Complex phase factor

Figure 2. Quantum classical cross-sections for the reaction D-I-Ha (r — l,j — 1) DH (v — l,/)-l-H at 1.8-eV total energy as a function of /. The solid line indicates results obtained without including the geometric phase effect. Boxes show the results with geometric phase effect included using either a complex phase factor (dashed) or a vector potential (dotted).

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). 

In the vector potential approach [6], the (real) electronic wave function (4>) is multiplied by a complex phase factor/(4>), defined such that... 

We have seen how the introduction of complex phase factors ensures a uniform description of a supersystem containing several noninteracting... 

An alternative solution to the gauge-independence problem in molecular calculations is to attach the complex phase factors directly to the atomic basis functions or atomic orbitals (AOs) rather than to the MOs. Thus, each basis function—which in modern calculations usually corresponds to a Gaussian-type orbital (GTO)—is equipped with a complex phase factor according to Eq. 87. A spherical-harmonic GTO may then be written in the from... 

Mead and Truhlar [52] introduced an elegant way of incorporating the geometric phase effect, namely the vector potential approach. In this method, the real electronic wave function 4>(a), where a is any internal angular coordinate describing the motion around the Cl, is multiplied by a complex phase factor c(a) to ensure the single-valuedness of the new complex electronic wave function ... 

When the complex phase factor takes the form of (2), the Laplacian of the nuclear Hamiltonian is modified according to... 

The periodicity of the nuclei in the system means that the square of the wave function must display the same periodicity. This is inherent in the Bloch theorem (eq. (3.75)), which states that the wave function value at equivalent positions in different cells are related by a complex phase factor involving the lattice vector t and a vector in the reciprocal space. 

Note that Rg has been replaced by Rb in the last bracket.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... 

Kendrick and Mead, it remains a nontrivial task. An alternative approach consists of using the ansatz in Eq. (14), where the complex phase factor is absorbed in the real nuclear wave functions.Although such an approach is especially convenient for Xs-type systems when adopting hyper-spherical coordinates (since the complex phase factor concerns in principle only the if hyperangle, it can be generalized to the asymmetric case (see Sec. 4.1.1). 

The effect of this complex phase factor is to move the global gauge origin O to the best gauge origin for the basis function located at center Rm, namely, the nucleus M to which the basis function is attached. 

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