Tangent formula

By just changing k into —k the generic triplet invariant may be rewritten as 4 h, i. = 4 i — 4 h k- For this invariant the Cochran formula states that 

Generally, each h reflection takes part in r triplets, for each of which the expectation  

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The tangent formula [3] is a key formula in direct methods that lets us refine phase values and determine new ones. Consider the situation in which we have a series of triplets with a common reflection They can be written... 

Consider also a situation in which all the phases on the RHS of all the equations above are known, at least approximately, then the tangent formula gives us an estimate for which in its simplest form is... 

The phases are refined to convergence using the tangent formula in one of its many variants. 

Deficiencies in intensities, which occur in x-ray powder dififiaction as well as in single crystal electron diffiaction, may cause problems even in early stages of ab initio structure analysis. Nevertheless, examples for successful use of the tangent formula or Sayre equation for structure determination from ED data have been worked out [14]. Other direct methods, like maximum entropy can provide us with an envelope of the molecules in the cell, which delivers an idea of its orientation [15]. An alternative approach to ab initio structure determination is the calculation of the gas phase conformation of an initial model for subsequent refinement by energy minimization [16]. 

Once a set of initial phases has been chosen, it must be refined against the set of structure invariants whose values are presumed to be known. So far, two optimization methods (tangent refinement and parameter-shift reduction of the minimal frmction) have proven useful for extracting phase information from the structure invariants. The tangent formula (Karle and Hauptman, 1956),... 

Conventional reciprocal-space direct methods, implementing tangent-formula refinement in... 

SnB uses the parameter-shift optimization of the minimal function by default whereas SHELXD uses the tangent formula. The complete direct-methods phasing process using the Shake-and-Bake algorithm is illustrated in flow-chart form in Fig. 9.2, and the main parameters are summarized in Table 9.4. 

The derivation of phases for noncentrosymmetric structures is more difficult, because the values are not restricted to 0° or 180°. In addition to the origin-defining phases, an enantiomorph-defining phase must be selected. Usually phases are derived by the tangent formula ... 

Probabilities are calculated that each triple product is positive and that each E value is positive. The tangent formula ... 

Tangent formula A formula of use in direct methods for developing additional relative phases and refining them. 

Usually each reflection forms more than one triplet and each of the triplets may be used for phase determination (estimation). In order to employ all triplets and thus, obtain the best agreement between phase angles that result from different triplets, Karle and Hauptmatf introduced a general expression for phase determination from triplets. This relationship is known today as the tangent formula ... 

The best possible agreement between phases is obtained by using a least squares refinement in combination with the tangent formula (Eq. 2.148). 

Rius J, Miravitlles C, Gies H, Amigo JM (1999) A tangent formula derived from Patterson-fimction arguments. VI. Stracture solution from powder patterns with systematic overlap. J Appl Ciystallogr 32 89-97... 

In their basic form DM exploit two types of prior information the positivity of the electron density map (this condition may be relaxed, e.g., for neutron diffraction, see Section 8.4.7), and the atomicity (the electrons are non-dispersed into the unit cell but concentrated around the nuclei). This information, apparently trivial, is very useful to succeed in all the steps of a modern DM procedure (1) scaling of the observed intensities and normalization of the structure factors (2) estimate of the structure invariants (3) application of the tangent formula (4) crystal structure completion and refinement. 

Extensive tests of phase refinement by the tangent formula of Haupt-... 

A. L. Spek and H. Krabbendam, The Application of the Generalized Tangent Formulae to Phase Refinement in Protein Crystallography, Report of CECAM Workshop, Paris (Orsay) (1972). 

The tangent formula provides a powerful equation for establishing relations between structure factors and phases and for refining a probabilistic set of phases till self-consistence this process is usually known as convergence. With the knowledge of a sufficiently large subset of phases, an electron density map can be calculated using eqn [11] to reveal the structure. These maps are called E-maps because they use the normalized structure factor, E hkl), as coefficients of the Fourier series rather than the structure factor, F hkl). 

The Hughes form of the Sayre equation (34) is for the centrosymmetric case in the noncenlro-symmetric case, at least as far as physical information is concerned, it corresponds to the tangent formula. In contrast to Equation (33), here the. sum is weighted with the normalized structure factors, and all pairs of structure factors possible in a special case are taken care of ... 

The angle < ), is the phase associated with the structure factor Fh. Equation (20.11) is called the tangent formula. For noncentrosymmetric crystals, certain phases are specified to establish an origin in the crystal. Certain symbols are also assigned for later evaluation, which is done by use of the theory of invariants and... 

The tangent formula composed of the sum of terms over k, both in the numerator and the denominator, is another equation... 

For noncentrosymmetric crystals the sum of angles and tangent formulas are, respectively. 

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