Nuclear hyperfine structure

Provided nuclear hyperfine interaction curves, Duo calculates field-free hyperfine structure of diatomic molecules. Currently, Duo supports two cases:

  1. only one of the nuclei possess non-zero nuclear spin, e.g., 14N16O and 24MgH.

  2. two identical nuclei with non-zero nuclear spin, e.g., H2 and D2.

Nuclear electric quadrupole interaction and nuclear magnetic dipole interactions including Fermi-contact, nuclear spin-electron spin dipole-dipole, nuclear spin-orbit, nuclear spin-rotation, nuclear spin-nuclear spin dipole-dipole, can be involved in the calculation. The hyperfine calculation is turned on with the following section:

hyperfine
    I 1
end

where the value after the keyword I indicates the nuclear spin of each individual nucleus.

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. F, 5. symmetry label, 6. total nuclear spin I, 7. J, 8. state, 9. v, 10. \Lambda, 11. \Sigma, 12. \Omega, 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

S = |\langle \psi(m,\tau, F)||T^{(1)}(\mu)||\psi_(m', \tau', F') \rangle|^2

where \psi(m,\tau, F) and \psi_(m', \tau', F') are the wavefunctions of the hyperfine states. T^{(1)}(\mu) is the tensor of transition electric dipole moment. Line strengths have the unit of Debye^2.

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. F_\mathrm{max} = J_\mathrm{max}-I_\mathrm{max}, where I_\mathrm{max} is the maximum value of total nuclear spin. 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 eight key words: hfcc-a, hfcc-bF, hfcc-c, hfcc-d, hfcc-ci, hfcc-eqq0, hfcc-eqq2 and hfcc-ii. 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 b_F(R) 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 c(R) and d(R) are defined by hfcc-c and hfcc-d sections. The b constant defined by Frosch and Foley can be calculated by b = b_F - c/3.

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 eQq_0(R) and eQq_2(R) 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 c_I(R) 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 ^1\Sigma 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

hfcc-ii

The nuclear spin-nuclear spin dipole-dipole interaction curve is defined by an hfcc-ii section. This term is used for homonuclear molecules with two equivalent non-zero nuclear spins.

hfcc-ii 1
name "<X1Sigma+|II|X1Sigma+>" (Nuclear spin-nuclear spin dipole-dipole)
factor 1.0
type polynom
values
    A0           0.1
end