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:

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

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 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:

.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