ESR/EPR
*******

The EPR (ESR) g-tensor, hyperfine interaction (A-tensor), nuclear
quadrupole interaction (Q-tensor), and zero-field splitting (ZFS,
D-tensor) can be calculated. Effects due to spin-orbit coupling can be
included. All-electron basis sets can be used.


ESR/EPR g-tensor and A-tensor
=============================

**A-tensor, no spin-orbit coupling**

   $AMSBIN/ams << eor
   ...
   Engine ADF
     ESR
     END
     SpinPolarization spinpolarization
     unrestricted Yes
     {Relativity
        level scalar
        formalism ZORA
     End}
     {NUCLEARMODEL gaussian}
     ...
   EndEngine
   eor

If spin-orbit coupling is neglected, the spin in the effective spin
Hamiltonian, which is commonly used for the interpretation of ESR
experiments, is the real electronic spin of the paramagnetic molecule.
In the spin-unrestricted DFT calculations one then uses eigenfunctions
of S_z. The A-tensor can then simply be calculated as the expectation
value of the corresponding operator; see Ref. [3].

The A-tensor will be calculated for all nuclei. Terms due to the spin-
polarization density at the nucleus are included in the evaluation of
the A-tensor. For an accurate evaluation of the spin-polarization
density at the nucleus it is important to use an all-electron basis
set for the nuclei that one is interested in, avoiding the frozen core
approximation. For heavy elements the incorporation of a Gaussian
finite nucleus model can be important. However, one should have really
large basis sets with tight basis functions to observe this effect in
calculations. One possibility is to use the
$AMSHOME/atomicdata/ADF/ZORA/QZ4P basis set, although even this large
basis set is not large enough sometimes. The basis sets in the
directories ZORA/TZ2P-J, ZORA/Z4P-J, and ZORA/jcpl are suitable for
finite nucleus calculations.

Since ADF2013, if a finite nuclear model is used for the charge
distribution, ADF also uses a finite distribution of the nuclear
   magnetic dipole moment when calculating the A-tensor.

**A-tensor, perturbative inclusion of spin-orbit coupling**

   $AMSBIN/ams << eor
   ...
   Engine ADF
     SpinPolarization spinpolarization
     unrestricted Yes
     Relativity
       Level Scalar
       Formalism ZORA
     End
     Symmetry NOSYM
     ...
   EndEngine
   eor

   $AMSBIN/cpl << eor
   adffile ams.results/adf.rkf
   hyperfine
    atoms 1 2 :: calculates A-tensor for atom 1 and 2, in input order
    SCF Converge=1e-7 {Iterations=25}
   end
   ...
   eor

The calculation of A-tensors is implemented in the CPL program as a
second derivative property (spin-orbit coupling and nuclear magnetic
field as perturbation) within the two-component relativistic zeroth-
order regular approximation (ZORA), see Ref. [4]. This implementation
allows for hybrid (only PBE0) DFT calculations, but not meta-GGAs and
not meta-hybrids.

Note that the CPKS convergence in CPL has to be set tightly (1e-7 or
1e-8) to get converged PSOSO terms for the A-tensor. For hyperfine
calculations the default value is 1e-7.

**g-tensor, perturbative inclusion of spin-orbit coupling**

   $AMSBIN/ams << eor
   ...
   Engine ADF
     SpinPolarization spinpolarization
     unrestricted Yes
     Relativity
       level scalar
       formalism ZORA
     End
     Symmetry NOSYM
     ...
   EndEngine
   eor

   $AMSBIN/nmr << eor
   adffile ams.results/adf.rkf
   nmr
    gfactors
    u1k best
    calc all
    out iso tens
   end
   end input
   eor

The calculation of g-tensors is implemented in the NMR program as a
second derivative property (spin-orbit coupling and external magnetic
field as perturbation) within the two-component relativistic zeroth-
order regular approximation (ZORA), see Ref. [5]. This implementation
allows for hybrid (B3LYP, PBE0, etc.) DFT calculations, but not meta-
GGAs and not meta-hybrids. This implementation requires the use of
all-electron basis sets.

For an older implementation of this method, see Refs. [1] [2] [8] [9].

**g-tensor and A-tensor, self-consistent spin-orbit coupling**

   $AMSBIN/ams << eor
   ...
   Engine ADF
     ESR
     END
     unrestricted Yes
     Relativity
       Level Spin-Orbit
       Formalism ZORA
       SpinOrbitMagnetization Collinear
     End
     Symmetry NOSYM
   EndEngine
   eor

In a spin-orbit coupled spin-unrestricted relativistic ZORA
calculation and the ESR block enabled, the g-tensor and the nuclear
magnetic dipole hyperfine interaction (A-tensor) will be calculated,
see also Refs. [3] [10]. In such a calculation degenerate perturbation
theory is used with the external magnetic field or nuclear magnetic
field as perturbation. The calculation must use the collinear
approximation, and symmetry must be NOSYM. This implementation does
allow for meta-GGA and (meta-)hybrid DFT calculations, but then GIAOs
are not used. There may be more than one unpaired electron. Terms due
to the spin-polarization density at the nucleus are included in the
evaluation of the A-tensor. However, one cannot set the number of
unpaired electrons; the 'spinpolarization' argument of the key CHARGE
will be ignored.

In the spin-unrestricted self-consistent spin-orbit coupled method for
the calculation of g- and A-tensors implemented in ADF it is assumed
that the solutions of the Kohn-Sham equations with a global
magnetization direction (collinear approximation) in a certain
direction are also eigenfunctions of the effective spin (present in
the effective spin Hamiltonian) in that same direction. In practice
this means that three self-consistent spin-orbit coupled calculations
are performed with the collinear approximation in the x-, y-, and
z-direction. The results of these three calculations are combined to
calculate the g- and A-tensors.

Note: in a spin-orbit coupled spin-restricted relativistic ZORA
calculation and the ESR block key, ADF will also calculate and print
the nuclear magnetic dipole hyperfine interaction, but the terms due
to the spin-polarization density at the nucleus are absent.
Furthermore, if there is more than one unpaired electron, the computed
results will simply be incorrect, without any warning from the
program. On the other hand, in case of one unpaired electron and very
large effects of spin-orbit coupling, the spin-restricted calculation
may be of interest, since it uses Kramers' symmetry exactly.


ESR/EPR Q-tensor
================

For the calculation of the ESR Q-tensor, see the key QTENS.


ESR/EPR Zero-field splitting (D-tensor)
=======================================

With the keyword ZFS the zero-field splitting (ZFS) of the ground
state can be calculated.

   ZFS
   {HARTREEFOCK}

Zero-field splitting is the breaking of degeneracies of the ground
state that is not described by a standard non-relativistic
Hamiltonian. ZFS as calculated by ADF is that exhibited by molecules
whose ground state has spin S > 1/2 and no spatial degeneracy. This
type of ZFS has two contributions, second-order spin-orbit coupling
and direct electron spin-spin coupling. The calculation of ZFS with
DFT is described in Ref. [6] [7] [11] [12]. With the keyword ZFS the
spin-orbit coupling term is included. If one also wants to calculate
the direct electron spin-spin term, one has to include the key
HARTREEFOCK, which calculates the Coulomb and (Hartree-Fock like)
exchange contributions to the direct electron spin-spin term. In the
spin-orbit coupling term no Hartree-Fock like exchange contributions
are included.

   HARTREEFOCK

ZFS can be calculated in combination with LDA and GGAs but not hybrid
or meta-GGA functionals. In order to calculate ZFS, the "Relativity ->
level" should be set to "Scalar" and "Relativity -> Formalism" to
"ZORA" (these are the default values).

The ZFS keyword alone is sufficient to calculate the zero-field
splitting. Several optional additional keywords can also be included.
The complete list is:

   ZFS {PEDERSON|NEESE} {ANALYSIS|FULLANALYSIS}

"PEDERSON|NEESE"
   PEDERSON: The available approaches for calculating ZFS with DFT
   each differ subtly from the others. We believe that the method
   proposed by van Wüllen and coworkers [11] [12] is the most
   theoretically complete but it may be that for certain systems the
   other approaches are more accurate. The van Wüllen formulation is
   the default but if the PEDERSON keyword is included then the
   equation proposed by Pederson and Khanna [6] is used. NEESE: If the
   NEESE keyword is included then the equation for ZFS proposed by
   Neese [7] is used.

"ANALYSIS|FULLANALYSIS"
   ANALYSIS: Neese has presented some interesting analyses of ZFS [7].
   If the ANALYSIS keyword is invoked then the contributions to the
   ZFS are divided into terms from alpha-beta, alpha-alpha, beta-beta
   and beta-alpha one-electron excitations. FULLANALYSIS: The output
   requested by the ANALYSIS keyword is further extended to analyze
   each of the alpha-beta, alpha-alpha, beta-beta and beta-alpha
   contributions in terms of the individual one-electron excitations.

[1] G. Schreckenbach and T. Ziegler, *Calculation of the G-tensor of
    electron paramagnetic resonance spectroscopy using Gauge-Including
    Atomic Orbitals and Density Functional Theory*, Journal of
    Physical Chemistry A 101, 3388 (1997)

[2] S. Patchkovskii and T. Ziegler, *Calculation of the EPR g-Tensors
    of High-Spin Radicals with Density Functional Theory*, Journal of
    Physical Chemistry A 105, 5490 (2001)

[3] E. van Lenthe, A. van der Avoird and P.E.S. Wormer, *Density
    functional calculations of molecular hyperfine interactions in the
    zero order regular approximation for relativistic effects*,
    Journal of Chemical Physics 108, 4783 (1998)

[4] 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)

[5] J. Autschbach and B. Pritchard, *Calculation of molecular
    g-tensors using the zeroth-order regular approximation and density
    functional theory: expectation value versus linear response
    approaches*, Theoretical Chemistry Accounts 129, 453 (2011)

[6] M.R. Pederson, S.N. Khanna, *Magnetic anisotropy barrier for spin
    tunneling in* Mn_12 O_12 *molecules*, Physical Review B 60, 9566
    (1999)

[7] F. Neese, *Calculation of the zero-field splitting tensor on the
    basis of hybrid density functional and Hartree-Fock theory*,
    Journal of Chemical Physics 127, 164112 (2007)

[8] S. Patchkovskii and G. Schreckenbach in Calculation of NMR and EPR
    parameters, ISBN13: 9783527307791, M. Kaupp, M. Bühl, V.G. Malkin,
    Editors, (Wiley, Weinheim, 2004).

[9] S. Patchkovskii, R.S. Strong, C.J. Pickard and S. Un, *Gauge
    invariance of the spin-other-orbit contribution to the g-tensors
    of electron paramagnetic resonance*, Journal of Chemical Physics
    122, 214101 (2005)

[10] E. van Lenthe, A. van der Avoird and P.E.S. Wormer, *Density
     functional calculations of molecular g-tensors in the zero order
     regular approximation for relativistic effects*, Journal of
     Chemical Physics 107, 2488 (1997)

[11] C. van Wüllen, *Magnetic anisotropy from density functional
     calculations. Comparison of different approaches:* Mn_12 O_12
     *acetate as a test case*, Journal of Chemical Physics 130, 194109
     (2009)

[12] S. Schmitt, P. Jost, C. van Wüllen, *Zero-field splittings from
     density functional calculations: Analysis and improvement of
     known methods*, Journal of Chemical Physics 134, 194113 (2011)
