Gauge origins

The first term is referred to as the diamagnetic contribution, while the latter is the paramagnetic part of the magnetizability. Each of the two components depend on the selected gauge origin however, for exact wave functions these cancel exactly. For approximate wave functions this is not guaranteed, and as a result the total property may depend on where the origin for the vector potential (eq. (10.61)) has been chosen. 

Here Iais the magnetic moment of nucleus A and Ra is the position (the nucleus is the natural Gauge origin). Adding this to the external vector potential in eq. (10.62) and expanding as in (10.63) gives... 

In the Hamiltonian conventionally used for derivations of molecular magnetic properties, the applied fields are represented by electromagnetic vector and scalar potentials [1,20] and if desired, canonical transformations are invoked to change the magnetic gauge origin and/or to introduce electric and magnetic fields explicitly into the Hamiltonian, see e.g. refs. [1,20,21]. Here we take as our point of departure the multipolar Hamiltonian derived in ref. [22] without recourse to vector and scalar potentials. 

Here Uj and Uj are Cartesian unit vectors, a) and j3) are localized orbitals that are doubly occupied in the HF ground state, jm) and n) are virtual orbitals. Rq is the position vector of the local gauge origin assigned to orbital a) and = (r — R ) x p is the angular momentum relative to Re- Superscript 1 denotes terms to first order in the fluctuation potential, and = [A — is the principal propagator at the zero energy... 

For low-pressure structures that are capable of withstanding pressures of not more than 1.5 psi (0.1 bar gauge), original design techniques were based on the Runes (pronounced Roo-ness) equation16 ... 

The magnetic field is independent of the choice of the gauge origin. So too are the computed magnetic properties if the wave function used is exact. Regrettably, we are not often afforded the opportunity to work with exact wave functions. For HF wave functions, one can also achieve independence of the gauge by using an infinite basis set, but that is hardly a practical option either. 

TABLE 8. Calculated diamagnetic, paramagnetic and total 29Si chemical shielding tensors a (in ppm, gauge origin at Si) compared with experimental chemical shifts, r... 

The basic computational method is that of coupled Hartree-Fock perturbation theory (14). At present we prefer the GIAO implementation mentioned above because of its computational efficiency and ease of use, but we have previously used a common gauge-origin method as implemented in the software SYSMO (15) as well as the random-phase approximation, localized orbital (RPA LORG) approach as implemented in the software RPAC (16). 

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