NMR spin-spin coupling constants
********************************

The NMR spin-spin coupling constants [1] [2] have been implemented in
a separate program CPL. It can be combined with ZORA and spin-orbit
treatment of relativistic effects to study heavy elements. The
original version of this part of the User's Guide was written by
Jochen Autschbach, primary author of the CPL code.

Two GUI tutorials on spin-spin coupling constant 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


Introduction
============

The CPL code of the Amsterdam Density Functional program system allows
the user to calculate *Nuclear Spin-spin Coupling Constants* (NSSCCs)
[1] [2]. NSSCCs are usually observed in NMR (Nuclear Magnetic
Resonance) spectroscopy and give rise to the splitting of the signals
of the NMR spectrum in multiplets. They contain a wealth of
information about the geometric and electronic structure of the
compound being investigated.

The calculation needs a standard adf.rkf (TAPE21) ADF output file. CPL
also reads an input key and optional settings from stdin (usually from
an input file). Technical parameters such as the maximum memory usage
can be set here as well.

One of the key features of the program is its ability to treat heavy
nuclei with the ZORA relativistic formalism. We refer the reader to
the literature for details about our implementation [1] [2], and the
general review on relativistic computations of NMR parameters [3].
Please use the information printed in the output header of the CPL
program in order to provide references of this work in scientific
publications.

The development of the CPL program started in 2000. CPL provides the
main functionality in order to evaluate NSSCCs based on DFT, as well
as a number of additional features in order to provide an analysis of
the results. Several analysis features for the coupling constant have
been added; see the CONTRIBUTIONS subkey. Please report bugs or
suggestions to SCM at support@scm.com.

**Theoretical and technical aspects**

Within the non-relativistic theory of nuclear spin-spin coupling,
there are four terms contributing to the NSSCC between two nuclei A
and B: the paramagnetic and diamagnetic orbital terms (OP and OD,
respectively), and the electron-spin dependent Fermi-contact (FC) and
spin-dipole term (SD). In the literature, the OP and OD terms are
often named PSO and DSO (for paramagnetic and diamagnetic spin-
orbital). In the more general ZORA formulation, very similar operators
are responsible for the NSSCC, therefore we use the same terminology
for the individual contributions. In general, the interpretation of
the results for a heavy atom system is basically equivalent to a non-
relativistic situation.

In most cases, the FC term yields the most important contribution to
the NSSCC. However, many exceptions are known for which one or each of
the other terms can be non-negligible or even dominant. We therefore
suggest that you always check, at least for a smaller but similar
model system, or by using a smaller basis set, which of the four terms
are negligible and which are dominant.

By default, the CPL program computes the FC coupling between the first
and all other nuclei of the molecule, respectively. Other couplings or
the computation of the OP, OD and SD terms can be requested by input
switches (see the 'Running CPL' section of this document for details).

All contributions to the NSSCC are evaluated with the help of the
numerical integration scheme implemented into ADF. In general, the
computation of the OD term is computationally very cheap, since only
integrals involving the electron density have to be evaluated. The
next expensive term is the OP term. For this contribution, the first-
order perturbed MOs have to be computed. With the available density
functionals in ADF, the OP term does not cause a change in the Kohn-
Sham potential, and the first-order MOs can be computed directly (i.e.
without an iterative procedure). This is equivalent to the approach
that has been implemented in the NMR code for ADF.

Both the FC and the SD terms induce electron spin-density to first-
order as a perturbation. Equivalent to the iterative solution of the
unperturbed Kohn-Sham equations, the first-order MOs depend on that
first-order spin-density, which in turn depends on the first-order
MOs. Therefore, in order to evaluate the FC and SD NSSCC
contributions, the CPL program carries out a SCF cycle. In the scalar
or non-relativistic case, the computational cost for the FC term is
comparable to an ADF single point calculation with a local density
functional. The evaluation of the SD term is more expensive. The
current implementation utilizes the CPL spin-orbit code to compute the
combined FC+SD contribution and therefore leaves some room for future
speed-ups. In most cases, the SD term yields a negligible NSSCC and
the much faster code for the scalar- or non-relativistic FC term can
be used. However, it is very important to include the SD term in the
computation if coupling anisotropies are to be evaluated.

In the case where the NSSCC computation is based on spin-orbit coupled
relativistic two component ZORA MOs, the SD term causes only a
marginal increase in computational time as compared to the FC term
alone. Generally, in this case the computational cost for the FC term
is already approximately one order of magnitude higher than in the
scalar or non-relativistic case, since the 3 (x, y, z) components of
the spin-density with respect to 3 components of the perturbation,
respectively, have to be determined self-consistently. The additional
presence of the SD term only shows up in a somewhat more costly
evaluation of the matrix elements of the perturbation operator.
However, CPL spends most of its computational time in the SCF cycle.
Therefore, in spin-orbit computations the computation of the FC+SD
terms is the default. The OP term has to be evaluated self-
consistently, too, in this case and is added as a perturbation in the
SCF cycle upon request.

We use the terminology 'perturbing' and 'responding nucleus' within
the CPL output. The 'perturbing' nucleus is the one for which the
first-order MOs have to be computed (self-consistently), while the
NSSCC is then determined by these first-order MOs and the FC, SD, and
OP matrix elements of the second, 'responding' nucleus. For the OD
term, this distinction makes no sense but is used in the output for
reasons of consistency.

Experimental NSSCCs between two nuclei A and B are usually reported as
J(A,B) in Hertz. From a computational point of view, the so-called
reduced NSSCCs K(A,B) are more convenient for comparisons. CPL outputs
both. The J's are set to zero in case the nuclear magneto-gyric ratio
of one of the nuclei A or B is not available at run time.

**Further technical aspects and current limitations**

In order to facilitate the future computation of rather large
molecules, all matrix elements of the perturbation operators FC, SD,
and OP are evaluated in the Slater AO basis that is specified as input
in the CREATE runs of ADF. The AO matrix elements are further
transformed to the basis of MOs and the calculation proceeds within
the MO basis. This allows for a convenient analysis of the results in
terms of contributions from individual occupied and virtual MOs. Such
an analysis can be requested by input.

The matrix elements themselves as well as the first-order
contributions to the potential are evaluated by numerical integration.
The CPL code, which is parallelized, can use multiple processors for
these steps of the computation. The accuracy setting for the numerical
integration is of high importance to obtain accurate matrix elements.
Furthermore, the basis set being employed needs to be flexible enough
to describe the perturbation correctly. This means usually that
modified basis sets have to be used in particular for heavy element
calculations.

The first-order potential is currently approximated by the VWN
functional. The X:math:*alpha* potential is available as an
alternative but usually leads to less accurate results. In ADF2009.01
the first-order potential of the PBE family of GGA functionals and the
hybrid PBE0 functional can be used.

Currently, only spin-restricted computations for systems with an even
number of electrons are supported. Further, the calculation does not
make use of symmetry and must be based on an ADF run with input
SYMMETRY=NOSYM. Non-Aufbau configurations are not supported. The atom
input list must not contain dummy atoms.

With the present version of CPL, the SD term and the FC/SD cross term
cannot be evaluated separately. Either the sum of FC + SD + cross
terms, or the FC term individually, are computed.

CPL is restartable after various time-consuming steps of the
computation.

In ADF2009.01 the hybrid PBE0 functional can be used in combination
with NMR spin-spin coupling calculations, see the documentation for
the extra keys that are needed. However, other hybrid functionals and
Hartree-Fock cannot or should not be used in combination with NMR
spin-spin coupling calculations.

In ADF2009.01 the effects of a finite size of a nucleus on the spin-
spin couplings can be calculated. A finite size of the nucleus can be
set with the NUCLEARMODEL key in the input for the ADF calculation.

**Bug fix in case more than 1 perturbing atom and DSO or PSO**

In the ADF2006.01b version a bug in the CPL module is fixed that gave
problems in ADF2006.01 and older versions. The problem in ADF2006.01
and older versions is: In case there is more than 1 perturbing atom
and the DSO or PSO term is calculated, only the results of the spin-
spin couplings for the first perturbing atom are correct, but the
results of the other spin-spin couplings may be incorrect.


Input file for CPL: adf.rkf
===========================

In order to run the CPL code, you need the general ADF output file
adf.rkf named TAPE21 being present in the directory where CPL is
running. Alternatively, one can use the key ADFFile to point to an
adf.rkf file. Most of the computation's specific settings will be
taken from adf.rkf (TAPE21), such as the integration accuracy, the
basis set, the density functional being employed, nuclear coordinates,
and so on. *That also means that nearly all of the aspects that affect
the quality of CPL's results are already determined in the input for
the ADF run*. Five aspects are of particular importance here:

1. The numerical integration accuracy: the perturbation operators are
   large in the vicinity of the nuclei. Therefore, you have to make
   sure that the integration grid is fine enough in the atomic core
   regions.

2. The basis set: NSSCCs are sensitive chemical probes, and therefore
   flexible basis sets have to be employed in order to yield a valid
   description of the MOs that determine the NSSCCs. We have found
   that it is imperative to use at least a TZ2P basis set. Additional
   polarization functions in the valence shell may be necessary.
   Furthermore, the FC perturbation usually requires additional steep
   1s functions (i.e. with exponents much higher than the nuclear
   charge) for a proper description. In the relativistic heavy element
   case, the use of additional steep basis functions as compared to
   the ZORA/TZ2P basis is mandatory. The use of steep functions is
   only of high importance for those nuclei, for which the NSSCC is to
   be evaluated.  In ADF2017 specialized large all electron basis sets
   have been added in the directory
   $AMSHOME/atomicdata/ADF/ZORA/TZ2P-J and
   $AMSHOME/atomicdata/ADF/ZORA/QZ4P-J, suitable for NSSCCs, where
   steep basis functions have been added, which is especially
   important for the FC term. Previously in ADF2009.01 for a limited
   number of elements basis sets suitable for NSSCCs have been added
   to the ADF basis set directory, in the directory
   $AMSHOME/atomicdata/ADF/ZORA/jcpl. These specialized basis sets are
   especially important for the heavy NMR nuclei. For the nuclei for
   which NSSCCs are to be evaluated, it is necessary to use all-
   electron basis sets. This is not a restriction due to the
   implementation, but we have found that, with the available frozen
   core basis sets, the flexibility of the basis in the vicinity of
   the nuclei is not sufficient. It is possible to use frozen core
   basis sets if you add enough basis functions in the core region
   such that the basis approaches the flexibility of at least a
   double-zeta all-electron basis there [1]. In that sense, the
   savings in computational time due to usage of a frozen core basis
   are not as pronounced as in standard ADF computations. Unless
   reliable frozen-core basis sets for the NSSCC computation are
   available we strongly discourage the use of frozen core basis sets
   with the CPL program!

3. The finite size of a nucleus: typically, the isotropic J-couplings
   are reduced in magnitude by about 10 to 15 % for couplings between
   one of the heaviest NMR nuclei and a light atomic ligand, and even
   more so for couplings between two heavy atoms, see Ref. [4].
   However, one should have really large basis sets with tight basis
   functions to observe this effect in calculations, see the previous
   point about basis sets. The TZ2P-J, QZ4P-J, and jcpl basis sets are
   suitable for finite nucleus calculations. A finite size of the
   molecule can be set in the ADF program with the key NUCLEARMODEL:

      NuclearModel Gaussian

4. The density functional: the results of the CPL code depend mostly
   on the shape of the MOs that have been determined by ADF, and their
   orbital energies. Both, in turn, depend on the density functional
   or Kohn-Sham potential that has been chosen for the ADF run (and
   the basis set quality). It is difficult to give a general advice
   here concerning the NSSCCs. So far we have found that the use of
   GGAs improves the NSSCCs with respect to experiment in most cases
   in comparison to LDA. Different GGAs often yield very similar
   results. Further, in particular for those cases for which the OP
   term is large or even dominant, both standard LDAs and GGAs
   sometimes do not provide an accurate enough description of the
   orbitals, and deviation of the CPL results as compared to
   experiment can be substantial. Future developments of density
   functionals might be able to cure these problems. For the time
   being, we recommend that you base the CPL run on different choices
   of density functionals in the ADF run, and investigate the
   convergence of the result with respect to basis set and integration
   accuracy. Note that CPL itself uses the VWN functional by default
   to determine the first-order perturbed MOs. There are enough
   indications to believe that this is a reasonable approximation for
   NMR purposes. In ADF2009.01 the first-order potential of the PBE
   family of GGA functionals and the first-order potential of the
   hybrid PBE0 functional can be used. See Refs. [6] [7] for
   applications of such first-order potentials. However, other hybrid
   functionals and Hartree-Fock cannot or should not be used in
   combination with NMR spin-spin coupling calculations.

5. Modeling the experimental setup: computing such sensitive numbers
   as NMR chemical shifts and in particular NSSCCs can result in
   substantial deviations from experimental data. The simple reason
   might be that the isolated system that has been computed at zero
   temperature is not at all a good approximation to the system that
   has been studied experimentally. We [8] [9] and other authors have
   found that in particular solvent effects can contribute very
   substantially to the NSSCC. In case you are comparing CPL results
   to experimental data obtained in strongly coordinating solvents we
   suggest that you consider solvent effects as a major influence. We
   have found that even weakly coordinating solvents can cause sizable
   effects on the NSSCCs for coordinatively unsaturated metal
   complexes. Other sources of errors can be the neglect of
   vibrational corrections to the NSSCCs (usually in the range of a
   few percent).

If the parameters of the underlying ADF computation are carefully
chosen and the density functional is able to provide an accurate
description of the molecule under investigation, it is possible to
compute NSSCCs by means of DFT with very satisfactory accuracy (please
note that for properties as sensitive as NSSCCs, agreement with
experimental results within about 10% error can be regarded as quite
good). Further, chemical trends will be correctly reproduced for a
related series of molecules in most cases. However, due to the
inherent approximate character of the density functionals currently
available with ADF, and necessary basis set limitations, great care
should be taken that the results are reliable. *CPL assumes Aufbau
configurations. Please make sure that there are no empty orbitals with
energies below the highest occupied MO (HOMO). In addition, the
SYMMETRY NOSYM key has to be used in the ADF computation.*


Running CPL
===========

**Main input switches**

With the ADF output TAPE21 present in the current working directory,
the CPL code is invoked by:

   $AMSBIN/cpl < input_file

where input_file contains the input for CPL. Best is supply also the
TAPE10 of the ADF calculation in addition to TAPE21 (specify SAVE
TAPE10 in the ADF input file).

input_file must contain at least one block-type input key in order to
start the CPL run. The input key is

   $AMSBIN/cpl << eor
     NMRCOUPLING
     END
   eor

This represents a minimal input file for CPL. The NMRCOUPLING key
hosts all optional keys that are relevant for the NSSCCs themselves.
In addition to the mandatory NMRCOUPLING key, CPL recognizes the
following input switches:

   $AMSBIN/cpl << eor
     ADFFile adffile
     TAPE10File tape10file
    ...
   eor

adffile is the path to an adf.rkf (TAPE21) file from which nmr reads
data and to which nmr writes data. Default: TAPE21. tape10file is the
path to a TAPE10 file from which nmr reads data. Default: TAPE10.

   $AMSBIN/cpl << eor
     GGA
     ...
   eor

See the separate section for this key, which influences the first
order potential that is used.

   $AMSBIN/cpl << eor
     RESTART /full/path/to/restart_file
     ...
   eor

Restart the computation from file restart_file (full pathname should
be given). This is the TAPE13 produced during a CPL run. By default,
TAPE13 is deleted after a successful completion of CPL. As with ADF
restarts, you cannot use the name TAPE13 for restart_file but you have
to rename it, e.g. to tape13.restart.

   $AMSBIN/cpl << eor
     SAVE TAPE13
     ...
   eor

Keep the restart file even after a successful completion of CPL.
TAPE13 is currently the only file that is meaningful as a parameter to
SAVE.

**NMRCOUPLING subkeys**

The available switches within a NMRCOUPLING/END block control the
computation of the NSSCCs. By default, the program will evaluate the
FC coupling contribution for the first nucleus being the perturbing
nucleus and all remaining nuclei responding.

Please **note** that the ordering of atoms in CPL is generally
different from the ADF input. The ordering of atoms is the one being
stored in TAPE21 and it is grouped by fragment types. In case you are
in doubt about the ordering of atoms, you can run CPL for a few
seconds. It will print a list of atoms with their coordinates. The
ordering is currently the same as required for the NMR program in the
ADF program system. On the other hand, note that for the subkeys
ATOMPERT and ATOMRESP the number of the atoms refer to the input
ordering in the ADF calculation. Curly brackets are not part of a key.

Available subkeys are:

   $AMSBIN/cpl << eor
     NMRCOUPLING
       NUCLEI {npert nresp1 nresp2}
       ATOMPERT {npert1 npert2 npert3}
       ATOMRESP {nresp1 nresp2 nresp3}
       GAMMA {nnuc gamma}
       DSO
       PSO
       SD
       FC
       SCF {ITERATIONS=25 | NOCYCLE | CONVERGE=1e-4 }
       XALPHA
       CONTRIBUTIONS {1E19} {LMO, SFO, LMO2, SFO2}
     END
     ...
   eor

"NUCLEI {npert nresp1 nresp2}"
   Use nucleus no. npert as the perturbing nucleus, and nuclei nresp1,
   nresp2, etc as responding nuclei. You can supply more than one
   NUCLEI keys, in which case CPL evaluates the first-order MOs for
   each perturbing nucleus that is specified and computes the NSSCCs
   between all specified responding nuclei. For each NUCLEI line in
   the input, CPL has to perform an SCF cycle. Note: for the numbers
   of the atoms the internal CPL numbering should be used.

"ATOMPERT {npert1 npert2 npert3}"
   See next subkey ATOMRESP.

"ATOMRESP {nresp1 nresp2 nresp3}"
   ATOMPERT: use nucleus no. npert1, npert2, etc. as the perturbing
   nuclei. ATOMRESP: use nucleus no. nresp1, nresp2, etc. as the
   responding nuclei. You can supply more than one ATOMPERT and (or)
   ATOMRESP key. CPL computes the NSSCCs for all pairs of combinations
   of perturbing atoms and responding atoms. For each perturbing atom
   CPL has to perform an SCF cycle, which is the expensive part in the
   calculation. Note: the numbers refer to the input ordering in the
   ADF calculation. Use the subkey NUCLEI to specify the nuclei
   according to the internal CPL numbers of the atoms.

"GAMMA {nnuc gamma}"
   Input a non-default magneto-gyric ratio of g = gamma for nucleus
   no. nnuc, in units of rad/(T s). Note that one should include the
   the typical 10^7  factor. CPL normally uses the g value of the most
   abundant NMR active isotope for a nucleus of a given charge by
   default. With the GAMMA keyword you can override this value or
   supply a value if CPL does not know about it. A list of g's that is
   used in the computation is printed in the output. You have to
   provide the GAMMA key for each nucleus you want to specify.

"DSO"
   Compute the diamagnetic orbital term for each NSSCC that is
   requested (not default)

"PSO"
   Compute the paramagnetic orbital term for each NSSCC (not default)

"SD"
   Compute the SD term for each NSSCC. This is only default for spin-
   orbit ADF runs. The output will contain the sum of the FC and SD
   contributions. *Please note that requesting this option results in
   a greatly increased computational cost in scalar or non-
   relativistic runs.* The option NOSD will turn the SD computation
   off in spin-orbit runs and has no effect otherwise.

"FC"
   Compute the FC contribution to the NSSCCs. This is the default
   option. Please note that it is currently not possible to compute
   the SD term without the FC term. Consult the 'practical aspects'
   section for instructions how to estimate the FC/SD cross term. The
   option NOFC will disable both the FC and SD computation.

"SCF {ITERATIONS=25 | NOCYCLE | CONVERGE=1e-4 }"
   Settings related to the SCF cycle that is carried out by CPL. Valid
   options are (with default values if applicable):

   "ITERATIONS 25"
      maximum number of iterations

   "NOCYCLE"
      perform no cycle, equivalent to ITERATIONS 0

   "CONVERGE 1e-4"
      convergence criterion, an input of e corresponds approximately
      to a convergence of log(-e) digits, i.e. the results will be
      converged to about four significant digits by default. The
      measurement for the convergence is based on the sum *S* of the
      magnitudes of all occupied-virtual matrix elements of the
      induced first-order exchange potential. Note that the actual
      convergence criterion being used in the computation is e times
      *S* of the first cycle, i.e. the convergence criterion is set
      relative to the initial value of *S*.

"XALPHA"
   Use first-order Xalpha potential instead of VWN potential
   (default). This will usually decrease the accuracy for couplings
   involving hydrogen, and does not have a large effect for couplings
   between heavier nuclei (not default). The key is mainly intended to
   ensure compatibility with our previously published results.

"CONTRIBUTIONS {1e19} {LMO, SFO, LMO2, SFO2}"
   Print contributions from individual orbitals to the FC and OP term
   of the NSSCCs that are larger in magnitude than a certain
   threshold. The threshold refers to the reduced coupling constant K
   in SI units (not default). Additionally, an analysis in terms of
   Boys localized MOs (see User's Guide and SFOs. At present, either
   each key LMO, SFO, LMO2, SFO2 can be used individually, or grouped
   as {LMO, SFO2} or {SFO2, LMO}. If you need all analyses or
   different combinations, it is recommended to restart the CPL
   calculation from TAPE13, and to specify 0 iterations in the SCF.
   This way, the only additional computational cost should be the
   analysis itself.

   The equation and an application for the analyses due to the LMO and
   SFO keys is described in Refs. [14] [15]. The other analysis is
   based on the same equation as in Ref. [5]. For an NBO analysis of
   the spin-spin couplings, see the  section on NBO analysis.

   In order for the LMO-based analyses to work, the MO → LMO
   transformation matrix needs to be stored on TAPE21. In the ADF
   input, you can achieve this with the option "STORE" to the LOCORB
   key, i.e.

      LOCORB STORE
         ... options
      END

**GGA key**

   $AMSBIN/cpl << eor
     GGA
     ...
   eor

"GGA"
   Use first-order GGA potential instead of the first-order VWN
   potential. Should only be used for the PBE family of GGA exchange-
   correlation functionals and for the hybrid functional PBE0. See
   Refs. [6] [7] for applications of calculating spin-spin couplings
   with PBE0. However, other hybrid functionals and Hartree-Fock
   cannot or should not be used in combination with this key GGA. For
   consistency reasons of the first-order potential one should use the
   keyword USESPCODE in the ADF calculation. An example input for ADF
   for the hybrid PBE0 would then contain:

   USESPCODE
   XC
    hybrid PBE0
   End


Practical Aspects
=================

**Minimal input**

The default settings for CPL are invoked by the simple minimal content
of the input file:

   $AMSBIN/cpl << eor
     NMRCOUPLING
     END
   eor

This is equivalent to

   $AMSBIN/cpl << eor
     NMRCOUPLING
      NUCLEI 1 2 3 4 5 6 7 8 ..(up to number of atoms)
      SCF CONVERGE 1e-4 ITERATIONS 25
      FC
     END
   eor

**Restarts**

CPL is restartable after the computation of each the complete set of
FC or FC/SD and OP matrix elements, and after their transformation to
the MO basis. Further, in spin-orbit runs or in scalar- or non-
relativistic computations involving the SD term, CPL is restartable
after each SCF cycle. As with ADF restarts, you need to supply a
proper input file for a restarted computation, and the restart file
TAPE13 (which needs to be renamed). Changing the input of a
calculation for a restart is not supported. In restarted runs, the
program will automatically continue at the latest possible point
before the execution stopped, and changing the input between restarts
can cause inconsistencies that may lead to a crash.

Unless you are computing a very large molecule, the most likely need
for a restart will probably occur during a computation of the FC/SD
SCF cycle. We have already mentioned that this is a very time
consuming part of the computation, and for this reason CPL can be
restarted after each completed SCF cycle. The convergence of the
results should not be affected by a restart. You can, e.g., use this
in order to complete a lengthy CPL computation in case you have tight
time limits in your queuing system, or after a power loss.

**How to avoid the unnecessary computation of many SCF cycles**

As already mentioned, once the first-order MOs with respect to the
perturbation by one of the nuclear spins have been determined, the
NSSCC between this and all other nuclei can be computed rather
quickly. For each nucleus that participates in at least one of the
coupling constants to be determined, the matrix elements of the FC,
SD, and OP operators have to be evaluated once (unless the computation
of the respective terms is disabled).

You can use this information in order to minimize the number of nuclei
for which an SCF cycle has to be performed. This can lead to a great
speedup of the computation. The final result, the NSSCC between A and
B, does not depend on which nucleus has been chosen as the
'perturbing' one, and which as the 'responding' one (convergence has
to be good enough, though). Suppose you want to compute the NSSCCs in
the water molecule, with O being nucleus no. 1. In that case,

   NUCLEI 1 2 3
   NUCLEI 2 3

yields the same O-H and H-H coupling constants as the input

   NUCLEI 2 1
   NUCLEI 1 3
   NUCLEI 3 2

but with less computational effort due to the fact that only 2 instead
of 3 SCF cycles will be performed. The example chosen here is trivial,
but in other cases it can be worthwhile to consider different
sequences of computations.

Alternatively you can use the ATOMPERT and ATOMRESP subkeys:

   ATOMPERT 1 2
   ATOMRESP 2 3

which will calculate the spin-spin coupling of the nuclei 1-2, 1-3,
and 2-3 (skips 2-2, since the nuclei are the same), which are the same
O-H and H-H couplings as before.

**Note**: The numbers of the nuclei for the subkeys ATOMPERT and
ATOMRESP refer to the input ordering in the ADF calculation, whereas
the numbers of the nuclei for the subkey NUCLEI refer to the internal
CPL numbers of the atoms.

**Computing individual terms in the coupling tensor**

As we have mentioned before, the FC, OP and OD terms can be calculated
individually, but not the SD term. In case the SD input option is
given, the FC+SD contribution is evaluated instead. This is NOT equal
to the sum of the individual FC and SD contributions since there is a
cross term between these two. Due to computational simplicity and
efficiency, CPL evaluates either the matrix elements for the FC
operator, or the combined ones for FC+SD. The final result therefore
contains either FC only, or FC, SD plus the cross terms. Only the
latter, in addition to the OP and OD contributions, should be compared
to experimental results. We will implement the computation of the
individual SD term in a future version of CPL in order to assist the
analysis of the CPL results.

Likewise, in a spin-orbit based relativistic computation, there exists
a cross term between the spin-dependent FC and SD terms, and the OP
term. In the scalar- or non-relativistic limit, this contribution is
always zero. With the PSO option present, CPL computes the FC, SD and
OP terms including all cross contributions. Even though the output
suggests that the individual OP and FC+SD terms are printed, they
contain additional cross terms if spin-orbit coupling is large. You
can run CPL with the options

   $AMSBIN/cpl << eor
     NMRCOUPLING
      NOFC
      NOSD
      PSO
     END
     ...
   eor

in order to evaluate the individual OP contribution(s). In a second
run, you can then compute just the FC+SD contributions. The
differences between these two CPL runs and a third one with all three
terms present yields the relativistic (FC+SD)/OP cross term.

**Two-bond and more-bond couplings**

CPL does not discriminate between one-bond and two-bond couplings etc.
in any technical sense. Even though we [1] [2]  [8] [9] have validated
the code mostly for one-bond NSSCCs, the coupling between any pair of
nuclei in the molecule can be computed. See Ref. [8] for an example.

**Principal axis system, the whole coupling tensor**

CPL evaluates the complete 3x3 coupling tensor with respect to the
Cartesian input coordinate system. Depending on the orientation of the
molecule, and the local symmetry, the coupling tensor has in fact
often only a small number of independent components. CPL evaluates the
'principal components' by the following procedure: the 3x3 matrix is
transformed into the basis of the eigenvectors of its symmetric part.
This diagonalizes the symmetric part of the coupling tensor. A set of
eigenvectors (= 'principal axis system') is also printed.

Main references [1] [2] [9] [8] [4] [6] [7] [3].

See also refs. [10] [11] [5] [12] [13].

[1] J. Autschbach and T. Ziegler, *Nuclear spin-spin coupling
    constants from regular approximate density functional
    calculations. I. Formalism and scalar relativistic results for
    heavy metal compounds*, Journal of Chemical Physics 113, 936
    (2000)

[2] J. Autschbach, and T. Ziegler, *Nuclear spin-spin coupling
    constants from regular approximate relativistic density functional
    calculations. II. Spin-orbit coupling effects and anisotropies*,
    Journal of Chemical Physics 113, 9410 (2000)

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

[4] J. Autschbach, *Magnitude of Finite-Nucleus-Size Effects in
    Relativistic Density Functional Computations of Indirect NMR
    Nuclear Spin-Spin Coupling Constants*, ChemPhysChem 10, 2274
    (2009)

[5] J. Khandogin, T. Ziegler, *A density functional study of nuclear
    magnetic resonance spin-spin coupling constants in transition
    metal systems*, Spectrochimica Acta Part A 55, 607 (1999)

[6] J. Autschbach, *Two-component relativistic hybrid density
    functional computations of nuclear spin-spin coupling tensors
    using Slater-type basis sets and density-fitting techniques*,
    Journal of Chemical Physics 129, 094105 (2008), Erratum: Journal
    of Chemical Physics 130, 209901 (2009)

[7] D.L. Bryce and J. Autschbach, *Relativistic hybrid density
    functional calculations of indirect nuclear spin-spin coupling
    tensors . Comparison with experiment for diatomic alkali metal
    halides*, Canadian Journal of Chemistry 87, 927 (2009)

[8] J. Autschbach and T. Ziegler, *A Theoretical Investigation of the
    Remarkable Nuclear Spin.Spin Coupling Pattern in* [(NC)_5 Pt-
    Tl(CN)]^-, Journal of the American Chemical Society 123, 5320
    (2001)

[9] J. Autschbach and T. Ziegler, *Solvent Effects on Heavy Atom
    Nuclear Spin.Spin Coupling Constants: A Theoretical Study of Hg.C
    and Pt.P Couplings*, Journal of the American Chemical Society 123,
    3341 (2001)

[10] N.F. Ramsey, *Electron Coupled Interactions between Nuclear Spins
     in Molecules*, Physical Review 91, 303 (1953)

[11] R.M. Dickson, T. Ziegler, *NMR Spin-Spin Coupling Constants from
     Density Functional Theory with Slater-Type Basis Functions*,
     Journal of Physical Chemistry 100, 5286 (1996)

[12] D.L. Bryce, R. Wasylishen, *Indirect Nuclear Spin-Spin Coupling
     Tensors in Diatomic Molecules: A Comparison of Results Obtained
     by Experiment and First Principles Calculations*, Journal of the
     American Chemical Society 122, 3197 (2000)

[13] G. Schreckenbach, S.K. Wolff, T. Ziegler, *Covering the Entire
     Periodic Table: Relativistic Density Functional Calculations of
     NMR Chemical Shifts in Diamagnetic Actinide Compounds*, in
     Modeling NMR chemical shifts, ACS Symposium Series, Vol 732, J.C.
     Facelli, A.C. de Dios, Editors (American Chemical Society,
     Washington DC, 1999), Chapter 7

[14] J. Autschbach, C.D. Igna, T. Ziegler, *A theoretical
     investigation of the apparently irregular behavior of Pt-Pt spin-
     spin coupling constants* Journal of the American Chemical Society
     125, 1028 (2003)

[15] B.L. Guennic, K. Matsumoto, J. Autschbach, *On the NMR properties
     of platinum thallium bonded complexes: Analysis of relativistic
     density functional theory results*, Magnetic Resonance in
     Chemistry 42, S99 (2004)
