4.9 Unconventional SCF Calculations 4.9.5 Non-Hermitian SCF with complex basis functions 5 Density Functional Theory

4.9.6 Relativistic Effects

(May 21, 2025)

Relativistic effects play a major role in several physical and chemical phenomena, such as the properties of heavy elements and the proper characterization of the most inner energy levels probed in X-Ray espectroscopy experiments. Solving the four component Dirac equation, which describes both electrons and its anti-particles (positrons), is computationally expensive. Since most chemical proceses can be explained by solely taking the electronic wavefunction into account, several ways of effectively decoupling the electronic and positronic degrees of freedom have been proposed.

The one-electron exact two-component (1eX2C) hamiltonian ⁵⁹¹ Ilias M., Saue T.
J. Chem. Phys.
(2007), 126, pp. 064102. Link ^, ⁸²¹ Liu W., Peng D.
J. Chem. Phys.
(2009), 131, pp. 031104. Link ^, ¹¹⁴⁸ Saue T.
ChemPhysChem
(2011), 12, pp. 3077. Link ^, ⁷⁹⁵ Li Z., Xiao Y., Liu W.
J. Chem. Phys.
(2012), 137, pp. 154114. Link provides one route for achieving such decoupling. The method relies on solving the more tractable one electron four-component Dirac Hamiltonian in a restricted kinetic balance (RKB) ⁷²² Kutzelnigg W.
Int. J. Quantum Chem.
(1984), 25, pp. 107. Link form to obtain the decoupling unitary transformations that will be used to modify the one-electron matrix elements, such as the kinetic energy and nuclear-attraction, to account for relativistic effects. A key ingredient to the X2C transformation matrices is to compute

W_{{\rm SF},\mu\nu}=\langle\phi_{\mu}|\vec{p}\cdot\left(V\vec{p}\right)|\phi_{% \nu}\rangle

(4.70)

where $W_{{\rm SF},\mu\nu}$ is a matrix elements for the small-component of the full Dirac Hamiltonian, only computing the scalar-relativistic component. This is accomplished by noting that the the momentum operator is the generator of translations and its effects on a basis function can be captured by taking appropriate derivatives of such functions. It should be noted that, in order to properly capture the effects of the small components to the electronic wavefunction through X2C, all-electron decontracted basis sets are required. Full details of the finite difference X2C algorithm are provided in Ref. 285 Cunha L. A. et al.
J. Phys. Chem. Lett.
(2022), 13, pp. 3438. Link . An example on how to include scalar relativistic effects to model K-edge X-Ray spectroscopy can be found in Section 7.13.4.

REL_X2C

REL_X2C
       Enables X2C scalar relativistic calculation
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Perform a regular, non-relativistic SCF calculation 1 Perform a scalar relativistic X2C calculation
RECOMMENDATION:
       Set to 1 if a scalar relativistic X2C calculation is desired.

REL_X2C_FD_DISPLACEMENT

REL_X2C_FD_DISPLACEMENT
       Controls finite difference step for calulating W
TYPE:
       INTEGER
DEFAULT:
       100
OPTIONS:
       $n$ Set finite difference step to $n\times 10^{-6}$
RECOMMENDATION:
       None

4.9.6.1 Including one-electron Spin-Orbit Coupling Effects

In principle, the full solution to the 1eX2C Hamiltonian can be achieved, which will also include a spin-orbit coupling term. This, however, requires a spin-generealized and complex-valued solution. This can be seen by the additional matrix $W_{{\rm SOC},\mu\ nu}$ required to obtain a full $W$ matrix:

W_{{\rm SOC},\mu\nu}=i\sigma\cdot\langle\phi_{\mu}|\vec{p}\times\left(V\vec{p}% \right)|\phi_{\nu}\rangle

(4.71)

which is complex-valued and requires a cross product of different pauli matrices, resulting in a spin-generalized solution. Thus SOC effects may only be included when REL_X2C = 1 is called in cGHF routine - other calculations (e.g., RHF, UHF, ROHF) will all result in SF-1eX2C and will not include SOC effects. Setting the rem varaible REL_X2C = 1 in cGHF class automatically computes $W_{{\rm SOC},\mu\nu}$ unless REL_SF_X2C_CGHF is also set to 1.

REL_SF_X2C_CGHF

REL_SF_X2C_CGHF
       Enables X2C scalar relativistic calculation in a cGHF routine.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Perform a full relativistic X2C calculation with SOC effects 1 Perform a scalar relativistic X2C calculation
RECOMMENDATION:
       Set to 1 if a scalar relativistic X2C calculation in a cGHF class is desired. Must be used with REL_X2C = 1. Otherwise, non-relativistic calculation is performed without X2C, and this rem variable is ignored.

4.9.6.2 Screened-Nuclear Spin-Orbit Coupling for two-electron SOC effects

Although two-electron SOC terms in the X2C term can in principle be computed, the relevant matrix elements are often highly complicated and difficult to be evaluated. Instead, a popular approach to approximation the two-electron SOC effects in X2C scheme is to empirically scale the spin-orbit dependent terms of the full 1eX2C hamiltonian. The family of this methods is called screened-nuclear spin-orbit approach (SNSO), first suggested by Boettger ¹³⁹ Boettger J. C.
Phys. Rev. B
(2000), 62, pp. 7809. Link . Since then, improved form of the coefficient which accommodate diffuseness/tightness of the $p$ -shell Gaussian-Type-Orbitals (GTOs) as well as occupied and virtual MOs ³⁷⁴ Filatov M., Zou W., Cremer D.
J. Chem. Phys.
(2013), 139, pp. 014106. Link ^, ¹⁴⁵³ Yoshizawa T., W Zou, Cremer D.
J. Chem. Phys.
(2016), 145, pp. 184104. Link were suggested. Particularly, a new parametrization of the original SNSO coefficients based on the full four-component Dirac-Coulomb-Breit (DCB) Hamiltonian orbital splitting results was provided in Ref 351 Ehrman J. et al.
J. Chem. Theory Comput.
(2023), 19, pp. 5785. Link .

REL_SNSO

REL_SNSO
       Enables 1eX2C-SNSO calculation in a cGHF routine.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Perform a full relativistic X2C calculation with unadjusted SOC Hamiltonian 1 Perform a full relativistic X2C calculation with SOC Hamiltonian adjusted by original Boetter coefficients 2 Perform a full relativistic X2C calculation with SOC Hamiltonian adjusted by modified SNSO coefficients by Yoshizawa and co-workers 3 Perform a full relativistic X2C calculation with SOC Hamiltonian adjusted by DC-SNSO coefficients from Ehrman and co-workers 4 Perform a full relativistic X2C calculation with SOC Hamiltonian adjusted by universal DCB-SNSO coefficients from Ehrman and co-workers 5 Perform a full relativistic X2C calculation with SOC Hamiltonian adjusted by row-dependent DCB-SNSO coefficients from Ehrman and co-workers
RECOMMENDATION:
       Set to 1 for most general cases. Universal DCB-SNSO coefficients are the most recent contribution to the SNSO family and is reported to give accurate doublet splitting of $L_{2,3}$ -edges.

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4.9.6 Relativistic Effects

4.9.6.1 Including one-electron Spin-Orbit Coupling Effects

4.9.6.2 Screened-Nuclear Spin-Orbit Coupling for two-electron SOC effects