Chemical Shifts

The chemical shift tensor is defined as:

$$\delta_{{\bf R}ij} = \frac{ \partial B^{\rm ind}_{{\bf R}i}}{ \partial B^{{\rm ext}}_j}$$

Here ${\bf R}$ denotes the atomic nuclear site, $i$ and $j$ denote cartesian indices, $B^{\rm ext}$ an applied DC external magnetic field and $B^{\rm ind}_{\bf R}$ the induced magnetic field at the nucleus. NMR experiments yield information on the symmetric part of the tensor. VASP can calculate chemical shifts for crystalline systems using the linear response method of Refs. [154,155].

The following five INCAR tags are relevant to linear response calculations of chemical shifts: LCHIMAG, DQ, ICHIBARE, LNMR_SYM_RED, and NLSPLINE. The defaults are:

LCHIMAG=.FALSE.   DQ=0.001   ICHIBARE=1   LNMR_SYM_RED=.FALSE.   NLSPLINE=.FALSE.

To switch linear response chemical shifts on, set: LCHIMAG=.TRUE. The other tags are related to the finite difference k-space derivatives (Eqs. 38, 40 and 47 of Ref. [155]).

• DQ is the step size for the finite difference k-space derivative. Typical values are in the range $0.001 \dots\ 0.003$. The default is often sufficient.
• ICHIBARE can have the values 1, 2 and 3. ICHIBARE sets the order of the stencils used to calculate the magnetic susceptibility (second order derivative in Eq. 47 of Ref. [155]). Often the default is sufficient. A higher ICHIBARE results in a substantial increase of the computational load.
• The star on which the k-space derivative is calculated is oriented along the cartesian directions in k-space. If the symmetry operations in k-space do not map this star onto itself, erroneous results can be obtained. To have VASP check for such operations, set LNMR_SYM_RED=.TRUE., and such operations will be discarded, resulting in a larger IBZ. In case of any doubt set LNMR_SYM_RED=.TRUE. Beware: It matters how the real space lattice vectors are set up relative to the cartesian coordinates in POSCAR. It determines the orientation of the k-space star and hence can affect the efficiency via the number of k-points in the IBZ.
• NLSPLINE=.TRUE. makes that the reciprocal space projectors are set up using a spline interpolation so that they are k-differentiable. This only slightly affects the chemical shifts themselves, but can have impact on the susceptibility contribution (the aforementioned Eq. 47). It is advised to set NLSPLINE=.TRUE., but only in case of calculation of chemical shift. As this option also gives slightly different total energies, it is advised to use the default NLSPLINE=.FALSE. for compatibility in all other calculations. Real space projectors are k-differentiable by construction.

The chemical shifts are calculated from the orbital magnetic response assuming the system is an insulator. It makes no sense to use smearing schemes intended for metals, indeed, doing so can generate nonsense. It is safe to use ISMEAR=0 and make SIGMA so small that no states have fractional occupancies.

The linear response calculation requires a high accuracy. Use EDIFF = 1E-10 or similar.

No special POTCARs are used. The GIPAW is applied using the projectors functions and partial waves that are in the regular POTCARs. A few remarks on accuracy in relation to POTCARs:

• Results sensitively depend on the quality, i.e. completeness of the partial wave/projector function set in the energy range needed for good chemical transferability. Result obtained with different POTCARs can be differ a few ppm for first and second row sp-bonded elements are possible (except for H).
• Use POTCARs generated with a consistent exchange-correlation functional. The PAW reconstruction with AE partial waves is crucial as the field on the nucleus needs to be calculated. So do not override LEXCH from POTCAR with an explicit GGA-tag in INCAR.
• Cutoffs ENCUT needed are typically higher than usual for PREC=A (it is advised to set PREC=A).

A typical INCAR could look like:

LCHIMAG = .TRUE.   # to switch on linear response for chemical shifts
ENCUT = 600.0      # typically higher cutoffs than usual are needed
EDIFF = 1E-10      # you'd need much smaller EDIFFs.
ISMEAR = 0; SIGMA= 0.1 # no fancy smearings, SIGMA sufficiently small
PREC = A           # nice
DQ = 0.001         # often the default is sufficient
ICHIBARE = 1       # often the default is sufficient
LNMR_SYM_RED = .TRUE. # be on the safe side
NSLPLINE = .TRUE.  # only needed if LREAL is NOT set.
LREAL = A          # helps for speed for large systems, not needed
NBANDS = ???       # to safe memory, ??? = NELECT/2

What to do in case of insufficient memory? VASP trades off memory savings against speed, opting for the latter. The response calculation is inherently parallel over k-points. This can be used to economize on memory: First do a regular self-consistent calculation at high accuracy for the full k-point mesh. Save the CHGCAR output. Next do a chemical shift calculation for each k-point in the IBZ separately, starting from CHGCAR, i.e. using ICHARG=11. Finally calculate the shifts as a k-point weighted average of the symmetrized shifts of the individual k-points.

At the end of OUTCAR VASP prints the chemical shift tensors both before and after symmetrization. These are the absolute tensors for the infinite lattice, excluding core contributions. Next lines ``Q=0 CONTRIBUTION TO CHEMICAL SHIFT'' are printed. This is a shift tensor arising solely from the ${\bf G} = 0$ component of the induced field. This component is related to the shape of the sample and depends only on the induced macroscopic surface currents (via the orbital magnetic susceptibility). It is printed for a spherical sample (for which is it nucleous independent), and calculated according to Eqs. 46-48 of Ref. [155], i.e. using the so-called $pGv$-approximation to the magnetic susceptibility. To obtain the full absolute tensor the contribution for $G=0$ has to be added to the nuclear shifts. The approximate susceptibility itself is also printed. Finally the isotropic chemical shift $\delta$, span $\Omega$ and skew $\kappa$ are printed [156]. Note that $\kappa$ is ill-defined if $\Omega = 0$.

All shifts are calculated from the only the valence electrons. Core contributions are rigid [157].

Beware: the treatment of the orbital magnetism is non-relativistic. This is fine for light nuclei.