The Gauge Problem

When the origin of our coordinate system is translated by a vector G, the vector potential of a constant external magnetic field transforms as A = x ra = — B X G. It follows that 

Field-dependent basis orbitals as vj (r, G, B) — Uj (f) exp(- (BxG)-f) should be used, resulting in a similar transformation of the electron field operators. Such field-dependent orbitals would guarantee gauge origin independence even in a finite basis. In this expression, G is the position of orbital Uj with respect to an arbitrary origin and the electron coordinate r is given in the same coordinate system. Usually the atomic orbitals are centered at the nuclear positions. In such cases, Rg takes the place of G for the orbital Vj f, Rg, By. 

1 It is a different question whether results obtained with small basis sets, even if origin-independent, are meaningful at all 

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Early application of theoretical methods to the calculation of magnetic peroperties was hampered by the so called "gauge problem". The gauge problem arises firom the fact that the Schrodinger equation contains the vector... 

In principle (and using infinite basis sets), NMR chemical shifts can be computed using a single origin. However, practical (i.e. finite) basis sets require that the gauge problem be addressed. The several ways of doing this are listed below ... 

On top of the basic and complicated theoretical problem, real practical problems exist in actually performing a shielding calculation. As discussed earlier, one must select a suitable basis set and a molecular geometry, decide how or indeed whether to deal with rovibrational effects, and devise a strategy for combating the gauge problem. We discuss these problems in the examples presented later in this chapter. 

Several methods have been proposed to alleviate this problem. The first is to freeze the core orbitals for some suitable state, such as the isolated atom, and perform the calculation in the valence space only, where the gauge problem is not as serious because the eigenvalues are smaller (van Lenthe et al. 1994). Such restrictions may prove to be too limiting for widespread applications— but no less serious than those made in pseudopotential or model potential methods, where the core is frozen. The second is to freeze the potential that appears in the denominator. To do this some approximations must be made. Several criteria for a valid frozen potential have been proposed by van Wiillen (1998) which are that the potential (a) has the correct behavior near the nuclei, (b) does not depend on the orbitals, and (c) has no contribution from distant atoms or molecules. In addition, the potential must represent the real system fairly well or it will be of no value. Probably the most obvious choice are to take a superposition of atomic potentials or to construct the potential from a superposition of atomic densities. These two differ for a density functional method only in the exchange-correlation potential. A third approach is to add terms from neighboring atoms to the potential in the... 

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