Authors: V. Subramanian and Syd Hall
Contact: Syd Hall, Crystallography Centre,
University of Western Australia, Nedlands 6009, Australia
REVIEW analyzes the agreement between triplet and/or quartet structure invariant relationships and a given phase set. It is intended primarily for testing and aiding in the development of direct methods software, but may also have uses for examining invariant relationships of unsuccessful solution runs. Agreement between the specified phase set and the invariant relationships is measured with a range of parameters.
REVIEW is intended principally to test and aid in the development of methods using structure invariant relationships. It applies a specified set of phases to triplet and/or quartet invariants (see the GENSIN documentation for definitions) and provides information on both the individual and bulk agreement of the phase relationships. Agreement parameters are based on a range of measures including the frequency of invariant violations and the root mean square deviation of invariant estimates from the specified phase set. A plot of ψ difference (between "found" and "expected") versus the probability factors A (for triplets), and B and XVsum (for quartets) may be printed in terms of the population of invariants.
Fragment contributions to the estimate of ψ may be included in or excluded from this analysis.
The phase set used in this analysis is extracted from the input bdf. These phases may have their origins in the calculations FC or CRYLSQ or GENTAN. Input phases are applied to each structure invariant relationship in order to calculate the values of and . The phase of each generator reflection is then estimated from these relationships and compared to the input phase set in two ways. The first is the frequency of invariant violations. A violation is said to occur if the ψ value of an individual invariant relationship exceeds the value of ψ "expected" from probability considerations by more than π/2. This frequency is measured for each generator reflection and the phase set as a whole. It provides general measure of the "correspondence" between the input phases and estimated phases.
A second method of measuring the correspondence between the input phases and the phases estimated from invariant relationships is based on the root mean square deviation between the ψ values and their expected values. This value is weighted according to the probability factors A or B. The general form of the weighted rmsd value is,
Relationships with Restricted ψ Values
Just as symmetry can restrict the phase values of a reflection to one of two values (e.g., 0 and π for centrosymmetric space groups), it can also restrict the value of ψ for a structure invariant relationship. This is of particular significance when the restriction requires ψ to be non-zero. Such occurrences in noncentrosymmetric space groups can be identified in this analysis. The user may exclude or include non-zero restricted ψ values from the agreement parameters (see field 7 of the REVIEW control line).
Expected ψ for Negative Quartets
Negative quartets are those quartets with the XVSUM less than XSLO (see GENSIN description for details). The expected value of ψ4 for negative quartets will vary according to whether fragment information (and therefore QPSI values) are applied. If fragment QPSI values are not applied, the expected value of ψ4 is π; otherwise, it is likely (depending on the precision of the fragment information) to be zero.
Four levels of printout are available with REVIEW. These are the pset, gsum, plot, and sinv options in fields 1 to 4 of the list control line.
This option prints the input phase set, generator number, h, k, l, E, σ(E), and phase angle. These are output three reflections to a line.
This option prints for each generator the following summary.
NGEN generator number
H K L reflection indices
|E| E - magnitude
SIGE error in E-magnitude
PHI -RST restricted phase (NR=no restriction)
PHI -INP input phase
PHI - T3 phase estimated from triplet invariants
ALPHA - T3 see Definition of Terms section for definition
ALPHA - Q4 see Definition of Terms section for quartets definition
ALPHA/EAL -T3 ratio of α to expected α for triplets
ALPHA/EAL -Q4 ratio of α to expected α for quartets
RMSDPSI-TS see equation (1)
RMSDPSI-Q4 see equation (2)
S.I. EXAMINED TS number of triplets applied
S.I. VIOLATED TS number of triplets in violation
S.I. EXAMINED PQ number of positive quartets applied
S.I. VIOLATED PQ number of positive quartets in violation
S.I. EXAMINED NQ number of negative quartets applied
S.I. VIOLATED NQ number of negative quartets in violation
This option uses one line per generator
This option prints a population distribution of invariants for ψ difference versus the probability factors A and B. For quartets, ψ difference versus the cross vector sum XVSUM.
This option prints for each structure invariant the following data,
PFAC probability factor (A or B)
SHF phase shift due to translational symmetry
N2 signed generator number of the 2nd invariant vector
N3 signed generator number of the 3rd invariant vector
N4 signed generator number of the 4th invariant vector
XVS cross-vector sum for the quartet relationship (XVSUM)
PSI -ACT actual invariant ψ calculated from input phases
PSI -RST restricted invariant ψ
PSI -FRG fragment estimate of invariant ψ
PSI -DIF ψ difference(=PSI ACT - PSI FRG)
PSI -RMS rms ψ accumulated from differences
ALPHA α from tangent components - running average
ALPHA/EAL ratio of α to expected α
PHI -ACT actual input phase
PHI -EST estimated phase from this invariant
PHI -DIF difference between PHI ACT - PHI EST
PHI -TAN phase from tangent formula - running average
The output is one line per invariant. Because of this caution should be exercised in order to save paper. The default listing will be for generator numbers 1-10 (see Example section to control this listing).
It is possible to list only the invariants which are violated (i.e., |ψ-<ψ>| > π/2) regardless of the list line arguments. This is done by entering reset psta 5
Phase Set Summary
This summary is always output at the end of each run.
PSET NUM phase set number
NPHI DET number of phases determined from invariants
SI EXAMINED T3 total number of triplets
SI EXAMINED PQ total number of positive quartets
SI EXAMINED NQ total number of negative quartets
SI VIOLATED T3 number of triplets in violation
SI VIOLATED PQ number of positive quartets in violation
SI VIOLATED NQ number of negative quartets in violation
ABS FOM T3 absolute figure of merit for triplet
ABS FOM Q4 absolute figure of merit for quartets
ALPHA T3 average α (per generator) for triplets
ALPHA Q4 average α (per generator) for quartets
RMSD PSI T3 average rmsd ψ3
RMSD PSI Q4 average rmsd ψ3
DPHI T3 average Δφ for triplets
DPHI Q4 average Δφ for quartets
Definition of Terms
ALPHAis the "observed" α, and defined as
where S and C are the numerator and denominator of the weighted tangent formula (see GENTAN description for details).
EAL is the "expected" α for the invariant phase, and is given by the modified Bessel function. (A)
PHI -TAN is the phase estimated from the application of the weighted tangent formula. This is a progressive average. The angle output modulo is 360o.
ABS FOM is the absolute figure-of-merit as defined in GENTAN.
DPHI is the mean difference between the input phase and the phase estimated from the structure invariant relationship. The mean is taken over the invariants in the phase set.
Reads a phase set from the input archive bdf
Reads structure invariant relationships from the bdf inv
REVIEW *4 NOQP
In this example any fragment contribution to the invariant ψ estimate is excluded; both triplet and quartet structure invariants will be reviewed.
REVIEW gt 2 *7 1
list gsum sinv plot
The GENTAN phase set 2 will be read from the bdf; invariants with non-zero restricted ψ will be excluded. Both triplet and quartet structure invariants will be analyzed. list options permit a listing of: (i) generator summary (one line per generator); (ii) all structure invariants of generator numbers 1-10 (one line per invariant); (iii) plots of ψ difference versus A, ψ difference versus B, and plot of ψ difference versus cross-vector sums XVsum.
REVIEW :refined phase data from bdf
list sinv *5 2 5 :list each generator numbers 2-5
quar no :only triplets processed