Small gauge transformations

If we define the vector potential of a homogeneous magnetic field B as A(r) = B X (r — Ro)/2, then the gauge origin Rq dependence can be understood as a gauge transformation with the gauge function defined by x = —(B X Ro) rl1 (compare section 2.4). This vector potential then produces terms that depend on the arbitrary position Rq- Only for exact wave functions do these terms vanish (complete basis set), while they carmot be neglected in any (small) finite one-electron basis set. 

It is apparent that the numbers and masses of the flavor and quark-lepton transforming gauge bosons are larger than those of the SU(5) minimal model. This means that the value of a is lower, and assuming that the duration of the inflationary period is fixed, the scale for the expansion of the universe is reduced. This means that there is the enhanced prospect for deviations from flatness. So one may presume that the universe started as a small 3-sphere with a large curvature, where the inflationary period flattened out the universe, but maybe not completely. This leaves open the prospect that if before inflation that if the universe were open or closed, k = 1, that the universe today still contains this structure on a sufficiently large scale. The closer to flatness the universe is, the tighter are the constraints on the masses of particles in the early universe. 

In calculations on small molecules the basis sets have in recent years approached the completeness condition and under these circumstances gauge invariance will be achieved automatically without introducing expheit transformations of the localized orbitals. Such procedures impose very heavy demands on computation time. 

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