Selecting a two-component Hamiltonian other than X2C¶
There are several two-component Hamiltonians besides the X2C Hamiltonian (see .X2C) implemented in DIRAC arising from the decoupling transformation of the one-electron DIRAC Hamiltonian. Likewise spin-orbit interaction terms can be left out and calculations can be performed in the spin-free mode (i.e. in boson symmetry).
Besides having spin-orbit effects from the decoupling transformation, there is the external AMFI spin-orbit operator which can contribute either with the one-electron spin-orbit operator (accurate to the first order in V, very poor) or with the more valuable mean-field contribution, u_so(1) (also in the first order), which is a reasonable approximation of the ‘Gaunt’ term at the four-component level.
The best two-component Hamiltonian is X2C+MFSSO (similar to BSS+MFSSO or IOTC+MFSSO) where one-electron scalar and spin-orbit effects are up to infinite order, and AMFI MFSSO contributions (mean-field spin-same orbit) provide a ‘screening’ of one-electron spin-orbit terms.
Both scalar and spin-orbit relativistic effects up to infinite order:
Scalar relativistic effects of type “from the beginning” up to the infinite order, no spin-orbit interaction:
.SPINFREE .BSS 109
Scalar relativistic effects “from the end” up to the infinite order:
.SPINFREE .BSS 009
Traditional scalar relativistic “from the beginning” second-order Douglas-Kroll-Hess Hamiltonian:
.SPINFREE .BSS 102
Scalar relativistic “from the end” second-order Douglas-Kroll-Hess Hamiltonian:
.SPINFREE .BSS 002
Douglas-Kroll-Hess Hamiltonian with first-order spin-orbit and second-order scalar relativistic effects:
Douglas-Kroll-Hess Hamiltonian with second-order spin-orbit and second-order scalar relativistic effects:
In the two-component variational scheme is possible to combine external AMFI spin-orbit terms, [Ilias2001], together with BSS integrals - see the *AMFI section. Note that AMFI provides only one-center atomic integrals.
Infinite order scalar and spin-orbit terms, (one-electron) mean-field spin-same-orbit (MFSSO2) from AMFI:
This is in fact the ‘maximum’ of two-component relativity, resembling Dirac-Coulomb Hamiltonian.
Second-order Douglas-Kroll-Hess spin-free from ‘the beginning’ with first order spin-orbit (SO1) term plus mean-field spin-same-orbit (MFSSO) from AMFI:
SO1 may come either from BSS-transformation or from AMFI. In the latter case it is only for one-center and for point nucleus.
Infinite order scalar and spin-orbit terms, (one-electron) mean-field spin-same and spin-other-orbit (MFSO2) from AMFI:
This mimics the Dirac-Coulomb-Gaunt Hamiltonian, as the spin-other-orbit (SOO) term comes from the Gaunt interaction term.
DIRAC allows to switch off AMFI spin-orbit contributions from various centers. See the keyword .NOAMFC.
Infinite order scalar terms “from the beginning”, (one-electron) spin orbit (SO1) and the mean-field spin-same-orbit (MFSSO2) terms exclusively from AMFI:
Second order (DKH) scalar terms “from the beginning”, (one-electron) spin orbit (SO1) and the mean-field spin-same-orbit (MFSSO2) terms exclusively from AMFI:
Infinite order scalar terms “from the end”, (one-electron) spin orbit (SO1) and the mean-field spin-same-orbit and spin-other-orbit (MFSO2) terms from AMFI:
Infinite order scalar terms “from the end”, and the (one-electron) spin orbit term (SO1) from AMFI:
AMFI one-electron spin-orbit terms - SO1 - are currently for the point nucleus.
For the BSS value ‘axyz’ of y=0, DIRAC employs spin-free picture change transformation of property operators, although the system is not in the boson (spin-free) symmetry for a>1.
Rough first order, DKH1 (not recommended for practical calculations):
For comparison purposes between BSS-SO1 and AMFI-SO1 atomic one-center integrals (point nucleus only) use: