Calculating NMR Spectra

While using the resolvent formulation for the Green's function

$$\begin{array}{rl}{\mathcal{G}}_{\eta }(\omega )& =\frac{1}{Z}\sum _n{\mathrm{e}}^{-\beta E_n}\frac{⟨n|{\hat{M}}^{-}|m⟩⟨m|{\hat{M}}^{+}|n⟩}{\omega -E_n-E_m+i\eta }\end{array}$$

is useful to resolve sharp features in an NMR spectrum, it does not address the problem of the exponential scaling of the Hamiltonian dimension. Evaluating this expression in a brute force approach would imply diagonalizing a Hamiltonian of dimension $2^M$ where $M$ is the number of spins in the molecule. This would restrict one to around 10 spins on a modern laptop. Therefore, different strategies need to be employed to evaluate NMR spectra for larger molecules.

Symmetry

First one can use the fact that the NMR Hamiltonian always conserves the $I^z$ quantum number, which means that we can identify a block diagonal structure in the Hamiltonian, where each block can be diagonalized individually. While this is always possible for the standard NMR Hamiltonian, it only restricts the dimension of the largest block to $\left(\genfrac{}{}{0ex}{}{M}{M/2}\right)$ which still grows rather quickly.

For some molecules one can also identify magnetically equivalent groups. Such a symmetry group is defined as a group of spins, where each individual spin has the same chemical shift and couples in the exact same way to the rest of the system. Identifying these groups is advantageous, as one can combine them into higher order spin representations. This allows to exploit the local SU(2) symmetry of these groups, splitting the Hamiltonian into even smaller blocks. As an example consider a propane molecule, which has eight hydrogen atoms. By identifying all symmetrically coupled groups in this molecule, the number of spins can be reduced to two. One being the CH₂ group which has a combined spin representation of one and the second being the two methyl groups each representing a spin 3/2 and adding up to a total spin representation of spin 3.

While these symmetry considerations are exact and can lead to a reasonable reduction in computational efford, they eventually break down when going to larger and larger molecules. Therefore, also approximate methods have to be used.

Clustering methods

The main approximation method used in the HQS NMR Tool is based on the observation that we do not just evaluate the full spectral function at once, but rather determine the contribution of each individual spin separately. We should therefore write the Green's function from which we obtain the spectral function as

$$\begin{array}{rl}{\mathcal{G}}_l(\omega )& =\frac{1}{Z}\sum _n{\mathrm{e}}^{-\beta E_n}\frac{⟨n|{\hat{M}}_l^{-}|m⟩⟨m|{\sum }_k{\hat{M}}_k^{+}|n⟩}{\omega -E_n-E_m+i\eta }\end{array}$$

where $l$ and $k$ are the indices of the individual spin contributions. This allows us to identify for each spin an effective Hamiltonian and evaluate the spectral function for it independently from the other spin contributions. It turns out that a good approximation for the Hamiltonian for a specific spin contribution is typically given by simply identifying the cluster of spins most strongly coupled to the spin of interest. A good measure for this coupling is the following weight matrix, which is motivated from perturbation theory and can be analyzed using methods from graph theory:

$$\begin{array}{r}{\Delta }_{x,y}=\frac{J_{xy}^2}{({\gamma }_x{\delta }_x-{\gamma }_y{\delta }_y)B}\end{array}$$

Here, $J_{xy}$ are the entries in the J coupling matrix connecting sites $x$ and $y$, ${\gamma }_{x/y}$ the gyromagnetic ratios, ${\delta }_{x/y}$ the chemical shift values, and $B$ the magnetic field strength.

Especially at high field, this method is extremely accurate allowing to choose cluster sizes around 8-12 spins for basically all molecules.