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:
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 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 |
||||
SPNORB |
Spatial spin-orbit integrals |
Anti-symmetric |
X1SPNORB |
|
Y1SPNORB |
||||
Z1SPNORB |
||||
QUADRUP |
Quadrupole moments integrals |
Symmetric |
XXQUADRU |
\(\Omega_1 = \frac{1}{4} (x^2-r^2)\) |
XYQUADRU |
\(\Omega_2 = \frac{1}{4} xy\) |
|||
XZQUADRU |
\(\Omega_3 = \frac{1}{4} xz\) |
|||
YYQUADRU |
\(\Omega_4 = \frac{1}{4} (y^2-r^2)\) |
|||
YZQUADRU |
\(\Omega_5 = \frac{1}{4} yz\) |
|||
ZZQUADRU |
\(\Omega_6 = \frac{1}{4} (z^2-r^2)\) |
|||
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 |
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 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'
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):
.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