# Parity-violation contribution to nuclear spin-rotation constants¶

## Introduction¶

In this tutorial we introduce the calculation of parity violation contribution to nuclear spin-rotation tensors as given in the DIRAC code.

The parity-violation contribution to nuclear spin-rotation (PVCNSR) tensor of a nucleus $$N$$ is given by

${\bf M}_N^{PV} = \; \frac{-\hslash}{h} \frac{G_F \, (1-4sen^2\theta_W)}{\sqrt{2} \, c} \, \lambda_N \; \langle \langle \rho_N({\bf r}) \, c \, {\bf \alpha} \; ; \; {\bf J}_e\rangle\rangle \; \cdot \; {\bf I}^{-1}$

where $$\rho_N({\bf r})$$ is the normalized nuclear charge density; $${\bf I}$$ is the inertia tensor of the molecule and $${\bf J}_e = \left({\bf r} - {\bf R}_{CM} \right) \times {\bf p}+{\bf S}_e$$ is the electronic total angular momentum.

## Application to the H2O2 molecule¶

As an example, we show a calculation of the PV contribution to the NSR constant of the oxygen nucleus at the Hydrogen peroxide molecule. The input file pvcsr.inp is given by

**DIRAC
.TITLE
Parity-violation contribution to spin-rotation
.WAVE FUNCTION
.PROPERTIES
**HAMILTONIAN
.URKBAL
**INTEGRALS
.SELECT
1
3
.UNCONTRACT
**WAVE FUNCTIONS
.SCF
*SCF
.EVCCNV
1.0E-7
**PROPERTIES
.PVCSR
.PRINT
2
*LINEAR RESPONSE
.THRESH
1.0D-7
*END OF


whereas the molecular input file H2O2.mol is

INTGRL
H2O2 molecule
cc-pVDZ basis
C   2    0
1.     2
H1    -0.690815           -1.726149            1.667775
H2     0.690815            1.726149            1.667775
LARGE BASIS cc-pVDZ
8.     2
O3     0.000000           -1.407846            0.000000
O4     0.000000            1.407846            0.000000
LARGE BASIS cc-pVDZ
FINISH


The calculation is run using:

pam --inp=pvcsr --mol=H2O2


As a result, PVCNSR are obtained at the coupled Hartree-Fock level. The code also works at the DFT level.

As the .PRINT flag in the input file is set to 1, the results are fully detailed and given in Hz.

The PVCNSR of the oxygen nucleus will look like:

Parity-violation contribution to nuclear spin-rotation constant (Hz) for the zero-value g-factor nucleus O3
-----------------------------------------------------------------------------------------------------------

WARNING: Nuclear g-value NOT available for isotope of mass   15.995

Total PVC to nuclear spin-rotation constant
-------------------------------------------

A       -0.17995229E-05  0.38632077E-05  0.55199909E-05
B       -0.58143612E-04  0.16007781E-05 -0.10392856E-03
C       -0.11600712E-03  0.11286829E-03 -0.17978966E-05

iso     -0.66554712E-06

M^LR-L(e-e)
-----------

A       -0.18709065E-05  0.38692922E-05  0.55349517E-05
B       -0.58168472E-04  0.15874564E-05 -0.10396608E-03
C       -0.11602249E-03  0.11290509E-03 -0.17906721E-05

iso     -0.69137409E-06

M^LR-S(e-e)
-----------

A        0.71236835E-07 -0.52291482E-08 -0.13478821E-07
B        0.13889115E-07  0.13475745E-07  0.14222802E-07
C       -0.65750397E-08 -0.11806627E-07 -0.73909112E-08

iso      0.25773890E-07

M^LR-L(e-p)
-----------

A        0.14586154E-09 -0.88205576E-09 -0.15419970E-08
B        0.11276044E-07 -0.15416722E-09  0.23348137E-07
C        0.22681711E-07 -0.25045439E-07  0.16651050E-09

iso      0.52734939E-10

M^LR-S(e-p)
-----------

A        0.96137869E-12  0.26634709E-10  0.60015457E-10
B       -0.30494980E-09  0.15492645E-12 -0.54368827E-10
C       -0.73067273E-09  0.57770647E-10 -0.81830548E-13

iso      0.34482486E-12


As it can be seen, the total SR constant of the oxygen nucleus is given. The linear response function is further separated in their (e-e) and (e-p) parts, as well as in their $$\mathbf{L}$$ and $$\mathbf{S}$$ parts.