# 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 | ||||

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 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:

```
.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
```