Gauge Dependence of Magnetic Properties

A more recent implementation, which completely eliminates the gauge dependence, is to make the basis functions explicitly dependent on the magnetic field by inclusion of a complex phase factor refening to the position of the basis function (usually the nucleus). 

Such orbitals are known as London Atomic Orbitals (LAO) or Gauge Including/ Invariant Atomic Orbitals (GIAO). The effect is that matrix elements involving GIAOs 

The kinetic energy is slightly more complicated, but it can be shown that the relation shown in eq. (10.96) holds. 

Note that Rg has been replaced by Rb in the last bracket. The use of GIAOs as basis functions makes all matrix elements, and therefore all properties, independent of the gauge origin. The wave function itself, however, is expressed in term of the basis functions, and therefore becomes gauge dependent, by means of a complex phase factor. The use of perturbation-dependent basis functions has the further advantage of greatly 

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To understand the difference between the spurious gauge terms (i), which are removed in the ideal case of optimal variational wavefunctions [65], and terms (ii), which account for the essential origin dependence of the property, let us first discuss die origin dependence of the quadrupole polarizability of magnetic susceptibility (22) within the conventional common-origin representation. 

Here (r - Rc) (r - Rq) is the dot product times a unit matrix (i.e. (r — Rg) (r — Rg)I) and (r - RG)(r — Rg) is a 3x3 matrix containing the products of the x,y,z components, analogous to the quadrupole moment, eq. (10.4). Note that both the L and P operators are gauge dependent. When field-independent basis functions are used the first-order property, the HF magnetic dipole moment, is given as the expectation value over the unperturbed wave funetion (for a singlet state) eqs. (10.18)/(10.23). 

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