Calculating the chemical shieldings

Considerations for 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 NMR category page and corresponding pages, so it will not be reiterated here.

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

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Chemical shielding

The chemical shielding tensor σ is calculated by linear response using LCHIMAG.

A typical INCAR file requires a few specific settings:

• A larger ENCUT value than is usually required, generally much higher than the value given by ENMAX in the POTCAR file.
• A small EDIFF is typically required to provide converged chemical shifts, e.g. 1E-8 eV.
• Tighter precision, e.g. PREC = Accurate.
• Non-spherical contributions to the gradient of the density inside PAW spheres, i.e. LASPH = .TRUE.

Two additional tags are unique to NMR calculations:

• LNMR_SYM_RED which ensure 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-deriviable.

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

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

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Electric field gradient

The electric field gradient is calculated using LEFG. The same tight settings for chemical shielding are required, alongside a stronger dependence on the structure and the chosen POTCAR used:

• Make sure to define the isotope-specific quadrupole moment for each species in your POSCAR file using QUAD_EFG.
• The structure is extremely important, so the experimental structure may sometimes be preferable to be used.
• The use of PAW potentials has a strong influence. GW POTCAR files often improve.

make less vague and fill out a bit more show convergence tests for example system

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

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Hyperfine coupling

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 - ref. Adam Gali paper

Make sure to define the nuclear gyromagnetic ratios for each element in your POSCAR file using NGYROMAG. The defaults are set to 1, which will return meaningless results for most systems.

 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

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Chemical shielding

Each nuclear isotope has a different gyromagnetic ratio. Even with the same isotope, the frequency can subtly differ based on the chemical environment. Electrons are also charged and so their movement in atoms, i.e. the electronic current, generates a magnetic field opposed to B_ext. This induced magnetic field B_ind reduces the magnetic field at the nucleus, decreasing the frequency measured in NMR. In this way, the electrons shield the nucleus from B_ext. Since the electron density of a molecule or crystal is determined by its molecular orbitals, its chemical environment can be probed by these subtle differences in frequency. This shielding relation between B_ext and B_ind is described by the chemical shielding tensor σ_ij:

$\textbf{B}_{ind}(\textbf{R}) = -\sigma(\textbf{R}) \textbf{B}_{ext}$

The chemical shielding itself cannot be measured in experiment, instead, it must be taken relative to a standard reference , which results in the chemical shift δ_ij:

$\delta_{ij}(\mathbf{R}) = \sigma_{ij}^{\mathrm{ref}} - \sigma_{ij}(\mathbf{R})$

where i and j are Cartesian axes.

The chemical shift (in ppm) is measurable and is related to the measured frequency ω_sample via the following equation:

$\delta = \frac{\omega_{ref} - \omega_{sample}}{\omega_{sample}} \times 10^6$

The chemical shielding tensor may be calculated by means of linear response:

$\sigma_{ij}(\textbf{R}) = - \frac{\partial B^{\mathrm{ind}}_i(\mathbf{R})}{\partial B^{\mathrm{ext}}_j}$

LCHIMAG calculates the chemical shieldings. These are for individual atoms and exclude the contribution due to the augmentation currents in other PAW spheres. If two-center terms are important, e.g. for H, then they may be included using LLRAUG.

Quadrupolar nuclei

Nuclei with I > ± ½ have an electronic quadrupolar moment. This means that, at the nucleus, there is a non-zero electric field gradient (EFG), i.e. the rate of change of the electric field with respect to position:

$V_{ij} = \frac{\partial^2 V}{\partial x_i \partial x_j}$

This comes from the quadrupolar nuclei being non-spherical an so having a non-uniform electric charge distribution. The electric quadrupolar moment couples with the EFG and so the chemical enviornment of the nucleus may be probed using nuclear quadrupole resonance (NQR) . The EFG is not directly measurable but the nuclear quadrupolar coupling constant C_q is, defined as:

$C_q = \frac{eQV_{zz}}{h}$

where e is the charge of an electron, Q is the isotope-specific quadrupole moment, and h is the Planck constant.

The EFG can be calculated using LEFG, which also calculates C_q so long as Q are defined using QUAD_EFG.

Hyperfine coupling

As well as the nuclei, electrons also have spin. Analogously to the nuclei, this may couple with B_ext to provide information about its environment. The interaction between internally generated magnetic fields and the magnetic dipole moment of the nucleus split otherwise degenerate energy levels. This splitting is known as hyperfine splitting.

In most stable systems, all electrons are paired together, spin-up and spin-down, resulting in overall no spin. If a system has unpaired electrons, e.g. radicals, metal oxides, defects, then these systems can be investigated, e.g. using electron paramagnetic resonance (EPR) .

The hyperfine tensor A^I describes the interaction between a nuclear spin S^I and the electronic spin distribution S^e (in most cases associated with a paramagnetic defect state):

$E=\sum_{ij} S^e_i A^I_{ij} S^I_j$

The hyperfine tensor can be calculated using LHYPERFINE.

How to

• Chemical shift tensors: LCHIMAG.
• Electric field gradient tensors: Electric Field Gradient. The main tags are:
  • LEFG to switch on the field gradient tensor calculation,
  • QUAD_EFG to specify the input nuclear quadrupole moments.
• Hyperfine tensors: LHYPERFINE.