Parity case basis states

Equations (3.2.70 and 3.2.71) give the explicit linear combination of signed-f2+, A+, fi, A, u>, A case (a+, a) basis states that corresponds to each signed-A+ case (a+, e) basis state. However (see Section 3.2.2) it is necessary to construct basis states that have well defined symmetry with respect to the (Tv operator, in particular overall parity,... 

The preceding discussion is focussed on the parity and e/f symmetry of case (a) basis states (Eq. 3.2.4a). Since it is often useful to construct basis sets of the form (ion-core) (Rydberg electron)) where the ion-core can be in cases a+, b+, or c+ and the Rydberg electron can be in cases a, b, c, d, or e (Watson 1999), the effects of cxv(xz) on each of these basis states are summarized here (Field, unpublished) ... 

Note that the parity of the Rydberg electron case (d) or (e) part of the basis state is determined by l alone, that the values of the rotation axis r projection quantum numbers Sr, Ir, sr, and jR do not enter into the specification of parity, and the signs of the rotation axis projection quantum numbers are not reversed upon application of crv(xz). 

A case (b+ d) basis state with definite overall parity, for example, may be constructed based on the effect of trv(xz) on the overall basis state,... 

Let us consider how independent /i(i ) 2 effects contribute to the v E) for the hydrogen halides, HX (X = I, Br, and Cl). The curves shown on Fig. 7.6 correspond to relativistic adiabatic potential energy curves (respectively 0 dotted, 0+ dashed, 1 and 2 solid) for HI obtained after diagonalization of the electronic plus spin-orbit Hamiltonians (see Section 3.1.2.2). The strong R-dependence of the electronic transition moment reflects the independence of the relative contributions of the case(a) A-S-Q basis states to each relativistic adiabatic II state. The independent experimental photodissociation cross sections are plotted as solid curves in Fig. 7.7 for HI and HBr. Note that, in addition to the independent variations in the A — S characters of each fl-state caused by All = 0 spin-orbit interactions, all transitions from the X1E+ state to states that dissociate to the X(2P) + H(2S) limit are forbidden in the separated atom limit because they are at best (2Pi/2 <— 2P3/2) parity forbidden electric dipole transitions on the X atom. In the case of the continuum region of an attractive potential, the energy dependence of the dissociation cross section exhibits continuity in the Franck-Condon factor density (see Fig. 7.18 Allison and Dalgarno, 1971 Smith, 1971 Allison and Stwalley, 1973). 

In this example, a has a special form that results in a simplification of (f2fc). Every eigenstate out of which I>(0) must be composed is a linear combination of basis states with definite parity (see Section 3.2.2). All eigenstates may be expressed as a linear combination of Hund s case (a) basis states (see Section 3.2.1), each of the symmetrized form... 

It is now necessary to construct linear combinations of the d-orbitals which transform according to the representations eg and t2g. Generally speaking, the construction of basis functions may be quite tedious, apart from a number of simple cases where it may be done practically by inspection (as for example, the one-dimensional representations). Basis sets for the common situations are tabulated in various places e. g., Koster et al. (34), Ballhausen (2), Griffith (21). It will be sufficient for our purpose to give the basis sets for eg and t2g and to demonstrate that they satisfy the necessary requirements. Since our discussion is confined to systems of d-electrons, all states will be of even parity or g-states. To simplify the notation we shall henceforth suppress the parity index, unless specifically needed. 

We consider now the case H(ls)H H" H(ls), i.e., the transition from a basis state in channel 1 to a basis state in channel 2. From parity considerations, we have... 

It formed the basis of the first solid state laser in I960. This emission consists of two sharp lines (the so-called R lines) in the far red (see Fig. 3.18). Since it is a line, it must be due to the transition -> A2 (Fig. 2.9) generally speaking the emission of transition metal ions originates from the lowest excited state. The life time of the excited state amounts to some ms, because the parity selection rule as welt as the spin selection rule apply. The emission line is followed by some weak vibronic transitions obviously this emission transition belongs to the weak-coupling case. 

As is done in the GL-theory for a single even-parity order parameter, we write the free energy density difference between the superconducting state and the norma state as an expansion in even powers of the complex gap function A(k), which is related to the anomalous thermal average of the microscopic theory [28] where c is the electron annihilation operator with wave vector k and spin t. However, for the multiple-order parameter case we must expand A(k) as a linear combination of the angular momentum basis functions Yj(k)),... 

---