Spin Hamiltonian

The stationary-state Hamiltonian for treatment of the interaction of an electron spin (S=1/2) with one nuclear spin () is formulated with a hf coupling term, a nuclear Zeeman term, and a nqi term (for , as follows:

where , , and  are the nuclear magneton, electron spin operator, and nuclear spin operator, respectively, is the nuclear g-value, A is the hf coupling tensor, and Q is the nqi tensor. The hf tensor has the principal components, , and is composed of an isotropic part , and a dipolar part.  In the point dipole approximation, the dipolar part is given by an axially symmetric dipolar tensor, , where , and , , and  are the electron g-value, Bohr magneton, and an effective distance between the electron and the nuclear spins, respectively.  The hf tensor  is rotated to the molecular frame, which is defined by a reference principal axis system (PAS) that is provisionally related to the electron g tensor PAS, by the following operation:

where    is a rotation matrix, which is defined by the Euler angles, , to rotate from the electron g tensor PAS to the hf coupling tensor PAS.

 The nqi tensor, Q, is defined by the nuclear quadrupole coupling constant, , and electron field gradient (efg) asymmetry parameter, η, where e, q, and Q are the elementary charge, the magnitude of the principal component of the efg tensor, and the nuclear quadrupole moment, respectively.  In its PAS, the traceless nqi tensor  is related to   and η by the following expressions:

where .  The hf and nqi PAS are not, in general, aligned.  In the toolbox, the orientation between the nqi tensor PAS and the hf tensor PAS is defined by the Euler angles , which consequently define a rotational matrix, , and nqi tensor Q in the molecular PAS (g tensor PAS) to be expressed as follows:

This definition of Q by using a two-stage rotation allows an additional constraint on the mutual orientations of A and Q during numerical optimization, which adds to the flexibility in the specification of the geometry model for multiple electron-nuclear interactions.

To summarize, the general coupled electron – single nucleus system is parameterized by using the following thirteen parameters: , , , , , , , , , η, , , and .  In practice, the coupled nucleus is assigned based on characteristic spectral features or by knowledge of the system, so that and  are fixed, which entails eleven adjustable parameters.  With the option to simplify the hf coupling to the point–dipole approximation, the Hamiltonian of one coupled nucleus is defined by ten parameters.  In the simulation, all of the parameters can be subjected to the numerical optimization.