One-electron operators

Syntax for specifying one-electron operators

In DIRAC, a general one-electron 4-component operator is generated from linear combinations of operators having the basic form

\[\hat{\boldsymbol{P}} = f \, \boldsymbol{M} \, \hat{\Omega},\]

where \(f\) is a scalar factor, \(\hat{\Omega}\) is a scalar operator, and \(\boldsymbol{M}\) is one of the following \(4 \times 4\) time-reversal symmetric matrices:

\[ \begin{align}\begin{aligned}& \boldsymbol{I}, \, \boldsymbol{\beta}, \, \boldsymbol{\gamma}^5, \, \boldsymbol{\beta}\boldsymbol{\gamma}^5,\\& i\boldsymbol{\alpha}_x, \, i\boldsymbol{\alpha}_y, \, i\boldsymbol{\alpha}_z, \, i\boldsymbol{\beta}\boldsymbol{\alpha}_x, \, i\boldsymbol{\beta}\boldsymbol{\alpha}_y, \, i\boldsymbol{\beta}\boldsymbol{\alpha}_z,\\& i\boldsymbol{\Sigma}_x, \, i\boldsymbol{\Sigma}_y, \, i\boldsymbol{\Sigma}_z, \, i\boldsymbol{\beta}\boldsymbol{\Sigma}_x, \, i\boldsymbol{\beta}\boldsymbol{\Sigma}_y, \, i\boldsymbol{\beta}\boldsymbol{\Sigma}_z.\end{aligned}\end{align} \]

One thing to notice is that \(\boldsymbol{I}\), \(\boldsymbol{\beta}\), \(\boldsymbol{\gamma}^5\), and \(\boldsymbol{\beta}\boldsymbol{\gamma}^5\) are time-reversal symmetric operators, and they are implemented with such symmetry in DIRAC. On the other hand, \(\boldsymbol{\alpha}_k\), \(\boldsymbol{\Sigma}_k\), \(\boldsymbol{\beta\alpha}_k\), and \(\boldsymbol{\beta\Sigma}_k\) (with \(k=x,y\), and \(z\)) are time-reversal antisymmetric operators. This information is saved by the program, but these matrices (i.e., \(\boldsymbol{\alpha}_k\), \(\boldsymbol{\Sigma}_k\), \(\boldsymbol{\beta\alpha}_k\), and \(\boldsymbol{\beta\Sigma}_k\)) are multiplied by the imaginary unit \(i=\sqrt{-1}\) in order to make them time-reversal symmetric and hence fit into the quaternion symmetry scheme of DIRAC (see [Saue1999] and [Salek2005] for more information).

Another important aspect to note is that DIRAC does not use the \(4 \times 4\) \(\boldsymbol{M}\) matrices in the standard representation. These matrices are reordered following a criterion described in Section Quaternion algebra and time-reversal symmetry. Further details about the implemented operators are provided in that Section, where the connection between the Dirac matrices and the quaternion algebra is given.

Operator specification

Operators are specified by the keyword .OPERATOR with the following arguments:

.OPERATOR
 'user defined operator name'
 operator type keyword
 one-electron scalar operator label for each component
 CMULT
 FACTORS
 factor for each component
 COMFACTOR
 common factor for all components

Note that the arguments following the keyword .OPERATOR must start with a blank.

The arguments are optional, except for the one-electron scalar operator label. If no operator type keyword is given, DIRAC assumes the operator type to be \(\boldsymbol{I}\), the diagonal \(4 \times 4\) matrix. Component factors under the keyword FACTORS, as well as the common factor under the keyword COMFACTOR, are all equal to one if not specified.

The keyword CMULT assures multiplication of the individual factors (or common factor) by the speed of light in vacuum, \(c\). This option has the further advantage that CMULT follows any user-specified modification of the speed of light, as provided by .CVALUE.

Operator type keywords

There are 22 basic operator types implemented in DIRAC. They are listed in the following Table, containing their keywords, operator forms, associated number of factors (NF), and time-reversal symmetry (TRS) without imaginary phase:

Keyword	Operator form	NF	TRS (without i)
DIAGONAL	\(f \boldsymbol{I} \hat{\Omega}\)	1	Symmetric
BETA	\(f \boldsymbol{\beta} \hat{\Omega}\)	1	Symmetric
GAMMA5	\(f \boldsymbol{\gamma}^5 \hat{\Omega}\)	1	Symmetric
BETAGAMM	\(f \boldsymbol{\beta} \boldsymbol{\gamma}^5 \hat{\Omega}\)	1	Symmetric
XALPHA	\(f i\boldsymbol{\alpha}_x \hat{\Omega}\)	1	Antisymmetric
YALPHA	\(f i\boldsymbol{\alpha}_y \hat{\Omega}\)	1	Antisymmetric
ZALPHA	\(f i\boldsymbol{\alpha}_z \hat{\Omega}\)	1	Antisymmetric
XAVECTOR	\(f_1 i\boldsymbol{\alpha}_y \hat{\Omega}_z - f_2 i\boldsymbol{\alpha}_z \hat{\Omega}_y\)	2	Antisymmetric
YAVECTOR	\(f_1 i\boldsymbol{\alpha}_z \hat{\Omega}_x - f_2 i\boldsymbol{\alpha}_x \hat{\Omega}_z\)	2	Antisymmetric
ZAVECTOR	\(f_1 i\boldsymbol{\alpha}_x \hat{\Omega}_y - f_2 i\boldsymbol{\alpha}_y \hat{\Omega}_x\)	2	Antisymmetric
ALPHADOT	\(f_1 i\boldsymbol{\alpha}_x \hat{\Omega}_x + f_2 i\boldsymbol{\alpha}_y \hat{\Omega}_y + f_3 i\boldsymbol{\alpha}_z \hat{\Omega}_z\)	3	Antisymmetric
XBETAALP	\(f i\boldsymbol{\beta} \boldsymbol{\alpha}_x \hat{\Omega}\)	1	Antisymmetric
YBETAALP	\(f i\boldsymbol{\beta} \boldsymbol{\alpha}_y \hat{\Omega}\)	1	Antisymmetric
ZBETAALP	\(f i\boldsymbol{\beta} \boldsymbol{\alpha}_z \hat{\Omega}\)	1	Antisymmetric
XSIGMA	\(f i\boldsymbol{\Sigma}_x \hat{\Omega}\)	1	Antisymmetric
YSIGMA	\(f i\boldsymbol{\Sigma}_y \hat{\Omega}\)	1	Antisymmetric
ZSIGMA	\(f i\boldsymbol{\Sigma}_z \hat{\Omega}\)	1	Antisymmetric
SIGMADOT	\(f_1 i\boldsymbol{\Sigma}_x \hat{\Omega}_x + f_2 i\boldsymbol{\Sigma}_y \hat{\Omega}_y + f_3 i\boldsymbol{\Sigma}_z \hat{\Omega}_z\)	1	Antisymmetric
XBETASIG	\(f i\boldsymbol{\beta} \boldsymbol{\Sigma}_x \hat{\Omega}\)	1	Antisymmetric
YBETASIG	\(f i\boldsymbol{\beta} \boldsymbol{\Sigma}_y \hat{\Omega}\)	1	Antisymmetric
ZBETASIG	\(f i\boldsymbol{\beta} \boldsymbol{\Sigma}_z \hat{\Omega}\)	1	Antisymmetric
iBETAGAM	\(f \boldsymbol{\beta} \boldsymbol{\gamma}^5 \hat{\Omega}\)	1	Antisymmetric

Note for advanced users: iBETAGAM corresponds to the time-reversal symmetric operator \(f \boldsymbol{\beta} \boldsymbol{\gamma}^5 \hat{\Omega}\), but has been assigned a time-reversal antisymmetry. This operator can be used, for example, for calculations of the molecular enhancement factors of the electron electric dipole moment (eEDM). For further details, see Section Quaternion algebra and time-reversal symmetry.

One-electron scalar operator labels

Operator label	Integral description	Components	Operators
KINENER	NR kinetic energy	KINENER	\(\hat{\Omega}_1 = -\frac{1}{2}\hat{\nabla}^2\)
MOLFIELD	Nuclear attraction	MOLFIELD	\(\hat{\Omega}_1 = \sum_K Z_K \|\boldsymbol{r}-\boldsymbol{R}_K\|^{-1}\)
OVERLAP	Overlap	OVERLAP	\(\hat{\Omega}_1 = 1\)
FERMI C	One-electron Fermi contact	FC NAMab	\(\hat{\Omega}_1 = \frac{4 \pi g_{e}}{3} \delta(\boldsymbol{r}-\boldsymbol{R}_K)\)
ANGMOM	Orbital angular momentum around CM	XANGMOM	\(\hat{\Omega}_1 = -[(\boldsymbol{r}-\boldsymbol{R}_{CM})\times\hat{\nabla}]_x\)
		YANGMOM	\(\hat{\Omega}_2 = -[(\boldsymbol{r}-\boldsymbol{R}_{CM})\times\hat{\nabla}]_y\)
		ZANGMOM	\(\hat{\Omega}_3 = -[(\boldsymbol{r}-\boldsymbol{R}_{CM})\times\hat{\nabla}]_z\)
DIPLEN	Dipole length	XDIPLEN	\(\hat{\Omega}_1 = x\)
		YDIPLEN	\(\hat{\Omega}_2 = y\)
		ZDIPLEN	\(\hat{\Omega}_3 = z\)
DIPVEL	Dipole velocity	XDIPVEL	\(\hat{\Omega}_1 = \frac{\partial}{\partial x}\)
		YDIPVEL	\(\hat{\Omega}_2 = \frac{\partial}{\partial y}\)
		ZDIPVEL	\(\hat{\Omega}_3 = \frac{\partial}{\partial z}\)
QUADRUP	Quadrupole moment	XXQUADRU	\(\hat{\Omega}_1 = \frac{1}{4} (x^2-r^2)\)
		XYQUADRU	\(\hat{\Omega}_2 = \frac{1}{4} xy\)
		XZQUADRU	\(\hat{\Omega}_3 = \frac{1}{4} xz\)
		YYQUADRU	\(\hat{\Omega}_4 = \frac{1}{4} (y^2-r^2)\)
		YZQUADRU	\(\hat{\Omega}_5 = \frac{1}{4} yz\)
		ZZQUADRU	\(\hat{\Omega}_6 = \frac{1}{4} (z^2-r^2)\)
SECMOM	Second moment	XXSECMOM	\(\hat{\Omega}_1 = - x^2\)
		XYSECMOM	\(\hat{\Omega}_2 = - xy\)
		XZSECMOM	\(\hat{\Omega}_3 = - xz\)
		YYSECMOM	\(\hat{\Omega}_4 = - y^2\)
		YZSECMOM	\(\hat{\Omega}_5 = - yz\)
		ZZSECMOM	\(\hat{\Omega}_6 = - z^2\)
THETA	Traceless quadrupole	XXTHETA	\(\hat{\Omega}_1 = - \frac{3}{2} (x^2-\frac{1}{3}r^2)\)
		XYTHETA	\(\hat{\Omega}_2 = - \frac{3}{2} xy\)
		XZTHETA	\(\hat{\Omega}_3 = - \frac{3}{2} xz\)
		YYTHETA	\(\hat{\Omega}_4 = - \frac{3}{2} (y^2-\frac{1}{3}r^2)\)
		YZTHETA	\(\hat{\Omega}_5 = - \frac{3}{2} yz\)
		ZZTHETA	\(\hat{\Omega}_6 = - \frac{3}{2} (z^2-\frac{1}{3}r^2)\)
SEFGMG	Magnetic term of Flambaum-Ginges self-energy potential	XSEFGMG
		YSEFGMG
		ZSEFGMG

where \(\boldsymbol{R}_{CM}\) and \(\boldsymbol{R}_K\) are the position vectors of the nuclear center of mass and the nucleus \(K\) with respect to the origin of coordinates, respectively. In addition, the operators \(x\), \(y\), and \(z\) are the Cartesian components of the electron position vector operator with respect to the origin of coordinates.

Note that in the one-electron Fermi-contact integrals \(g_e\) is the electronic g-factor (i.e., \(g_e = 2\) unless QED effects are taken into account), ‘NAM’ are the three first letters in the name of atom \(K\) as given in the .mol file, and ‘ab’ stands for the number of the symmetry-adapted nucleus.

Other one-electron operator labels are:

Keyword	Description
ANGLON	Angular momentum around the nuclei
CARMOM	Cartesian moments (symmetric) \((l + 1)(l + 2)/2\) components (\(l = i + j + k\))
CM1	First order magnetic field derivatives of electric field
CM2	Second order magnetic field derivatives of electric field
DARWIN	Darwin-type integrals
DIASUS	Angular London orbital contribution to diamagnetic susceptibility
DSO	Diamagnetic spin-orbit integrals
DSUSCGO	Diamagnetic susceptibility with common gauge origin
DSUSLH	Angular London orbital contribution to diamagnetic susceptibility
DSUSNOL	Diamagnetic susceptibility without London contribution
ELFGRDC	Electric field gradient at the individual nuclei, cartesian
ELFGRDS	Electric field gradient at the individual nuclei, spherical
EXPIKR	Cosine and sine
HBDO	Half B-differentiated overlap matrix
HDO	Half-derivative overlap integrals
HDOBR	Ket-differentiation of HDO-integrals with respect to magnetic field
LONMOM	London orbital contribution to angular momentum
MAGMOM	One-electron contributions to magnetic moment
MASSVEL	Mass velocity
NEFIELD	Electric field at the individual nuclei
NSTCGO	Nuclear shielding with common gauge origin
NUCPOT	Potential energy of the interaction of electrons with individual nuclei, divided by the nuclear charge
NUCSHI	Nuclear shielding tensor
NUCSLO	London orbital contribution to nuclear shielding tensor
NUCSNLO	Nuclear shielding integrals without London orbital contribution
PSO	Paramagnetic spin-orbit
S1MAG	Second order contribution from overlap matrix to magnetic properties
S1MAGL	Bra-differentiation of overlap matrix with respect to magnetic field
S1MAGR	Ket-differentiation of overlap matrix with respect to magnetic field
SDFC	Spin-dipole + Fermi contact
SOLVENT	Electronic solvent integrals
SPHMOM	Spherical moments (real combinations), symmetric, (\(2l + 1\)) components (\(m = 0, -1, +1, ..., -l, +l\))
SPIN-DI	Spin-dipole integrals
SPNORB	Spatial spin-orbit
SQHDO	Half-derivative overlap integrals not to be antisymmetrized
SQHDOR	Half-derivative overlap integrals not to be anti-symmetrized
SQOVLAP	Second order derivatives overlap integrals
UEHLING	Uehling potential
SEFGLF	Low-frequency term of Flambaum-Ginges self-energy potential
SEFGHF	High-frequency term of Flambaum-Ginges self-energy potential
SEPZ	Pyykko-Zhao self-energy model potential

Examples of using various operators

We give here several concrete examples on how to construct operators for various properties.

Kinetic part of the Dirac Hamiltonian

The kinetic part of the electronic Dirac Hamiltonian is given (in a.u.) by

\[ \begin{align}\begin{aligned}\hat{H}^K & = c\vec{\boldsymbol{\alpha}}\cdot\hat{\vec{p}} = c\left(\boldsymbol{\alpha}_x\hat{p}_x+\boldsymbol{\alpha}_y\hat{p}_y+\boldsymbol{\alpha}_z\hat{p}_z\right)\\ & = -c \left[ i\boldsymbol{\alpha}_x \frac{\partial}{\partial x} + i\boldsymbol{\alpha}_y \frac{\partial}{\partial y} + i\boldsymbol{\alpha}_z \frac{\partial}{\partial z} \right].\end{aligned}\end{align} \]

This operator may be specified by:

.OPERATOR
 'Kin energy'
 ALPHADOT
 XDIPVEL
 YDIPVEL
 ZDIPVEL
 COMFACTOR
 -137.03599926085

where 137.03599926085 is the CODATA22 value for \(c\).

The speed of light \(c\) is an important parameter in relativistic theory, but its explicit value in atomic units is not necessarily remembered. A simpler way to specify the kinetic energy operator is therefore:

.OPERATOR
 'Kin energy'
 ALPHADOT
 XDIPVEL
 YDIPVEL
 ZDIPVEL
 CMULT
 COMFACTOR
 -1

Magnetic interactions

Another example is the operator that describes the electromagnetic interaction between electrons and a uniform external magnetic field: \(\hat{H}^B=-\frac{c}{2}\vec{B}\cdot\vec{\boldsymbol{\alpha}}\times\hat{\vec{r}}\).

The operator to use in this case is \(-\frac{c}{2}\vec{\boldsymbol{\alpha}}\times\vec{r}\), and in DIRAC each component is calculated individually. For example, the \(x\)-component of this operator is \(-\frac{c}{2}\left(\boldsymbol{\alpha}_y z - \boldsymbol{\alpha}_z y\right)\), and can be requested using:

.OPERATOR
 'B_x'
 XAVECTOR
 ZDIPLEN
 YDIPLEN
 CMULT
 COMFACTOR
 -0.5

The program will assume all operators to be Hermitian. The operator XAVECTOR is defined as \(f_1 i\boldsymbol{\alpha}_y \hat{\Omega}_z-f_2 i\boldsymbol{\alpha}_z \hat{\Omega}_y\) (see Operator type keywords), and in this particular case the scalar operators are \(\hat{\Omega}_z=z\) and \(\hat{\Omega}_y=y\) (see ZDIPLEN and YDIPLEN in One-electron scalar operator labels). Additionally, \(f_1=f_2=-\frac{c}{2}\). Then, DIRAC will assume the operator to be \(-\frac{c}{2}\left(i\boldsymbol{\alpha}_y z - i\boldsymbol{\alpha}_z y\right)\).

The imaginary phase \(i\) makes the operator time-reversal symmetric.

Diagonal operators

If no other argument is given, the program assumes the operator to be diagonal and expects the operator name to be the component label, for instance:

.OPERATOR
 OVERLAP

Finite-field perturbation: Electric dipole moment

Another example is a finite-perturbation calculation where the perturbed Hamiltonian operator arising from the interaction between the electrons and a uniform external electric field oriented along the \(z\)-axis (\(\hat{H}'=-q E_z \hat{z}=e E_z \hat{z}\)) is added to the unperturbed Hamiltonian \(\hat{H}_0\) defined in **HAMILTONIAN (default: Dirac-Coulomb).

If we take \(e E_z = 0.01\) (in a.u.), the \(\hat{z}\) dipole length operator will be used to build the total Hamiltonian \(\hat{H} = \hat{H}_0 + 0.01 \, \hat{z}\) by adding the perturbed Hamiltonian \(\hat{H}'\) to the Dirac-Coulomb Hamiltonian \(\hat{H}_0\) using:

**HAMILTONIAN
.OPERATOR
 ZDIPLEN
 COMFACTOR
 0.01

Note: Don’t forget to decrease the symmetry of your system.

Fermi-contact integrals

Here is an example where the Fermi-contact (FC) integrals for a certain nucleus \(K\) are added to the Hamiltonian in a finite-field calculation.

Let’s assume we are looking at a PbX dimer (order in the .mol file: 1. Pb, 2. X) and we want to add (under **HAMILTONIAN) the FC interaction for the Pb nucleus to the unperturbed Hamiltonian \(\hat{H}_0\) as a perturbation with a given field strength of \(10^{-9}\) a.u..

The FC integrals are called using the label 'FC NAMab' (see One-electron scalar operator labels), and the total perturbed Hamiltonian \(\hat{H}=\hat{H}_0+\hat{H}'\) (where, in a.u., \(\hat{H}'=10^{-9} \, \frac{4\pi}{3} \, g_e \delta(\boldsymbol{r}-\boldsymbol{R}_\text{Pb})\)) can be requested using:

**HAMILTONIAN
.OPERATOR
 'Density at Pb nucleus'
 DIAGONAL
 'FC Pb 01'
 FACTORS
 0.000000001

Important note: The values obtained after the fit of the finite-field energies need to be scaled by \(\frac{3}{4 \pi g_{e}} = \frac{1}{8.3872954891254192}\) in order to get the raw density \(\delta(\boldsymbol{r}-\boldsymbol{R}_\text{Pb})\).

The next example shows how to calculate the electronic density at the different nuclei in the molecule, as the expectation values \(\langle -\delta(\boldsymbol{r}-\boldsymbol{R}_K) \rangle\) (for each nucleus \(K\) in the system) for a Dirac-Coulomb Hartree-Fock wave function including a decomposition of the molecular orbital contributions to the density:

**DIRAC
.WAVE FUNCTION
.PROPERTIES
**HAMILTONIAN
**WAVE FUNCTION
.SCF
**PROPERTIES
.RHONUC
*EXPECTATION
.ORBANA
*END OF

For further details, see .RHONUC.

Cartesian moment expectation value

In the following example we calculate the cartesian multipole moment expectation value \(\langle x^1 y^2 z^3 \rangle\) for a Levy-Leblond Hartree-Fock wave function:

**DIRAC
.WAVE FUNCTION
.PROPERTIES
**HAMILTONIAN
.LEVY-LEBLOND
**WAVE FUNCTION
.SCF
**PROPERTIES
*EXPECTATION
.OPERATOR
 CM010203
*END OF

Quaternion algebra and time-reversal symmetry

The sixteen \(4 \times 4\) matrices listed in Section Syntax for specifying one-electron operators, i.e., \(\boldsymbol{I}\), \(\boldsymbol{\beta}\), \(\boldsymbol{\gamma}^5\), \(\boldsymbol{\beta}\boldsymbol{\gamma}^5\), \(i\boldsymbol{\alpha}_x\), \(i\boldsymbol{\alpha}_y\), \(i\boldsymbol{\alpha}_z\), \(i\boldsymbol{\beta}\boldsymbol{\alpha}_x\), \(i\boldsymbol{\beta}\boldsymbol{\alpha}_y\), \(i\boldsymbol{\beta}\boldsymbol{\alpha}_z\), \(i\boldsymbol{\Sigma}_x\), \(i\boldsymbol{\Sigma}_y\), \(i\boldsymbol{\Sigma}_z\), \(i\boldsymbol{\beta}\boldsymbol{\Sigma}_x\), \(i\boldsymbol{\beta}\boldsymbol{\Sigma}_y\), and \(i\boldsymbol{\beta}\boldsymbol{\Sigma}_z\), are usually expressed in Molecular Physics using the standard representation where, for example,

\[\begin{split}\tilde{\boldsymbol{\beta}} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}, \; \tilde{\boldsymbol{\gamma}}^{5} = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}, \; i\tilde{\boldsymbol{\alpha}}_z = \begin{pmatrix} 0 & 0 & i & 0 \\ 0 & 0 & 0 & -i \\ i & 0 & 0 & 0 \\ 0 & -i & 0 & 0 \end{pmatrix}.\end{split}\]

Here we will use the notation \(\tilde{\boldsymbol{U}}\) to refer to any of these sixteen matrices when they are expressed in the standard representation.

It can be seen that all these matrices can be written as the Kronecker product of two \(2 \times 2\) matrices \(\boldsymbol{A}\) and \(\boldsymbol{B}\), such that \(\tilde{\boldsymbol{U}} = \boldsymbol{A} \otimes \boldsymbol{B}\). In particular, \(\boldsymbol{A}\) is one of the real matrices \(\boldsymbol{I}\), \(\boldsymbol{\sigma}_x\), \(i \boldsymbol{\sigma}_y\), or \(\boldsymbol{\sigma}_z\), whereas \(\boldsymbol{B}\) is \(\boldsymbol{I}\), \(i \boldsymbol{\sigma}_x\), \(i \boldsymbol{\sigma}_y\), or \(i \boldsymbol{\sigma}_z\). Here, \(\boldsymbol{\sigma}_x\), \(\boldsymbol{\sigma}_y\), and \(\boldsymbol{\sigma}_z\) are the Pauli matrices, and \(i\) is the imaginary unit.

The individual matrix elements of \(\tilde{\boldsymbol{U}}\) are given as

\[\tilde{U}_{pq} = A_{ij} \, B_{kl} \, ,\]

where \(p=2(i-1)+k\) and \(q=2(j-1)+l\).

In the DIRAC code, the structure of the Dirac equation with respect to time-reversal symmetry is displayed by using reordered Dirac 4-spinors, where the components are grouped on spin labels (\(\alpha\), \(\beta\)) rather than large (upper) and small (lower) components (\(L\), \(S\)),

\[\begin{split}\begin{pmatrix} \psi^{L\alpha} \\ \psi^{L\beta} \\ \psi^{S\alpha} \\ \psi^{S\beta} \end{pmatrix} \rightarrow \begin{pmatrix} \psi^{L\alpha} \\ \psi^{S\alpha} \\ \psi^{L\beta} \\ \psi^{S\beta} \end{pmatrix}.\end{split}\]

The same reordering applies to all the matrices \(\tilde{\boldsymbol{U}}\), which are then transformed accordingly. In the original ordering (i.e., in the standard representation), these matrices are given as

\[\begin{split}\tilde{\boldsymbol{U}} = \begin{pmatrix} \tilde{U}_{L\alpha,L\alpha} & \tilde{U}_{L\alpha,L\beta} & \tilde{U}_{L\alpha,S\alpha} & \tilde{U}_{L\alpha,S\beta} \\ \tilde{U}_{L\beta,L\alpha} & \tilde{U}_{L\beta,L\beta} & \tilde{U}_{L\beta,S\alpha} & \tilde{U}_{L\beta,S\beta} \\ \tilde{U}_{S\alpha,L\alpha} & \tilde{U}_{S\alpha,L\beta} & \tilde{U}_{S\alpha,S\alpha} & \tilde{U}_{S\alpha,S\beta} \\ \tilde{U}_{S\beta,L\alpha} & \tilde{U}_{S\beta,L\beta} & \tilde{U}_{S\beta,S\alpha} & \tilde{U}_{S\beta,S\beta} \end{pmatrix} = \boldsymbol{A} \otimes \boldsymbol{B}.\end{split}\]

However, after reordering one clearly has the transformed matrices, which satisfy

\[\begin{split}\boldsymbol{M} = \begin{pmatrix} M_{L\alpha,L\alpha} & M_{L\alpha,S\alpha} & M_{L\alpha,L\beta} & M_{L\alpha,S\beta} \\ M_{S\alpha,L\alpha} & M_{S\alpha,S\alpha} & M_{S\alpha,L\beta} & M_{S\alpha,S\beta} \\ M_{L\beta,L\alpha} & M_{L\beta,S\alpha} & M_{L\beta,L\beta} & M_{L\beta,S\beta} \\ M_{S\beta,L\alpha} & M_{S\beta,S\alpha} & M_{S\beta,L\beta} & M_{S\beta,S\beta} \end{pmatrix} = \boldsymbol{B} \otimes \boldsymbol{A}.\end{split}\]

In other words, \(M_{PQ} = B_{kl} \, A_{ij}\), with \(P=2(k-1)+i\) and \(Q=2(l-1)+j\). This reordering guarantees that all the matrices \(\boldsymbol{M}\) are time-reversal symmetric.

Additionally, following the arguments given in Ref. [Saue1999], it is possible to make the following mapping involving the quaternion units:

\[ \begin{align}\begin{aligned}&\boldsymbol{I} &\rightarrow & \qquad 1\\&i \boldsymbol{\sigma}_z &\rightarrow & \qquad \check{\imath}\\&i \boldsymbol{\sigma}_y &\rightarrow & \qquad \check{\jmath}\\&i \boldsymbol{\sigma}_x &\rightarrow & \qquad \check{k},\end{aligned}\end{align} \]

or, in a general way, \(\boldsymbol{B} \rightarrow b\), where \(b\) is a quaternion unit arising from the relationships given above.

Thus, any linear combination of the 16 reordered matrices \(\boldsymbol{M}\) can be mapped to a \(2 \times 2\) matrix \(\boldsymbol{m}\), which is a linear combinations of the quaternion units.

To make the proposed discussion clearer, in the following we will discuss two examples. In first place, we will transform \(\tilde{\boldsymbol{\beta}}\). We start by writting it as the following Kronecker product of two \(2 \times 2\) matrices: \(\tilde{\boldsymbol{\beta}}=\boldsymbol{\sigma}_z \otimes \boldsymbol{I}\). In the reordered representation, this matrix will be expressed as \(\boldsymbol{\beta}=\boldsymbol{I} \otimes \boldsymbol{\sigma}_z\). Finally, the \(\boldsymbol{B}\) matrix (in this case, \(\boldsymbol{I}\)) is transformed to a quaternion unit (in the current example, \(1\)). In this way, the \(4 \times 4\) reordered matrix \(\boldsymbol{\beta}\) can be written in quaternion form as the \(2 \times 2\) matrix \(\boldsymbol{\sigma}_z\). In other words:

\[\begin{split}\tilde{\boldsymbol{\beta}} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} \quad\rightarrow\quad \boldsymbol{\beta} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} \quad\rightarrow\quad \boldsymbol{\beta}^\text{quat} = \boldsymbol{\sigma}_z.\end{split}\]

As a second example, we take \(i\tilde{\boldsymbol{\alpha}}_z\). This \(4 \times 4\) matrix, in its original ordering (according to the standard representation), can be written as the Kronecker product \(i\tilde{\boldsymbol{\alpha}}_z=\boldsymbol{\sigma}_x \otimes i \boldsymbol{\sigma}_z\). This means that in the reordered representation it will be written as \(i\boldsymbol{\alpha}_z = i \boldsymbol{\sigma}_z \otimes \boldsymbol{\sigma}_x\). In this case, the matrix \(\boldsymbol{B}\) is \(i \boldsymbol{\sigma}_z\), so that the quaternion unit \(b\) will be \(\check{\imath}\). Therefore, the quaternion form of this reordered matrix will be \(i\boldsymbol{\alpha}_z^\text{quat} = \check{\imath} \boldsymbol{\sigma}_x\). Equivalently:

\[\begin{split}i\tilde{\boldsymbol{\alpha}}_z = \begin{pmatrix} 0 & 0 & i & 0 \\ 0 & 0 & 0 & -i \\ i & 0 & 0 & 0 \\ 0 & -i & 0 & 0 \end{pmatrix} \quad\rightarrow\quad i\boldsymbol{\alpha}_z = \begin{pmatrix} 0 & i & 0 & 0 \\ i & 0 & 0 & 0 \\ 0 & 0 & 0 & -i \\ 0 & 0 & -i & 0 \end{pmatrix} \quad\rightarrow\quad i\boldsymbol{\alpha}_z^\text{quat} = \check{\imath} \boldsymbol{\sigma}_x.\end{split}\]

The relationship between \(\tilde{\boldsymbol{U}}\) (i.e., the sixteen \(4 \times 4\) matrices in original ordering, which corresponds to the standard representation) and the quaternion form of their reordered equivalent \(2 \times 2\) matrices \(\boldsymbol{m}\) is given in the following Table (they follow the transformation rule \(\tilde{\boldsymbol{U}} = \boldsymbol{A} \otimes \boldsymbol{B}\) \(\rightarrow\) \(\boldsymbol{M} = \boldsymbol{B} \otimes \boldsymbol{A}\) \(\rightarrow\) \(\boldsymbol{m} = b \boldsymbol{A}\)):

\(\tilde{\boldsymbol{U}}\)	\(\boldsymbol{A} \otimes \boldsymbol{B}\)	\(\boldsymbol{m}\)
\(\boldsymbol{I}\)	\(\boldsymbol{I} \otimes \boldsymbol{I}\)	\(\boldsymbol{I}\)
\(i\boldsymbol{\Sigma}_z\)	\(\boldsymbol{I} \otimes i \boldsymbol{\sigma}_z\)	\(\check{\imath}\, \boldsymbol{I}\)
\(i\boldsymbol{\Sigma}_y\)	\(\boldsymbol{I} \otimes i \boldsymbol{\sigma}_y\)	\(\check{\jmath}\, \boldsymbol{I}\)
\(i\boldsymbol{\Sigma}_x\)	\(\boldsymbol{I} \otimes i \boldsymbol{\sigma}_x\)	\(\check{k}\, \boldsymbol{I}\)
\(\boldsymbol{\beta}\)	\(\boldsymbol{\sigma}_z \otimes \boldsymbol{I}\)	\(\boldsymbol{\sigma}_z\)
\(i\boldsymbol{\beta}\boldsymbol{\Sigma}_z\)	\(\boldsymbol{\sigma}_z \otimes i \boldsymbol{\sigma}_z\)	\(\check{\imath}\, \boldsymbol{\sigma}_z\)
\(i\boldsymbol{\beta}\boldsymbol{\Sigma}_y\)	\(\boldsymbol{\sigma}_z \otimes i \boldsymbol{\sigma}_y\)	\(\check{\jmath}\, \boldsymbol{\sigma}_z\)
\(i\boldsymbol{\beta}\boldsymbol{\Sigma}_x\)	\(\boldsymbol{\sigma}_z \otimes i \boldsymbol{\sigma}_x\)	\(\check{k}\, \boldsymbol{\sigma}_z\)
\(\boldsymbol{\beta}\boldsymbol{\gamma}^5\)	\((i\boldsymbol{\sigma}_y) \otimes \boldsymbol{I}\)	\((i\boldsymbol{\sigma}_y)\)
\(i\boldsymbol{\beta}\boldsymbol{\alpha}_z\)	\((i\boldsymbol{\sigma}_y) \otimes i \boldsymbol{\sigma}_z\)	\(\check{\imath}\, (i\boldsymbol{\sigma}_y)\)
\(i\boldsymbol{\beta}\boldsymbol{\alpha}_y\)	\((i\boldsymbol{\sigma}_y) \otimes i \boldsymbol{\sigma}_y\)	\(\check{\jmath}\, (i\boldsymbol{\sigma}_y)\)
\(i\boldsymbol{\beta}\boldsymbol{\alpha}_x\)	\((i\boldsymbol{\sigma}_y) \otimes i \boldsymbol{\sigma}_x\)	\(\check{k}\, (i\boldsymbol{\sigma}_y)\)
\(\boldsymbol{\gamma}^5\)	\(\boldsymbol{\sigma}_x \otimes \boldsymbol{I}\)	\(\boldsymbol{\sigma}_x\)
\(i\boldsymbol{\alpha}_z\)	\(\boldsymbol{\sigma}_x \otimes i \boldsymbol{\sigma}_z\)	\(\check{\imath}\, \boldsymbol{\sigma}_x\)
\(i\boldsymbol{\alpha}_y\)	\(\boldsymbol{\sigma}_x \otimes i \boldsymbol{\sigma}_y\)	\(\check{\jmath}\, \boldsymbol{\sigma}_x\)
\(i\boldsymbol{\alpha}_x\)	\(\boldsymbol{\sigma}_x \otimes i \boldsymbol{\sigma}_x\)	\(\check{k}\, \boldsymbol{\sigma}_x\)

It is important to note that although the time-reversal antisymmetric operators \(\boldsymbol{\alpha}_k\), \(\boldsymbol{\Sigma}_k\), \(\boldsymbol{\beta\alpha}_k\), and \(\boldsymbol{\beta\Sigma}_k\) (with \(k=x,y\), and \(z\)) become time-reversal symmetric once they are multiplied by the imaginary unit \(i\) and written in quaternion form as described above, DIRAC saves the information that these operators were originally T-reversal antisymmetric (before adding the imaginary phase).

For further details on the quaternion symmetry scheme used in DIRAC, see Ref. [Saue1999].

Note: For complete consistency, the names of the operators related to \(i\boldsymbol{\alpha}_k\), \(i\boldsymbol{\Sigma}_k\), \(i\boldsymbol{\beta\alpha}_k\), and \(i\boldsymbol{\beta\Sigma}_k\) (\(k=x,y,z\)) could have been XiALPHA, YiALPHA, ZiALPHA, XiAVECTOR, YiAVECTOR, ZiAVECTOR, iALPHADOT, XiBETAALP, YiBETAALP, ZiBETAALP, XiSIGMA, YiSIGMA, ZiSIGMA, XiBETASIG, YiBETASIG, ZiBETASIG, and iSIGMADOT (see Operator type keywords). However, due to historical reasons, this convention is not used in DIRAC.

Note for advanced users: The operator iBETAGAM is implemented in the code just as \(\beta\boldsymbol{\gamma}^5\), but its time-reversal symmetry without imaginary phase is set as odd.