Manual:One-electron operators

From Diracwiki

Jump to: navigation, search

Contents


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 Ω, and where M_{4 \times 4} is one of the following 4 \times 4 matrices:

I_{4 \times 4}, \gamma_5,

αx,αy,αz,

Σx,Σy,Σz,

βΣx,βΣy,βΣz,

iβαx,iβαy,iβαy

Operator types

There is 19 basic operator types used in DIRAC.

Operator types
Keyword Operator form Nr. Factors
DIAGONAL f I_{4 \times 4} \Omega 1
GAMMA5 fγ5Ω 1
XALPHA fαxΩ 1
YALPHA fαyΩ 1
ZALPHA fαzΩ 1
XSIGMA fΣxΩ 1
YSIGMA fΣyΩ 1
ZSIGMA fΣzΩ 1
XAVECTOR f1αyΩz − f2αzΩy 2
YAVECTOR f1αzΩx − f2αxΩz 2
ZAVECTOR f1αxΩy − f2αyΩx 2
ALPHADOT f1αxΩx + f2αyΩy + f3αzΩz 3
XBETASIG fβΣxΩ 1
YBETASIG fβΣyΩ 1
ZBETASIG fβΣzΩ 1
XiBETAAL fiβαxΩ 1
YiBETAAL fiβαyΩ 1
ZiBETAAL fiβαzΩ 1
BETAGAMMA5 fiβγ5Ω 1

Operator specification

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

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

List of one-electron operators

Keyword Components
MOLFIELD Nuclear attraction integrals (Symmetric) MOLFIELD \Omega_1 = \sum_K \hat{V}_{iK}
OVERLAP Overlap integrals (Symmetric) OVERLAP Ω1 = 1
BETAMAT Overlap integrals, only SS-block (Symmetric) BETAMAT Ω1 = 1
DIPLEN Dipole length integrals (Symmetric) XDIPLEN Ω1 = x
YDIPLEN Ω2 = y
ZDIPLEN Ω3 = z
DIPVEL Dipole velocity integrals (Anti-symmetric) XDIPVEL \Omega_1 = \frac{\partial}{\partial x}
YDIPVEL \Omega_2 = \frac{\partial}{\partial y}
ZDIPVEL \Omega_3 = \frac{\partial}{\partial z}
QUADRUP Quadrupole moments integrals (Symmetric) XXQUADRU \Omega_1 = \frac{\partial^2}{\partial x \partial x}
XYQUADRU \Omega_2 = \frac{\partial^2}{\partial x \partial y}
XZQUADRU \Omega_3 = \frac{\partial^2}{\partial x \partial z}
YYQUADRU \Omega_4 = \frac{\partial^2}{\partial y \partial y}
YZQUADRU \Omega_5 = \frac{\partial^2}{\partial y \partial z}
ZZQUADRU \Omega_6 = \frac{\partial^2}{\partial z \partial z}
SPNORB Spatial spin-orbit integrals (Anti-symmetric) X1SPNORB \Omega_1 = \sum_K \frac{l_x}{r_{iK}^3}
Y1SPNORB \Omega_2 = \sum_K \frac{l_y}{r_{iK}^3}
Z1SPNORB \Omega_3 = \sum_K \frac{l_z}{r_{iK}^3}
SECMOM Second moments integrals (Symmetric) XXSECMOM Ω1 = xx
XYSECMOM Ω1 = xy
XZSECMOM Ω1 = xz
YYSECMOM Ω1 = yy
YZSECMOM Ω1 = yz
ZZSECMOM Ω1 = zz
THETA Traceless theta quadrupole integrals
CARMOM Cartesian moments integrals,Symmetric integrals,(l + 1)(l + 2)/2 components ( l = i + j + k) ; see also the example further below Ωl = xiyjzk
SPHMOM Spherical moments integrals (real combinations), symmetric integrals, (2l+1) components, ( m = + 0, - 1, + 1,..., + l) \Omega_l = R_{l \pm _m}
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

add more from http://dirac.chem.sdu.dk/dirlab/Dirlabdoc/Master.html ...

Examples of using various operators

We give here several concrete examples on how to construct operators for various properties.

XAVECTOR

An example: 

.OPERATOR
 'B_x'
 XAVECTOR
 ZDIPLEN
 YDIPLEN
 COMFACTOR
 -68.517999904721

Kinetic part of the Dirac Hamiltonian

As another example 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 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 parturbation

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

... and here is an 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
.DHF
**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
.DHF
**PROPERTIES
*EXPECTATION
.OPERATOR
 CM010203
*END OF
Retrieved from "http://wiki.chem.vu.nl/dirac/index.php/Manual:One-electron_operators"
Personal tools