Relativistic effects
********************

Scalar relativistic effects using the ZORA formalism are included by
default in ADF calculations. Different relativistic options, such as
spin-orbit coupling, can be included using the "Relativity" key.

Note:

  Starting from the 2020 version, ZORA scalar relativistic effects are
  included by default. In 2019.3 and previous versions of ADF,
  relativistic effects were **not** included by default.

   Relativity
      Formalism [Pauli | ZORA | X2C | RA-X2C]
      Level [None | Scalar | Spin-Orbit]
      Potential [MAPA | SAPA]
      SpinOrbitMagnetization [NonCollinear | Collinear | CollinearX | CollinearY | CollinearZ]
   End

"Relativity"
   Type:
      Block

   Description:
      Options for relativistic effects.

   "Formalism"
      Type:
         Multiple Choice

      Default value:
         ZORA

      Options:
         [Pauli, ZORA, X2C, RA-X2C]

      Description:
         Note that if Level is None, no relativistic effects are taken
         into account, irrespective of the chosen formalism.
         Pauli stands for the Pauli Hamiltonian.
         ZORA means the Zero Order Regular Approximated Hamiltonian,
         recommended.                      X2C and RA-X2C both stand
         for an exact transformation of the 4-component Dirac equation
         to 2-components.                      X2C is the modified
         Dirac equation by Dyall.                      RA-X2C is the
         regular approach to the modified Dirac equation.

   "Level"
      Type:
         Multiple Choice

      Default value:
         Scalar

      Options:
         [None, Scalar, Spin-Orbit]

      GUI name:
         Relativity

      Description:
         None: No relativistic effects.                      Scalar:
         Scalar relativistic.                      This option comes
         at very little cost.                      Spin-Orbit: Spin-
         orbit coupled.                      This is the best level of
         theory, but it is (4-8 times) more expensive than a normal
         calculation. Spin-orbit effects are generally quite small,
         unless there are very heavy atoms in your system, especially
         with p valence electrons (like Pb).                      See
         also the SpinOrbitMagnetization subkey.

   "Potential"
      Type:
         Multiple Choice

      Default value:
         MAPA

      Options:
         [MAPA, SAPA]

      Description:
         Starting from ADF2017 instead of SAPA (the Sum of neutral
         Atomic potential Approximation) MAPA is used by default for
         ZORA. The MAPA (the Minimum of neutral Atomic potential
         Approximation) at a point is the minimum of the neutral
         Atomic potentials at that point. Advantage of MAPA over SAPA
         is that the gauge dependence of ZORA is reduced. The ZORA
         gauge dependency is small for almost all properties, except
         for the electron density very close to a heavy nucleus. The
         electron density very close to a heavy nucleus can be used
         for the interpretation of isomer shifts in Mossbauer
         spectroscopy.

   "SpinOrbitMagnetization"
      Type:
         Multiple Choice

      Default value:
         CollinearZ

      Options:
         [NonCollinear, Collinear, CollinearX, CollinearY, CollinearZ]

      Description:
         Relevant only for spin-orbit coupling and if unrestricted key
         has been activated.                      Most XC functionals
         have as one ingredient the spin polarization in case of
         unrestricted calculations. Normally the direction of the spin
         quantization axis is arbitrary and conveniently chosen to be
         the z-axis.                      However, in a spin-orbit
         calculation the direction matters, and it is arbitrary to put
         the z-component of the magnetization vector into the XC
         functional. There is also the exotic option to choose the
         quantization axis along the x or y axis.
         It is also possible to plug the size of the magnetization
         vector into the XC functional. This is called the non-
         collinear approach.                      - NonCollinear: the
         non-collinear method.                      - CollinearXYZ:
         use the x, y, or z component as spin polarization for the XC
         functional.                      - Collinear: the same as
         CollinearZ.

**MAPA and SAPA**

The SAPA method is described in Ref. [1] for the BAND program. The
same potential was used in the ADF program. However, starting from
ADF2017, instead of SAPA (the Sum of neutral Atomic Potential
Approximation) MAPA is used by default for ZORA. The MAPA (the Minimum
of neutral Atomic Potential Approximation) at a point is the minimum
of the neutral atomic potentials at that point. The advantage of MAPA
over SAPA is that the gauge dependence of ZORA is reduced. The ZORA
gauge dependence is small for almost all properties, except for the
electron density very close to a heavy nucleus. The electron density
very close to a heavy nucleus can be used for the interpretation of
isomer shifts in Mössbauer spectroscopy.

**Recommendations:**

   Relativity
      Level Scalar
      Formalism ZORA
      Potential MAPA
   End

or

   Relativity
      Level Spin-Orbit
      Formalism ZORA
      Potential MAPA
   End


Pauli
=====

Specification of the Pauli formalism means that the first-order
relativistic corrections (the Pauli Hamiltonian) will be used [2] [3]
[4] [5] [6] [7] [8] [9] [10] [11]. In a *scalar* relativistic run ADF
employs the single point-group symmetry and only the so-called
*scalar* relativistic corrections, Darwin and mass-velocity. The
treatment is not strictly first-order, but is *quasi*-relativistic, in
the sense that the first-order scalar relativistic Pauli Hamiltonian
is diagonalized in the space of the non-relativistic solutions, i.e.
in the non-relativistic basis set.

The quasi-relativistic approach improves results considerably over a
first-order treatment. There are, however, theoretical deficiencies
due to the singular behavior of the Pauli Hamiltonian at the nucleus.
This would become manifest in a complete basis set, but results are
reasonable with the normally employed basis sets. However, this aspect
implies that it is not recommended to apply this approach with an all-
electron basis set for the heavy atoms, and for very heavy elements
even a frozen core basis set often fails to give acceptable results.
The problems with the quasi-relativistic approach of the Pauli
Hamiltonian are discussed, for example, in Ref. [12].


ZORA
====

The ZORA approach gives generally better results than the Pauli
formalism. For all-electron calculations, and in fact also for
calculations on very heavy elements (actinides), the Pauli method is
absolutely unreliable. Therefore the ZORA method is the recommended
approach for relativistic calculations with ADF.

ZORA refers to the Zero-Order Regular Approximation [12] [17] [18]
[19] [20]. This formalism requires special basis sets, primarily to
include much steeper core-like functions; applying the ZORA method
with other, non-adapted basis sets, gives unreliable results. The ZORA
basis sets can be found in subdirectories under the
"$AMSHOME/atomicdata/ADF/ZORA" directory.

The ZORA formalism can also be used in geometry optimizations.
However, there is a slight mismatch between the energy expression and
the potential in the ZORA approach, which has the effect that the
geometry where the gradients are zero does not exactly coincide with
the point of lowest energy. The differences are very small, but not
completely negligible, on the order of 0.0001 angstrom.


X2C and RA-X2C
==============

X2C stands for an exact transformation of the 4-component Dirac
equation to 2 components [21] [22]. RA-X2C also stands for an exact
transformation of the 4-component Dirac equation to 2 components, in
this case using a regular approach to calculate the transformation
matrix. In practice, however, approximations are often made, and this
is also true for the implementation in ADF. In the X2C and RA-X2C
method implemented in ADF, first the 4-component Dirac equation for a
model potential (MAPA) of the molecule is calculated for the given ADF
basis set, using the modified Dirac equation by Dyall [13] for X2C, or
using the regular approach [14] to the modified Dirac equation for RA-
X2C.

With the method used in ADF, in the basis set limit X2C and RA-X2C
should lead to the same results for the model potential (MAPA).
However, since in practice always a finite basis set is used, even for
the model potential the results for X2C and RA-X2C will differ. Next
the (electronic) 4-component Dirac solutions are transformed exactly
to 2 components [15]. The transformation used is also used to
calculate an effective one-electron 2-component kinetic energy
operator in a basis set representation. This kinetic energy matrix is
then assumed to be constant in further ADF SCF calculation on the
molecule. The full Hamiltonian is approximated using the electron
density in the 2-component picture, the so-called Foldy-Wouthuysen
picture.

In a similar way, a spin-free (scalar relativistic) form of X2C and
RA-X2C is calculated. The spin-free form of X2C and RA-X2C will be
different even in the basis set limit, see Ref. [16].

X2C and RA-X2C in ADF can be used in single-point calculations only.
An all-electron basis set is needed. For bond energies,
polarizabilities, and TD-DFT excitation energies the Foldy-Wouthuysen
picture will be used. For the calculation of the EFG, ESR g-value, and
ESR A-tensor, the Dirac picture will be used. X2C and RA-X2C are not
implemented for frozen cores, optimization of nuclear coordinates,
frequencies, or NMR properties.


Spin-Orbit coupling
===================

The Spin-Orbit option uses double-group symmetry. The symmetry-adapted
orbitals are labeled by the quantum number *J* rather than *L* and any
references in input to subspecies, such as a specification of
occupation numbers, must refer to the double group labels.

Create runs must *not* use the Spin-Orbit formalism. The SFO analysis
of Molecular Orbitals for a Spin-Orbit calculation is only implemented
in the case of a scalar relativistic fragment file, which is the whole
molecule.

In a Spin-Orbit run each level can allocate 2 electrons (times the
dimension of the irreducible representation) as in a normal restricted
calculation. However, contrary to the normal case these two electrons
are not directly associated with spin-\alpha and spin-\beta, but
rather with the more general Kramers' symmetry. Using the unrestricted
feature in order to assign different numbers of electrons to alpha and
beta spin respectively cannot be applied as such. However, one can use
the unrestricted option in combination with the collinear or non-
collinear approximation. In that case one should use symmetry NOSYM,
and each level can allocate 1 electron.


Partial ZORA
------------

Expert option. The partial ZORA method is an analysis method to
suppress spin-orbit coupling locally, such that one can analyze the
effect of spin-orbit coupling into atomic contributions, see Ref.
[23].

Intended for use with atomic fragments. In spin-orbit (SO) coupled
ZORA calculations, the keyword can be used to suppress the SO
interaction locally at selected atomic fragment types (internal
ordering). There is a matching setup in the NMR program without the
need for additional input. For example, if in the ADF input the
following keyword is given:

   PartialZora atom 2 4

this will disable SO effects at atomic fragment types 2 and 4
(internal ordering of atomic fragments). This is achieved via a
modification of the atomic potentials that are used to assemble the
ZORA SO integrals, as explained in the publication cited above. When
used for NMR calculations, for example, this can be used to suppress
spin-orbit 'heavy atom' effects on light atom NMR shielding (HALA) for
purposes of analysis.

[1] P.H.T. Philipsen, E. van Lenthe, J.G. Snijders and E.J. Baerends,
    *Relativistic calculations on the adsorption of CO on the (111)
    surfaces of Ni, Pd, and Pt within the zeroth-order regular
    approximation*, Physical Review B 56, 13556 (1997)

[2] J.G. Snijders and E.J. Baerends, *A perturbation theory approach
    to relativistic calculations. I. Atoms*, Molecular Physics 36,
    1789 (1978)

[3] J.G. Snijders, E.J. Baerends and P. Ros, *A perturbation theory
    approach to relativistic calculations. II. Molecules*, Molecular
    Physics 38, 1909 (1979)

[4] T. Ziegler, J.G. Snijders and E.J. Baerends, *Relativistic effects
    on bonding*, Journal of Chemical Physics 74, 1271 (1981)

[5] R.L. DeKock, E.J. Baerends, P.M. Boerrigter and J.G. Snijders, *On
    the nature of the first excited states of the uranyl ion*,
    Chemical Physics Letters 105, 308 (1984)

[6] R.L. DeKock, E.J. Baerends, P.M. Boerrigter and R. Hengelmolen,
    *Electronic structure and bonding of* Hg(CH_3 )_2 , Hg(CN)_2 ,
    Hg(CH_3 )(CN), Hg(CCCH_3 )_2 , *and* Au(PMe)_3 (CH_3 ), Journal of
    the American Chemical Society 106, 3387 (1984)

[7] P.M. Boerrigter, *Spectroscopy and bonding of heavy element
    compounds*, 1987, Vrije Universiteit.

[8] P.M. Boerrigter, M.A. Buijse and J.G. Snijders, *Spin-Orbit
    interaction in the excited states of the dihalogen ions* F_2 ^+ ,
    Cl_2 ^+ and Br_2 ^+, Chemical Physics 111, 47 (1987)

[9] P.M. Boerrigter, E.J. Baerends and J.G. Snijders, *A relativistic
    LCAO Hartree-Fock-Slater investigation of the electronic structure
    of the actinocenes* M(COT)_2 , *M=Th, Pa, U, Np and Pu*, Chemical
    Physics 122, 357 (1988)

[10] T. Ziegler, V. Tschinke, E.J. Baerends, J.G. Snijders and W.
     Ravenek, *Calculation of bond energies in compounds of heavy
     elements by a quasi-relativistic approach*, Journal of Physical
     Chemistry 93, 3050 (1989)

[11] J. Li, G. Schreckenbach and T. Ziegler, *A Reassessment of the
     First Metal-Carbonyl Dissociation Energy in* M(CO)_4 (M = Ni, Pd,
     Pt), M(CO)_5 (M = Fe, Ru, Os), and M(CO)_6 (M = Cr, Mo, W) *by a
     Quasirelativistic Density Functional Method*, Journal of the
     American Chemical Society 117, 486 (1995)

[12] E. van Lenthe, A.E. Ehlers and E.J. Baerends, *Geometry
     optimization in the Zero Order Regular Approximation for
     relativistic effects*, Journal of Chemical Physics 110, 8943
     (1999)

[13] K.G. Dyall, *An exact separation of the spin-free and spin-
     dependent terms of the Dirac-Coulomb-Breit Hamiltonian*, Journal
     of Chemical Physics 100, 2118 (1994)

[14] A.J. Sadlej and J.G. Snijders, *Spin separation in the regular
     Hamiltonian approach to solutions of the Dirac equation*,
     Chemical Physics Letters 229, 435 (1994)

[15] E. van Lenthe, E.J. Baerends, and J.G. Snijders, *Construction of
     the Foldy-Wouthuysen transformation and solution of the Dirac
     equation using large components only*, Journal of Chemical
     Physics 105, 2373 (1996)

[16] L. Visscher and E. van Lenthe, *On the distinction between scalar
     and spin-orbit relativistic effects*, Chemical Physics Letters
     306, 357 (1999)

[17] E. van Lenthe, E.J. Baerends and J.G. Snijders, *Relativistic
     regular two-component Hamiltonians*, Journal of Chemical Physics
     99, 4597 (1993)

[18] E. van Lenthe, E.J. Baerends and J.G. Snijders, *Relativistic
     total energy using regular approximations*, Journal of Chemical
     Physics 101, 9783 (1994)

[19] E. van Lenthe, J.G. Snijders and E.J. Baerends, *The zero-order
     regular approximation for relativistic effects: The effect of
     spin-orbit coupling in closed shell molecules*, Journal of
     Chemical Physics 105, 6505 (1996)

[20] E. van Lenthe, R. van Leeuwen, E.J. Baerends and J.G. Snijders,
     *Relativistic regular two-component Hamiltonians*, International
     Journal of Quantum Chemistry 57, 281 (1996)

[21] K.G. Dyall, *Interfacing relativistic and nonrelativistic
     methods. I. Normalized elimination of the small component in the
     modified Dirac equation*, Journal of Chemical Physics 106, 9618
     (1997)

[22] W. Kutzelnigg and W. Liu, *Quasirelativistic theory equivalent to
     fully relativistic theory*, Journal of Chemical Physics 123,
     241102 (2005)

[23] J. Autschbach, K. Sutter, L.A. Truflandier, E. Brendler, J.
     Wagler, *Atomic Contributions from Spin-Orbit Coupling to* ^29Si
     *NMR Chemical Shifts in Metallasilatrane Complexes*, Chemistry -
     A European Journal 18, 12533 (2012)
