One-electron operators¶

Syntax for the specification of one-electron operators¶

A general one-electron property operator in 4-component calculations is generated from linear combinations of the basic form:

\[\hat{P} = f M_{4 \times 4} \hat{\Omega}\]

with the scalar factor \(f\) and the scalar operator \(\hat{\Omega}\), and where

\[M_{4 \times 4}\]

is one of the following \(4 \times 4\) matrices:

\[ \begin{align}\begin{aligned}I_{4 \times 4}, \gamma_5, \beta \gamma_5,\\i\alpha_x, i\alpha_y, i\alpha_z\\\Sigma_x, \Sigma_y, \Sigma_z\\\beta \Sigma_x, \beta \Sigma_y, \beta \Sigma_z\\i \beta \alpha_x, i \beta \alpha_y, i \beta \alpha_z\end{aligned}\end{align} \]

One thing to notice is that an imaginary \(i\) is added to the time-antisymmetric Dirac \(\boldsymbol{\alpha}\) - matrices and their derivatives to make them time symmetric and hence fit into the quaternion symmetry scheme of DIRAC (see [Saue1999] and [Salek2005] for more information).

Operator types¶

There are 21 basic operator types used in DIRAC, listed in this Table:

Keyword	Operator form	Nr. factors
DIAGONAL	\(f I_{4 \times 4} \Omega\)	1
XALPHA	\(f \alpha_x \Omega\)	1
YALPHA	\(f \alpha_y \Omega\)	1
ZALPHA	\(f \alpha_z \Omega\)	1
XAVECTOR	\(f_1 \alpha_y \Omega_z - f_2 \alpha_z \Omega_y\)	2
YAVECTOR	\(f_1 \alpha_z \Omega_x - f_2 \alpha_x \Omega_z\)	2
ZAVECTOR	\(f_1 \alpha_x \Omega_y - f_2 \alpha_y \Omega_x\)	2
ALPHADOT	\(f_1 \alpha_x \Omega_x + f_2 \alpha_y \Omega_y + f_3 \alpha_z \Omega_z\)	3
GAMMA5	\(f \gamma_5 \Omega\)	1
XSIGMA	\(f \Sigma_x \Omega\)	1
YSIGMA	\(f \Sigma_y \Omega\)	1
ZSIGMA	\(f \Sigma_z \Omega\)	1
XBETASIG	\(f \beta \Sigma_x \Omega\)	1
YBETASIG	\(f \beta \Sigma_y \Omega\)	1
ZBETASIG	\(f \beta \Sigma_z \Omega\)	1
XiBETAAL	\(f i \beta \alpha_x \Omega\)	1
YiBETAAL	\(f i \beta \alpha_y \Omega\)	1
ZiBETAAL	\(f i \beta \alpha_z \Omega\)	1
BETA	\(f \beta \Omega\)	1
SIGMADOT	\(f_1 \Sigma_x \Omega_x + f_2 \Sigma_y \Omega_y + f_3 \Sigma_z \Omega_z\)	1
iBETAGAMMA5	\(f i \beta \gamma_5 \Omega\)	1

Operator specification¶

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

.OPERATOR
 'operator name'
 operator type keyword
 operator labels for each component
 FACTORS
 factors for each component
 CMULT
 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 operator label. Component factors as well as the common factor are all one if not specified.

List of one-electron operators¶

Operator label	Description	Symmetry	Components	Operators
MOLFIELD	Nuclear attraction integrals	Symmetric	MOLFIELD	\(\Omega_1 = \sum_K V_{iK}\)
OVERLAP	Overlap integrals	Symmetric	OVERLAP	\(\Omega_1 = 1\)
BETAMAT	Overlap integrals, only SS-block	Symmetric	BETAMAT	\(\Omega_1 = 1\)
DIPLEN	Dipole length integrals	Symmetric	XDIPLEN	\(\Omega_1 = x\)
			YDIPLEN	\(\Omega_2 = y\)
			ZDIPLEN	\(\Omega_3 = z\)
DIPVEL	Dipole velocity integrals	Anti-symmetric	XDIPVEL
			YDIPVEL
			ZDIPVEL
QUADRUP	Quadrupole moments integrals	Symmetric	XXQUADRU
			XYQUADRU
			XZQUADRU
			YYQUADRU
			YZQUADRU
			ZZQUADRU
SPNORB	Spatial spin-orbit integrals	Anti-symmetric	X1SPNORB
			Y1SPNORB
			Z1SPNORB
SECMOM	Second moments integrals	Symmetric	XXSECMOM	\(\Omega_1 = xx\)
			XYSECMOM	\(\Omega_2 = xy\)
			XZSECMOM	\(\Omega_3 = xz\)
			YYSECMOM	\(\Omega_4 = yy\)
			YZSECMOM	\(\Omega_5 = yz\)
			ZZSECMOM	\(\Omega_6 = zz\)

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

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 Dirac Hamiltonian may be specified by:

.OPERATOR
 'Kin energy'
 ALPHADOT
 XDIPVEL
 YDIPVEL
 ZDIPVEL
 COMFACTOR
 -68.51799475

where -68.51799475 is \(-c/2\).

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

.OPERATOR
 'Kin energy'
 ALPHADOT
 XDIPVEL
 YDIPVEL
 ZDIPVEL
 CMULT
 COMFACTOR
 -0.5

where the keyword CMULT assures multiplication of the common factor -0.5 by \(c\). This option has the further advantage that CMULT follows any user-specified modification of the speed of light, as provided by .CVALUE.

XAVECTOR¶

Another example:

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

The program will assume all operators to be Hermitian and will therefore insert an imaginary phase i if necessary (applies to antisymmetric scalar operators).

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

.OPERATOR
 OVERLAP

Dipole moment as finite field perturbation¶

Another example is the finite perturbation calculation with the \(\hat{z}\) dipole length operator added to the Hamiltonian (don’t forget to decrease the symmetry of your system):

\[\hat{H} = \hat{H}_0 + 0.01 \cdot \hat{z}\]

.OPERATOR
 ZDIPLEN
 COMFACTOR
 0.01

Fermi-contact integrals¶

Here is an example where the Fermi-contact (FC) integrals for a certain nucleus are added to the Hamiltonian in a finite-field calculation. Let’s assume you are looking at a PbX dimer (order in the .mol file: 1. Pb, 2. X) and you want to add to the Dirac-Coulomb **HAMILTONIAN the FC integrals for the Pb nucleus as a perturbation with a given field-strength (FACTORS).

Important note: The raw density values obtained after the fit of your finite-field energies need to be scaled by \(\frac{1}{\frac{4 \pi g_{e}}{3}} = \frac{1}{8.3872954891254192}\), a factor that originates from the definition of the operator for calculating the density at the nucleus:

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

Here is next example of how-to calculate the electron density at the nucleus as an expectation value \(\langle 0 \vert \delta(r-R) \vert 0 \rangle\) for a Dirac-Coulomb HF 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

Cartesian moment expectation value¶

In the following example I am calculating a cartesian moment expectation value \(\langle 0 \vert x^1 y^2 z^3 \vert 0 \rangle\) for a Levy-Leblond HF wave function:

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