Category:NMR

Density-functional theory (DFT) can be used to calculate various nuclear properties of a crystal measurable through nuclear magnetic resonance (NMR) and associated methods. In the Projector Augmented Wave (PAW) approach, a frozen core approximation is used. The Kohn-Sham (KS) states near the nucleus are correctly described using all-electron (AE) partial waves.

Chemical shielding

The chemical shift δ is a property that describes the shielding of an applied, external magnetic field B felt by a nucleus with non-zero spin. The chemical shift is the difference in chemical shielding σ relative to a reference σ_ref.

${\displaystyle \delta_{ij} = \sigma_{ij}^{\mathrm{ref}} - \sigma_{ij} }$

The chemical shielding tensor can be calculated using the Gauge-Including PAW (GIPAW) method, using the LCHIMAG tag ^[1][2]. The chemical shielding is calculated via the induced current (cf. WRT_NMRCUR) ^[1][2] and has contributions from the plane-wave grid and one-center contributions (the induced field at the center of a PAW sphere due to the augmentation current inside that sphere). 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 should be included using LLRAUG ^[3][4].

Magnetic susceptibility

The macroscopic magnetic susceptibility ${\displaystyle \chi}$ is the degree of magnetization of a material in response to an applied magnetic field ^[5].

${\displaystyle \textbf{B}_{\textrm{in}}^{(1)}(\textbf{G}=0) = \frac{8 \pi}{3} \chi \textbf{B} }$

where ${\displaystyle \textbf{B}}$ is the external magnetic field and ${\displaystyle \textbf{B}_{\textrm{in}}^{(1)}}$ is the induced magnetic field.

The orbital magnetic susceptibility is calculated via a finite-differences approach, where a key variable Q_ij is approximated in two ways. The so-called pGv-approximation is used by default ^[2], where p is momentum, v is velocity, and G is a Green's function. An alternative approach, the vGv-approximation is available to calculate the susceptibility ^[6], which can be switched off using LVGVCALC. With LVGVAPPL one can force VASP to use the vGv result for the ${\displaystyle \mathbf{G=0}}$ contribution instead of the pGv used as default. With LVGVCALC one can suppress the calculation of the vGv magnetic susceptibility.

Like the chemical shielding, the magnetic susceptibility is calculated by linear response using LCHIMAG ^[1][2], so they will both be shown in the same OUTCAR file.

Quadrupolar nuclei - electric field gradient

Nuclei with I > ± ½ have a non-zero electric field gradient (EFG) and an electronic quadrupolar moment. The electric quadrupolar moment couples with the EFG and so the chemical enviornment of the nucleus may be probed using nuclear quadrupole resonance (NQR) ^[7]. The all-electron (AE) nature of the PAW approach makes it particularly suitable for calculating the EFG. The EFG is the second derivative of the potential ${\displaystyle V}$:

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

which is a sum of three parts along the Cartesian i,j axes:

${\displaystyle V_{i,j} = \tilde{V}_{i,j} -\tilde{V}_{i,j}^1 + V_{i,j}^1 }$

where ${\displaystyle \tilde{V}_{i,j}}$ is the plane-wave part of the AE potential, ${\displaystyle \tilde{V}_{i,j}^1}$ is the one-cener expansion of the pseudopotential method, and ${\displaystyle V_{i,j}^1}$ is the one-center expansion of the AE potential.

In VASP, the EFG is calculated using the LEFG tag. The commonly reported nuclear quadrupolar coupling constant C_q is then printed using isotope-specific quadrupole moment defined using QUAD_EFG ^[8].

Hyperfine coupling

The hyperfine tensor ${\displaystyle A^I}$ describes the interaction between a nuclear spin ${\displaystyle S^I}$ and the electronic spin distribution ${\displaystyle S^e}$ (in most cases associated with a paramagnetic defect state measureable by electron paramagnetic resonance (EPR) ^[9]):

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

The hyperfine tensor is split into two terms, isotropic (or Fermi contact) ${\displaystyle A^I_{iso}}$ and anisotropic (or dipolar contributions) ${\displaystyle A^I_{ani}}$:

${\displaystyle A^I_{i,j} = (A^I_{iso})_{i,j} + (A^I_{ani})_{i,j} }$.

Within the PAW approach, ${\displaystyle A^I_{iso}}$ is calculated from the spin-density ${\displaystyle sigma}$, which is split into three terms, the pseudo-spin-density ${\displaystyle \tilde{\sigma}}$, and the one-center expansions of the true- and pseudo-spin density, ${\displaystyle \sigma^1}$ and ${\displaystyle \tilde{\sigma}^1}$, respectively ^[10]:

${\displaystyle \sigma = \tilde{\sigma} + \sigma^1 - \tilde{\sigma}^1 }$.

${\displaystyle A^I_{ani}}$ is calculated from the equivalent three dipolar-dipolar contribution terms ${\displaystyle W_{i,j}(\textbf{R})}$:

${\displaystyle W_{i,j}(\textbf{R}) = \tilde{W}_{i,j}(\textbf{R}) + W_{i,j}^1(\textbf{R}) - \tilde{W}_{i,j}^1(\textbf{R}) }$

Both the Fermi contact and dipolar contribution terms are related to the nuclear gyromagnetic moment γ, which are defined in NGYROMAG. The hyperfine tensor calculation itself is defined using LHYPERFINE.

References