Calculating the chemical shieldings

Performing NMR calculations Introduction to NMR calculations

There are several different options available to calculate NMR properties. It is possible to calculate the chemical shielding, the two-center contributions, the electric field gradient, and the hyperfine coupling constant. The theory is already covered in THE NMR category page and corresponding pages, so it will not be reiterated here.

For all calculations, tighter convergence settings than typical are required, e.g. for a structure relaxation. No additional files are required besides the four standard POSCAR, POTCAR, INCAR, and KPOINTS, unless specifically mentioned. It is important to have a well-converged structure, as all of these calculations described below can be very sensitive to it. For each of the following calculations, the NMR property is calculated post-SCF.

Chemical shielding

The chemical shielding tensor σ is the relation between the induced and external magnetic fields and describes how much the electrons shield the nuclei from an external field. The absolute chemical shielding is calculated by linear response using LCHIMAG . This is directly related to the chemical shift δ (cf. NMR category page and LCHIMAG page for details) and, indirectly, to the resonance frequency.

Two additional tags are unique to NMR calculations:

• LNMR_SYM_RED which ensures that all symmetry operations for the k-space derivatives are consistent when calculating chemical shifts.
• NLSPLINE which constructs PAW projectors in reciprocal space to ensure that they are k-derivable.

An example INCAR file is given below:

 ENCUT = 400              # Plane-wave energy cutoff in eV
 ISMEAR = 0; SIGMA = 0.01 # Defines the type of smearing; smearing width in eV
 EDIFF = 1E-8             # Energy cutoff criterion for the SCF loop, in eV
 PREC = Accurate          # Sets the "precision" mode
 LASPH = .TRUE.           # Non-spherical contributions to the gradient of the density in the PAW spheres 
 
 LCHIMAG = .TRUE.         # Turns on linear response for chemical shifts
 LNMR_SYM_RED = .TRUE.    # Consistent symmetry with star and k-space derivatives
 NLSPLINE = .TRUE.        # Differentiable projectors in reciprocal space

Output

The isotropic chemical shieldings are printed to the OUTCAR file. The reference shift experienced by the core is given first:

  Core NMR properties

  typ  El   Core shift (ppm)
 ----------------------------
    1  C     -200.5098801
 ----------------------------

  Core contribution to magnetic susceptibility:     -0.31  10^-6 cm^3/mole
 --------------------------------------------------------------------------

The isotropic chemical shielding for each atom excluding and including G=0 contributions, as well as the span and skew, follow. Finally, core contributions are taken into account for the ISO_SHIFT, SPAN, and SKEW:

 ---------------------------------------------------------------------------------
  CSA tensor (J. Mason, Solid State Nucl. Magn. Reson. 2, 285 (1993))
 ---------------------------------------------------------------------------------
             EXCLUDING G=0 CONTRIBUTION             INCLUDING G=0 CONTRIBUTION
         -----------------------------------   -----------------------------------
  ATOM    ISO_SHIFT        SPAN        SKEW     ISO_SHIFT        SPAN        SKEW
 ---------------------------------------------------------------------------------
  (absolute, valence only)
     1      77.7746      0.0000      0.0000       66.5779      0.0000      0.0000
     2      77.7746      0.0000      0.0000       66.5779      0.0000      0.0000
 ---------------------------------------------------------------------------------
  (absolute, valence and core)
     1    -122.7353      0.0000      0.0000     -134.3162      0.0000      0.0000
     2    -122.7353      0.0000      0.0000     -134.3162      0.0000      0.0000
 ---------------------------------------------------------------------------------
  IF SPAN.EQ.0, THEN SKEW IS ILL-DEFINED
 ---------------------------------------------------------------------------------

The chemical shielding tensor itself is found earlier in the OUTCAR file. The UNSYMMETRIZED TENSORS and SYMMETRIZED TENSORS can be found underneath Absolute Chemical Shift tensors. Additionally, the magnetic susceptibility is printed shortly after and can be found by searching for ORBITAL MAGNETIC SUSCEPTIBILITY.

Recommendations and advice

A typical INCAR file requires a few specific settings:

• A larger ENCUT value than usual, generally much higher than the value given by ENMAX in the POTCAR file, e.g. 800 eV for C.
• A small EDIFF is typically required to provide converged chemical shifts, e.g. 1E-8 eV.
• Tighter precision, e.g. PREC = Accurate.

Two additional terms may make a difference depending on your system, which should be tested with and without to determine their importance:

• Non-spherical contributions to the gradient of the density inside PAW spheres, i.e. LASPH = .TRUE.
• Occasionally, e.g. for systems containing H or first-row elements, and short bonds, the two-center contributions are important. In this case, LLRAUG = .TRUE. should be used .

For each system, make sure to test that the chemical shieldings calculated are converged with respect to ENCUT, EDIFF, and KPOINTS mesh. Convergence is considered to be typically within 0.1 ppm.

Electric field gradient

Nuclei with a spin > ± ½ are called quadrupolar nuclei. They have a non-spherical shape and therefore a non-zero electric field gradient (EFG) at the nucleus. The EFG is calculated using LEFG . By including the quadrupole moment of the isotopes, the quadrupole coupling constants C_q can be calculated (multiple definitions exist in the literature, ensure that you are correctly comparing). These are measured using nuclear quadrupole resonance (NQR) spectroscopy, a type of zero- to ultralow-field (ZULF) NMR.

There is one additional keyword that must be defined:

• QUAD_EFG defines the isotope-specific quadrupole moment for each species in your POSCAR file, taken from an online database, e.g. here .

A typical INCAR file is given below:

 ENCUT = 400              # Plane-wave energy cutoff in eV
 ISMEAR = 0; SIGMA = 0.01 # Defines the type of smearing; smearing width in eV

 EDIFF = 1E-8             # Energy cutoff criterion for the SCF loop, in eV
 PREC = Accurate          # Sets the "precision" mode
 LASPH = .TRUE.           # Non-spherical contributions to the gradient of the density in the PAW spheres

 LEFG = .TRUE.            # Electric field gradient calculations
 QUAD_EFG = 0. -696. 20.44 0. 2.860  # Nuclear quadrupolar moments for Pb I N O D

Important: Make sure to replace the QUAD_EFG in the INCAR with the values for the isotopes in your system.

Output

The EFG is listed atom-wise after the SCF cycle has been completed. First, as calculated:

  Electric field gradients (V/A^2)
 ---------------------------------------------------------------------
  ion       V_xx      V_yy      V_zz      V_xy      V_xz      V_yz
 ---------------------------------------------------------------------
     1        -         -         -         -         -         -       

And then afterwards following diagonalization:

  Electric field gradients after diagonalization (V/A^2)
  (convention: |V_zz| > |V_xx| > |V_yy|)
 ----------------------------------------------------------------------
  ion       V_xx      V_yy      V_zz     asymmetry (V_yy - V_xx)/ V_zz
 ----------------------------------------------------------------------
     1       -         -         -             -         

The corresponding eigenvectors are then printed. Finally, the quadrupolar parameters are presented, which, unlike the EFG, may be measured in experiment.

            NMR quadrupolar parameters

  Cq : quadrupolar parameter    Cq=e*Q*V_zz/h
  eta: asymmetry parameters     (V_yy - V_xx)/ V_zz
  Q  : nuclear electric quadrupole moment in mb (millibarn)
 ----------------------------------------------------------------------
  ion       Cq(MHz)       eta       Q (mb)
 ----------------------------------------------------------------------
    1        -             -         -                      

Recommendations and advice

The same tight settings for chemical shielding are required, alongside a stronger dependence on the structure and the chosen POTCAR used:

• A larger ENCUT value than usual, generally much higher than the value given by ENMAX in the POTCAR file, e.g. 800 eV for C.
• A small EDIFF is typically required to provide converged chemical shifts, e.g. 1E-8 eV.
• Tighter precision, e.g. PREC = Accurate.
• The structure is extremely important, so using the experimental structure can improve results.
• The use of PAW potentials has a strong influence, GW POTCAR files often improve values.
• Semi-core electrons can be important (check the POSCAR files with *_pv or *_sv) as well as explicit inclusion of augmentation channels with $d$-projectors.

In addition, test whether non-spherical contributions are important for your system:

• Non-spherical contributions to the gradient of the density inside PAW spheres, i.e. LASPH = .TRUE.

Be aware of some specifics relevant to the implementation used:

• Several definitions of $C_q$ are used in the NMR community, ensure that you are comparing between the same definitions in calculation and experiment.
• For heavy nuclei inaccuracies are to be expected because of an incomplete treatment of relativistic effects.

For each system, make sure to test that the chemical shieldings calculated are converged with respect to ENCUT, EDIFF, and KPOINTS mesh. Convergence is considered to be typically within 3 significant figures for the EFG in the z-direction V_zz (though this may be infeasible for heavier elements).

Hyperfine coupling

The hyperfine coupling constants are calculated using LHYPERFINE, i.e. the hyperfine splitting . Specifically, the coupling between the nuclear magnetic dipole moment and the magnetic field generated by the electrons (or nuclear spin-electron spin coupling) is referred to. For the interaction between the nuclear quadrupole moment and the electric field gradient, see LEFG and the description in insert link back to efg above when hyperfine made into its own how-to page. The hyperfine splitting can be measured using electron paramagnetic resonance (EPR), also called electron spin resonance (ESR), as well as in atomic spectroscopy.

There is one additional keyword that must be defined:

• NGYROMAG defines the nuclear gyromagnetic ratios for each element in your POSCAR file. The defaults are set to 1, which will return meaningless results. Reasonable values may be found here.

An example INCAR file is given here:

 ENCUT = 500              # Plane-wave energy cutoff in eV
 ISMEAR = 0; SIGMA = 0.01 # Defines the type of smearing; smearing width in eV

 EDIFF = 1E-8             # Energy cutoff criterion for the SCF loop, in eV
 PREC = Accurate          # Sets the "precision" mode

 LHYPERFINE = .TRUE.      # Turns on calculating the hyperfine coupling tensor
 NGYROMAG = 10.7084 42.577478461 # Specifies the nuclear gyromagnetic ratios for the ions - C and H in this case
 ISPIN = 2                # Turns on spin-polarization - noncollinear can also be used

Important: Make sure to replace the NGYROMAG in the INCAR with the values for the isotopes in your system.

Output

You can find the output for the hyperfine calculation in the OUTCAR file after the SCF cycle finishes. The total magnetic moment is listed, then the Fermi contact term:

 Total magnetic moment S=     1.0000000

 Fermi contact (isotropic) hyperfine coupling parameter (MHz)
 -------------------------------------------------------------
  ion      A_pw      A_1PS     A_1AE     A_1c      A_tot
 -------------------------------------------------------------
   1        -         -         -         -         -
   2        -         -         -         -         -
 -------------------------------------------------------------

Note the A_tot does not include the core contribution term A_1c . The dipolar hyperfine coupling parameter comes next and finally the total hyperfine coupling parameter

 Dipolar hyperfine coupling parameters (MHz)
 ---------------------------------------------------------------------
  ion      A_xx      A_yy      A_zz      A_xy      A_xz      A_yz
 ---------------------------------------------------------------------
   1        -         -         -         -         -         -
   2        -         -         -         -         -         -
 ---------------------------------------------------------------------


 Total hyperfine coupling parameters after diagonalization (MHz)
 (convention: |A_zz| > |A_xx| > |A_yy|)
 ----------------------------------------------------------------------
  ion      A_xx      A_yy      A_zz     asymmetry (A_yy - A_xx)/ A_zz
 ----------------------------------------------------------------------
   1        -         -         -              -         
   2        -         -         -              -          
 ---------------------------------------------------------------------

Recommendations and advice

Distinct from the chemical shielding and EFG, the hyperfine constant is less dependent on EDIFF and ENCUT, generally converging relatively quickly with respect to both. However, it is extremely strongly influenced by the method used. HSE06 was found to give values close to experimental values for molecular systems .