Hamiltonian zero-field

As an illustration consider then a zero-field Hamiltonian for S = 4 in which we have retained only the familiar axial D- and rhombic E-term plus the cubic terms that split the non-Kramer s doublets in first order ... 

The bistable character of the zero-field Hamiltonian will be destroyed by a sufficiently large field. The critical field for n h is called the anisotropy field, and is expressed by... 

Using the zero field n(m states we calculate the matrix elements (/i m Ez n m ) of the Stark perturbation to the zero field Hamiltonian. Writing the matrix element in spherical coordinates and choosing the z axis as the axis of quantization,... 

We have chosen to use the hyperfine-coupled representation, where for 12CH, F is equal to J 1 /2. An appropriate basis set is therefore t], A N, A S, J, /, F), with MF also important when discussing Zeeman effects. As usual the effective zero-field Hamiltonian will be, at the least, a sum of terms representing the spin-orbit coupling, rigid body rotation, electron spin-rotation coupling and nuclear hyperfine interactions, i.e. 

Problem 13.4. The zero-field Hamiltonian (Hzf) of two-interacting triplet molecules can be represented by... 

The energies ( ) of the zero-field Hamiltonian of the singlet, triplet, and qui calculated with the following formulae [18] ... 

Among the off-diagonal elements for the Zeeman terms (Hz) in the eigenbase of the zero field Hamiltonian, the following ones are only non-vanishing ... 

Although the determination of the zero field eigenvalues and wavefunctions is described in many standard texts on rotational spectroscopy, we will briefly recall the principles and give some results for later reference. From Eq. (IV. 59 a) and IV.59b), the zero field Hamiltonian of an asymmetric top molecule is given by (from now on quantum mechanical operators will be denoted by underlining) ... 

The zero-field Hamiltonian H0 includes the spin-orbit and spin-spin interactions and eventually the crystal field operators the first-order Zeeman term n comprises the orbital and spin contributions. The matrix elements of the full Hamiltonian in the chosen basis set are... 

In the presence of the magnetic field the Zeeman matrices are added to the zero-field Hamiltonian matrix. As the z-component of the Zeeman interaction enters the diagonal, factorisation to the 2 x 2 secular equations is possible and from the analytical roots the identification of the van Vleck coefficients is possible... 

In this fortunate case the zero-field Hamiltonian is diagonal, giving rise to energy levels... 

The zero-field Hamiltonian appropriate for an exchange interaction in the dinuclear system adopts the form of... 

In a certain fortunate case the molecular-state zero-field Hamiltonian adopts a diagonal form. Then a vector addition model assumes the validity. 

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