4.27. GENTAN: Tangent phasing

4.27.2.1. Selecting Origin Defining Reflections (ODR)

The first step in the GENTAN calculation is to specify sufficient phases to define uniquely the cell origin. The formal conditions required to do this have been detailed in the GENSIN documentation. The specification of phases to fix the cell origin is to be done automatically unless the user intervenes via the phi or assign control lines. All ODR phases entered are checked for validity by the program. If they are incorrect or insufficient, additional or new phases will be automatically selected to satisfy the origin fixing requirements.

4.27.2.2. permute and magic: Selecting starting phases

The selection of starting phases which are involved in reliable structure invariant relationships is critical to the success of these methods. In GENTAN, the generator reflections are sorted by a convergence-type process (Woolfson, 1976) that maximizes the connections through structure invariant relationships between ODR phases and additional starting phases. This is called the MAXCON(nection) procedure. If the structure is noncentrosymmetric this procedure is also used to specify one or more EDR (enantiomorph defining reflection) phases to fix the enantiomorphic form of the structure. This is done automatically by the MAXCON procedure unless the EDR phase is selected manually by the user. Additional phases are specified in the MAXCON procedure as requested (see field 2, select line). The MAXCON procedure sorts generator reflections in order of descending connectivity and this sort order is used in all subsequent operations. The sorted order of generators is referred to as the phase path. The origin, enantiomorph, and any other MAXCON-selected phases are at the beginning of this path.

In addition to the starting phases selected for maximum connectivity, phases are specified to optimize the rate of "phase extension" to the remaining generators. This is referred to as the MAXEXT(ension) procedure. Extra phases are often needed in the starting set to accelerate phase propagation along the sorted phase path, rather than maximizing the connections between the initial starting set. It is important to emphasise that the MAXCON approach insures that there are strong links between the initial starting set and other strong generators, while the MAXEXT procedure provides additional phases which enhance the rate at which new phases are generated in one pass down the phase path. The criteria used in the MAXEXT procedure to measure the rate of "phases extension" is specified in field 6 of the select control line.

Both the MAXCON and MAXEXT procedures use all available triplet and quartet invariant relationships unless instructed otherwise (see field 5, select line). This option is useful if the automatically selected starting phases fail to provide a solution. Note that field 1 of the invar line has precedence over this option.

The permutation of phases assigned to starting reflections is performed in several ways. In the perm ute mode (see field 1, select line), each starting phase (other than the ODRs) is assigned a value according to whether it is restricted or unrestricted. Restricted phases can have two values (separated by ) and unrestricted phases are usually assigned four different values (separated by /2). In this mode therefore, the total number of phase sets tested is increased by a factor of two, or four, for each restricted, or unrestricted, phase specified. The number of phase sets increases very rapidly with the number of unrestricted starting phases.

An alternative approach to phase permutation is available using the magic integer procedure of White and Woolfson (1975). The magic option (field 1, select line) permutes the unrestricted starting phases so that the number of phase sets increases at a much lower rate. If a large number of unrestricted phases is needed to start an analysis, the magic integer permutation approach can provide a considerable reduction in computing time. The magic permutation procedure does increase the initial rms error of phases but for large analyses this is is usually a worthwhile tradeoff for benefits of more unrestricted starting phases.

4.27.2.3. random: Specifying random starting phases

The inherent fragility of a phasing process started with a limited set of known phases and extended sequentially to all other phases has already been discussed. The success of this procedure hinges on the reliability of a few individual structure invariants. This is particularly critical in the early stages of the extension process when new phase estimates are often determined from one or two relationships. An incorrect phase estimate at this stage will frequently cause the phasing procedure to fail.

The random phase approach of Yao Jia-xing (1981) is an alternative to using a limited set of permuted phases. In the random mode (field 1, select line) random phases are assigned to all generators except the origin defining reflections. The use of random starting phases lessens the dependence on a small number of critical relationships by insuring that all invariants are immediately involved in the phase refinement process. Weights of each refined phase are used to filter the phase extension process. When the weight w(h) of phase estimate (h) exceeds its starting value (default is 0.25 - see field 9, select line), this phase replaces the random starting value. This mode is particularly useful for large structures in low symmetry where there are few relationships among generators, and for strongly non-random structures where invariant relationships with large probability factors may still be suspect. Strong enantiomorph definition is also possible through the application of random starting phases.

In the random start mode, tangent refinements are performed on different random starting sets until either a correct solution is identified, or the phase set maximum is reached. The user may specify a random number generator "seed" (field 10, select line) and in this way insure that repeat runs employ different random starting phases. Alternatively, the default seed will insure that the same random phases will be generated, if this procedure is desired in a re-run.

4.27.3.1. cascade: Phase estimation DURING an iteration

In the cascade mode phases are evaluated in sequential order along the phase path, and accepted or rejected immediately on estimation. Each phase estimate is made separately and tested for acceptability, based upon ²(h) and w, before proceeding to the next phases estimate. An accepted phase is then available for phasing reflections further down the phase path. In this way phase information cascades down the phase path and provides the most rapid phase extension possible per iteration.

The two main characteristics of this process are that it is very dependent on the order of the phase path, and that phase values change during an iteration. This is important because three or four phases are dependent on a common triplet or quartet. If the of this invariant changes substantially during the course of a single iteration the phase estimates tend to be dominated by phases near the end of the phase path. As these phases are usually of lower reliability, some instability in the refinement process can result. This may show itself as phase oscillations from iteration to iteration. This will be a problem if the value of is strongly non-zero; as may be the case in heavy-atom or strongly planar structures.

4.27.3.2. block: Phase estimation AFTER a tangent iteration

The block procedure differs from the cascade method in that known phase values remain fixed during a tangent iteration. New phase estimates are only calculated at the completion of an iteration. This insures that phase estimates are made from fixed phase values, and the values of invariants remain constant for the entire iteration. Phases are therefore determined as a block defined by the current length of the phase path. This procedure will not propagate phases as rapidly as the cascade mode but it is less susceptible to phase instability when values of depart significantly from zero. It is the recommended procedure for strongly non-random structures.

4.27.4.1. w1: Probabilistic weights

For unrestricted phases the weight w(h) based on equation (8) is approximately linear with respect to (h) (see p92-94, Stewart and Hall, 1971). It is possible, therefore, to approximate w(h) as K(h) where K is set a fixed fraction. This approach has the disadvantage, however, of not providing weights normalized about 1.0. This normalization is important for the correct evaluation of (h) from equation (9). In addition, for very large structures, all (h) values will be small and weights based only on (h) alone must also be small. The converse is true for small structures. For this reason, GENTAN uses a modified probabilistic weight which is essentially structure-independent and has values restrained to the range of 0 to 1.

4.27.4.2. w2: Hull-Irwin Statistical Weights

The probabilistic weight w1 is not suitable for application to some structural types. It tends to increase rapidly to 1.0 even for a small number of invariant relationships. As a consequence, it can be insensitive to significant variations in phase agreement. In addition, the w1 weighting scheme has no provision for over-correlated phase sets which are characterized by unexpectedly high (h) values.

To overcome these deficiencies, Hull and Irwin (1978) have suggested a weighting scheme based on the ratio of (h) and the expectation value <(h)>. It has the general functional form

W2(h) = ƒ{ [(h) / <(h)>]²} (14)

<(h)> = _m X_j P_j (15)

where P_j = I₁(X_j)/I₀(X_j) for nonrestricted ; (16)

P_j = tanh(X_j) for restricted (17)

The precise functional form of ƒ{} is given by Hull and Irwin.

The w2 weighting scheme has several important properties. First, it depends on the individual expectation value <(h)> calculated from the actual number of invariants involved in the current estimate of (h). Secondly, the value of weight w2 decreases if the phase agreement exceeds that expected for that stage of the refinement. In this way, it reduces the contribution of over-correlated phase estimates and enhances values that are close to that expected. Thirdly, w2 takes into account the importance of phase correlation effects to weighting for different structure analyses (see the plot of w2 above). At the same time the overall magnitude of w2 is essentially independent of structure size, and the scaling procedures used in w1 are not needed.

There is, however, a fundamental limitation to w2 which is due to its implicit dependence on a reliable estimate of the expectation value. For strongly non-random structures the estimates for <(h)>, based on equations (16) and (17), may be inaccurate and this can lead to incorrect weights. In these cases the w1 weights, which are based solely on phase agreement (i.e., (h)), will tend to be more reliable.

4.27.4.3. w3: Modified H-I Statistical Weights

This weighting scheme is identical to w2 except that it is a function of x rather than x². That is,

w3 (h) = ƒ{(h) / <(h)>} (18)

w3 is not as sensitive as w2 to variations in the ratio of (h) and <(h)>. In some structures this is advantageous, particularly when invariant relationships are sparse and there is a tendency for phase oscillations. The weights due to w3 are more heavily damped than w2 but not as insensitive as w1 to variations in phase agreement. w3 may, for this reason, be considered a compromise between w1 and w2 .

4.27.5.1. Relative Figure-of-Merit (RFOM)

This parameter is the inverse of the ABSFOM parameter of the MULTAN program (Main et al., 1980) and has the form

RFOM = [_h< (h)> - _{h r} ]/ [ _h (h) - _{h r}] (19)

where all 's are the mean values for the calculated , the expected (see equation (15)) and the random (i.e., if all phases were randomly distributed). For a correct phase set the value of (h) should approach that of <(h)> and RFOM should tend to 1.0. Incorrect phase sets will deviate significantly from 1.0, random phases towards 2.0, and overcorrelated phases towards 0.0. In general, however, phase sets with small RFOMs are more likely to be correct than those with larger RFOMs. The actual range of RFOMs for a given GENTAN run will vary according to the validity of the estimate of <(h)>. For this reason RFOM tends to be less reliable for strongly non-random structures.

4.27.5.2. R-factor Figure-of-Merit (RFAC)

The RFAC parameter is similar to the residual FOM calculated in MULTAN (Main et al., 1980) except for a scale that takes into account the relative dominance of heavy atoms in the structure.

RFAC = _h |(h) - <(h)>| / _h <(h)> (20)

RFAC is a minimum when there is close correspondence between the refined (h) and the expected <(h)>. In this respect it is very similar to the R-factor of Karle and Karle (1966). RFAC is, like RFOM, dependent on the reliable estimate of <(h)>.

4.27.5.3. Psi(zero) Figure-of-Merit (PSI0)

The (0) triplet invariants of Cochran and Douglas (1957) provide a sensitive figure-of-merit which is largely independent of the triplet and quartet invariants used in the tangent refinement. A (0) triplet relates two generator reflections (with |E| > EMIN) to a third which is selected to have an |E|-value as close as possible to zero (see the GENSIN documentation). The phases estimated from a series of (0) triplets are expected to be random when the contributing phases from the other two large-|E| reflections are correct. When this is the case the resulting values of ² are significantly lower than if the distribution of contributing phases was biased or incorrect. These invariants are used to form the figure-of-merit

PSI0 = _k (k) / _k <(k)> for (0) triplets (21)

PSI0 should be smallest for the correct phase sets. PSI0 is, along with NEGQ, one of the most sensitive and independent methods of measuring the relative likelihood of success.

4.27.5.4. Negative Quartet Figure-of-Merit (NEGQ)

Quartet structure invariant relationships are classified according to the magnitude of their crossvector |E| values. When the crossvector sum is very low there is a high probability that the invariant phase will tend to have a value of rather than 0. These invariants are referred to as negative quartets. In GENTAN negative quartets are usually not used applied to the tangent refinement process but are retained as a test of the phase sets (unless the active code is entered in the invar line). The negative quartets are considered independent because, unlike the positive quartets, they cannot be represented by a series of triplet invariants. The negative quartets provide, therefore, a separate estimate of the phases. A direct comparison of these phases provides the basis for the figure-of-merit,

NEGQ = |(h) - (h)| /n for n negative quartets (22)

where (h) is the tangent refined phase, (h) is the phase estimated from negative quartets alone. Correct phase sets should have low values of NEGQ ranging from 0^o for centrosymmetric structures, to 20-60^o for noncentrosymmetric structures. Note that if fragment QPSI values are used, the value of is automatically set to 0^o and the NEGQ test will remain valid. This FOM is a very powerful discriminator of phase sets provided that sufficient negative quartets are available.

4.27.5.5. Combined Figure-of-Merit (CFOM)

The combined FOM is a scaled sum of the four FOM parameters RFOM, RFAC, PSI0 and NEGQ.

CFOM = [WFOM_i (FOMMAX_i-FOM_i)/(FOMMAX_i-FOMMIN_i)] i=1,4 (23)

The FOM weights WFOM_i may be specified on the setfom control line. These values are subsequently scaled so that the maximum value of CFOM is 1.0. It is important to stress that CFOM is a relative parameter and serves mainly to highlight which is the best combination of FOMs for a given run. It does not indicate whether these FOMs will provide a solution.

4.27.5.6. Absolute Measure-of-Success Parameter (AMOS)

The AMOS parameter is a structure-independent gauge of the correctness of a phase set. It uses pre-defined estimates of the optimal values for the FOM parameters RFOM, RFAC, PSI0 and NEGQ. OPTFOM values may be user defined (see setfom line). Rejection values for the four FOM parameters are derived from the OPTFOM values as REJFOM=2*OPTFOM. The default values are as follows,

AMOS = [WFOM_i ( REJFOM_i-FOM_i) / OPTFOM_i ] i=1,4 (24)

where the WFOM values are scaled so that AMOS ranges from 0 to 100. In addition to being used to sort phase sets in order of correctness, the AMOS values provide a realistic gauge of the correctness of phase sets. As a rule of thumb, AMOS values can be interpreted in the following way:

These classifications are only approximations. The predictability of optimal FOM values can be perturbed by a variety of structure dependent factors and by the FOM weighting. Nevertheless, the AMOS value provides the user with a concise overview of the phase sets.

4.27.8.1. Psi Calculated from Fragment Information

If structure information of types 3 and 4 (oriented and positioned) is entered into the GENEV calculation, and the qpsi option is invoked in GENSIN, then the phase estimate of the invariant may be used in the extension and refinement process. This can provide a significant improvement to the phasing process, particularly for planar and heavy-atom structures, and provides a valuable second line of attack for less tractable problems. The rule of thumb is: "if oriented or positional structural information is available, use it!"

4.27.8.2. Applying Structure Factor Phases as Starting Phases

An alternative approach to partial structure information is to use calculated structure factor phases as input starting phases (Karle, 1976). This approach has the advantage over the group structure factor method of not requiring the repeat of GENEV and GENSIN calculations. In practice this method is limited because it fails to take into account the expected change to the values which is available from knowledge of the structure. Because of this, strongly non-random structural features tend to dominate the phases. Nevertheless, careful application of the parameters on the partsf line can provide an important alternative to applying known structural information.