GENTAN : Tangent phasing

Author: Syd Hall, Crystallography Centre, University of Western Australia, Nedlands WA 6907, Australia

GENTAN uses triplet and/or quartet structure invariant relationships in a general tangent formula to propagate and refine structure factor phases. The phases required to define the cell origin and enantiomorph are specified automatically, or may be selected (wholly or partly) by the user.

Tangent Formula

The tangent formula approach of Karle and Hauptman (1956) is the most widely used procedure for the extension and refinement of structure factor phases. Users who are relatively unfamiliar with this field are well advised to read summaries of these methods by Karle and Karle (1966) and Stewart and Hall (1971). Practical guidelines and background information on the application of the tangent formula is detailed in the proceedings of the 1975 IUCr Computing School (Ahmed, 1976).

The tangent formula is a straightforward computational approach to summing phases estimated with the triplet invariant relationship

 (4.44)

The conditions for which has a value close to zero have been discussed in the GENSIN introduction. If there are m triplet relationships then the cyclic nature of the contributing phases requires that the mean value of ( ) has the trigonometric form

 (4.45)

Because contributing phases can vary in reliability, a weighted tangent formula is required to average the phases as follows,

 (4.46)

The quantity is a measure of the joint reliability of the two phases contributing to the i'th triplet invariant. This approach can be applied to any order of structure invariant. A general expression for structure invariants of order n is

 (4.47)

It follows from equation (4.46) that the general tangent formula suitable for application to these invariants has the form shown in equation (4.48) (Hall, 1978). X is A, B, . . . according to the order of the invariant.

 (4.48)

The general tangent formula also permits the use of non-zero values. This is particularly important if oriented or positioned fragment information is used to estimate from the group structure factors (see the GENEV and GENSIN documentation and the section on partial structure information below). In strongly non-random structures can depart significantly from zero even for invariants with large X values.

It has also been discussed in GENSIN that may be expected to be closer to than 0 for negative quartets (see the GENSIN documentation for definition). The negative quartets are used by GENTAN for FOM purposes only, unless the  active  option is entered on the invar line (then negative quartets are used actively and not for the FOM).

Starting the tangent process

The tangent phasing process is usually initiated with a few "known" phases. From these phases additional values are determined through the application of the tangent formula to connecting invariant relationships. In turn, new phases are used to expand the phasing process further.

The tangent refinement of a phase set stops when the estimated phases converge to a constant value. Convergence occurs when the refined phases are self-consistent with the structure invariant relationships and the refinement constraints applied (e.g., weighting scheme). Due to the pyramidal nature of the phasing process (i.e., a few phases determine many), the final phase set is also strongly dependent on the starting phases. To a large extent the success or failure of the multisolution method is determined by the actual values of the starting phases. This is why a great deal of the computational effort goes into the selection of starting reflections.

Where do the "known" starting phases come from? Some may be specified directly in order to fix the origin in the cell. These are known as origin defining reflections (ODR). ODR phases alone are usually insufficient to reliably start the phasing process. The larger the number of starting phases, the less dependence there is on the reliability of a few invariant relationships and the higher the likelihood of obtaining a correct solution.

Where do the additional "known" phases come from? Because phase values are usually not known before a solution, one way is to assign trial phase values to a limited number of reflections and permute these in a series of separate tangent refinements. Permuted starting phases form the basis for the multisolution approach to the tangent phasing process.

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.

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.

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.

Tangent phase extension and refinement

The extension and refinement of phases is simultaneous in the tangent process. When a structure invariant relationship contains only one unknown phase then this phase is estimated from the relationship. This is referred to as a phase extension. If all phases in an invariant are known, then each member phase is determined by a combination of the others. When a phase occurs in more than one such invariant, the estimate of its value is averaged or refined. Repeated averaging of phases in this way is referred to as phase refinement. As discussed earlier, the generator reflections and related invariants are sorted into an optimal phase path during the selection of starting phases. Reflections and invariants are subsequently processed in this sequence and one pass down this list is known as a single refinement iteration. The maximum number of refinement iterations may be specified by the user via the refine control line (field 3). The default value is 30.

The reliability of each phase estimate (h) is gauged by the calculated value of (h) (see equation (4.52) below). Phases are accepted for phase extension if the value of (h) is above a threshold (see field 5, refine line). This threshold is reduced with each iteration (see field 7) thus insuring that the most reliable phases are used in the early iterations.

Only the top sorted generators (those used in the MAXEXT selection process (see field 4, select line)) are phased in the early iterations. When the average value of (h) changes less than a specified percentage (see field 7, refine line) additional generators (50, or 25% of total, whichever is greater) are added to the phasing process. The weight of each phase, is also used to control phase extension and refinement. A weight threshold is set for the duration of the refinement (see field 4, refine line) and is used to reject less reliable phases.

There are two different methods for propagating phases in GENTAN;  cascade  and  block . These two modes differ only in the point in the tangent iteration where the phase is estimated. The difference in procedure can, however, have a profound effect on phase convergence and stability.

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 (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.

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.

Weighting

The tangent formula weight is defined by the joint probabilities of the phases contributing to the RHS of equation (4.48). If the individual weight of a phase is defined as

 (4.49)

and the joint tangent weight as

 (4.50)

The variance of an unrestricted phase is given by the following expression (Karle and Karle, 1966) where and are modified Bessel functions dependent on ,

 (4.51)

is a measure of agreement between contributing estimates of within the tangent formula and is defined in the following equation where T and B are the tangent formula numerator (top) and denominator (bottom), respectively:

 (4.52)

It follows that the variance of may be calculated from equations (4.51) and (4.52). Computationally, however, this is prohibitive so in practice it is necessary to approximate w(h). GENTAN provides for three different weighting schemes: probabilistic, Hull-Irwin statistical, and modified statistical.

w1 : Probabilistic weights

For unrestricted phases the weight w(h) based on equation (4.51) 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 (4.52). 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.

Acentric structures

 (4.53)

where ' = min ( 5., X ) and X is mean A (triplet) or B (quartet).

Centric structures

The value of w(h) calculated from equation (4.51) is not valid for restricted phases because it assumes a continuous phase distribution. For restricted phases the probability that cos (h) is positive is given by the following expression:

 (4.54)

It follows that weight of an individual phase has the form

 (4.55)

Although w(h) calculated from equation (4.55) will lie in the range 0.0 to 1.0, it has the same deficiencies of unrestricted weights based solely on (h). For this reason GENTAN uses the relative weight expression for restricted phases.

 (4.56)

where ' = min(2., X) and X = mean of A (triplets) or B (quartets)

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

 (4.57)

 (4.58)

where

 (4.59)

 (4.60)

The precise functional form of 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 (4.59) and (4.60), 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.

w3 : Modified H-I Statistical Weights

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

 (4.61)

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.

Identifying the correct phase set

It is very desirable in multisolution tangent methods to have some method of detecting "correct" phase sets prior to computing a Fourier transform (i.e., E-map). An a priori assessment of phase sets is made in GENTAN using two 'measure of success' parameters CFOM and AMOS. These parameters are based principally on the four figures-of-merit, RFOM, RFAC, PSI0, and NEGQ.

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

 (4.62)

where all 's are the mean values for the calculated , the expected (see equation (4.58)) and the random (i.e., if all phases were randomly distributed). For a correct phase set the value of should approach that of 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 . For this reason RFOM tends to be less reliable for strongly non-random structures.

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.

 (4.63)

RFAC is a minimum when there is close correspondence between the refined and the expected . 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 .

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

The 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 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 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 are significantly lower than if the distribution of contributing phases was biased or incorrect. These invariants are used to form the figure-of-merit

 (4.64)

for ψ(0) triplets.

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.

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,

 (4.65)

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

Combined Figure-of-Merit (CFOM)

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

 (4.66)

The FOM weights 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.

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,

 RFOM RFAC PSI0 NEGQ OPTFOM 1.0 0.25 .75 60. REJFOM 2.0 0.5 1.5 120.

The absolute measure-of-success parameter is calculated from all active FOMs as

 (4.67)

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:

 AMOS 100-81% high probability of being correct set 80-61% good chance of being correct set 60-41% possibility of being correct set 40-21% low probability of being correct set 20-0% unlikely to be correct set

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.

Rejection Of Phase Sets

Phase sets must satisfy certain criteria before being considered for possible output to the bdf. Each phase set is tested at three stages in the tangent process and is rejected if the FOM values and other parameters are outside acceptable limits. In this way time is not spent on phase sets that have little or no chance of being correct - a very desirable feature for a multisolution procedure.

The first rejection test is made following the sixth tangent iteration and involves only the top block of sorted phase estimates. This is referred to as the PRETEST of FOMs and the rejection criteria 011, 012, and 014 are applied. The user may disable the PRETEST of FOMs with the setfom line (field 1). The second rejection test is made after the last tangent iteration and a phase set is rejected according to the criteria 021, 022, 023, 024, and 025. The final rejection test occurs during the sorting of phase sets and the calculation of the AMOS value. Phase sets are rejected if criteria 036 and 037 are not satisfied.

PRETEST Rejection Criteria

 011 reject if RFOM > 2*REJFOM(1) 012 reject if RFAC > REJFOM(2) and RFOM > REJFOM(1) 014 reject if RFOM < 0.25

Last Iteration Rejection Criteria

 021 reject if RFOM > REJFOM(1) 022 reject if RFAC > REJFOM(2) 023 reject if PSI0 > REJFOM(3) 024 reject if RFOM < 0.25 025 reject if convergence limit not reached after maximum refinementiterations (see refine line)

Final Rejection Criteria

 036 [a] reject if |av.φ - | exceeds 45° 037 reject if AMOS < 5% [a] The value of av. is 90° for centrosymmetric structures and 150-180° for noncentrosymmetric structures. This test rejects "all-plus catastrophe" phase sets.

Automatic Termination

Tests for phase correctness are made at the same time as the second rejection test. These are made on the basis of the optimum FOM values, OPTFOM. If all tests are satisfied, the tangent phasing process is terminated and the program enters the sort mode. In this way GENTAN insures that computing time is not wasted on generating further phase sets when a correct set of phases has already been calculated. This is particularly important when the  random  start option is invoked. Tangent cycling is terminated

if RFOM
< OPTFOM(1) and RFAC < OPTFOM(2) and PSI0 <
OPTFOM(3) and NEGQ < OPTFOM(4) and av. φ >
45°

If inadvertant termination occurs, the user can adjust the OPTFOM values used in the above criteria, or switch off this test entirely (field 10, setfom line).

Application Of Partial Structure Data

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!"

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.

File Assignments

• Reads |E| values from input archive bdf

• Writes estimated phases to the output archive bdf

• Reads structure invariants from file inv

Examples

GENTAN


This run automatically selects starting phases. Phases are extended and refined in the  block  mode using weight scheme 2. The top four phase sets are written to the output bdf. All FOM rejection and cycle termination tests will be applied.

GENTAN pout 10                   :output the top 10 phase sets
invar trip                          :use triplet invariants only

select magic                   :use magic integer permuted phases

assign odr 7 23 5 per 1 17            :assign ODR/permute phases

refine block w3                   :block mode with mod H-I weights

setfom nopr                   :do not pretest FOM values


GENTAN pset 128                   :permit 128 phase sets
invar *7 allxv                   :include all quartets
refine cascade w2 30            :max iterations to 30
phi 5 2 -3 0. odr                   :define ODR
phi 1 5 7 0. odr                   :define ODR
phi 3 2 2 0. odr                   :define ODR
select random                   :select all phases with random
values
archiv -1601 -1603                   :delete items 1601 and 1603
from bdf


References

• Ahmed, F.R. and Hall, S.R. 1976. Computer Application of the Symbolic Addition and Tangent Procedure. Crystallographic Computing Techniques, Eds. F.R. Ahmed, K. Huml, B. Sedlacek Munksgaard: Copenhagen, 71-84.

• Cochran, W. and Douglas, A.S. 1955. The Use of a High Speed Digital Computer for the Direct Determination of Crystal Structures I. Proc. Roy. Soc., A277, 486-500.

• Hall, S.R. 1978. Paper 15.1-2, Collected Abstracts. 11th IUCr Congress, Warsaw.

• Hull, S.E. and Irwin, M.J. 1978. On the Application of Phase Relationships to Complex Structures. XIV. The Additional Use of Statistical Information in Tangent-Formula Refinement. Acta Cryst., A34, 863-870.

• Karle, J. 1976. Structures and Use of the Tangent Formula and Translation Functions. Crystallographic Computing Techniques. Eds. F.R. Ahmed, K. Huml, B. Sedlacek. Munksgaard: Copenhagen, 155-164.

• Karle, J. and Hauptman, H. 1956. Theory of Phase Determination for the Four Types of Non-Centrosymmetric Space Groups 1P222, 2P222, 3P12, 3P22. Acta Cryst., 9, 635.

• Karle, J. and Karle, I.L. 1966. The Symbolic Addition Procedure for Phase Determination for Centrosymmetric and Noncentrosymmetric Crystals. Acta Cryst., 21, 849.

• Main, P., Fiske, S.J., Hull, S.E., Lessinger, L., Germain, G., Declercq, J.P. and Woolfson, M.M. 1980. MULTAN-80 Program Writeup, Dept. of Physics, University of York, York, England.

• Stewart, R.F. and Hall, S.R. 1971. X-ray Diffraction: Determination of Organic Structures by Physical Methods. Eds. F.C. Nachod and J.J. Zuckerman. Academic Press: New York, 74-132.

• White, P.S. and Woolfson, M.M. 1975. The Application of Phase Relationships to Complex Structures VII. Magic Integers, Acta Cryst., A31, 53-56.

• Woolfson, M.M. 1976. Doing Without Symbols - MULTAN. Crystallographic Computing Techniques. Eds. F.R. Ahmed, K. Huml, B. Sedlacek.Munksgaard: Copenhagen, 85-96.

• Yao Jai-xing. 1981. On the Application of Phase Relationships to Complex Structures XVIII. RANTAN - Random MULTAN, Acta Cryst., A37, 642-644.