Gauge dependence

The measurements are predicted computationally with orbital-based techniques that can compute transition dipole moments (and thus intensities) for transitions between electronic states. VCD is particularly difficult to predict due to the fact that the Born-Oppenheimer approximation is not valid for this property. Thus, there is a choice between using the wave functions computed with the Born-Oppenheimer approximation giving limited accuracy, or very computationally intensive exact computations. Further technical difficulties are encountered due to the gauge dependence of many techniques (dependence on the coordinate system origin). 

The most reliable results are obtained using ah initio methods with moderate-to large-sized polarized basis sets. The use of gauge-independent atomic orbitals (GIAO) removes gauge dependency problems. 

A diaphragm gauge depends for its indication on the deflection of a diaphragm, usually metalhc, when subjected to a difference of pressure between the two faces. These gauges are available for the same... 

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). 

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). 

It should be stressed that the weight functions h(1> and hi2) are gauge-dependent. 

A compromise between the opposing interpretations of Pople and Mason is offered by Webb and co-workers (46). These authors conclude that it is improper to equate the diamagnetic terms used by the two authors, since in contrast to Pople s o A, Flygare s contains significant and variable nonlocal contributions and is comparable rather to the gauge-dependent term in the original Ramsey expression (29). 

Each cell is provided with an individually controlled air valve. Air pressure is between 108 and 124 kN/m2 (7 and 23 kN/m2 gauge) depending on the depth and size of the machine and the pulp density. Typical energy requirements for this machine range from 3.1 kW/m3 of cell volume for a 2.8 m3 unit to 1.2 kW/m3 for a 42 m3 unit. 

In (2.47) we have again chosen to work in Feynman gauge, which is technically simplest to handle. This choice does not introduce any gauge dependence, as demonstrated in [19]. Thus E is a well-defined functional of the (ti -P and e -. The corresponding exchange potential can be evaluated using the ROPM in the order (see Sections). 

The role of gauge dependence, relativistic and correlation effects in electronic transitions... 

Role of gauge dependence in electronic transitions 30.2 Relativistic and correlation effects, gauge dependence... 

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