Condon-Shortley convention

Firstly, these /8 parameters are rather ill-defined but most immediately, the qualitative trends between the values has been shown merely to reflect a greater reduction in Fi than in from the corresponding free-ion values. In short these parameter trends furnish little or no evidence for differential orbital expansion. Further, even if such differentiation is really there - by which we presumably mean within a conventional m.o. sheme - it must manifest itself only within the numerical values of (and ligand field parameters). While working within a (f basis, therefore, there is no inconsistency whatever by recognizing the non-spherical hgand field within Fl.f. of (1-5) at the same time as representing the interelectron repulsions - l/r,y of (1-5) - in the usual free-ion Condon-Shortley theory. 

The phase choice (sign) is referred to the Condon-Shortley convention the factor —1 occurs only for positive odd values of m. Another phase choice, according to the Fano-Racah convention, can be met in literature and hence... 

The spherical harmonics are complex functions difficult to visualise and also their handling is impractical. A simple unitary transformation exists, yielding the real, normalised and orthogonal functions—angular wave functions Yfa for m positive only. They are collected in Table 1.9, using the Condon-Shortley phase convention. 

Note + is drawn with (---) and without (—) Condon-Shortley convention (see... 

Since a quantum mechanical treatment will be used later, we introduce now, in addition to the normalization factor, the Condon-Shortley convention which leads to the following equations. 

To make all types of polarization conform with the normalization factor, a point of high importance for MCD, we consider the following schemes. The Condon-Shortley convention, which is very inconvenient for the purpose of visualization and does not contribute to a better understanding, is not taken into consideration in the following figures. However, calculations will obviously have to take it into account. 

Note that our w operator includes Condon-Shortley convention and is in agreement with the m i operator reported in Piepho and Schatz 1983, p. 79. 

There are a tew definitions of the spherical harmonics in the literature [see E.O.Steinbom and KRuedenberg, Advan.Quantum Chem., 7,1 (1973)]. The Condon-Shortley convention often is used, and is related to the definition given above in the following way = em. Yj — (—l)" [f/], where sm = j... 

This transformation is in accordance with the Condon-Shortley phase conventions for the spherical basis functions [7]. In fact, our initial Hamiltonian matrix in Eq. (7.21) was constructed in this way. The resulting vector corresponds to the triplet spin functions, which we used in Sect. 6.4. The total spinor product space has dimension 4. The remainder after extraction of the three triplet functions corresponds to the spin singlet, which is invariant and transforms as a scalar. Spinors are thus the fundamental building blocks of 3D space. Their transformation properties were known to Rodrigues as early as 1840. It was some ninety years before Pauli realized that elementary particles, such as electrons, had properties that could be described... 

Now consider the commutator of J3 with the isotopic spin raising operator T+ = Ti -I- iT2 which, in the Condon-Shortley phase convention (Condon and Shortley, 1963) has the following effect ... 

For integer values of / the spherical harmonics take the following form where, in particular, the Condon-Shortley phase convention has been adopted ... 

The sign convention is that prescribed by Condon and Shortley (5) and is not that often seen in quantum-mechanics texts. It is required to give correct results when using the raising and lowering operators /+ and /. ... 

By standard procedures the tx states may be diagonalized with respect to the z operators, yielding / mr> kets (with / = 1). If one adopts a Condon and Shortley phase convention [9] these eigenkets read ... 

It is important to note that different authors use different phase conventions. Those of Condon and Shortley [CSh35] will be employed here, requiring [STa63]... 

Phase conventions have been chosen to be consistent with those of Condon and Shortley.13 In terms of tensor operators, the square modulus of f becomes... 

There are some small differences between conventions for spherical harmonics Yim(6,(p) in different texts (following most chemists, we use the so-called "Condon and Shortley" [10] convention). Note These functions are also the solutions to the Rayleigh problem of the normal modes of waves on a flooded planet (s, p, d,f functions), and they also occur in the study of earthquakes. 

There is a phase convention implicit in these two equations, the so-called Condon and Shortley convention [9], which is universally adopted. 

Brown and Howard (1976) have suggested that all molecule-fixed matrix elements be evaluated in terms of space-fixed operator components. The reasons for this are the space-fixed components of all operators obey normal commutation rules it is natural to adopt the Condon and Shortley phase convention for... 

This phase convention is similar to what is often called the Condon and Shortley phase convention, which specifies that ... 

The choice of a phase convention is a matter of taste. However, the convention adopted must be internally consistent. See Brown and Howard (1976) for a discussion of the Condon and Shortley (1953) phase convention and molecule-versus space-fixed angular momentum components see Larsson (1981) for a brief but comprehensive summary of all of the most frequently encountered phase conventions. Throughout this book, an attempt has been made to use the phase conventions of Eq. (3.2.82), Eq. (3.2.85a), Eq. (3.2.86), and Eq. (3.2.87), and... 

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