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# Relativistic Effects

Relativistic quantum chemistry

K.G. Dyall and K. Faegri Jr., Introduction to Relativistic Quantum Chemistry (Oxford Univ. Press, 2007)
M. Reiher and A. Wolf, Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science (Wiley, 2014)

Reviews

W. Kutzelnigg, Perturbation Theory Based on Quasi-Relativistic Hamiltonians Chapter 1, in Relativistic Electronic Structure Theory Part I. Fundamentals edited by P. Schwerdtfeger (Elsevier, Amsterdam, 2002) p. 664
L. Cheng, S. Stopkowicz, and J. Gauss, Analytic energy derivatives in relativistic quantum chemistry, Int. J. Quant. Chem. 114, 1108-1127 (2014); review on analytic derivatives in relativistic quantum chemsitry

Single-point energy calculations

perturbation theory based on mass-velocity and Darwin terms (MVD1 and MVD2)

R.D. Cowan and D.C. Griffin, Approximate relativistic corrections to atomic radial wave functions, J. Opt. Soc. Am. 66, 1010 (1976)

W. Klopper, Simple recipe for implementing computation of first‐order relativistic corrections to electron correlation energies in framework of direct perturbation theory, J. Comp. Chem. 18, 20-27 (1997)

Direct perturbation theory (DPT)

W. Kutzelnigg, E. Ottschofski, and R. Franke, Relativistic Hartree–Fock by means of stationary direct perturbation theory. I. General theory, J. Chem. Phys. 102, 1740 (1995); general theory at HF level
E. Ottschofski and W. Kutzelnigg, Relativistic Hartree–Fock by means of stationary direct perturbation theory. II. Ground states of rare gas atoms, J. Chem. Phys. 102, 1752 (1995); results using STOs for rare gas atoms
W. Klopper, Simple recipe for implementing computation of first‐order relativistic corrections to electron correlation energies in framework of direct perturbation theory, J. Comp. Chem. 18, 20-27 (1997); use of analytic gradients in DPT
S. Stopkowicz and J. Gauss, Direct perturbation theory in terms of energy derivatives: Fourth-order relativistic corrections at the Hartree–Fock level, J. Chem. Phys. 134, 064114 (2011); DPT4 at HF level including SO corrections
W. Schwalbach, S. Stopkowicz, L. Cheng, and J. Gauss, Direct perturbation theory in terms of energy derivatives: Scalar-relativistic treatment up to sixth order, J. Chem. Phys. 135, 194114 (2011); SF-DPT6 at HF level
S. Stopkowicz and J. Gauss, A one-electron variant of direct perturbation theory for the treatment of scalar-relativistic effects, Mol. Phys. 117, 1242-1251 (2019); 1el variant of DPT (DPT2-1e)

Spin-free (SF) four-compononent (SF-Dirac Coulomb) approaches

K. Dyall, An exact separation of the spin‐free and spin‐dependent terms of the Dirac–Coulomb–Breit Hamiltonian, J. Chem. Phys. 100, 2118 (1994); spin separation, can be also found in earlier work of Kutzelnigg
L. Cheng and J. Gauss, Analytical evaluation of first-order electrical properties based on the spin-free Dirac-Coulomb Hamiltonian, J. Chem. Phys. 134, 244112 (2011); implementation for HF, MP2, and CC. Note that this implementation is different from the one in DIRAC

(Spin-free) X2C-1e calculations

K. G. Dyall, Interfacing relativistic and nonrelativistic methods. I. Normalized elimination of the small component in the modified Dirac equation, J. Chem. Phys. 106, 9618 (1997); here named normalized elimination of the small component (NESC)
W. Kutzelnigg and W. Liu, Quasirelativistic theory equivalent to fully relativistic theory, J. Chem. Phys. 123, 241102 (2005); general X2C strategy
W. Liu and D. Peng, Infinite-order quasirelativistic density functional method based on the exact matrix quasirelativistic theory, J. Chem. Phys. 125, 044102 (2006); X2C-mf variant
M. Ilias and T. Saue, An infinite-order two-component relativistic Hamiltonian by a simple one-step transformation, J. Chem. Phys. 126, 064102 (2007); one-electron variant of X2C
J. Sikkema, L. Visscher, T. Saue, and M. Ilias, The molecular mean-field approach for correlated relativistic calculations, J. Chem. Phys. 131, 124116 (2009); X2C-mf variant
W. Liu and D. Peng, Exact two-component Hamiltonians revisited, J. Chem. Phys. 131, 031104 (2009); proper choice of R matrix

X2C-1e calculations with spin-orbit coupling

J. Li and L. Cheng, An atomic mean-field spin-orbit approach within exact two-component theory for a non-perturbative treatment of spin-orbit coupling, J. Chenm. Ohys. 148, 144108 (2018)
A. Astana, J. Li, and L. Cheng, Exact two-component equation-of-motion coupled-cluster singles and doubles method using atomic mean-field spin-orbit integrals, J. Chem. Phys. 150, 074101 (2019)

Geometry optimizations

using perturbation theory based on mass-velocity and Darwin terms (MVD1 and MVD2)

C. Michauk and J. Gauss, Perturbative treatment of scalar-relativistic effects in coupled-cluster calculations of equilibrium geometries and harmonic vibrational frequencies using analytic second-derivative techniques, J. Chem. Phys. 127, 044106 (2007)

using direct perturbation theory (DPT)

C. Michauk and J. Gauss, unpublished; see C. Michauk, Ph.D. thesis, University of Mainz, 2008

using SFX2c-1e

 L. Cheng and J. Gauss, Analytic energy gradients for the spin-free exact two-component theory using an exact block diagonalization for the one-electron Dirac Hamiltonian, J. Chem. Phys. 135, 084114 (2011)
 see also, W. Zou, M. Filatov, and D. Cremer, Development and application of the analytical energy gradient for the normalized elimination of the small component method, J. Chem. Phys. 134, 244117 (2011)

First-order properties

using direct perturbation theory (DPT)

S. Stopkowicz and J. Gauss, Relativistic corrections to electrical first-order properties using direct perturbation theory, J. Chem. Phys. 129, 164119 (2008); DPT2
S. Stopkowicz and J. Gauss, Fourth-order relativistic corrections to electrical first-order properties using direct perturbation theory, J. Chem. Phys. 134, 204106 (2011); numerical differentiation at DPT4 level including SO effects

using a spin-free Dirac-Coloumb approach

L. Cheng and J. Gauss, Analytical evaluation of first-order electrical properties based on the spin-free Dirac-Coulomb Hamiltonian, J. Chem. Phys. 134, 244112 (2011)

using SFX2c-1e

L. Cheng and J. Gauss, Analytic energy gradients for the spin-free exact two-component theory using an exact block diagonalization for the one-electron Dirac Hamiltonian, J. Chem. Phys. 135, 084114 (2011)

using SFX2c-mf

T. Kirsch, F. Engel, and J. Gauss, Analytic evaluation of first-order properties within the mean-field variant of spin-free exact two-component theory, J. Chem. Phys. 150, 204115 (2019)

Second-order properties

using direct perturbation theory (DPT)

S. Stopkowicz, D.P. O'Neill, and J. Gauss, unpublished; see S. Stopkowicz, diploma thesis, University of Mainz, 2008

using SFX2c-1e

L. Cheng and J. Gauss, Analytic second derivatives for the spin-free exact two-component theory, J. Chem. Phys. 135, 244104 (2011)

Perturbative treatments of spin-orbit effects

within DPT

S. Stopkowicz and J. Gauss, Direct perturbation theory in terms of energy derivatives: Fourth-order relativistic corrections at the Hartree–Fock level, J. Chem. Phys. 134, 064114 (2011)

on top of a SFDC calculation

L. Cheng, S. Stopkowicz, and J. Gauss, Spin-free Dirac-Coulomb calculations augmented with a perturbative treatment of spin-orbit effects at the Hartree-Fock level, J. Chem. Phys. 139, 214114 (2013)

on top of a SFX2c-1e calculation

L. Cheng and J. Gauss, Perturbative treatment of spin-orbit coupling within spin-free exact two-component theory, J. Chem. Phys. 141, 164107 (2014)
 L. Cheng, F. Wang, J.F. Stanton, and J. Gauss, Perturbative treatment of spin-orbit-coupling within spin-free exact two-component theory using equation-of-motion coupled-cluster methods, J. Chem. Phys. 148, 044108 (2018)

Spin-orbit splittings

using EOM-CC theory

K. Klein and J. Gauss, Perturbative calculation of spin-orbit splittings using the equation-of-motion ionization-potential coupled-cluster ansatz, J. Chem. Phys. 129, 194106 (2008); first implementation for EOMIP-CC
E. Epifanovsky, K. Klein, S. Stopkowicz, J. Gauss, and A.I. Krylov, Spin-orbit couplings within the equation-of-motion coupled-cluster framework: Theory, implementation, and benchmark calculations, J. Chem. Phys. 143, 064102 (2015); generalization to EOMEA-, EOMEE-, and EOMSF-CC

using Mk-MRCC theory

L.A. Mück and J. Gauss, Spin-orbit splittings in degenerate open-shell states via Mukherjee's multireference coupled-cluster theory: A measure for the coupling contribution, J. Chem. Phys. 136, 111103 (2012)

Two-component methods

two-component CCSD and CCSD(T) approaches using ECPs with SO couplings

F. Wang, J. Gauss, and C. van Wüllen, Closed-shell coupled-cluster theory with spin-orbit coupling, J. Chem. Phys. 129, 064113 (2008)

gradients for two-component CCSD and CCSD(T) approaches with SO couplings

F. Wang and J. Gauss, Analytic energy gradients in closed-shell coupled-cluster theory with spin-orbit coupling, J. Chem. Phys. 129, 174110 (2008)

second derivatives for two-component CCSD and CCSD(T) approaches with SO couplings

F. Wang and J. Gauss, Analytic second derivatives in closed-shell coupled-cluster theory with spin-orbit coupling, newwin% J. Chem. Phys. 131, 164113 (2009)

two-component EOMIP-CCSD approach

 Z. Tu, F. Wang, and X. Li, Equation-of-motion coupled-cluster method for ionized states with spin-orbit coupling, J. Chem. Phys. 136, 174102 (2012)