# Relativistic all-electron 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 two-component functions traditionally known as the large and small components; these may further be broken down into the spin components.

The implementation of approximate all-electron relativistic methods in quantum chemical codes requires the removal of the negative energy states and the factoring out of the spin-free terms. Both of these may be achieved using a transformation of the Dirac Hamiltonian known in general as a Foldy-Wouthuysen (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 Kroll1 and developed by Hess23. 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)4567 and modification of the Dirac equation by Dyall8, and involves an exact FW transformation on the atomic basis set level910.

Since these approximations only modify the integrals, they can in principle be used at all levels of theory. At present the Douglas-Kroll and ZORA implementations can be used at all levels of theory whereas Dyall’s approach is currently available at the Hartree-Fock 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
[DOUGLAS-KROLL [<string (ON||OFF) default ON> \
<string (FPP||DKH||DKFULL||DK3||DK3FULL) default DKH>]  ||
ZORA [ (ON || OFF) default ON ] ||
DYALL-MOD-DIRAC [ (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 sub-directives that can be specified within the RELATIVISTIC block.

### Douglas-Kroll approximation¶

The spin-free and spin-orbit one-electron Douglas-Kroll approximation have been implemented. The use of relativistic effects from this Douglas-Kroll approximation can be invoked by specifying:

 DOUGLAS-KROLL [<string (ON||OFF) default ON> \
<string (FPP||DKH||DKFULL|DK3|DK3FULL) default DKH>]


The ON|OFF string is used to turn on or off the Douglas-Kroll approximation. By default, if the DOUGLAS-KROLL keyword is found, the approximation will be used in the calculation. If the user wishes to calculate a non-relativistic quantity after turning on Douglas-Kroll, 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 free-particle projection operators11 whereas the DKH and DKFULL approximations are based on external-field projection operators12. The latter two are considerably better approximations than the former. DKH is the Douglas-Kroll-Hess approach and is the approach that is generally implemented in quantum chemistry codes. DKFULL includes certain cross-product integral terms ignored in the DKH approach (see for example Häberlen and Rösch13). The third-order Douglas-Kroll approximation has been implemented by T. Nakajima and K. Hirao1415. This approximation can be called using DK3 (DK3 without cross-product integral terms) or DK3FULL (DK3 with cross-product 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 Douglas-Kroll Hamiltonian. Basis sets that were contracted using the non-relativistic (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 Douglas-Kroll 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 Douglas-Kroll method. Additional flexibility is available to the user by explicitly specifying a Douglas-Kroll fitting basis set. This basis set must be named D-K basis (see Basis Sets).

### Zeroth Order regular approximation (ZORA)¶

The spin-free and spin-orbit one-electron zeroth-order regular approximation (ZORA) have been implemented. ZORA can be accessed only via the DFT and SO-DFT modules. The use of relativistic effects with ZORA can be invoked by specifying:

 ZORA [<string (ON||OFF) >


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 non-relativistic 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 1d-30
end


To invoke the relativistic ZORA model potential approach due to van Wullen (references 16 and 17).

For model potentials constructed from 4-component densities:

 relativistic
zora on
zora:cutoff 1d-30
modelpotential  1
end


For model potentials constructed from 2-component densities:

 relativistic
zora on
zora:cutoff 1d-30
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 α2. The advantage of the modification is that the operators now resemble the operators of the Breit-Pauli Hamiltonian, and can be classified in a similar fashion into spin-free, spin-orbit and spin-spin terms. It is the spin-free 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 Foldy-Wouthuysen (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 energy-dependent. The second approximation therefore performs the elimination on an atom-by-atom 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 two-electron Coulomb interaction, so that the only corrections that need to be made are in the one-electron integrals. This is the equivalent of the Douglas-Kroll(-Hess) approximation as it is usually applied.

The use of these approximations can be invoked with the use of the DYALL-MOD-DIRAC directive in the RELATIVISTIC directive block. The syntax is as follows.

 DYALL-MOD-DIRAC [ (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 DYALL-MOD-DIRAC keyword is found, the approximation will be used in the calculation. If the user wishes to calculate a non-relativistic 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 two-electron approximations are available NESC1E || NESC2E, and both have analytic gradients. The one-electron approximation is the default. The two-electron approximation specified by NESC2E has some sub options which are placed on the same logical line as the DYALL-MOD-DIRAC directive, with the following syntax:

 NESC2E [ (SS1CENT [ (ON || OFF) default ON ] || SSALL) default SSALL ]
[ (SSSS [ (ON || OFF) default ON ] || NOSSSS) default SSSS ]


The first sub-option gives the capability to limit the two-electron corrections to those in which the small components in any density must be on the same center. This reduces the (LL|SS) contributions to at most three-center integrals and the (SS|SS) contributions to two centers. For a case with only one relativistic atom this option is redundant. The second controls the inclusion of the (SS|SS) integrals which are of order α4. 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 one-electron 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 two-electron 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 one-electron 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 one-electron 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 DYALL-MOD-DIRAC¶

Some examples follow. The first example sets up the data for relativistic calculations on water with the one-electron approximation and the two-electron approximation, using the library basis sets.

 start h2o-dmd
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 cc-pvdz_pt_sf_fw
hydrogen library cc-pvdz_pt_sf_fw
end
basis "large"
oxygen library cc-pvdz_pt_sf_lc
hydrogen library cc-pvdz_pt_sf_lc
end
basis "large2" rel
oxygen library cc-pvdz_pt_sf_lc
hydrogen library cc-pvdz_pt_sf_lc
end
basis "small"
oxygen library cc-pvdz_pt_sf_sc
hydrogen library cc-pvdz_pt_sf_sc
end
set "ao basis" fw
set "large component" large
set "small component" small
relativistic
dyall-mod-dirac
end
set "ao basis" large2
unset "large component"
set "small component" small
relativistic
dyall-mod-dirac nesc2e
end


The second example has oxygen as a relativistic atom and hydrogen nonrelativistic.

 start h2o-dmd2
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 cc-pvdz_pt_sf_fw rel
hydrogen library cc-pvdz
end
basis "large component"
oxygen library cc-pvdz_pt_sf_lc
end
basis "small component"
oxygen library cc-pvdz_pt_sf_sc
end
relativistic
dyall-mod-dirac
end


### X2C: exact two-component relativistic Hamiltonian¶

The exact two-component Hamiltonian1819 has been implemented in NWChem202122.

 X2C [<string (ON||OFF) 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 non-relativistic 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 1d-15
end


## References¶

1. Douglas, M.; Kroll, N.M. (1974). “Quantum electrodynamical corrections to the fine structure of helium”. Annals of Physics 82: 89-155. DOI:10.1016/0003-4916(74)90333-9

2. Hess, B.A. (1985). “Applicability of the no-pair equation with free-particle projection operators to atomic and molecular structure calculations”. Physical Review A 32: 756-763. DOI:10.1103/PhysRevA.32.756

3. Hess, B.A. (1986). “Relativistic electronic-structure calculations employing a two-component no-pair formalism with external-field projection operators”. Physical Review A 33: 3742-3748. DOI:10.1103/PhysRevA.33.3742

4. Chang, C; Pelissier, M; Durand, M (1986). “Regular Two-Component Pauli-Like Effective Hamiltonians in Dirac Theory”. Physica Scripta 34: 394. DOI:10.1088/0031-8949/34/5/007

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

6. Faas, S.; Snijders, J.G.; van Lenthe, J.H.; van Lenthe, E.; Baerends, E.J. (1995). “The ZORA formalism applied to the Dirac-Fock equation”. Chemical Physics Letters 246: 632-640. DOI:10.1016/0009-2614(95)01156-0

7. Nichols, P.; Govind, N.; Bylaska, E.J.; de Jong, W.A. (2009). “Gaussian Basis Set and Planewave Relativistic Spin-Orbit Methods in NWChem”. Journal of Chemical Theory and Computation 5: 491-499. DOI:10.1021/ct8002892

8. Dyall, K.G. (1994). “An exact separation of the spin-free and spin-dependent terms of the Dirac–Coulomb–Breit Hamiltonian”. The Journal of Chemical Physics 100: 2118-2127. DOI:10.1063/1.466508

9. 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: 9618-9626. DOI:10.1063/1.473860

10. Dyall, K.G.; Enevoldsen, T. (1999). “Interfacing relativistic and nonrelativistic methods. III. Atomic 4-spinor expansions and integral approximations”. The Journal of Chemical Physics 111: 10000-10007. DOI:10.1063/1.480353

11. Hess, B.A. (1985). “Applicability of the no-pair equation with free-particle projection operators to atomic and molecular structure calculations”. Physical Review A 32: 756-763. DOI:10.1103/PhysRevA.32.756

12. Hess, B.A. (1986). “Relativistic electronic-structure calculations employing a two-component no-pair formalism with external-field projection operators”. Physical Review A 33: 3742-3748. DOI:10.1103/PhysRevA.33.3742

13. Haeberlen, O.D.; Roesch, N. (1992). “A scalar-relativistic extension of the linear combination of Gaussian-type orbitals local density functional method: application to AuH, AuCl and Au2”. Chemical Physics Letters 199: 491-496. DOI:10.1016/0009-2614(92)87033-L

14. Nakajima, T.; Hirao, K. (2000). “Numerical illustration of third-order Douglas-Kroll method: atomic and molecular properties of superheavy element 112”. Chemical Physics Letters 329: 511-516. DOI:10.1016/S0009-2614(00)01035-6

15. Nakajima, T.; Hirao, K. (2000). “The higher-order Douglas–Kroll transformation”. The Journal of Chemical Physics 113: 7786-7789. DOI:10.1063/1.1316037

16. 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 first-order relativistic calculations”. The Journal of Chemical Physics 109: 392-399 DOI:10.1063/1.476576

17. van Wullen, C.; Michauk, C. (2005). “Accurate and efficient treatment of two-electron contributions in quasirelativistic high-order Douglas-Kroll density-functional calculations”. The Journal of Chemical Physics 123, 204113 DOI:10.1063/1.2133731

18. Liu, W.; Peng, D. (2009). J. Chem. Phys. 2009, 131, 031104 DOI:10.1063/1.3159445

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

20. Autschbach, J.; Peng, D; Reiher, M. (2012). J. Chem. Theory Comput. 2012, 8, 4239–4248 DOI:10.1021/ct300623j

21. Peng, D.; Reiher, M. (2012). Theor. Chem. Acc. 131, 1081 DOI:10.1007/s00214-011-1081-y

22. 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