20. NMR Spectroscopy and Magnetic Properties

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1 20. NMR Spectroscopy and Magnetic Properties Nuclear Magnetic Resonance (NMR) Spectroscopy is a technique used largely by organic, inorganic, and biological chemists to determine a variety of physical and chemical properties of atoms and the molecules they are part of. NMR instruments use a powerful magnet to create a strong magnetic field. Nuclei that are spin active, such as protons or carbon-13 nuclei, absorb electromagnetic radiation at a frequency specific to that isotope. One dimensional NMR spectra display a number of signals (peaks) equivalent to the number of inequivalent groups of one kind of nucleus. For example, methane has four hydrogens (protons), but they are all equivalent. This means that if a proton is exchanged for another group, it is impossible to tell which of the four protons was removed. The proton spectrum for methane is thus one single peak NMR Shielding Shielding refers to have guarded a nucleus is against the magnetic field created by the NMR magnet. As electron density surrounding a nucleus increases, that nucleus becomes more shielded against the magnetic field. Conversely, nuclei are deshielded as electron density decreases. This results in a shift in the placement of a peak on the x-axis of an NMR spectrum. The more to the right a peak appears, the more shielded that nucleus is. The more to the left a peak appears, the more deshielded that nucleus is. Do note that shifts can be negative, but nearly all carbon-13 and proton peaks will appear at a positive ppm. The x-axis of NMR spectra is in parts per million, ppm. This is actually the ratio of the frequency of the spin active nucleus in the magnet to the frequency of the magnet itself multiplied by one million. The University of Wyoming has a 400 MHz NMR and a 600 MHz NMR. If the resonant frequency of a proton is 400 Hz on the 400 MHz NMR, then (400 Hz/400 MHz)*10 6 = 1 ppm. The same proton in the same molecule on the 600 MHz NMR would still appear extremely close

2 to 1 ppm, but its resonant frequency would now be approximately 600 Hz. Using ppm instead of simply hertz allows chemists to use different magnet strengths, but still compare each other's data because equivalent nuclei always give the same chemical shift in ppm regardless of the magnet strength. Adding electronegative groups, such as halogens, singly bonded oxygens, or doubly bonded oxygens, to a carbon atom removes electron density from that carbon and from any protons still bonded to the carbon. By removing electron density from these nuclei, the nuclei are less shielded from the magnetic field and will be deshielded. They will appear at a higher ppm than a similar hydrocarbon. Chemical shifts are always relative to a standard molecule, most of that of tetramethyl silane, TMS. Outputs of calculations will give absolute chemical shifts, but in order to compare them to experimental values, calculate the absolute chemical shift for TMS and subtract the absolute chemical shift of your specified molecule from the absolute chemical shift of TMS.

3 Tetramethylsilane (TMS) Predicting NMR Properties 1. Chemical Shifts. Simply adding "NMR" in the route section will allow shielding values to be calculated. #T RHF/6-31G(d) NMR Test You should optimize the geometry first as always. NMR calculations benefit greatly from accurate geometries and larger basis sets. The following is an example calculation for the chemical shift of the carbon nucleus in methane: %Chk=NMRmethane #T B3LYP/6-31G(d) Opt Test Opt 0 1 C H H H H Link1-- %Chk=NMRmethane #T RHF/6-31G(d) NMR Geom=Check Guess=Read Test NMR 0 1

4 The output will include the following: GIAO Magnetic shielding tensor (ppm): 1 C Isotropic = The chemical shift of experimental methane does not remotely match this chemical shift, however, because this is an absolute chemical shift (or, more precisely, the absolute value of the magnetic shielding). We need to calculate the absolute chemical shift of TMS next. Of course, it has to be calculated at the same level of theory by now this should be obvious why. The TMS output will appear as follows: GIAO Magnetic shielding tensor (ppm): 1 C Isotropic = By subtracting the value of the standard (TMS) from the one for your sample (methane), the desired chemical shift is obtained; = -3.9 ppm The experimental value is -7.0 ppm. Note that a negative sign indicates that the specified molecule is more shielded than the reference molecule while a positive sign indicates that the specified molecule is less shielded than the reference molecule. All experimental chemical shifts are relative. 2. Spin-Spin Coupling. Nuclei themselves possess a small magnetic field and can therefore influence the frequency of nearby nuclei. This pair of nuclei are said to be coupled. Coupling constants are expressed in Hz and typically range from a few hertz up to 20 Hz. The most common type of coupling is scalar coupling which occurs through chemical bonds. Two nuclei are less likely to be coupled as the number of bonds between them increases, with coupling between nuclei more than three bonds apart being fairly rare. Splitting patterns result from coupling in proton NMR spectra. When a proton is note coupled to any other proton, it appears as a singlet: one tall peak. When a proton is coupled to one and only one other proton, the peak appears as a doublet: two identical, or nearly identical, peaks in terms of height and peak area. When a proton is coupled to two equivalent protons, its peak appears as a triplet: three peaks with a height ratio of 1:2:1. Simple splitting patterns follow the n+1 rule, where n is the number of equivalent protons that a proton is split by. The heights of the peaks that result follow patterns present in Pascal's Triangle, ie doublets are 1:1, triplets 1:2:1, quartets 1:3:3:1, etc. If a proton is split by two or more groups of equivalent protons, for example protons on carbon 2 of n-pentane, the splitting pattern becomes much more complicated.

5 The protons in green are not coupled to any other proton, as the closest non-equivalent proton is 5 bonds away. They appear as a single peak, a singlet, as a result. The red protons are coupled to the three blue protons, so the red protons appear as a quartet with a 1:3:3:1 relative height ratio. The blue protons are split only by the 2 red protons, so the blue protons appear as a triplet. The coupling constant(s) for a peak, if it has any, is the distance in Hz that a peak was split into. To calculate coupling constants, add NMR=SpinSpin to the route section. If SpinSpin is included in the previous methane calculations, an additional output is included as follows: Total nuclear spin-spin coupling J (Hz): D D D D D D D D D D D D D D D+00

6 20.3. Technical Issues with Calculating NMR Properties One of the problems with calculating magnetic properties, including NMR, is the so-called origin-gauge dependence. That means that what we calculate is generally dependent on where we pick the origin of our coordinate systems. Obviously, the real properties cannot depend on that. The reason why the calculated properties change with the selection of the gauge origin is that we have an approximate, not exact, wave function and that we approximate it using a finite basis set. It can be shown that in the limit of an infinite bases set the properties become gauge invariant. Unfortunately, working with an infinite basis is hardly an option. To reduce artifacts associated with the gauge origin, two different approaches have seen extensive use in the literature. The older method employs gauge-including atomic orbitals (GIAOs) as a basis set. By a clever incorporation of the gauge origin into the basis functions themselves, all matrix elements involving the basis functions can be arranged to be independent of it. An alternative is the individual gauge for localized orbitals (IGLO) method, where different gauge origins are used for each localized MO in order to minimize error introduced by having the gauge origin far from any particular MO. Of the two methods, modern implementations of GIAO are probably somewhat more robust, but it is possible to obtain good results with either. One should also be aware of issues with using the Effective Core Potentials (ECPs). If the core electrons of the heavy atom are represented by an ECP, then it is not in general possible to predict the chemical shift for that nucleus, since the remaining basis functions will have incorrect behavior at the nuclear position (note that it is mostly the tails of the valence orbitals at the nucleus that influence the chemical shift, not the core orbitals themselves, since they are filled shells). However, ECPs may be an efficient choice if the only chemical shifts of interest are computed for other nuclei. You also remember that ECPs can deal with relativistic effects. Relativistic effects are also an important consideration when predicting chemical shits. In terms of computing absolute chemical shifts, they can be very large in heavy elements. For relative chemical shifts the error is greatly reduced, because relativistic effects are primarily associated with core orbitals, and core orbitals do not change much from one chemical environment to the next. Nevertheless, accurate calculations involving atoms beyond the first row of transition metals are challenging Performance of NMR Calculations For molecules composed of only first-row atoms, heavy-atom chemical shifts can be computed with a fair degree of accuracy. Even HF gives acceptable accuracy in most instances, although some improvements are available in favorable instances from DFT (note, however, that LDA and B3LYP do particularly badly see the Table below) and MP2. The MP2 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 they may corrected by using empirical factors, in the same spirit as the scaling of vibrational frequencies. For example, scaling 13 C shieldings computed at the B3LYP/MIDI! level by 1.16 and adding ppm provides an RMS error of only 3.6 ppm over a diverse test set of

7 experimental values measured in solution. Note that the presence of multiple bonds makes the chemical shifts of the atoms involved quite sensitive to the level of theory, particularly for nitrogen and oxygen atoms. The following tables (from Cramer: Essentials of Computational Chemistry) give you an idea what degree of accuracy you can expect from NMR shielding (chemical shift) calculations. Note that proton ( 1 H) chemical shifts are among the toughest to calculate precisely, because they span a fairly narrow range - perhaps 15 ppm. Table 1. Absolute chemical shieldings a

8 The calculation of spin spin coupling is more difficult than that of a 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, which are again amongst the hardest to predict because they tend to be small in magnitude and the absolute errors are correspondingly magnified when considered in a relative sense. Some representative calculations are provided in the following table. Computed coupling constants are quite sensitive to basis set, and accurate predictions require very flexible bases. As a rule, DFT is much more robust than HF theory for predicting coupling constants, and the HF should be avoided for this type of calculation. Table 2: Spin-spin coupling constant from LDA calculations and experiments

9 20.4. Hyperfine Coupling Constants of Radicals Molecules with unpaired electrons carry a non-zero electronic spin, which interacts with the (non-zero) spins of the individual nuclei. The energy difference between the two the electronic and nuclear spins being either aligned or opposed in the z-direction can be measured by electron spin resonance (ESR) spectroscopy and defines the isotropic hyperfine splitting (h.f.s.) or hyperfine coupling constant. To compute this quantity the molecular Hamiltonian is modified to include a spin magnetic dipole at a particular nuclear position. The integral that results used to evaluate the necessary perturbation is known as a Fermi contact integral. For any open-shell molecule these coupling constants are calculated automatically as part of the Population Analysis section labeled as Fermi contact analysis. These values are given in atomic units and thus need to be converted into frequency, in this case as MHz. The following expression accomplishes this: bf = (16 /3)(g/2)gIKBF where: g = observed free electron factor, K = MHz is a composite conversion factor, BF is the atomic unit value of the hyperfine coupling calculated by Gaussian and gi is the gyromagnetic ratio for a nucleus, which is the magnetic moment divided by the spin Atom Spin Magnetic Moment Proton 1/ Carbon-13 1/ Nitrogen Make sure to include the keyword Density so that the population analysis uses the proper electron density. Calculation of hyperfine coupling requires that the localization of excess spin must be accurately determined at HF level the ROHF methodology is therefore not very useful, because it cannot properly account for spin polarization. UHF on the other hand suffers from spin contamination, which can lead to bad results. Projection (annihilation) of the spin contaminants is usually and improvement. An important consideration is that the Fermi contact interaction (that s where contact comes from) arises from the electrons basis functions overlapping the nucleus. Unfortunately, the core orbitals which have the highest overlap are usually treated in the most approximate way, and the GTOs also have a wrong shape at the nucleus (the STO is the right shape). For this reason, specific basis sets were developed for calculations of h.f.s. that correct for this. Overall, DFT is generally very good in computing h.f.s, except where delocalization is a problem, which tends to occur in radicals. If that is the case, MP2 is usually the method that offers the best price/performance ratio though it is considerably more expensive than DFT.

10 References Cramer, C. J.: Essential of Computational Chemistry: Theories and Methods, Wiley, Foresman, JB. Frisch E. Exploring Chemistry With Electronic Structure Methods, Gaussian, inc, 1993 Lewars: Computational chemistry: introduction to the theory and applications of molecular and quantum mechanics, Springer, Ramachandran, Gopakumar, Namboori: Computational Chemistry and Molecular Modeling: Principles and Applications, Springer, 2008

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