LCHIMAG: Difference between revisions

Latest revision as of 11:02, 11 March 2025

LCHIMAG = .TRUE. | .FALSE.
Default: LCHIMAG = .FALSE.

Description: Calculate the chemical shifts and magnetic susceptibility within linear response theory.

For LCHIMAG = True, the chemical shift tensors and magnetic susceptibility is computed. The implementation^[1] is based on linear response theory using the gauge-invariant PAW method of Yates, Pickard, and Mauri ^[2]^[3], that is an extension to the standard PAW method to account for the effects of a vector gauge field $A$ . The NMR response currents are computed and the induced B field is calculated based on the Biot-Savart law.

See also WRT_NMRCUR to write the currents and NUCIND to compute the nuclear-independent chemical shielding.

Warning: This method only works for non-metallic systems, i.e., that have a finite bandgap.

Definitions

The chemical shielding tensor is defined as:

{\displaystyle \sigma_{ij}(\mathbf{R}) = - \frac{ \partial B^{\mathrm{in}}_i(\mathbf{R})}{ \partial B^{\mathrm{ext}}_j} }

Here $\mathbf{R}$ denotes the atomic nuclear site, $i$ and $j$ denote Cartesian indices, $\mathbf{B}^{\mathrm{ext}}$ an applied DC external magnetic field and $\mathbf{B}^{\mathrm{in}}(\mathbf{R})$ the induced magnetic field at the nucleus.

NMR experiments yield information on the shielding relative to a reference compound:

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

Here, $\sigma_{ij}^{\mathrm{ref}}$ is the isotropic shielding of the nucleus in the reference compound. $\delta_{ij}(\mathbf{R})$ is the chemical shift tensor. In order to compare numerical results with the experimental data, one usually considers a series of compounds and references that to the experimental series.

Mind: If VASP reports the chemical shift (LNMRSHIELD = False) it is simply the negative shielding,

\delta_{ij}(\mathbf{R})= - \sigma_{ij}(\mathbf{R})

. The reference must still be added in the postprocessing to analyze the data.

VASP calculates chemical "shifts" for non-metallic crystalline systems using the linear response method of Yates, Pickard, and Mauri ^[2]^[3].

The isotropic chemical "shift" $\sigma_{iso}$ , span $\Omega$ , and skew $\kappa$ are also reported, according to the following Herzfeld-Berger convention ^[4]:

{\displaystyle \sigma_{iso} = (\sigma_{11} + \sigma_{22} + \sigma_{33})/3 }

{\displaystyle \Omega = \sigma_{33} - \sigma_{11} }

{\displaystyle \kappa = 3(\sigma_{iso} - \sigma_{22})/\Omega. }

The orbital magnetic susceptibility $\chi$ is calculated according to a finite-differences approach:

{\displaystyle \chi_{\textrm{bare}} = \lim_{q\to0} \frac{F(q) 2F(q) + F(-q)}{q^2} }

where $F_{ij}(q)=(2-\delta_{ij})Q_{ij}(q)$ .

Q_ij is approximated in two ways. The so-called pGv-approximation is used by default ^[3], where p is momentum, v is velocity, and G is a Green's function. An alternative approach, the vGv-approximation is also used to calculate an alternative susceptibility since VASP 6.4.0 ^[5]. Q is defined for the pGv-approximation as:

{\displaystyle Q(q) = - \frac{1}{c^2 N_k V_c} \sum_{i=x,y,z} \sum_{o,\textbf{k}} \textrm{Re}[\langle \bar{u}^{(0)}_{o,\textbf{k}} | \hat{\textbf{u}}_i \times (-i \nabla + \textbf{k}) \times \mathcal{G}_{\textbf{k} + \textbf{q}_i}(\epsilon_{o,\textbf{k}}) \hat{\textbf{u}}_i \times \textbf{v}_{\textbf{k} + \textbf{q}_i, \textbf{k}}(\epsilon_{o,\textbf{k}}) | \bar{u}^{(0)}_{o,\textbf{k}} \rangle] }

and for the vGv-approximation as:

{\displaystyle Q(q) = - \frac{1}{c^2 N_k V_c} \sum_{i=x,y,z} \sum_{o,\textbf{k}} \textrm{Re}[\langle \bar{u}^{(0)}_{o,\textbf{k}} | \hat{\textbf{u}}_i \times \textbf{v}_{\textbf{k} + \textbf{q}_i, \textbf{k}}(\epsilon_{o,\textbf{k}}) \times \mathcal{G}_{\textbf{k} + \textbf{q}_i}(\epsilon_{o,\textbf{k}}) \hat{\textbf{u}}_i \times \textbf{v}_{\textbf{k} + \textbf{q}_i, \textbf{k}}(\epsilon_{o,\textbf{k}}) | \bar{u}^{(0)}_{o,\textbf{k}} \rangle] }

.

Output

The isotropic chemical shieldings are printed towards the end of the OUTCAR file, after the self-consistent calculation has finished. The chemical shift tensors both before and after space group symmetrization. These are the absolute tensors for the infinite lattice, excluding core contributions. They can be searched for under the UNSYMMETRIZED TENSORS and SYMMETRIZED TENSORS after Absolute Chemical Shift tensors. Additionally, the magnetic susceptibility is printed shortly after and found under ORBITAL MAGNETIC SUSCEPTIBILITY.

Magnetic susceptibility

The magnetic susceptibility is found at the start of the ORBITAL MAGNETIC SUSCEPTIBILITY, excluding core contribution. The magnetic susceptibility is split into that obtained by the pGv-approximation and obtained by the vGv-approximation:

-------------------------------------------------------------
  ORBITAL MAGNETIC SUSCEPTIBILITY, excluding core contribution
 -------------------------------------------------------------
  Approximate magnetic susceptibility, pGv (10^-6 cm^3/mole)
     1        -70.928534         -0.000000          0.000000
     2         -0.000000        -70.928534          0.000000
     3          0.000000          0.000000        -70.928534
 -------------------------------------------------------------
  Approximate magnetic susceptibility, vGv (10^-6 cm^3/mole)
     1        -63.412095         -0.000000          0.000000
     2         -0.000000        -63.412095          0.000000
     3          0.000000          0.000000        -63.412095

         principal value                      axis
       (10^-6 cm^3/mole)           x,          y,          z
      --------------------------------------------------------
              -63.412095      0.1652     -0.9863      0.0000
              -63.412095     -0.9863     -0.1652      0.0000
              -63.412095      0.0000      0.0000      1.0000
 -------------------------------------------------------------

Chemical shielding

To obtain the full absolute tensors requires adding both the $\mathbf{G=0}$ contribution (cf. G=0 CONTRIBUTION TO CHEMICAL SHIFT) and the contributions due to the core electrons. The latter consists of contributions for each chemical species separately (depending on POTCAR) and a global $\mathbf{G=0}$ susceptibility contribution.

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
 --------------------------------------------------------------------------

Important: The isotropic chemical shift

\delta_{\mathrm{iso}}\mathrm{[VASP]}

(ISO_SHIFT) is the negative of the isotropic shielding. To make it a real shift one should add the reference shielding.

Next, the tensor is processed and its chemical shielding anisotropy (CSA) characteristics are printed in the OUTCAR. The tensor is symmetrized ( $\sigma_{ij} = \sigma_{ji}$ is enforced) and diagonalized. From the three diagonal values the isotropic chemical "shift" $\delta_{\mathrm{iso}}\mathrm{[VASP]}$ , span $\Omega$ , and skew $\kappa$ are calculated and printed see Ref. ^[4] for unambiguous definitions. Note that $\kappa$ is ill-defined if $\Omega = 0$ . Units are ppm, except for the skew. A typical output is given below:

                                                                                                          
   ---------------------------------------------------------------------------------
    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    4598.8125      0.0000      0.0000     4589.9696      0.0000      0.0000
       2     291.5486      0.0000      0.0000      282.7058      0.0000      0.0000
       3     736.5979    344.8803      1.0000      727.7550    344.8803      1.0000
       4     736.5979    344.8803      1.0000      727.7550    344.8803      1.0000
       5     736.5979    344.8803      1.0000      727.7550    344.8803      1.0000
   ---------------------------------------------------------------------------------
    (absolute, valence and core)
       1   -6536.1417      0.0000      0.0000    -6547.9848      0.0000      0.0000
       2   -5706.3864      0.0000      0.0000    -5718.2296      0.0000      0.0000
       3   -2369.4015    344.8803      1.0000    -2381.2446    344.8803      1.0000
       4   -2369.4015    344.8803      1.0000    -2381.2446    344.8803      1.0000
       5   -2369.4015    344.8803      1.0000    -2381.2446    344.8803      1.0000
   ---------------------------------------------------------------------------------
    IF SPAN.EQ.0, THEN SKEW IS ILL-DEFINED
   ---------------------------------------------------------------------------------

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

Important:

The columns excluding the $\mathbf{G=0}$ contribution are useful for supercell calculations on molecules.
The columns including the $\mathbf{G=0}$ contribution are for crystals.
The upper block gives the shielding due to only the electrons included in the SCF calculation.
The lower block has the contributions due to the frozen PAW cores added. These core contributions are rigid ^[6]. They depend on POTCAR and are isotropic, i.e. affect neither SPAN nor SKEW.

References

[dewijs:laskowski:jcp:2017-1] G. A. de Wijs, R. Laskowski, P. Blaha, R. W. A. Havenith, G. Kresse, and M. Marsman, NMR shieldings from density functional perturbation theory: GIPAW versus all-electron calculations, J. Chem. Phys. 146, 064115 (2017).

[pickard:prb:2001-2] C. J. Pickard and F. Mauri, All-electron magnetic response with pseudopotentials: NMR chemical shifts, Phys. Rev. B 63, 245101 (2001).

[yates:prb:2007-3] J. R. Yates, C. J. Pickard, and F. Mauri, Calculation of NMR chemical shifts for extended systems using ultrasoft pseudopotentials, Phys. Rev. B 76, 024401 (2007).

[mason:ssn:1993-4] J. Mason, Conventions for the reporting of nuclear magnetic shielding (or shift) tensors suggested by participants in the NATO ARW on NMR shielding constants at the University of Maryland, College Park, July 1992, Solid State Nucl. Magn. Reson. 2, 285 (1993).

[avezac:prb:2007-5] M. d'Avezac, N. Marzari, and F. Mauri, Spin and orbital magnetic response in metals: Susceptibility and NMR shifts, Phys. Rev. B 76, 165122 (2007).

[gregor:jcp:1999-6] T. Gregor, F. Mauri, and R. Car, A comparison of methods for the calculation of NMR chemical shifts, J. Chem. Phys. 111, 1815 (1999).

[1]

[2]

[3]

[4]

[5]

[6]

@@ Line 1: / Line 1: @@
 {{TAGDEF|LCHIMAG|.TRUE. {{!}} .FALSE. | .FALSE.}}
-Description: calculate the chemical shifts by means of linear response.
+Description: Calculate the chemical shifts and magnetic susceptibility within linear response theory.
 ----
-For {{TAG|LCHIMAG}}=.TRUE., VASP calculates the chemical shift tensors.
+For {{TAGO|LCHIMAG|True}}, the chemical shift tensors and magnetic susceptibility is computed. The implementation{{Cite|dewijs:laskowski:jcp:2017}} is based on linear response theory using the gauge-invariant PAW method of Yates, Pickard, and Mauri {{Cite|pickard:prb:2001}}{{Cite|yates:prb:2007}}, that is an extension to the standard [[PAW method]] to account for the effects of a vector gauge field <math>A</math>. The NMR response currents are computed and the induced B field is calculated based on the Biot-Savart law.
+See also {{TAG|WRT_NMRCUR}} to write the currents and {{TAG|NUCIND}} to compute the nuclear-independent chemical shielding.
+{{NB|warning|This method only works for non-metallic systems, i.e., that have a finite bandgap.}}
+== Definitions ==
 The chemical shielding tensor is defined as:
@@ Line 10: / Line 15: @@
 </math>
-Here <math>\mathbf{R}</math> denotes the atomic nuclear site, <math>i</math> and <math>j</math> denote cartesian indices, <math>B^{\mathrm{ext}}</math> an applied DC external magnetic field and <math>B^{\mathrm{in}}(\mathbf{R})</math> the induced magnetic field at the nucleus.
+Here <math>\mathbf{R}</math> denotes the atomic nuclear site, <math>i</math> and <math>j</math> denote Cartesian indices, <math>\mathbf{B}^{\mathrm{ext}}</math> an applied DC external magnetic field and <math>\mathbf{B}^{\mathrm{in}}(\mathbf{R})</math> the induced magnetic field at the nucleus.
-NMR experiments yield information on the symmetric part of the tensor. Typical NMR experiments yield information on the shielding relative to that of a reference compound:
+NMR experiments yield information on the shielding relative to a reference compound:
 :<math>
 \delta_{ij}(\mathbf{R}) = \sigma_{ij}^{\mathrm{ref}} - \sigma_{ij}(\mathbf{R})
 </math>
-In this (approximate) formula <math>\sigma_{ij}^{\mathrm{ref}}</math> is the isotropic shielding of the nucleus in the reference compound.  <math>\delta_{ij}(\mathbf{R})</math> is the chemical shift tensor.
+Here, <math>\sigma_{ij}^{\mathrm{ref}}</math> is the isotropic shielding of the nucleus in the reference compound.  <math>\delta_{ij}(\mathbf{R})</math> is the chemical shift tensor. In order to compare numerical results with the experimental data, one usually considers a series of compounds and references that to the experimental series.
+{{NB|mind|If VASP reports the chemical shift ({{TAGO|LNMRSHIELD|False}}) it is simply the negative shielding, <math>\delta_{ij}(\mathbf{R})= - \sigma_{ij}(\mathbf{R})</math>. The reference must still be added in the postprocessing to analyze the data.}}
-In VASP the chemical "shift" tensor is calculated as:
-:<math>
-\delta_{ij}(\mathbf{R})\mathrm{[VASP]} = \frac{ \partial B^{\mathrm{in}}_i(\mathbf{R})}{ \partial B^{\mathrm{ext}}_j}
-</math>
-This is minus the shielding tensor. It is not the true chemical shift tensor. To convert it to the real shift tensor one should add the
-reference shielding:
-:<math>
-\delta_{ij}(\mathbf{R}) = \sigma_{ij}^{\mathrm{ref}} + \delta_{ij}(\mathbf{R})\mathrm{[VASP]}
-</math>
 VASP calculates chemical "shifts" for non-metallic crystalline systems using the linear response method of Yates, Pickard, and Mauri {{Cite|pickard:prb:2001}}{{Cite|yates:prb:2007}}.
@@ Line 43: / Line 40: @@
 </math>
-== Input ==
+The orbital magnetic susceptibility <math>\chi</math> is calculated according to a finite-differences approach:
-The chemical shift is calculated via the induced current (cf. {{TAG|WRT_NMRCUR}}) {{Cite|pickard:prb:2001}} {{Cite|yates:prb:2007}}.
-It 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). Two-center contributions (induced fields due to augmentation currents in other PAW spheres) are neglected by default. These contributions can be switched on using {{TAG|LLRAUG}}.
+:<math>
+\chi_{\textrm{bare}} = \lim_{q\to0} \frac{F(q) 2F(q) + F(-q)}{q^2}
+</math>
+where <math>F_{ij}(q)=(2-\delta_{ij})Q_{ij}(q)</math>.
+''Q<sub>ij</sub>'' is approximated in two ways. The so-called ''pGv''-approximation is used by default {{Cite|yates:prb:2007}}, where ''p'' is momentum, ''v'' is velocity, and ''G'' is a Green's function. An alternative approach, the ''vGv''-approximation is also used to calculate an alternative susceptibility since VASP 6.4.0 {{Cite|avezac:prb:2007}}. ''Q'' is defined for the ''pGv''-approximation as:
+:<math>
+Q(q) = - \frac{1}{c^2 N_k V_c} \sum_{i=x,y,z} \sum_{o,\textbf{k}} \textrm{Re}[\langle \bar{u}^{(0)}_{o,\textbf{k}} | \hat{\textbf{u}}_i \times (-i \nabla + \textbf{k})  \times \mathcal{G}_{\textbf{k} + \textbf{q}_i}(\epsilon_{o,\textbf{k}}) \hat{\textbf{u}}_i \times \textbf{v}_{\textbf{k} + \textbf{q}_i, \textbf{k}}(\epsilon_{o,\textbf{k}}) | \bar{u}^{(0)}_{o,\textbf{k}} \rangle]
+</math>
+and for the ''vGv''-approximation as:
+:<math>
+Q(q) = - \frac{1}{c^2 N_k V_c} \sum_{i=x,y,z} \sum_{o,\textbf{k}} \textrm{Re}[\langle \bar{u}^{(0)}_{o,\textbf{k}} | \hat{\textbf{u}}_i \times \textbf{v}_{\textbf{k} + \textbf{q}_i, \textbf{k}}(\epsilon_{o,\textbf{k}})  \times \mathcal{G}_{\textbf{k} + \textbf{q}_i}(\epsilon_{o,\textbf{k}}) \hat{\textbf{u}}_i \times \textbf{v}_{\textbf{k} + \textbf{q}_i, \textbf{k}}(\epsilon_{o,\textbf{k}}) | \bar{u}^{(0)}_{o,\textbf{k}} \rangle]
+</math>.
 == Output ==
@@ Line 51: / Line 64: @@
 ===Magnetic susceptibility===
-By default the orbital '''magnetic susceptibility''' is calculated using the so-called ''pGv''-approximation, i.e. Eqs. 46-48 of Yates '' et al.'' {{Cite|yates:prb:2007}}. As of vasp.6.4.0 also the ''vGv''-approximation of the susceptibility is calculated. By default, the ''pGv'' result is applied for the <math>\mathbf{G=0}</math> contribution to the shifts. With {{TAG|LVGVAPPL}} one can force VASP to use the ''vGv'' result for the <math>\mathbf{G=0}</math> contribution instead. With {{TAG|LVGVCALC}} one can suppress the calculation of the ''vGv'' susceptibility. For details see {{TAG|LVGVCALC}}.
 The magnetic susceptibility is found at the start of the <code>ORBITAL MAGNETIC SUSCEPTIBILITY, excluding core contribution</code>. The magnetic susceptibility is split into that obtained by the ''pGv''-approximation and obtained by the ''vGv''-approximation:
@@ Line 94: / Line 105: @@
   --------------------------------------------------------------------------
 </pre>
-{{NB|important|The chemical shieldings calculated are the negative of the chemical shift. Note that the isotropic chemical shift <math>\delta_{\mathrm{iso}}\mathrm{[VASP]}</math> (ISO_SHIFT) is actually minus the isotropic shielding. To make it a ''real shift'' one should add the reference shielding.}}
+{{NB|important|The isotropic chemical shift <math>\delta_{\mathrm{iso}}\mathrm{[VASP]}</math> (ISO_SHIFT) is the negative of the isotropic shielding. To make it a ''real shift'' one should add the reference shielding.}}
 Next, the tensor is processed and its chemical shielding anisotropy (CSA) characteristics are printed in the {{FILE|OUTCAR}}. The tensor is symmetrized (<math>\sigma_{ij} = \sigma_{ji}</math> is enforced) and diagonalized. From the three diagonal values the isotropic chemical "shift" <math>\delta_{\mathrm{iso}}\mathrm{[VASP]}</math>, span <math>\Omega</math>, and skew <math>\kappa</math> are calculated and printed see Ref. {{Cite|mason:ssn:1993}} for unambiguous definitions. Note that <math>\kappa</math> is ill-defined if <math>\Omega = 0</math>. Units are ppm, except for the skew. A typical output is given below:
@@ Line 130: / Line 141: @@
 *The upper block gives the shielding due to only the electrons included in the SCF calculation.
 *The lower block has the contributions due to the frozen PAW cores added. These core contributions are rigid  {{Cite|gregor:jcp:1999}}. They depend on {{FILE|POTCAR}} and are isotropic, i.e. affect neither SPAN nor SKEW.}}
 <!--
 As of VASP.? on OUTCAR also a summary of the tensors per ion is printed. This is done both excluding and including the '''G'''=0 contribution.
@@ Line 136: / Line 146: @@
 Next the principal components and the principal axes are printed (from the symmetrized tensor).
 They are ordered following Mason {{Cite|mason:ssn:1993}}, i.e. σ<sub>11</sub> < σ<sub>22</sub> < σ<sub>33</sub>.
-Finally a line is printed with (again) the isotropic shielding σ<sub>iso</sub>, the span Ω & skew κ (Herzfeld-Berger, Mason sections 2.2 and 2.3) and the shielding anistropy Δ & asymmetry η (Haeberlen, Mason section 2.6).
+Finally a line is printed with (again) the isotropic shielding σ<sub>iso</sub>, the span Ω & skew κ (Herzfeld-Berger, Mason sections 2.2 and 2.3) and the shielding anisotropy Δ & asymmetry η (Haeberlen, Mason section 2.6).
+<pre>
+ ---------------------------------------------------------------------------------------
+  CSA tensor (J. Mason, Solid State Nucl. Magn. Reson. 2, 285 (1993))
+  for chemical shielding  (including the isotropic core contribution)
+  BDIR labels direction of applied magnetic field (i.e. B0)
+  For BDIR==1, B0 along x-axis, etc.
+  Induced field listed along cartesian directions for each BDIR
+ ---------------------------------------------------------------------------------------
+                 EXCLUDING G=0 CONTRIBUTION             INCLUDING G=0 CONTRIBUTION
+            ------------------------------------   ------------------------------------
+  ion  BDIR            X           Y           Z              X           Y           Z
+          (   iso_shield        span        skew) (  iso_shield        span        skew)
+ ---------------------------------------------------------------------------------------
+     1     142.5256     -0.0000      0.0000       229.8645     -0.0000      0.0000
+      -0.0000    142.5256      0.0000        -0.0000    229.8645     -0.0000
+      -0.0000      0.0000    142.5256        -0.0000      0.0000    229.8645
+          (     142.5256      0.0000      0.0000) (    229.8645      0.0000      0.0000)
+     1     142.5256     -0.0000      0.0000       229.8645     -0.0000      0.0000
+      -0.0000    142.5256      0.0000        -0.0000    229.8645     -0.0000
+      -0.0000      0.0000    142.5256        -0.0000      0.0000    229.8645
+          (     142.5256      0.0000      0.0000) (    229.8645      0.0000      0.0000)
+ ---------------------------------------------------------------------------------------
+  IF SPAN.EQ.0, THEN SKEW IS ILL-DEFINED
+ ---------------------------------------------------------------------------------------
+</pre>
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