Rotation functions

P, Jy, and J , are the components of the total orbital angular momentum J of the nuclei in the IX frame. The Euler angles a%, b, cx appear only in the P, P and P angular momentum operators. Since the results of their operation on Wigner rotation functions are known, we do not need then explicit expressions in temis of the partial derivatives of those Euler angles. 

When the above analysis is applied to a diatomic species such as HCl, only k = 0 is present since the only vibration present in such a molecule is the bond stretching vibration, which has a symmetry. Moreover, the rotational functions are spherical harmonics (which can be viewed as D l, m, K (Q,< >,X) functions with K = 0), so the K and K quantum numbers are identically zero. As a result, the product of 3-j symbols... 

TO Yeates. The asymmetric regions of rotation functions between Patterson functions of arbitrarily high symmetry. Acta Crystallogr A 49 138-141, 1993. 

It is reasonably well established that the non BO coupling term involving second derivatives of the electronic wavefunction contributes less to the coupling than does the term (-ih-3 rk/dRa) (-ib k 9Ra)/nia having first derivatives of the electronic and vibration-rotation functions. Hence, it is only the latter terms that will be discussed further in this paper. 

Before returning to the non-BO rate expression, it is important to note that, in this spectroscopy case, the perturbation (i.e., the photon s vector potential) appears explicitly only in the p.i f matrix element because this external field is purely an electronic operator. In contrast, in the non-BO case, the perturbation involves a product of momentum operators, one acting on the electronic wavefimction and the second acting on the vibration/rotation wavefunction because the non-BO perturbation involves an explicit exchange of momentum between the electrons and the nuclei. As a result, one has matrix elements of the form (P/ t)Xf > in the non-BO case where one finds lXf > in the spectroscopy case. A primary difference is that derivatives of the vibration/rotation functions appear in the former case (in (P/(J.)x ) where only X appears in the latter. 

Translational functions in eq. (100) cancel out because of the approximation adopted, the vibrational functions also cancel. The logarithm of the ratio of rotation functions possesses a constant value of —.246. The contribution of the exponential term is considerably higher and therefore determines the value of the disproportionation constant. Theoretical values of log K listed in Table X... 

Likewise, let QL(t) denote the orientation of the emission dipole in the lab frame, and iK t) — [ (/), (/)] its orientation in the molecular frame. Using the transformation property of the Wigner rotation functions,... 

The convention employed here for the rotation functions matches that of Wigner(75) and Edmonds176 and differs from that employed previously in this laboratory(23, 2 81, x2 85) which corresponds to defining all Euler rotations... 

J = 1,3,5 — are antisymmetric with respect to the nuclear coordinates. It follows that homonuclear diatomic molecules with anti-symmetric nuclear spin wave functions (nuclei with half-integer I = 1/2, 3/2...) can combine only with symmetric rotational functions (even J = 0,2,4...), while those with symmetric nuclear spin wave functions (even I) can combine only with antisymmetric rotational functions... 

Notice from Equation 4.113 for I = 1 (deuterium) it is the combination of the symmetric nuclear spin with the symmetric rotational functions which has the higher statistical weight. At high temperature (ortho/para)DEUTERiuM = 2/1, and at low temperature the ortho form predominates. 

Storoni LC, McCoy AJ, Read RJ. 2004. Likelihood-enhanced fast rotation functions. Acta Cryst D60 432-438. 

Analyse results make sure you understand the symmetry of the rotation function group, the translation solution (the y coordinate is undetermined in P2(l) etc.). Get a feeling of the (maximum) height of the signal you can expect. 

Before executing the real MR protocol, we strongly recommend running a test case. Understanding the conventions of the rotation function will be greatly facilitated by reading a recent... 

The need to find a better score for the rotation function the Phaser euphoria... 

Many people have recognized that the rotation function suffers from some drawbacks and have tried to improve the score by using origin-removed Patterson functions, normalized structure factors E-values, etc. (Briinger, 1997). 

Briinger and colleagues developed a direct rotation function, which is just a correlation coefficient between Eq s Ejy,Q (Omega), the normal-... 

Another idea is to use different runs of the same program with slightly different models it is most aptly described as an application of a consistency principle, namely it is required that the solution should appear consistently in all runs, even with a rather low score. Special algorithms have been developed to cluster similar solutions in eulerian angles space and convincing results have shown that it is indeed possible to increase the signal-to-noise ratio of the rotation function in this way (Urzhumtsev and Urzhumtseva, 2002). 

The need for automated protocols is apparent from the strategy adopted by AMoRe to circumvent the problem that the score of the rotation function (RF) is far from being perfect and does not always rank the solutions correctly (Navaza, 2001). Indeed, it is often observed that the true solution is not the top solution, with many false positives. Hence, AMoRe runs a translation function (TF) for each of, typically, the top 50 or 100 solutions of the rotation function. This is actually quite rapid as TF is based on FFT then, the first 10 solutions of each of these TF runs is in turn refined using a very effective implementation of rigid-body refinement (Navaza, 2001). 

If there is NCS in the crystal, all molecules of the asymmetric unit must be searched in turn every time a potential solution has been found, it is possible to use this information to increase the signal-to-noise ratio of the searches for the other molecules. But then, the combinatorics of testing the 50 top solutions of the rotation function and then the 10 top solutions of each associated translation function for rigid-body refinement cannot be done by hand as in the previous case, as soon as there is more than two molecules in the asymmetric unit. In NCS-MR, depending on the number of molecules present in the asymmetric unit, there are thousands of possibilities to be searched. Also, as one is searching with only a fraction of the asymmetric unit, the signal to be expected is intrinsically lower. 

As we mentioned already, the problem of the rotation function is its score, leading to a difficult energy landscape to be searched we can now describe another way to tackle this problem. Since the Translation Function score is much more sensitive, one might try to run a translation for every possible rotation angle, therefore exploring the 6D space exhaustively. The space to be searched in eule-rian angles depends on the space group of the crystal and can be found in Rao et al. (1980). It turns out that it is doable in most cases within reasonable cpu time with a normal workstation. 

Another special case where the signal of the rotation function could be enhanced concerns crystals where the self-rotation fimction can be interpreted without ambiguity in this case, the so-called locked cross-rotation fimction (Tong and Rossmaim, 1997) allows to search for cross rotations which are compatible with the self rotation function. This usually results in a much better signal-to-noise ratio. 

Crowther, R. A. (1972). The fast rotation function. In The Molecular Replacement Method, Rossmann, M. G., ed. Gordon and Breach, New York, pp. 173-185. 

DeLano, W. L. and Brunger, A. T. (1995) The direct rotation function Patterson correlation search applied to molecular replacement. Acta Crystallogr. D 51, 740-748. 

Navaza, J. (2001). Rotation functions. In International Tables of Crystallography, vol. F, p. 269-274. KluwerAcademic Publishers. Dordrecht, Boston, London. 

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