How to construct NMR spectra from chemical shift tensors?

If I know the chemical shift tensor (from ab initio methods), $\sigma$, for each atom in my structure, is it possible to construct an NMR spectra for my system? If yes, how and what mathematical implementations do I need?

Yes, it is possible. It is done often. The solution depends on the dynamics and orientation(s) of the spins in the ensemble (as will be shown bellow). I will assume you want to calculate the spectrum in frequency space.

$$\omega_{cs} = \sigma_{zz}^{LF}(-\gamma \vec{B}_{0})$$

We can hide a few variables and rewrite this in a much more useful way as: $$\omega_{CS} = (\vec{b}^t_0 \overset {\leftrightarrow}{\omega}\vec{b}_0)^{LF} = (\vec{b}^t_0 \overset {\leftrightarrow}{\omega}\vec{b}_0)^{PAS} =(\vec{b}^t_0 \overset {\leftrightarrow}{\omega}\vec{b}_0)^{RF}$$

where,

$$\vec{b_0} = \frac{\vec{B_0}}{B_0}.$$

The above allows us to calculate the chemical shift in many reference frames. The frame where $\omega$ is diagonal (the PAS) is often convenient.

$$\omega_{CS} = \omega_{xx}^{PAS}(\vec{b}^{PAS}_{0,x})^2 + \omega_{yy}^{PAS}(\vec{b}^{PAS}_{0,x})^2 + \omega_{zz}^{PAS}(\vec{b}^{PAS}_{0,x})^2$$

We can define $\theta$ as the angle between $Z^{PAS}$ and $b_0$ and $\phi$ as the angle between $X^{PAS}$ and the x-component of the $b_0$ field. This lets us write:

$$\omega_{CS} = \omega_{xx}^{PAS} \cos^2\phi \sin^2\theta + \omega_{yy}\sin^2\phi \sin^2\theta+ \omega_{zz}^{PAS}\cos^2\theta$$

The above is the main equation you will need to calculate a chemical shift spectrum in frequency space. Numerically you will sum over all the angles in your ensemble. If your sample is a powder you could use angles sampled from a Gaussian distribution on a sphere for example. Typically people then apply convolution with a Lorentzian function (to account for relaxation and smooth the numerical artifacts). If your sample is a solution and the molecular reorientation time is fast compared to the inverse of the anisotropy, everything but the isotropic component will be averaged away. To put this in context (and to get to some possibly familiar equations) it will be useful to introduce some convention. In the Haeberlen-convention:

$$|\delta_{zz} - \delta_{iso}| \ge |\delta_{xx} - \delta_{iso}| \ge |\delta_{yy} - \delta_{iso}|$$ $$\delta_{iso} = \frac{1}{3}(\delta_{xx} + \delta_{yy}+ \delta_{zz})$$ $$\delta = \delta_{zz} =\delta_{iso}$$ $$\eta = (\delta_{yy}-\delta_{xx})/\delta$$

Where $\delta$ is called the anisotropy and $\eta$ is called asymmetry. Graphically:

Using this convention and some geometry you can get to: $$\omega _{CS} = \frac{\delta}{2} (3\cos ^2\theta -1 + \eta \sin^2 \theta \cos 2\phi) + \delta_{iso}$$

Also, ab initio chemical shift calculations normally are reported in shielding values, you will probably want to convert to shift values using a sutable reference.

Reference: Multidimensional Solid-State NMR and Polymers, by Klaus Schmidt-Rohr and Hans Wolfgang Spiess, chapter 2

• Great answer. Indeed some software include sets of pre-calculated reference shifts (e.g., TMS) computed with common methods and basis sets. Otherwise you'll need to run a particular reference compound to report shifts that people will understand. – Geoff Hutchison Jan 23 '15 at 21:28