neither LDA nor B3LYP) and MP2. The latter level of theory is quite accurate, but at relatively high cost in terms of demand for computational resources. Various groups have demonstrated that errors from levels having lower accuracy are sufficiently systematic that errors may be significantly reduced by application of a simple linear regression equation (Sebag, Forsyth, and Plante 2001; Giesen and Zumbulyadis 2002). Thus, for instance, scaling 13C shieldings computed at the B3LYP/MIDI! level by −1.16 and adding 225.1 ppm provides an RMS error of only 3.6 ppm over a diverse test set of experimental values measured in solution.

Note that the mean unsigned errors listed in Table 9.5 for absolute chemical shifts are larger than the errors for relative chemical shifts, as expected. The errors in the relative shifts must be considered to be rather good given the range of experimental values spanned. Note also that the high anisotropy of multiple bonds makes the chemical shifts of the atoms involved quite sensitive to the level of theory, particularly for nitrogen and oxygen atoms.

Table 9.5 contains a relative paucity of data for 1H. This nucleus is somewhat more difficult to work with because it spans a fairly modest spectral range, perhaps 15 ppm in typical chemical environments. Rablen, Pearlman, and Finkbiner (1999), however, have carried out calculations of 1H chemical shifts for 80 organic molecules, and demonstrated reasonable results from various DFT functionals with large basis sets; they also identified scaling factors that improved agreement with experiment (for a similar study focusing on aromatic proton chemical shifts, see Wang et al. 2001). Wang, Hinton, and Pulay (2002) similarly reported good success from both HF and DFT calculations for the prediction of 1H chemical shifts in eight cyclic amides for which experimental data in both DMSO and D2O were available. Finally, Patchkovskii and Thiel (1999) have reported a reparameterization for H, C, N, and O within the MNDO model with the goal of better predicting chemical shifts; they applied their modification B with three-center terms (MB3) MNDO to 384 common organic molecules and obtained errors consistent with those listed in Table 9.5, which must be regarded as fairly good given the tremendous efficiency of the model (of course, making predictions for other nuclei would require further reparameterization).

Computed and experimental data for the chemical shifts of heavy elements have been less extensively compared. Table 9.6 lists some results for 77Se that are illustrative of the wide range of chemical shifts typically possible for such nuclei (here more than 2000 ppm) as well as the degree to which the chemical phase may affect the comparisons. The calculations are gas phase, although in Chapters 11 and 12 we will discuss techniques for including condensed-phase effects in computational predictions.

a Unless otherwise indicated, a quadruple-ζ basis set with double polarization functions is used (Cheeseman et al. 1996; the quoted experimental data are mostly taken from this reference as well). b MNDO modification B including three-center terms (Patchkovskii and Thiel 1999). c Using a basis set of STOs triple-ζ in the valence region and double-ζ in the core (Schreckenbach and Ziegler 1995). d Using the 6-311++G(2d,p) basis set (Adamo, Cossi, and Barone 1999). e Using the IGLO IV basis set and the multiplicative Kohn–Sham (MKS; Wilson and Tozer 2001) method to compute the chemical shifts (Wilson, Bradley, and Tozer 2001). f Mean unsigned error in heavy-atom

absolute shieldings. g Mean unsigned error in heavy-atom shieldings relative to CH4 (13C), NH3 (15N), H2O (17O), and HF (19F).

a Using a basis set of STOs triple-ζ in the valence region and double-ζ in the core unless otherwise indicated (Schreckenbach et al. 1996). b Using a (11s7p2d/6s2p)[7s6p2d/4s2p] basis set (Sychrovsky, Grafenstein,¨ and Cremer 2000). c Using a basis set with the core expanded to triple-ζ . d (Pseudo)rotationally averaged.

The calculation of spin–spin coupling is less straightforward than the calculation of chemical shift, in part because of the additional complications associated with two local magnetic moments, as opposed to one moment and one external, uniform field. Moreover, the most commonly reported couplings in the experimental literature are proton–proton couplings in organic and biological molecules, and these are amongst the more difficult to predict because they tend to be small in magnitude, so absolute errors are magnified when considered in a relative sense. Some representative calculations are provided in Table 9.7.

Computed coupling constants show moderate to large sensitivity to basis set, and accurate predictions require very flexible bases (see, for example, the hydrogen fluoride (HF) data in Table 9.7). In addition, DFT is much more robust than HF theory for predicting coupling constants, and the latter level of theory simply should not be used for this purpose.

9.5 Case Study: Matrix Isolation of Perfluorinated p-Benzyne

Synopsis of Wenk et al. (2001) ‘Matrix Isolation of Perfluorinated p-Benzyne’.

The class of antitumor-antibiotics known as enediynes undergo in vivo Bergman cyclization of the enediyne functionality to generate p-benzyne reactive intermediates that damage genetic material. Because the damage results in double-stranded DNA cleavage, they are extraordinarily cytotoxic, and this has sparked interest in better understanding p-benzynes in general (this species has already been discussed at some length in Chapters 7 and 8). One issue associated with the parent p-benzyne is that it is thermochemically unstable relative to its enediyne precursor, making its isolation more challenging. In this case, Wenk and coworkers sought to identify a precursor not suffering from this problem, and determined from DFT and CASSCF calculations that perfluorinated p-benzyne was roughly 8 kcal mol−1 more stable than the enediyne that would be produced from retro-Bergman ring opening, and moreover that the barrier to that ring opening was nearly 38 kcal mol−1, this being nearly double the barrier in the unfluorinated case. Girded with this thermochemical armor, they set out to synthesize the diradical by UV photolysis of 1,4-diiodo-2,3,5,6-tetrafluorobenzene (Figure 9.9).

When this precursor is photolyzed at 3 K in a neon matrix, IR spectroscopy indicates rapid formation of a new species A. Prolonged photolysis creates a second product B whose IR bands are distinct from the first. And, if the matrix containing the second product is irradiated with UV light of somewhat longer wavelength, IR analysis indicates that a third product C is generated. All of the IR bands observed for A, B, and C are listed in Table 9.8. These bands are compared to frequencies computed at the B3LYP/6-311G(d,p) level for 4-iodo-2,3,5,6-tetrafluorophenyl radical (ITFP) and perfluorohex-3-en-1,5-diyne (PFHED) and to frequencies computed at the CASSCF(8,8)/cc-pVDZ level of theory and scaled by 0.91 for perfluorinated p-benzyne (PFPB). The authors do not explain their recourse to two different levels of theory, but presumably they were not comfortable with the DFT model, even used unrestricted, for the multiconfigurational p-benzyne.

In any event, the generally excellent agreement between the experimental and computed spectra permits the secure assignment of the bands for A to ITFP, the bands for B to PFPB, and the bands for C to PFHED (note that scaling of the DFT bands by the scale factor for

Figure 9.9 Bergman cycloaromatization reactions for hex-3-en-1,5-diyne and its perfluorinated congener, as well as a photochemical reaction scheme for generating the perfluorinated diradical from an iodinated precursor. What spectral features would be expected to be most diagnostic of the different intermediates? What levels of theory would be appropriate for predicting these spectral signatures? (Note that equilibrium arrows of unequal length indicate which species predominates at equilibrium.)

Table 9.8 Experimental and computed IR spectra (cm−1) for A, B, and

C, and ITFP, PFPB, and PFHED, respectively

B3LYP/6-31G(d) in Table 9.3 would result in slightly improved agreement for A and C) as would be suggested by the synthetic scheme in Figure 9.9. Intensity data were also used, although those are not shown here; importantly, the ‘missing’ bands in the experimental IR spectra are all predicted to be of very low intensity in the computed spectra. Interestingly, both CASSCF and unrestricted B3LYP predict the singlet and triplet states of the diradical to be essentially degenerate, leaving the question open as to which (if either) is the lower in energy.

The use of computed spectra to bolster structural assignments has seen heavy use in matrix isolation experiments. This is a slightly atypical example, insofar as the species involved actually require some careful attention to non-dynamical correlation, but represents an excellent example of how theory can aid experiment in the identification of short-lived reactive species.

Bibliography and Suggested Additional Reading

Bachrach, S. 1994. ‘Population Analysis and Electron Densities from Quantum Mechanics’, in Reviews in Computational Chemistry , Vol. 5, Lipkowitz, K. B. and Boyd, D. B. Eds., VCH: New York, 171.

Cramer, C. J. 1991. ‘Dependence of Isotropic Hyperfine Coupling in the Fluoromethyl Radical Series on Inversion Angle’ J. Org. Chem., 56, 5229.

Hehre, W. J., Radom, L., Schleyer, P. v. R., and Pople, J. A. 1986. Ab Initio Molecular Orbital Theory , Wiley: New York.

Helgaker, T., Jaszunski, M., and Ruud, K. 1999. ‘Ab Initio Methods for the Calculation of NMR Shielding and Indirect Spin – spin Coupling Constants’ Chem. Rev., 99, 293.

Jensen, F. 1999. Introduction to Computational Chemistry , Wiley: Chichester.

Koch, W. and Holthausen, M. C. 2000. A Chemist’s Guide to Density Functional Theory , Wiley-VCH: Weinheim.

Kubinyi, H. 2003. ‘Comparative Molecular Field Analysis (CoMFA)’, in Handbook of Chemoinformatics. From Data to Knowledge, Vol. 4, Gasteiger, J., Ed., Wiley-VCH: Weinheim, 1555.

Leach, A. R. 2001. Molecular Modelling, 2nd Edn., Prentice Hall: London. Levine, I. N. 1975. Molecular Spectroscopy , Wiley, New York.

Malkin, V. G., Malkina, O. L., Eriksson, L. A., and Salahub, D. R. 1995. ‘The Calculation of NMR and ESR Spectroscopy Parameters Using Density Functional Theory’, in Modern Density Functional Theory; A Tool for Chemistry , Politzer, P. and Seminario, J., Eds., Elsevier: Amsterdam, 273.

Thompson, J. D., Cramer, C. J., and Truhlar, D. G. 2003. ‘Parameterization of Charge Model 3 for AM1, PM3, BLYP, and B3LYP’, J. Comput. Chem., 24, 1291.

Wiberg, K. B. and Rablen, P. R. 1993. ‘Comparison of Atomic Charges Derived via Different Procedures’ J. Comput. Chem., 14, 1504.

Wilson, E. B., Jr., Decius, J. C., and Cross, P. C. 1955. Molecular Vibrations , Dover: New York.

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