Oneelectron operators¶
Syntax for the specification of oneelectron operators¶
A general oneelectron property operator in 4component calculations is generated from linear combinations of the basic form:
with the scalar factor \(f\) and the scalar operator \(\hat{\Omega}\), and where
is one of the following \(4 \times 4\) matrices:
One thing to notice is that an imaginary \(i\) is added to the timeantisymmetric 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 oneelectron 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 SSblock 
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 
Antisymmetric 
XDIPVEL 

YDIPVEL 

ZDIPVEL 

SPNORB 
Spatial spinorbit integrals 
Antisymmetric 
X1SPNORB 

Y1SPNORB 

Z1SPNORB 

QUADRUP 
Quadrupole moments integrals 
Symmetric 
XXQUADRU 
\(\Omega_1 = \frac{1}{4} (x^2r^2)\) 
XYQUADRU 
\(\Omega_2 = \frac{1}{4} xy\) 

XZQUADRU 
\(\Omega_3 = \frac{1}{4} xz\) 

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

YZQUADRU 
\(\Omega_5 = \frac{1}{4} yz\) 

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

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 
Traceless quadrupole integrals 
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 spinorbit 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 
Oneelectron Fermi contact integrals 
HBDO 
Half Bdifferentiated overlap matrix 
HDO 
Halfderivative overlap integrals 
HDOBR 
Ketdifferentiation of HDOintegrals with respect to magnetic field 
KINENER 
Electronic (nonrelativistic) kinetic energy (operator: \(\nabla^2\)) 
LONMOM 
London orbital contribution to angular momentum 
MAGMOM 
Oneelectron 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 spinorbit integrals 
S1MAG 
Second order contribution from overlap matrix to magnetic properties 
S1MAGL 
Bradifferentiation of overlap matrix with respect to magnetic field 
S1MAGR 
Ketdifferentiation of overlap matrix with respect to magnetic field 
SDFC 
Spindipole + Fermi contact integrals 
SOLVENT 
Electronic solvent integrals 
SPHMOM 
Spherical moments integrals (real combinations), symmetric integrals, (2l + 1) components ( m = +0, 1, +1, …, +l) 
SPINDI 
Spindipole integrals 
SQHDO 
Halfderivative overlap integrals not to be antisymmetrized 
SQHDOR 
Halfderivative overlap integrals not to be antisymmetrized 
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 userspecified 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):
.OPERATOR
ZDIPLEN
COMFACTOR
0.01
Fermicontact integrals¶
Here is an example where the Fermicontact (FC) integrals for a certain nucleus are added to the Hamiltonian in a finitefield 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 DiracCoulomb **HAMILTONIAN the FC integrals for the Pb nucleus as a perturbation with a given fieldstrength (FACTORS).
Important note: The raw density values obtained after the fit of your finitefield 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 howto calculate the electron density at the nucleus as an expectation value \(\langle 0 \vert \delta(rR) \vert 0 \rangle\) for a DiracCoulomb 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 LevyLeblond HF wave function:
**DIRAC
.WAVE FUNCTION
.PROPERTIES
**HAMILTONIAN
.LEVYLEBLOND
**WAVE FUNCTION
.SCF
**PROPERTIES
*EXPECTATION
.OPERATOR
CM010203
*END OF