Relativistic allelectron approximations¶
Overview¶
All methods which include treatment of relativistic effects are ultimately based on the Dirac equation, which has a four component wave function. The solutions to the Dirac equation describe both positrons (the “negative energy” states) and electrons (the “positive energy” states), as well as both spin orientations, hence the four components. The wave function may be broken down into twocomponent functions traditionally known as the large and small components; these may further be broken down into the spin components.
The implementation of approximate allelectron relativistic methods in quantum chemical codes requires the removal of the negative energy states and the factoring out of the spinfree terms. Both of these may be achieved using a transformation of the Dirac Hamiltonian known in general as a FoldyWouthuysen (FW) transformation. Unfortunately this transformation cannot be represented in closed form for a general potential, and must be approximated. One popular approach is that originally formulated by Douglas and Kroll^{1} and developed by Hess^{2}^{3}. This approach decouples the positive and negative energy parts to second order in the external potential (and also fourth order in the fine structure constant, α). Other approaches include the Zeroth Order Regular Approximation (ZORA)^{4}^{5}^{6}^{7} and modification of the Dirac equation by Dyall^{8}, and involves an exact FW transformation on the atomic basis set level^{9}^{10}.
Since these approximations only modify the integrals, they can in principle be used at all levels of theory. At present the DouglasKroll and ZORA implementations can be used at all levels of theory whereas Dyall’s approach is currently available at the HartreeFock level. The derivatives have been implemented, allowing both methods to be used in geometry optimizations and frequency calculations.
RELATIVISTIC directive¶
The RELATIVISTIC
directive provides input for the implemented
relativistic approximations and is a compound directive that encloses
additional directives specific to the
approximations:
RELATIVISTIC
[DOUGLASKROLL [<string (ONOFF) default ON> \
<string (FPPDKHDKFULLDK3DK3FULL) default DKH>] 
ZORA [ (ON  OFF) default ON ] 
DYALLMODDIRAC [ (ON  OFF) default ON ] 
[ (NESC1E  NESC2E) default NESC1E ] ] 
X2C [ (ON  OFF) default ON ]
[CLIGHT <real clight default 137.0359895>]
END
Only one of the methods may be chosen at a time. If both methods are found to be on in the input block, NWChem will stop and print an error message. There is one general option for both methods, the definition of the speed of light in atomic units:
CLIGHT <real clight default 137.0359895>
The following sections describe the optional subdirectives that can be
specified within the RELATIVISTIC
block.
DouglasKroll approximation¶
The spinfree and spinorbit oneelectron DouglasKroll approximation have been implemented. The use of relativistic effects from this DouglasKroll approximation can be invoked by specifying:
DOUGLASKROLL [<string (ONOFF) default ON> \
<string (FPPDKHDKFULLDK3DK3FULL) default DKH>]
The ONOFF string is used to turn on or off the DouglasKroll
approximation. By default, if the DOUGLASKROLL
keyword is found, the
approximation will be used in the calculation. If the user wishes to
calculate a nonrelativistic quantity after turning on DouglasKroll,
the user will need to define a new RELATIVISTIC
block and turn the
approximation OFF
. The user could also simply put a blank RELATIVISTIC
block in the input file and all options will be turned off.
The FPP
is the approximation based on freeparticle projection
operators^{11} whereas the DKH
and DKFULL
approximations are based on
externalfield projection operators^{12}. The latter two are
considerably better approximations than the former. DKH
is the
DouglasKrollHess approach and is the approach that is generally
implemented in quantum chemistry codes. DKFULL
includes certain
crossproduct integral terms ignored in the DKH
approach (see for
example Häberlen and Rösch^{13}). The thirdorder DouglasKroll
approximation has been implemented by T. Nakajima and K.
Hirao^{14}^{15}. This approximation can be called using DK3
(DK3
without crossproduct integral terms) or DK3FULL
(DK3 with crossproduct
integral terms).
The contracted basis sets used in the calculations should reflect the relativistic effects, i.e. one should use contracted basis sets which were generated using the DouglasKroll Hamiltonian. Basis sets that were contracted using the nonrelativistic (Schödinger) Hamiltonian WILL PRODUCE ERRONEOUS RESULTS for elements beyond the first row. See appendix A for available basis sets and their naming convention.
NOTE: we suggest that spherical basis sets are used in the calculation. The use of high quality cartesian basis sets can lead to numerical inaccuracies.
In order to compute the integrals needed for the DouglasKroll
approximation the implementation makes use of a fitting basis set (see
literature given above for details). The current code will create this
fitting basis set based on the given ao basis
by simply uncontracting
that basis. This again is what is commonly implemented in quantum
chemistry codes that include the DouglasKroll method. Additional
flexibility is available to the user by explicitly specifying a
DouglasKroll fitting basis set. This basis set must be named
DK basis
(see Basis Sets).
Zeroth Order regular approximation (ZORA)¶
The spinfree and spinorbit oneelectron zerothorder regular approximation (ZORA) have been implemented. ZORA can be accessed only via the DFT and SODFT modules. The use of relativistic effects with ZORA can be invoked by specifying:
ZORA [<string (ONOFF) >
The ON
OFF
string is used to turn on or off ZORA. No default is present, therefore
ZORA
keyword need to be followed by ON
in order for the approximation to be used in the
calculation. If the user wishes to calculate a nonrelativistic quantity
after turning on ZORA, the user will need to define a new RELATIVISTIC
block and turn the approximation OFF. The user can also simply put a
blank RELATIVISTIC
block in the input file and all options will be
turned off.
To increase the accuracy of ZORA calculations, the following settings may be used in the relativistic block
relativistic
zora on
zora:cutoff 1d30
end
To invoke the relativistic ZORA model potential approach due to van Wullen (references ^{16} and ^{17}).
For model potentials constructed from 4component densities:
relativistic
zora on
zora:cutoff 1d30
modelpotential 1
end
For model potentials constructed from 2component densities:
relativistic
zora on
zora:cutoff 1d30
modelpotential 2
end
Both approaches are comparable in accuracy and depends on the system.
Dyall’s Modified Dirac Hamitonian approximation¶
The approximate methods described in this section are all based on Dyall’s modified Dirac Hamiltonian. This Hamiltonian is entirely equivalent to the original Dirac Hamiltonian, and its solutions have the same properties. The modification is achieved by a transformation on the small component, extracting out σ⋅p/2mc. This gives the modified small component the same symmetry as the large component, and in fact it differs from the large component only at order α^{2}. The advantage of the modification is that the operators now resemble the operators of the BreitPauli Hamiltonian, and can be classified in a similar fashion into spinfree, spinorbit and spinspin terms. It is the spinfree terms which have been implemented in NWChem, with a number of further approximations.
The first is that the negative energy states are removed by a normalized elimination of the small component (NESC), which is equivalent to an exact FoldyWouthuysen (EFW) transformation. The number of components in the wave function is thereby effectively reduced from 4 to 2. NESC on its own does not provide any advantages, and in fact complicates things because the transformation is energydependent. The second approximation therefore performs the elimination on an atombyatom basis, which is equivalent to neglecting blocks which couple different atoms in the EFW transformation. The advantage of this approximation is that all the energy dependence can be included in the contraction coefficients of the basis set. The tests which have been done show that this approximation gives results well within chemical accuracy. The third approximation neglects the commutator of the EFW transformation with the twoelectron Coulomb interaction, so that the only corrections that need to be made are in the oneelectron integrals. This is the equivalent of the DouglasKroll(Hess) approximation as it is usually applied.
The use of these approximations can be invoked with the use of the
DYALLMODDIRAC
directive in the RELATIVISTIC
directive block. The
syntax is as follows.
DYALLMODDIRAC [ (ON  OFF) default ON ]
[ (NESC1E  NESC2E) default NESC1E ]
The ON
OFF
string is used to turn on or off the Dyall’s modified Dirac
approximation. By default, if the DYALLMODDIRAC
keyword is found, the
approximation will be used in the calculation. If the user wishes to
calculate a nonrelativistic quantity after turning on Dyall’s modified
Dirac, the user will need to define a new RELATIVISTIC
block and turn
the approximation OFF
. The user could also simply put a blank
RELATIVISTIC
block in the input file and all options will be turned off.
Both one and twoelectron approximations are available NESC1E

NESC2E
, and both have analytic gradients. The oneelectron approximation
is the default. The twoelectron approximation specified by NESC2E
has
some sub options which are placed on the same logical line as the
DYALLMODDIRAC
directive, with the following
syntax:
NESC2E [ (SS1CENT [ (ON  OFF) default ON ]  SSALL) default SSALL ]
[ (SSSS [ (ON  OFF) default ON ]  NOSSSS) default SSSS ]
The first suboption gives the capability to limit the twoelectron corrections to those in which the small components in any density must be on the same center. This reduces the (LLSS) contributions to at most threecenter integrals and the (SSSS) contributions to two centers. For a case with only one relativistic atom this option is redundant. The second controls the inclusion of the (SSSS) integrals which are of order α^{4}. For light atoms they may safely be neglected, but for heavy atoms they should be included.
In addition to the selection of this keyword in the RELATIVISTIC
directive block, it is necessary to supply basis sets in addition to the
ao basis
. For the oneelectron approximation, three basis sets are
needed: the atomic FW basis set, the large component basis set and the
small component basis set. The atomic FW basis set should be included in
the ao basis
. The large and small components should similarly be
incorporated in basis sets named large component
and small component
,
respectively. For the twoelectron approximation, only two basis sets
are needed. These are the large component and the small component. The
large component should be included in the ao basis
and the small
component is specified separately as small component
, as for the
oneelectron approximation. This means that the two approximations can
not be run correctly without changing the ao basis
, and it is up to the
user to ensure that the basis sets are correctly specified.
There is one further requirement in the specification of the basis sets.
In the ao basis, it is necessary to add the rel
keyword either to the
basis directive or the library tag line (See below for examples). The
former marks the basis functions specified by the tag as relativistic,
the latter marks the whole basis as relativistic. The marking is
actually done at the unique shell level, so that it is possible not only
to have relativistic and nonrelativistic atoms, it is also possible to
have relativistic and nonrelativistic shells on a given atom. This would
be useful, for example, for diffuse functions or for high angular
momentum correlating functions, where the influence of relativity was
small. The marking of shells as relativistic is necessary to set up a
mapping between the ao basis and the large and/or small component basis
sets. For the oneelectron approximation the large and small component
basis sets MUST be of the same size and construction, i.e. differing
only in the contraction coefficients.
It should also be noted that the relativistic code will NOT work with basis sets that contain sp shells, nor will it work with ECPs. Both of these are tested and flagged as an error.
Examples for DYALLMODDIRAC¶
Some examples follow. The first example sets up the data for relativistic calculations on water with the oneelectron approximation and the twoelectron approximation, using the library basis sets.
start h2odmd
geometry units bohr
symmetry c2v
O 0.000000000 0.000000000 0.009000000
H 1.515260000 0.000000000 1.058900000
H 1.515260000 0.000000000 1.058900000
end
basis "fw" rel
oxygen library ccpvdz_pt_sf_fw
hydrogen library ccpvdz_pt_sf_fw
end
basis "large"
oxygen library ccpvdz_pt_sf_lc
hydrogen library ccpvdz_pt_sf_lc
end
basis "large2" rel
oxygen library ccpvdz_pt_sf_lc
hydrogen library ccpvdz_pt_sf_lc
end
basis "small"
oxygen library ccpvdz_pt_sf_sc
hydrogen library ccpvdz_pt_sf_sc
end
set "ao basis" fw
set "large component" large
set "small component" small
relativistic
dyallmoddirac
end
task scf
set "ao basis" large2
unset "large component"
set "small component" small
relativistic
dyallmoddirac nesc2e
end
task scf
The second example has oxygen as a relativistic atom and hydrogen nonrelativistic.
start h2odmd2
geometry units bohr
symmetry c2v
O 0.000000000 0.000000000 0.009000000
H 1.515260000 0.000000000 1.058900000
H 1.515260000 0.000000000 1.058900000
end
basis "ao basis"
oxygen library ccpvdz_pt_sf_fw rel
hydrogen library ccpvdz
end
basis "large component"
oxygen library ccpvdz_pt_sf_lc
end
basis "small component"
oxygen library ccpvdz_pt_sf_sc
end
relativistic
dyallmoddirac
end
task scf
X2C: exact twocomponent relativistic Hamiltonian¶
The exact twocomponent Hamiltonian^{18}^{19} has been implemented in NWChem^{20}^{21}^{22}.
X2C [<string (ONOFF) default ON>
The ON
OFF
string is used to turn on or off X2C. By default, if the
X2C
keyword is found, the approximation will be used in the
calculation. If the user wishes to calculate a nonrelativistic quantity
after turning on X2C, the user will need to define a new RELATIVISTIC
block and turn the approximation OFF. The user can also simply put a
blank RELATIVISTIC
block in the input file and all options will be
turned off.
To increase the accuracy of X2C calculations, the following settings may be used in the relativistic block
relativistic
x2c on
x2c:cutoff 1d15
end
References¶

Douglas, M.; Kroll, N.M. (1974). “Quantum electrodynamical corrections to the fine structure of helium”. Annals of Physics 82: 89155. DOI:10.1016/00034916(74)903339. ↩

Hess, B.A. (1985). “Applicability of the nopair equation with freeparticle projection operators to atomic and molecular structure calculations”. Physical Review A 32: 756763. DOI:10.1103/PhysRevA.32.756. ↩

Hess, B.A. (1986). “Relativistic electronicstructure calculations employing a twocomponent nopair formalism with externalfield projection operators”. Physical Review A 33: 37423748. DOI:10.1103/PhysRevA.33.3742. ↩

Chang, C; Pelissier, M; Durand, M (1986). “Regular TwoComponent PauliLike Effective Hamiltonians in Dirac Theory”. Physica Scripta 34: 394. DOI:10.1088/00318949/34/5/007. ↩

van Lenthe, E (1996). “The ZORA Equation” (in English). ↩

Faas, S.; Snijders, J.G.; van Lenthe, J.H.; van Lenthe, E.; Baerends, E.J. (1995). “The ZORA formalism applied to the DiracFock equation”. Chemical Physics Letters 246: 632640. DOI:10.1016/00092614(95)011560. ↩

Nichols, P.; Govind, N.; Bylaska, E.J.; de Jong, W.A. (2009). “Gaussian Basis Set and Planewave Relativistic SpinOrbit Methods in NWChem”. Journal of Chemical Theory and Computation 5: 491499. DOI:10.1021/ct8002892. ↩

Dyall, K.G. (1994). “An exact separation of the spinfree and spindependent terms of the Dirac–Coulomb–Breit Hamiltonian”. The Journal of Chemical Physics 100: 21182127. DOI:10.1063/1.466508. ↩

Dyall, K.G. (1997). “Interfacing relativistic and nonrelativistic methods. I. Normalized elimination of the small component in the modified Dirac equation”. The Journal of Chemical Physics 106: 96189626. DOI:10.1063/1.473860. ↩

Dyall, K.G.; Enevoldsen, T. (1999). “Interfacing relativistic and nonrelativistic methods. III. Atomic 4spinor expansions and integral approximations”. The Journal of Chemical Physics 111: 1000010007. DOI:10.1063/1.480353. ↩

Hess, B.A. (1985). “Applicability of the nopair equation with freeparticle projection operators to atomic and molecular structure calculations”. Physical Review A 32: 756763. DOI:10.1103/PhysRevA.32.756. ↩

Hess, B.A. (1986). “Relativistic electronicstructure calculations employing a twocomponent nopair formalism with externalfield projection operators”. Physical Review A 33: 37423748. DOI:10.1103/PhysRevA.33.3742. ↩

Haeberlen, O.D.; Roesch, N. (1992). “A scalarrelativistic extension of the linear combination of Gaussiantype orbitals local density functional method: application to AuH, AuCl and Au2”. Chemical Physics Letters 199: 491496. DOI:10.1016/00092614(92)87033L. ↩

Nakajima, T.; Hirao, K. (2000). “Numerical illustration of thirdorder DouglasKroll method: atomic and molecular properties of superheavy element 112”. Chemical Physics Letters 329: 511516. DOI:10.1016/S00092614(00)010356. ↩

Nakajima, T.; Hirao, K. (2000). “The higherorder Douglas–Kroll transformation”. The Journal of Chemical Physics 113: 77867789. DOI:10.1063/1.1316037. ↩

van Wullen, C. (1998). “Molecular density functional calculations in the regular relativistic approximation: Method, application to coinage metal diatomics, hydrides, fluorides and chlorides, and comparison with firstorder relativistic calculations”. The Journal of Chemical Physics 109: 392399 DOI:10.1063/1.476576 ↩

van Wullen, C.; Michauk, C. (2005). “Accurate and efficient treatment of twoelectron contributions in quasirelativistic highorder DouglasKroll densityfunctional calculations”. The Journal of Chemical Physics 123, 204113 DOI:10.1063/1.2133731 ↩

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Saue, T. (2011). “Relativistic Hamiltonians for Chemistry: A Primer”. ChemPhysChem, 12, 3077–3094 DOI:10.1002/cphc.201100682 ↩

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Autschbach, J. (2021). Quantum Theory for Chemical Applications:From Basic Concepts to Advanced Topics, Oxford, University Press, Chapter 24 DOI:10.1093/oso/9780190920807.001.0001 ↩