Exercise Week 9: NMR

Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in the presence of a magnetic field absorb and re-emit electromagnetic energy. This energy is at a specific resonance frequency which depends on the strength of the magnetic field and the magnetic properties of the isotope of the elements.

The energy levels studied in NMR spectroscopy are the spin eigenstates of chemically bonded nuclei in the presence of an external magnetic field. The nuclear magnetic moment I of an NMR active nucleus can align with an externally applied magnetic field in only 2I + 1 ways, either reinforcing or opposing it. The precise resonant frequencies are associated with transitions between these energy levels, which depend on the effective magnetic field at the nucleus. The effective magnetic field is the superposition of the external magnetic field and that of the field induced by the electrons of the molecule itself, at the position of the nucleus. The induced field and hence the shielding tensor at the nucleus is characteristic for the electron distribution, and hence the structure of the molecule. This makes NMR such a powerful analytical tool.

The NMR shielding relates the external (applied) magnetic field, B_ext, and the induced (opposing) field, B_ind. In the NMR experiment, the local magnetic field at the nucleus, i.e. the difference between the applied and the induced magnetic field, is the quantity which determines the resonance of the nucleus.

Since the NMR shielding represents the linkage between two vectors, each having three components, it is generally a 3-by-3 tensor (the shielding tensor). Of particular importance is however the nuclear magnetic shielding constant, σ, often called the isotropic NMR shielding constant. It is the average of the σ₁₁, σ₂₂ and σ₃₃. i.e. \(σ = \frac{1}{3} (σ_{11} + σ_{22} + σ_{33})\)

The NMR shielding constant can not be measured experimentally; what one measures in an NMR experiment, is the deviation of the nuclear shielding in the molecule of interest from the nuclear shielding in a reference substance, that is, the so-called NMR chemical shift, δ, i.e. \(\delta = \sigma(ref) - \sigma\)

The use of NMR chemical shifts instead of shielding constants brings a couple of important advantages: The NMR spectrometer can be calibrated with a reference substance; errors arising from the determination of the nuclear magnetic shieldings can be avoided (or at least reduced), because only the difference between the shieldings is determined and used for analysis. The latter is particularly relevant for the theoretical NMR studies: for a proper NMR calculation, accurate description of the electronic density at the nucleus is very important, but can be hampered e.g. by insufficient quality of the basis sets. The errors are however systematic, that is, they are rather similar for the molecule of interest and the reference substance. Since the calculated shielding constants for the two systems are subtracted from each other, the error in the thus-determined chemical shift δ will be significantly reduced. In organic chemistry, the standard NMR reference substance is tetramethylsilane (TMS), as it provides easily measurable signals for the ¹³C and ¹H nuclei, and the intermolecular interactions are small.

We use CP2K to calculate the shielding constant. First one needs to optimize the molecular structure as we did in exercise 1. You could find the atomic structure online and then optimize it. After having the optmized structure, then you could use the following templete to calculate the isotropic NMR shielding constant.

Here is an example of performing all-electron calculations with GAPW to compute absolute nuclear magnetic shieldings of ¹H in CH₄ molecule in CP2K.

The atomic coordinates should be replaced in your molecule.

After running the CP2K calculation, you will get a file named like ch4-gapw-SHIFT-1.data. The isotropic NMR shielding constant is there. Note that in the above CH₄ example, we didn’t apply symmetry to the system. Hence the isotropic NMR shielding constants for four same H in CH₄ are slightly different with each other. One could take the average value of them.