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:


Scalar relativistic effects “from the end” up to the infinite order:


Traditional scalar relativistic “from the beginning” second-order Douglas-Kroll-Hess Hamiltonian:


Scalar relativistic “from the end” second-order Douglas-Kroll-Hess Hamiltonian:


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 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 only between BSS-SO1 and AMFI-SO1 atomic one-center integrals (point nucleus only) use:


Spin-orbit states from the COSCI method

This tutorial’s part demonstrates the importance of the effective mean-field spin-orbit scrrening on spin-orbit states of the open-shell system.

In the DIRAC test we calculate the spin-orbit components of the \(^{2}P\) state of Fluorine, using a COSCI wavefunction and with several different Hamiltonians.

Hamiltonian Splitting/cm-1
DC 434.511758
BSS+MFSSO 438.792872
BSS_RKB+MFSSO 438.793184
DKH2+MFSSO 438.792782
BSSsfBSO1+MFSSO 438.868634
DKH2sfBSO1+MFSSO 438.868738
BSSsfESO1+MFSSO 438.866098
DKH2sfESO1+MFSSO 438.866201
BSS 583.459766
BSS_RKB 583.459995
DKH2 583.459700
BSSsfESO1 583.533060
DKH2sfESO1 583.533187
BSSsfBSO1 583.535908
DKH2sfBSO1 583.536036
DC2BSS_RKB(DF) 585.906861

The closest results to those with the four-component Dirac-Coulomb (DC) Hamiltonian are with the infinite-order two-component Hamiltonian containing mean-field spin-orbit screening (those with +MFSSO), see for example, Refs. [Ilias2001], [Ilias2007].

Spin-orbit states of the \(Rn^{77+}\) cation

Let us proceed to the isoelectronic, simple heavy system: Fluorine-like (9 electrons), highly charged \(Rn^{77+}\) cation (Z=86). Its calculated \(^{2}P\) splittings are in the following table:

Hamiltonian Splitting/eV
DC 3700.081
BSS+MFSSO 3796.844
DKH2+MFSSO 3777.837
DC2BSS_RKB(DF) 3810.190
BSS 3808.859
BSS_RKB 3810.273
DKH2 3790.044
DKH2sfBSO1+MFSSO 4047.324
DKH2sfBSO1 4056.349

Again, the two-component infinite order Hamiltonian with the AMFI screening (BSS+MFSSO), together with the screened second-order Douglas-Kroll-Hess operator (DKH2+MFSSO) are reproducing four-component DC results. “Naked” (non-screened) two-component Hamiltonians (BSS, DKH2) are not preforming so well with respect to the screened ones. Likewise the one-electron spin-orbit terms must be at least of order two (starting from the DKH2 Hamiltonian).