# Nuclear hyperfine structure¶

Provided nuclear hyperfine interaction curves, Duo calculates field-free hyperfine structure of diatomic molecules. Currently, Duo supports cases where one of the nuclei possess nuclear spin, e.g., 14N16O and 24MgH. Nuclear electric quadrupole interaction and nuclear magnetic dipole interactions including Fermi-contact, nuclear spin-electron spin dipole-dipole, nuclear spin-orbit, nuclear spin-rotation, can be involved in the calculation. The hyperfine calculation is turn on with the following section:

```
hyperfine
I 1
end
```

where the value after the keyword `I`

indicates the nuclear spin.

Two output files are generated after calculation.
The one named `hyperfine.states`

contains nuclear hyperfine resolved states.
The columns in this file are:
1. counting number, 2. energy [cm-1], 3. total degeneracy,
4. , 5. ,
6. parity, 7. , 8. state, 9. ,
10. , 11. ,
12. , respectively.

The other named `hyperfine.trans`

contains the hyperfine transitions.
This file has five columns which are:
1.counting number of the upper state,
2.counting number of the lower state,
3.Einstein-A coefficient,
4.transition wavenumber [cm-1],
5.line strength.
Line strengths in this file are calculated by

where and are the wavefunctions of the hyperfine states. is the tensor of transition electric dipole moment. Line strengths have the unit of .

An electric dipole moment curve should be defined in a `dipole`

section in the first place.

```
dipole 1 1
name "<X,2Pi|DMC|X,2Pi>"
spin 0.5 0.5
lambda 1 1
factor 1 (0, 1 or i)
type polynom
values
A0 1
end
```

You may turn off the hyperfine calculation by:

```
hyperfine none
I 1
end
```

The global setup of `J`

,

```
jrot 0.5 - 3.5
```

affects the maximum of `F`

.
.
The minimum of `F`

is always 0 or 1/2.

Currently, Duo does not support refinement of hyperfine curves.
Thus, `fitting`

and `hyperfine`

sections cannot be activated simultaneously.

The nuclear hyperfine interaction curves are introduced with the following seven key words:
`hfcc-a`

, `hfcc-bF`

, `hfcc-c`

, `hfcc-d`

, `hfcc-ci`

, `hfcc-eqq0`

and `hfcc-eqq2`

.
The default units are `angstrom`

and `cm-1`

.
Hyperfine couplings between electronic states are not allowed at present.

`hfcc-a`

¶

Nuclear spin - orbit interaction curve is defined by an `hfcc-a`

section.
This term is important when the electron orbital angular momentum is non-zero.

```
hfcc-a 1 1
name "<X2Pi|NSO|X2Pi>" (Nuclear spin-orbit)
spin 0.5
factor 1.0
type polynom
values
A0 0.1
end
```

`hfcc-bF`

¶

The Fermi-contact interaction curve is defined by an `hfcc-bF`

section.
This term is important when the electron spin angular momentum is non-zero.

```
hfcc-bf 1 1
name "<X2Pi|FC|X2Pi>" (Fermi-contact)
spin 0.5
factor 1.0
type polynom
factor 1
values
A0 0.1
end
```

`hfcc-c`

and `hfcc-d`

¶

The electron spin - nuclear spin dipole-dipole
interaction curves and are defined by `hfcc-c`

and `hfcc-d`

sections.
The constant defined by Frosch and Foley can be
calculated by
.

```
hfcc-c 1 1
name "<X2Pi|SDND_C|X2Pi>" (Electron spin - nuclear spin dipole-dipole, c)
spin 0.5
factor 1.0
type polynom
values
A0 0.1
end
hfcc-d 1 1
name "<X2Pi|SDND_D|X2Pi>" (Electron spin - nuclear spin dipole-dipole, d)
spin 0.5
factor 1.0
type polynom
values
A0 0.1
end
```

`hfcc-eqq0`

and `hfcc-eqq2`

¶

The nuclear electric quadrupole
interaction curves and are defined by `hfcc-eqq0`

and `hfcc-eqq2`

sections.
These terms are active when
the nuclear spin is not less than 1.

```
hfcc-eqq0 1 1
name "<X2Pi|eQq0|X2Pi>" (Electric quadrupole eQq0)
spin 0.5
factor 1.0
type polynom
values
A0 0.1
end
hfcc-eqq2 1 1
name "<X2Pi|eQq2|X2Pi>" (Electric quadrupole eQq2)
spin 0.5
factor 1.0
type polynom
values
A0 0.1
end
```

`hfcc-ci`

¶

The nuclear spin - rotation curve is defined by an `hfcc-ci`

section.
This term is usually negligible compared with other nuclear hyperfine interactions.
Nevertheless,
when all the other hyperfine couplings are inactive,
this term becomes important, e.g.
for a state of a nuclear spin 1/2 molecule.

```
hfcc-ci 1 1
name "<X2Pi|NSR|X2Pi>" (Nuclear spin - rotation)
spin 0.5
factor 1.0
type polynom
values
A0 0.1
end
```