One-electron operators

Syntax for the specification of one-electron operators

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

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

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

\[M_{4 \times 4}\]

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

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

Operator types

There are 19 basic operator types used in DIRAC, liste in this Table:

Keyword Operator form Nr. factors
DIAGONAL \(f I_{4 \times 4} \Omega\) 1
GAMMA5 \(f \gamma_5 \Omega\) 1
XALPHA \(f \alpha_x \Omega\) 1
YALPHA \(f \alpha_y \Omega\) 1
ZALPHA \(f \alpha_z \Omega\) 1
XSIGMA \(f \Sigma_x \Omega\) 1
YSIGMA \(f \Sigma_y \Omega\) 1
ZSIGMA \(f \Sigma_z \Omega\) 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
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
BETAGAMMA5 \(f i \beta \gamma_5 \Omega\) 1

Operator specification

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

 'operator name'
 operator type
 labels for each component
 factors for each component
 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.

List of one-electron operators

Keyword 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  
QUADRUP Quadrupole moments integrals Symmetric XXQUADRU  
SPNORB Spatial spin-orbit integrals Anti-symmetric X1SPNORB  
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 origin
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 at the nuclei
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:

 'Kin energy'

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:

 'Kin energy'

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.


Another example:


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:


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):

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

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:

 'Density at nucleus'
 'FC Pb 01'

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:


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: