Parity-violation contribution to electric field gradient
Introduction
In this tutorial we introduce the calculation of parity violation contribution to the electric field gradient tensor as given in the DIRAC code.
As described in PROPERTIES, the parity-violation contribution to the electronic part of the electric field gradient tensor (PVCEFG) of a nucleus \(K\) is given by
where the contribution from the K nucleus to the nuclear spin-independent parity-violating operator
is expressed in terms of the Fermi coupling constant \(G_F=2.222516\times 10^{-14}E_ha_0^3\) (this value corresponds to CODATA 2022, but it can change if you use other CODATA set), the weak nuclear charge \(Q_{w,K}\) and the normalized nuclear charge density \(\rho_K\) (in units of the inverse of cube distances). Results are given in a.u. Further details can be found in [Aucar2025_PVEFG].
Application to the H2O2 molecule
As an example, we show a calculation of the PV contribution to the EFG tensor of the oxygen and hydrogen nuclei at the Hydrogen peroxide molecule. The input file efg_dc.inp is given by
**DIRAC
.TITLE
PVC to EFG
.WAVE FUNCTION
.PROPERTIES
**INTEGRALS
.SELECT
2
1
3
**GENERAL
.CODATA
CODATA22
**WAVE FUNCTIONS
.SCF
*SCF
.ERGCNV
5.0E-9
**PROPERTIES
.PVC
.PVCEFG
.ABUNDANCIES
0.0001
.PRINT
1
*END OF
whereas the molecular input file H2O2.mol is
INTGRL
H2O2 molecule
3-21G basis
C 2 0
1. 2
H1 -0.690815 -1.726149 1.667775 isotope=1
H2 0.690815 1.726149 1.667775 isotope=1
LARGE BASIS 3-21G
8. 2
O3 0.000000 -1.407846 0.000000 isotope=17
O4 0.000000 1.407846 0.000000 isotope=17
LARGE BASIS 3-21G
FINISH
The calculation is run using:
pam --inp=efg_dc --mol=H2O2
As a result, PVCEFG are obtained at the RPA (i.e., coupled Hartree-Fock) level of approach. The code also works at the DFT level.
Reading the output file
As the .PRINT flag in the input file is set to 1, extra details are printed. The results are given in au. The PVCEFG to the oxygen and hydrogen nuclei will look like:
**********************************************************************
**************** PARITY-VIOLATION CONTRIBUTION TO EFG ****************
**********************************************************************
According to J. J. Aucar and A. F. Maldonado,
Parity Violation Effects on the Electric Field Gradient,
Phys. Chem. Chem. Phys., 27, 7594 (2025)
--------------------------------------------------------
@ Nuclear species:
===============================================================
@ name charge isotope mass abundance Qw
---------------------------------------------------------------
@ H1 1 1 1.000000 99.985 0.0751
@ 2 2.000000 0.015 -0.9249
@ O3 8 1 16.000000 99.760 -7.3990
@ 2 18.000000 0.200 -9.3990
@ 3 17.000000 0.040 -8.3990
-------------------------------------------------------------
@ Selected nuclear species:
===============================================================
@ name charge isotope mass abundance Qw
---------------------------------------------------------------
@ H1 1 1 1.000000 99.985 0.0751
@ O3 8 3 17.000000 0.040 -8.3990
-------------------------------------------------------------
===========================================================================================
Nucleus: H1
Total PV contributions to EFG tensor (a.u.):
--------------------------------------------
x y z
x -0.254391E-22 0.239946E-22 -0.230101E-22
y 0.239946E-22 -0.222935E-22 -0.121943E-22
z -0.230101E-22 -0.121943E-22 0.477326E-22
e-p and e-e response contributions (a.u.):
------------------------------------------
x y z x y z
x 0.398466E-25 -0.817881E-25 0.414748E-26 -0.254789E-22 0.240764E-22 -0.230143E-22
y -0.817881E-25 -0.531662E-25 -0.174272E-25 0.240764E-22 -0.222404E-22 -0.121769E-22
z 0.414748E-26 -0.174272E-25 0.133196E-25 -0.230143E-22 -0.121769E-22 0.477193E-22
===========================================================================================
Nucleus: O3
Total PV contributions to EFG tensor (a.u.):
--------------------------------------------
x y z
x 0.770225E-17 -0.122332E-17 -0.182415E-17
y -0.122332E-17 -0.697042E-17 -0.454296E-17
z -0.182415E-17 -0.454296E-17 -0.731824E-18
e-p and e-e response contributions (a.u.):
------------------------------------------
x y z x y z
x 0.144288E-19 0.482375E-20 -0.146931E-19 0.768782E-17 -0.122814E-17 -0.180945E-17
y 0.482375E-20 0.157480E-21 0.326208E-20 -0.122814E-17 -0.697058E-17 -0.454623E-17
z -0.146931E-19 0.326208E-20 -0.145862E-19 -0.180945E-17 -0.454623E-17 -0.717238E-18
===========================================================================================
As it can be seen, the total PVC to the EFG tensor elements is further separated in their (e-e) and (e-p) parts.
Warning
The basis set used here is rather small for demonstration purposes; larger basis sets are recommended for production calculations.