# 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 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

$$f_1 \alpha_x \Omega_x + f_2 \alpha_y \Omega_y + f_3 \alpha_z \Omega_z$$

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

$$f_1 \Sigma_x \Omega_x + f_2 \Sigma_y \Omega_y + f_3 \Sigma_z \Omega_z$$

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

SPNORB

Spatial spin-orbit integrals

Anti-symmetric

X1SPNORB

Y1SPNORB

Z1SPNORB

Symmetric

$$\Omega_1 = \frac{1}{4} (x^2-r^2)$$

$$\Omega_2 = \frac{1}{4} xy$$

$$\Omega_3 = \frac{1}{4} xz$$

$$\Omega_4 = \frac{1}{4} (y^2-r^2)$$

$$\Omega_5 = \frac{1}{4} yz$$

$$\Omega_6 = \frac{1}{4} (z^2-r^2)$$

SECMOM

Second moments integrals

Symmetric

XXSECMOM

$$\Omega_1 = - x^2$$

XYSECMOM

$$\Omega_2 = - xy$$

XZSECMOM

$$\Omega_3 = - xz$$

YYSECMOM

$$\Omega_4 = - y^2$$

YZSECMOM

$$\Omega_5 = - yz$$

ZZSECMOM

$$\Omega_6 = - z^2$$

THETA

Symmetric

XXTHETA

$$\Omega_1 = - \frac{3}{2} (x^2-\frac{1}{3}r^2)$$

XYTHETA

$$\Omega_2 = - \frac{3}{2} xy$$

XZTHETA

$$\Omega_3 = - \frac{3}{2} xz$$

YYTHETA

$$\Omega_4 = - \frac{3}{2} (y^2-\frac{1}{3}r^2)$$

YZTHETA

$$\Omega_5 = - \frac{3}{2} yz$$

ZZTHETA

$$\Omega_6 = - \frac{3}{2} (z^2-\frac{1}{3}r^2)$$

Keyword

Description

ANGLON

Angular momentum around the nuclei

ANGMOM

Electronic angular momentum around the molecular center of mass

CARMOM

Cartesian moments integrals, symmetric integrals, (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 integrals

FERMI C

One-electron Fermi contact integrals

HBDO

Half B-differentiated overlap matrix

HDO

Half-derivative overlap integrals

HDOBR

Ket-differentiation of HDO-integrals with respect to magnetic field

KINENER

Electronic (non-relativistic) kinetic energy (operator: $$\nabla^2$$)

LONMOM

London orbital contribution to angular momentum

MAGMOM

One-electron contributions to magnetic moment

MASSVEL

Mass velocity integrals

NEFIELD

Electric field at the individual nuclei

NSTCGO

Nuclear shielding integrals with common gauge origin

NUCPOT

Potential energy of the interaction of electrons with individual nuclei, divided by the nuclear charge

NUCSHI

Nuclear shielding tensor integrals

NUCSLO

London orbital contribution to nuclear shielding tensor integrals

NUCSNLO

Nuclear shielding integrals without London orbital contribution

PSO

Paramagnetic spin-orbit integrals

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 integrals

SOLVENT

Electronic solvent integrals

SPHMOM

Spherical moments integrals (real combinations), symmetric integrals, (2l + 1) components ( m = +0, -1, +1, …, +l)

SPIN-DI

Spin-dipole integrals

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

## 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'
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'
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