NMR Chemical Shifts
*******************

NMR chemical shifts have been implemented [1] [2] [3] [4] [5] in a
separate property program NMR. It requires the adf.rkf (TAPE21) result
file from an ADF calculation. The NMR module can be combined with the
ZORA treatment for relativistic effects and with spin-orbit effects,
making it suitable for treatment of heavy elements. See also the
general review on relativistic computations of NMR parameters [6].

Two GUI tutorials on NMR chemical shift calculations are currently
available:

* Chemical shifts and nuclear spin-spin coupling constants with NBO
  analysis: Analysis of NMR parameters with Localized Molecular
  Orbitals

* Chemical shifts and nuclear spin-spin coupling constants with
  visualization of the spectrum in the GUI: H-NMR spectrum with spin-
  spin coupling

The NMR chemical shift \delta_i is defined as the difference of the
isotropic NMR chemical shielding of a reference compound \sigma_{ref}
minus that of the interesting compound \sigma_i:

   \delta_i = \sigma_{ref} - \sigma_i

The NMR program calculates NMR chemical shieldings \sigma_i.


Important notes
===============

**adf.rkf (TAPE21) and TAPE10**

NMR requires an ASCII input file and adf.rkf (TAPE21) and TAPE10
result files from an ADF calculation on the molecule to be analyzed.
The ADF result files adf.rkf (TAPE21) and TAPE10 must be present with
names TAPE21 and TAPE10 in the directory where you execute NMR, or
alternatively you can use the keys ADFFile and TAPE10File to point to
an adf.rkf file and TAPE10 file. Use the keywords SAVE TAPE10 in the
ADF calculation in order to obtain a TAPE10 result file.

**Recalculation of TAPE10 by NMR**

Warning: the NMR property program will not always give the correct
result for every SCF potential in the ADF calculation, for example the
SAOP potential, or if one uses COSMO in the ADF calculation, if one
lets the NMR recalculate TAPE10. This is due to the GIAO method used
in this program, which requires the calculation of the SCF potential,
which is not done correctly for potentials other than the standard LDA
and GGA potentials. To obtain correct results, you should, in addition
to the use of TAPE21, also use TAPE10 that ADF generates by using the
keywords SAVE TAPE10, and use it as input for the NMR property
program. On TAPE10 the SCF potential is written, which is read in by
the NMR program.

**Atomic calculation**

NMR calculations on one atom must have symmetry NOSYM.

**Spin-orbit coupling**

NMR calculations on systems computed by ADF with spin-orbit
relativistic effects included must have used NOSYM symmetry in the ADF
calculation. NMR can also be combined with ADF ZORA calculations. The
NMR program reads from adf.rkf (TAPE21) the relativistic option that
is used in the ADF calculation, and will use the same relativistic
option in the NMR calculations.

**Kernel used in spin-orbit coupling**

One can include an improved exchange-correlation kernel, as was
implemented by J. Autschbach [7], but this is not used by default; see
subkey USE. This is an important option that should be seriously
considered and has been advocated in Ref. [7].

**Bug spin-orbit part ADF2008 - ADF2013**

In ADF2008.01 a bug was introduced in the spin-orbit part of the
calculated chemical shielding, which caused the calculated chemical
shielding to be gauge dependent. This bug is relevant for spin-orbit
coupled calculations for ADF versions ADF2008-ADF2013. In ADF2014 this
bug has been fixed.

**Unscaled ZORA default ADF2014**

There is gauge dependence if the scaled ZORA method is used in the
calculation of NMR chemical shieldings. Therefore the default method
for NMR chemical shielding calculations is changed in ADF2014 to use
the unscaled ZORA method.

**SAOP**

The use of the model SAOP potential leads to isotropic chemical shifts
which are substantially improved over both LDA and GGA functionals,
and of similar accuracy as results with a self-interaction-corrected
functional (SIC), see [8]. SAOP is computationally expedient and
routinely applicable to all systems, requiring virtually the same
computational effort as LDA and GGA calculations.

**NICS**

The Nucleus-Independent Chemical Shift (NICS) can be calculated at any
point in the molecule.

**Hybrids**

Starting from ADF2009.01 Hartree-Fock and the hybrid potentials can be
used in combination with NMR chemical shielding calculations, see
Refs. [9] [10]. Use SAVE TAPE10 in the ADF calculation. The use of
frozen cores and hybrids gives gauge dependent results for the NMR
chemical shieldings, therefore the NMR program will stop in this case.

**Meta-GGAs and meta-hybrids**

Meta-GGAs and meta-hybrids should not be used in combination with NMR
chemical shielding calculations. The results are wrong due to an
incorrect inclusion of GIAO terms.

**Range-separated Hybrids**

Starting from ADF2026.01 range-separated hybrid potentials can be used
in combination with NMR chemical shielding calculations. Range-
separated functionals with a meta-GGA part cannot be used.

**LibXC**

Starting from ADF2026.01 many of the LibXC functionals can be used in
combination with NMR chemical shielding calculations. XC functionals
with a meta-GGA part cannot be used.


Input options
=============

The input file for NMR uses the block key NMR, with several (optional)
subkeys, each having a series of options. For analysis, a separate
block key can be used.

   $AMSBIN/nmr << eor
   ADFFile adffile
   TAPE10File tape10file
   ZSOAO2007
   RECALCULATETAPE10
   NMR
   End
   Analysis
   End
   eor

**ADFFile adffile**

Path to adf.rkf (TAPE21) file from which nmr reads data and to which
nmr writes data. Default is TAPE21.

**TAPE10File tape10file**

Path to TAPE10 file from which nmr reads data. Default is TAPE10.

**ZSOAO2007 keyword**

In the ADF2008.01 a bug was introduced in the spin-orbit part of the
calculated chemical shielding, which caused the calculated chemical
shielding to be gauge dependent. This bug is relevant for spin-orbit
coupled calculations for ADF versions ADF2008-ADF2013. Workaround in
ADF versions ADF2008-ADF2013 is to include the keyword ZSOAO2007 in
the NMR part of the input, which causes a one-center approximation to
be used. The bug has been fixed in ADF2014 by introducing an extra
gauge-correction term for the spin-orbit coupled part. One can still
get the (slightly) incorrect results in ADF2014 by using the keyword
WRONGSOGAUGE, and not including ZSOAO2007.

**RECALCULATETAPE10 keyword**

If there is no TAPE10 present, the NMR program will stop. One can use
the key RECALCULATETAPE10 such that TAPE10 will be recalculated by the
NMR module. This is not recommended. Better use "SAVE TAPE10" in the
ADF calculation, and use this TAPE10 as input for NMR.

**NMR block key**

   NMR
     Out  OutOptions
     Calc CalcOptions
     Use UseOptions
     U1K  U1KOptions
     Nuc  NucOptions
     Atoms  AtomsOptions
     Ghosts  GhostsOptions
     Analysis  AnalysisOptions
   End

"Out OutOptions"
   The subkey Out controls printed output. Its options specify the
   details by their (optional) presence. The following OutOptions are
   recognized (Default ISO TENS):

   "All"
      Implies all the other options

   "ISO"
      Isotropic shielding constants

   "Tens"
      Shielding tensors

   "Eig"
      Eigenvectors

   "U1"
      The U1 matrix

   "F1"
      The first order change in the Fock matrix

   "S1"
      The first order change in the Overlap matrix

   "AOP"
      The paramagnetic AO matrix (= the matrix in the representation
      of elementary atomic basis functions)

   "AOD"
      The diamagnetic AO matrix

   "AOF"
      The Fermi-contact AO matrix

   "REFS"
      Literature references

   "INFO"
      General information

"Calc CalcOptions"
   The subkey Calc controls what is actually calculated. The following
   options are available (Default ALL):

   "All"
      Implies all of the other options to this key

   "Para"
      The paramagnetic part

   "Dia"
      The diamagnetic part

   "FC"
      The Fermi-contact part in case of the Pauli Hamiltonian

   "SO"
      The Fermi-contact part in case of the ZORA Hamiltonian

"Use UseOptions"
   The subkey Use controls some options (default none)

   "FXC"
      Improves the exchange-correlation kernel used, as was
      implemented by J. Autschbach [7]. Important only in case of
      spin-orbit coupled calculations. This may give some (small)
      gauge dependent results when using this. Important option that
      should be seriously considered and has been advocated in Ref.
      [7].

   "SCALED"
      Implies the scaled ZORA method, which gives (slightly) gauge
      dependent results, as was shown in Ref. [7]. Note that for the
      ZORA Hamiltonian the default is the unscaled ZORA method. For
      chemical shifts, only compare results with the same options.

   "SO1C"
      Before ADF2008.01, in the spin-orbit term a 1-center
      approximation was used, which does not suffer from gauge
      dependence. This 1-center approximation can be used with USE
      SO1C.

"U1K U1KOptions"
   The subkey U1K determines which terms are included in the
   calculation of the U1 matrix (first order changes in MO
   coefficients). Options (Default Best):

   "Best"
      The best (recommended) options for each relativistic option are
      included for this subkey. Implies 'None' for non-relativistic
      and scalar relativistic ZORA, 'SO' + 'SOFULL' for spin-orbit
      coupled ZORA, and 'MV' + 'Dar' for the Pauli Hamiltonian.

   "None"
      Implies none of the other options to this key.

   "All"
      Implies all the other options to this key.

   "MV"
      The mass-velocity term.

   "Dar"
      The Darwin term.

   "ZMAN"
      The Spin-Zeeman term. Can be included only in case of spin-orbit
      coupled Pauli Hamiltonian.

   "SO"
      ZORA spin-orbit part.

   "SOFULL"
      ZORA spin-orbit part.

Note: for chemical shifts, only compare results with the same options.
If the subkey U1K is used with the option ALL in the ZORA calculation,
then the scaled ZORA orbital energies are used in the making of the U1
matrix, which is not recommended. Recommended is to use 'U1K Best' in
all cases, which uses plain ZORA orbital energies in the making of the
U1 matrix.

"NUC NucOptions"
   This subkey NUC specifies for which nuclei the NMR shielding is
   calculated. By default, all nuclei are calculated, i.e. as for
   omitting the subkeys ATOMS and NUC. Alternatively you may use this
   option by simply typing Nuc in the NMR block (without any further
   data); this means: for no nuclei at all. You may also type the
   index of the atom(s) you want to see analyzed.

   Example:

      NUC 2 1

   The numbers refer to the internal numbering of the nuclei as it
   appears somewhere early in the general ADF output. This internal
   numbering is also the internal NMR numbering, but it is not
   necessarily the same as the input ordering. Use the subkey ATOMS to
   specify the nuclei according to this input ordering in the ADF
   calculation.

   Note that the number of nuclei has a significant consequence for
   the total CPU time.

"Atoms AtomsOptions"
   This subkey ATOMS specifies for which nuclei the NMR shielding is
   calculated. By default, all nuclei are calculated, i.e. as for
   omitting the subkeys ATOMS and NUC.

   Example:

      ATOMS 2 1

   The numbers refer to the input ordering in the ADF calculation. Use
   the subkey NUC to specify the nuclei according to the internal NMR
   numbers of the atoms.

"GHOSTS"
   The subkey GHOSTS is a block type subkey. The format is:

      Ghosts
        xx1 yy1 zz1
        xx2 yy2 zz2
        ...
      End

   With this key, the user can specify ANY point(s) within the
   molecule at which the shielding is to be calculated (whatever the
   physical meaning of this shielding is). One can think of those
   points as neutrons within the molecule. There is a publication by
   P. Schleyer et al. using a similar feature (J. Am. Chem. Soc.
   **118**, 6317, 1996). They call it NICS, Nucleus-Independent
   Chemical Shift. Note that the NICS value is minus 1 times the
   isotropic part of the shielding tensor that is calculated at these
   points:

      xx1 yy1 zz1

   real numbers that specify the Cartesian coordinates of 'ghost' 1,
   etc.

   The coordinates have to be specified in Angstrom. The same set of
   coordinates has to be specified as 'point charges with charge zero'
   using the subsubkey MultipolePotential in the
   ElectrostaticEmbedding subkey of the key System in the AMS block.
   This is necessary in order to allow the appropriate distribution of
   integration points around the ghosts.

   E.g., if you want to have two 'ghosts' with the coordinates xx1 yy1
   zz1 and xx2 yy2 zz2 then you **must** also have in the input the
   key MultipolePotential key as follows:

      System
        ...
        ElectrostaticEmbedding
          MultipolePotential
            Coordinates
              xx1 yy1 zz1   0.0
              xx2 yy2 zz2   0.0
            End
          End
        End
      End

   (the last number is the charge at these coordinates - zero).

   Eventually, this step should be programmed internally but for now
   the procedure outlined above works. No check is done to verify
   whether those 'point charges' are taken care of or not, but their
   omission leads to unpredictable results.

   Only Cartesian coordinates are possible for ghosts, even if the
   atoms were originally specified using internal coordinates. This
   should not be a problem, though (e.g., one could start an ADF run
   of the molecule of interest, and get very soon the Cartesian
   coordinates of the atoms in the output. This run would then be
   aborted, and restarted with the ghosts specified as desired.) The
   ghosts are numbered in the output as NNUC+1, NNUC+2 ... where NNUC
   is the total number of nuclei in this molecule. Default: no ghosts.

"Analysis AnalysisOptions"
   The subkey Analysis controls the MO analysis. After the word
   (subkey) Analysis you type an integer, which then specifies that
   the first so many MOs are to be analyzed. Default: no Analysis. The
   value of this analysis subkey in the block key NMR is somewhat
   limited. The separate ANALYSIS block key can give more analysis of
   the NMR chemical shielding.

**Analysis block key**

The NMR shielding tensor can be analyzed in detail; see Refs. [12]
[13] [6]. For the analysis option with the ANALYSIS block key there
are some restrictions. In the ADF calculation, all-electron basis sets
should have been used, and SYMMETRY NOSYM. It cannot be used in case
of non-relativistic calculations. The ADF calculation should use
relativistic scalar ZORA or relativistic spin-orbit ZORA. In case of
scalar relativistic ZORA the keyword FAKESO should be added to the NMR
input (outside of the NMR or Analysis block keys). The analysis
utilizes the ZORA spin-orbit branch of the NMR code. For scalar ZORA,
the NMR analysis contributions will appear in equivalent pairs for
spin-orbitals even if the ADF calculation is closed-shell spin
restricted. The MO numbering then also reflects this doubling of MOs.
In the analysis, canonical MOs number 1 and 2 are the alpha and beta
spin ADF MO 1, canonical MOs number 3 and 4 correspond to the alpha
and beta spin ADF MO 2, and so on. In case of spin-orbit relativistic
ZORA the keyword FAKESO should not be included. Curly brackets are not
part of the key. For an NBO analysis of NMR, see the section on NBO
analysis.

   Analysis
     print threshold
     canonical
     {components}
   End
   {FakeSO}

"print threshold"
   The print keyword selects printout of contributions relative to the
   total diamagnetic, paramagnetic. For example in case of 'print
   0.01' only contributions greater than 1% are printed. Set to zero
   to print ALL contributions.

"canonical"
   It enables an analysis of the shielding in terms of the canonical
   MOs.

"components"
   The components keyword is optional and enables an analysis not only
   of the isotropic shielding but also of the diagonal Cartesian
   components of the tensor XX, YY, and ZZ. In order to analyze the
   principal shielding tensor components with canonical MOs you can
   calculate the shielding tensor first with the NMR code, rotate the
   molecule such that the principal axes system aligns with the
   Cartesian coordinate system, and then repeat the NMR calculation
   with the analysis features switched on.


Paramagnetic NMR Chemical Shifts
********************************

Knowledge of the g-tensor and hyperfine A-tensor can be used in the
prediction and analysis of paramagnetic NMR shifts, see Refs. [11]
[14] [15]. Because of the dependence on the g-tensor, prediction of
paramagnetic NMR (pNMR) shifts is not straightforward, as the
dependence of the pNMR shift on excess \alpha or \beta electron spin
density at a nucleus is to be combined with the sign and magnitude of
the isotropic g-value.  Ref. [11] describes in detail how to calculate
pNMR contact chemical shifts and pNMR pseudo-contact chemical shifts,
using the ADF program. In ADF2024 (part of) this procedure can be
included more automatically.

Warning:

  Procedure to calculate paramagnetic NMR chemical shifts contains a
  bug in AMS2024.101 - AMS2024.106. Fixed in later releases.

For doublets (effective spin S=1/2), or for effective spin S higher
than 1/2 where one neglects effects due to zero-field splitting (ZFS,
ESR D-tensor) the paramagnetic NMR shielding \sigma can be written as
[Ref. [16]]:

   \sigma = \sigma_{orb} - \frac{\mu_B}{g_N \mu_N} \frac{S(S+1)}{3k_B
   T} \mathbf{g} \cdot \mathbf{A}

Here \sigma_{orb} is the so called orbital dependent part of the NMR
chemical shielding, \mu_B is the Bohr magneton, \mu_N is the nuclear
magneton, g_N is the nuclear magnetic g-factor, k_B is the Boltzmann
constant, and T is the temperature in K (Kelvin). In addition S is the
effective spin, \mathbf{g} is the g-tensor, and \mathbf{A} is the
A-tensor that are used in the effective spin Hamiltonian, which is
commonly used for the interpretation of ESR experiments.

For open shell molecules in ADF the so called orbital dependent part
of the NMR chemical shielding \sigma_{orb} can only be calculated
without taken into account spin-orbit coupling.

   # ADF calculation on the molecule

   $AMSBIN/ams << eor
     ...
     Engine ADF
       SpinPolarization value
       Unrestricted Yes
       Relativity
         Level Scalar
       End
       Basis
         Core None
       End
       Symmetry NOSYM
       SAVE TAPE10
     EndEngine
   eor

   # Calculation of the ESR A-tensors

   $AMSBIN/cpl << eor
     adffile path/to/adf.rkf
     tape10file path/to/TAPE10
     hyperfine
       atoms add_all_atoms
     END
   eor
   rm logfile TAPE15

   # Calculation of the ESR g-tensor

   $AMSBIN/nmr << eor
     adffile path/to/adf.rkf
     tape10file path/to/TAPE10
     nmr
       gfactors
       out iso tens
       u1k best
       calc all
     END
   eor
   rm logfile TAPE15

   # Calculation of the orbital dependent part of the NMR chemical shielding

   $AMSBIN/nmr << eor
     ADFFile path/to/adf.rkf
     TAPE10File path/to/TAPE10
     ALLINONE
     nmr
        u1k best
        calc all
        out iso tens
     end
     Temperature temperature
   eor

Note that one can only do this at the scalar relativistic ZORA level,
one needs symmetry NOSYM, one needs all electron basis sets, and one
needs to include the key ALLINONE in the input for NMR. If one does
not specify the temperature, a default temperature of 298.15 K is
used.

      **See also:**
      Examples: pNMR chemical shielding

[1] G. Schreckenbach and T. Ziegler, *The calculation of NMR shielding
    tensors using GIAO's and modern density functional theory*,
    Journal of Physical Chemistry 99, 606 (1995)

[2] G. Schreckenbach and T. Ziegler, *The calculation of NMR shielding
    tensors based on density functional theory and the frozen-core
    approximation*, International Journal of Quantum Chemistry 60, 753
    (1996)

[3] G. Schreckenbach and T. Ziegler, *Calculation of NMR shielding
    tensors based on density functional theory and a scalar
    relativistic Pauli-type Hamiltonian. The application to transition
    metal complexes*, International Journal of Quantum Chemistry 61,
    899 (1997)

[4] S.K. Wolff and T. Ziegler, *Calculation of DFT-GIAO NMR shifts
    with inclusion of spin-orbit coupling*, Journal of Chemical
    Physics 109, 895 (1998)

[5] S.K. Wolff, T. Ziegler, E. van Lenthe and E.J. Baerends, *Density
    functional calculations of nuclear magnetic shieldings using the
    zeroth-order regular approximation (ZORA) for relativistic
    effects: ZORA nuclear magnetic resonance* , Journal of Chemical
    Physics 110, 7689 (1999)

[6] J. Autschbach and S. Zheng, *Relativistic computations of NMR
    parameters from first principles: Theory and applications*, Annual
    Reports on NMR Spectroscopy 67, 1 (2009)

[7] J. Autschbach, *The role of the exchange-correlation response
    kernel and scaling corrections in relativistic density functional
    nuclear magnetic shielding calculations with the zeroth-order
    regular approximation*, Molecular Physics 111, 2544 (2013)

[8] J. Poater, E. van Lenthe and E.J. Baerends, *Nuclear magnetic
    resonance chemical shifts with the statistical average of orbital-
    dependent model potentials in Kohn.Sham density functional
    theory*, Journal of Chemical Physics 118, 8584 (2003)

[9] M. Krykunov, T. Ziegler and E. van Lenthe, *Hybrid density
    functional calculations of nuclear magnetic shieldings using
    Slater-type orbitals and the zeroth-order regular approximation*,
    International Journal of Quantum Chemistry 109, 1676 (2009)

[10] M. Krykunov, T. Ziegler and E. van Lenthe, *Implementation of a
     hybrid DFT method for calculating NMR shieldings using Slater-
     type orbitals with spin-orbital coupling included. Applications
     to* ^187 Os, ^195 Pt and ^13 C *in heavy metal complexes*,
     Journal of Physical Chemistry A 113, 11495 (2009)

[11] J. Autschbach, S. Patchkovskii, and B. Pritchard, *Calculation of
     Hyperfine Tensors and Paramagnetic NMR Shifts Using the
     Relativistic Zeroth-Order Regular Approximation and Density
     Functional Theory*, Journal of Chemical Theory and Computation 7,
     2175 (2011)

[12] J. Autschbach, *Analyzing NMR shielding tensors calculated with
     two-component relativistic methods using spin-free localized
     molecular orbitals*, Journal of Chemical Physics 128, 164112
     (2008)

[13] J. Autschbach and S. Zheng, *Analyzing Pt chemical shifts
     calculated from relativistic density functional theory using
     localized orbitals: The role of Pt 5d lone pairs*, Magnetic
     Resonance in Chemistry 46, S45 (2008)

[14] S. Moon, and S. Patchkovskii, *First-principles calculations of
     paramagnetic NMR shifts*, in Calculation of NMR and EPR
     parameters, ISBN13: 9783527307791, M. Kaupp, M. Bühl, V.G.
     Malkin, Editors, (Wiley, Weinheim, 2004).

[15] P. Hrobárik, Ro. Reviakine, A.V. Arbuznikov, O.L. Malkina, V.G.
     Malkin, F.H. Köhler, and M. Kaupp, *Density functional
     calculations of NMR shielding tensors for paramagnetic systems
     with arbitrary spin multiplicity: Validation on 3d metallocenes*,
     Journal of Chemical Physics 126, 024107 (2007)

[16] B.Martin, J. Autschbach, *Temperature dependence of contact and
     dipolar NMR chemical shifts in paramagnetic molecules*, Journal
     of Chemical Physics 142, 054108 (2015)
