Reduced matrix elements, rotational

From the WET, Eq. [166], it is obvious that the reduced matrix element (RME) depends on the specific wave functions and the operator, whereas it is independent of magnetic quantum numbers m. The 3/ symbol depends only on rotational symmetry properties. It is related to the corresponding vector... 

The first reduced matrix element in (8.212) is now expanded by using a first-rank rotation matrix to transform from the space- to the molecule-fixed axis system, and retaining only the wholly diagonal matrix elements ... 

In order to evaluate the reduced matrix element of T1 (L), we first rotate into the molecule-fixed axis system, q, using a first rank rotational matrix, so that... 

The intensities of the spectral lines and the depolarization coefficients are functions of the reduced matrix elements of the polarizability tensor calculated by vibronic functions. In order to estimate the possibility of observation of the pure rotational Raman spectra under consideration, one has to consider in more detail the polarizability operator. Its components belonging to the line y of representation f can be presented in the form of a power series with respect to the displacement qriri active in the Jahn-Teller effect (the other components can be neglected as not active in the pure rotational Raman spectrum under consideration) ... 

The form of the rotation for the symmetric ( )s and antisymmetric q components of the rotation can be written in terms of reduced matrix elements and the sum over Mp and F performed using standard techniques. When this is done we find. 

The application of angular-momentum theory to atomic spectroscopy is not limited to bringing eqs. (3)-(7) into play. In their book, Condon and Shortley (1935, ch. 3) developed the theory with particular attention to operators T that are vectors. They did this by specifying the commutation relations of Twith respect to J rather than by stating the transformation properties of the components of T under rotations. They considered angular momenta J built from two parts S and L) and obtained formulas for the matrix elements of operators that behave as a vector with respect to one part (say L) and a scalar with respect to the other (S, in this case). These formulas involve proportionality constants that would be called reduced matrix elements today. Condon and Shortley systematized the methods that had come into current use but which were often only hinted at, if that, by many theorists. For example. Van Vleck (1932) gave the formula... 

Effective collision cross sections are related to the reduced matrix elements of the linearized collision operator It and incorporate all of the information about the binary molecular interactions, and therefore, about the intermolecular potential. Effective collision cross sections represent the collisional coupling between microscopic tensor polarizations which depend in general upon the reduced peculiar velocity C and the rotational angular momentum j. The meaning of the indices p, p q, q s, s and t, t is the same as already introduced for the basis tensors In the two-flux approach only cross sections of equal rank in velocity (p = p ) and zero rank in angular momentum (q = q = 0) enter die description of the traditional transport properties. Such cross sections are defined by... 

These factors are relatively invariable with regard to k. Equation (62) can be considered as a geometrical rotational-invariant average for the maximum crystal-field splitting in any point symmetry. This approximation is good if the D are stationary in k, i.e. if the Z7, IT and if have nearly the same value (or at least the same order of magnitude). Based on the stationarity of the squares of the reduced matrix elements, Auzel (Auzel and Malta 1983) has selected for the lanthanide ions levels which should theoretically provide the best crystal-field probe H4 and for Pr ", I13/2 and I9/2 forNd ", I13/2 and Sr,5/2 for Sm +, Fj for Eu, Fj for Tb +, %n, %U2, %V2, His/2 and... 

Let us now consider the second mechanism, namely, the appearance of the electronic contribution gj due to the interaction with the paramagnetic electronic states. In particular, the singlet terms 1II and of one parity (either u u or g - g) interact because of the non-zero matrix elements of the electron-rotation operator [—l/(2/iro)](J+L- + J L+), where // is the reduced mass, ro is the internuclear distance (in atomic units) and the cyclic components of the vectors are defined in the same way as in [267] = Lx iLy, = Jx iJy connecting the x and y... 

The rotation matrix elements reduce to spherical harmonics when j is an integer and m or n is zero ... 

The rotational factor is further reduced to the matrix element of the operator product (TV2) (TV2) by the closure rule. The contribution can then be expressed in terms of an operator of the form... 

The agreement between experiment and theory is now much better than before, the discrepancy having been reduced from 5.444 to 0.182 MHz, but it is still poor compared with the experimental accuracy which is quoted as 0.01 MHz. However, our theory is still approximate because the electron spin spin interaction mixes N = 2 with N = 4, which introduces more hyperfine matrix elements off-diagonal in both N and J. The nuclear spin-rotation term, equation (8.271), does not contribute to the first-order energy of the N = 0 level, and makes a negligible second-order contribution. We will not pursue this analysis any further, our aim having been to illustrate the complexity of the fitting process moreover this was achieved for 13 different vibrational levels. 

The target states are expressed, according to equn. (7.35) for the full potential matrix elements, in terms of orbitals a) and P). The quantity that relates the orbitals to the target states i ) and i) is the m-scheme density matrix i alap i). Its transformation properties under rotations are important in finding the reduced potential matrix elements. 

The expectation values represented by the double sums in Eq. (4.3) depend on the potential function in Eq. (3.27). For a given harmonic frequency in the basis set, the matrix elements Zy and Zy are fixed but the t and tjv depend on the value of B in the dimensionless potential of Eq. (3.27). For a single-minimum potential there is a high degree of correlation between the values and the value of B, each of which leads to curvature in the rotational-constant variation with vibrational state1S). Since there are ten adjustable parameters, namely, three coefficients for each of the rotational constants plus one potential constant, B, in the reduced potential, it is necessary to determine the rotational constants in a large number of vibrational states if microwave data alone are used. 

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