Proc. Natl. Acad. Sci. USAVol. 77, No. 1, pp. 5-9, January 1980Applied Physical and Mathematical Sciences

Triplet phase invariants: Formula for acentric case from fourth-order determinantal joint probability distributions

(crystal structure/phase determination/triplet selection/determinantal distributions)

JEROME KARLELaboratory for the Structure of Matter, Naval Research Laboratory, Washington, D.C. 20375

Contributed by Jerome Karle, October 15, 1979

ABSTRACT A conditional probability distribution fortriplet invariants is derived for noncentrosymmetric crystalsfrom fourth-order determinantal joint probability distributions.The formula makes use of the entire data set in the computationsfor each invariant. Test calculations indicate that the formulacan be used to help select invariants whose values are distrib-uted close to zero and also to evaluate special invariants asso-ciated with seminvariant phases.

The main approach to direct phase determination is based uponthird-order determinantal inequalities that arise from thenonnegativity of the electron distribution in crystals and theirprobabilistic features. From the recognition of the role playedby the determinantal inequalities and the fact that the ine-qualities become more restrictive in their implications as theirorder increases, it has seemed worthwhile to investigate thepossible implications of fourth and higher order determinantalinequalities to phase determination. To facilitate this, generaldeterminantal joint probability distributions were derived fordeterminants of all orders (1). Applications of this theory havealready been made for centrosymmetric crystals (2) for thefourth-order determinantal probability distributions. Numeroustest calculations of the resulting formula (2) have shown thatmany hundreds of invariants can be selected by use of the for-mula with essential certainty that their value is equal to zero.Several invariants whose value is equal to ir can also be selectedon occasion with very high reliability.

In the fourth order, information can be obtained concerningthe phase invariants Okl-k2 + k-kl+k3 + Ok2-k3 given that theassociated structure factor magnitudes 16kl.k2 , 16_kl+k31, and1 k2-k3I and many sets of | kI |, |42l, and | k4,, are known.Related to this is the past work on the B3,0 formula (3-8), aninvestigation by Messager and Tsoucaris (9) concerning theproperties of conditional probability distributions involvingfourth-order determinants, and, more recently, studies byGiacovazzo, who derived conditional probability distributionsfor the triplet phase invariants for the space groups P1(10) andP1(11).The probability distributions that form the background for

this investigation have novel features. One such feature is theoccurrence of a function denoted by the symbol V that can beinterpreted as an approximation to the higher order terms inthe probability distributions and that plays a key role in theapplicability of the probability distributions. In this respect,although the probability distribution for the acentric case infourth-order resembles the result of Messager and Tsoucaris (9)in mathematical form, it differs mainly in the presence of thefunction V.

In the centric case (2), it was possible to proceed withoutapproximation from the fourth-order determinantal jointprobability distribution to obtain a conditional distribution that

gives information concerning the phase invariant Okk-k2 +(k5-1+k3 + ¢k2-k1 , given the six magnitudes 16 1-k2, I6-k1+k31,I42-k31, 16ki , 1421, and |6k3l. The same is true for theacentric case. With the assumption of independence, it is pos-sible to combine the many distributions arising from the manysets, |I4kII, | k21, and 16k,1, associated with a given invariant.Some applications of these theoretical results are presented inorder to illustrate their computational characteristics andprovide the basis for some suggested uses.

TRIPLET PROBABILITY DISTRIBUTION,ACENTRIC CASE

This analysis begins with expression 15 of ref. 1, which givesthe joint probability distribution of the variates in an mth orderdeterminant, Dmp, for the acentric case. For the fourth order,we have

P4,0(1411|,14k21,1431,16k1-k21,16-k1+k3J,J|5k2-k31,Okl, Xk2, ¢>k3, ¢'ki-k2, ¢'-kl+k3,Xk2-k3)

=N4,0 exp 6jD4,P~i A4,p 1 [1A1,4,p,g 2,4,p,gwhere the 4k = 16klexp(igkk) are quasinormalized structurefactors with magnitudes 141k and phases 4k, N4,0 is a normal-izing constant, 60 -n M,

60 6-k I6-k2 6-k3

D4,p = 6k1 60 6ki-k2 4kI-k3 , [24s2 6-kl+k2 60 4k2-k343 6-kl+k3 6-k2+k3 60

p refers to a specific set of k1, k2, and kIs, and the As in the av-erage term of the exponential function are principal minors ofD4,p. The manner of formation of the As is described in detailin ref. 1. The average term in Eq. 1 is a constant whose valueis increased as the 16 Is involved in Eq. 2 are increased in value.It may be written

(A1,4,p.q24,p~q )q = V4p [3]where the term 6-4 is formed by factoring out all the diagonalterms from the As. In the calculations made in this paper, thetriplet invariants occurring in the average term of Eq. 3 wereall assumed to have the value of zero.The smallest value for V4,p is unity, occurring when the

magnitudes of the off-diagonal 6 approach zero. Its value in-creases as the magnitudes increase, and larger values mightrange from 1.3 to 1.5.The right side of Eq. 1 can be written

N'4,exp[a3cos(0k1 + 40-k2 + 13)+ a6COS((kk + q-k3 + 16) + a9cos(Okk2 + 4k3 + 39)]

X exp(2V4,pQ31 6kl-k26-kl+kc36k2-k31 cosc), [4]where

5

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a3m = a3m-2 + a3m-1 + 2a3m-2a3m-lcos(/3m-2- /3m-1), m = 1,2,3 [5]

33m = tan-1 a3m-2sinO3m-2 + a3m-lsin/3m-1a3m-2C05/33m-2 + aam-lcos/B3m-1'

m = 1,2,3 [6]a1 = 2V4,pQ3lekl-k26-kl,+ k2,la2 = -2V4,p(2Q2 -Q4)Ie6kl 3-k1-k2ek2-j3,a4 = 2V4,pQ3jekeI(-k36-kl+3j, 9a5 = -2V4,p(2Q2- Q4)j kk6k2-kj6-k36-k2+j3j, [10]a7 = 2V4,pQ3lj-k26k35k2-k3|j [11]a8 = -2V4p(2Q2 - Q4)1-k26-k1+k2ek36ki-k3j, [12]

[13]

32 = '-k,+k3 + Ok2-k, [14]/4 = '-kj+k3, [15]

05 = 1-kl+k2 + &-k2+k3, [16137 = /k2-k3, [17]

/8 = Ski-k3 + q5-k,+k2, [18]

*I+= Oki-k2 + 0-kj+k3 + 'tk2-kd3 [19]Qn = Un/an/2 [20]

N= E Z7 [21]

,=1

and Zj is the atomic number of the jth atom in a unit cell con-taining N atoms. In Eqs. 7-12, Q3, and -(2Q3-Q) have re-placed the coefficients 6-1 and 6-2 of the triplet and quartetterms, respectively, in order to account for the possible presenceof atoms of unequ~il atomic number (1, 2).

It is possible to obtain from expression 4 the conditionalprobability distribution for the triplet phase invariant 4), giventhe values for the six magnitudes, 141|,1421,|31,Iekj-k21|,6-kj+k3I|, and | k2-k3 |. To obtain this result it isnecessary to integrate expression 4 over all possible values ofkk1, kk2, and 0ki.. This is readily accomplished if the first ex-ponential function in 4 is expressed as a product of three ex-ponential functions and each of the three resulting functionsis replaced by its infinite series expansion in terms of Besselfunctions of imaginary argument. After integration, the re-sulting expression for the conditional probability distributionof the triplet invariant is

P(D; I 4k,11 gk21lls1kI3IIk4-k2lI -kl+k31,lI k2-k3I)

= N"40[I0 (a3)I (a6)Io (a9) +2 E In (an)In (a6)In (a9)n=1

X cosn(4) - T12 - T45 T78)]X exp(2V4,pQ31Q k4-k26-kl+k6k2-I3j cos4'), [22]

where

T = tan-l acsin4) [23]af +ajcost2

and the Im are Bessel functions of imaginary argument.There are many sets of k1, k2, k3 represented by K for a given

triplet defined by k1 - k2, -k1 + Ic3, k2 - k3. In order to takethis into account and obtain a conditional probability distri-bution for 4) given the 1k associated with the many sets K, itis assumed that the individual contributions from the bracketedterm in Eq. 22 are sufficiently independent that it can be re-

placed by a product of such terms over K. It is further assumedthat independence applies to that part of the contribution ofeach set k1, k2, k3 to the exponential term of Eq. 22 that dif-fers from the value of the exponential function when only16kl-k2 1j |j-kj+kI,j1Gk2-k31 are known. The conditionalprobability distribution becomesP(d; 1 kl:k E Kj,|k4-k21 X |6-k+k3j, |k2-k31)= Nŵ l[I (t3)Io(a6)Io(a-g) + 2 E In(as)In(a6)In(ag)

X cosn(4 -T12-45 - r78)]}

X exp{2Q3j kj-k2e-kj+k36k2-k31

X cos4jV3,p + F(V4,p - V3,p)J} [24]where I1 6:k EC K}means the set I 4k1 where k is contained inthe set K, N4w is a normalizing constant, and

( Ap, 6 . [25]1,3,p,q A2,3,p,qfq

The appropriate D3,p for Eq. 25 is formed by crossing out thefirst row and first column of Eq. 2 and the As are defined in ref.1.

It was noted previously for the centric case (2) that when4k1-k2J , 6-k1+ksJ, and 2-kkj are all large and, in addition,16k4I, 16k21, and 16k|I are also all large, the value of zero isfavored for the triplet invariant 4). If, in contrast, the magni-tudes of the 6 associated with the triplet invariant are again alllarge but one of 14k l,I 6k2I | 6k'J is small while the other twoare large, the value of 7r is favored for the triplet invariant 4).Numerical tests with Eq. 22 show similar trends for the tripletinvariant 4) in the acentric case. However, Eq. 24 can presenta maximum value for 4) anywhere in the range between 0 and7r.

SOME CALCULATIONSThe computation of the probability formula 24 for a particularcosine invariant, cos4) = cos(4kl-k2 + 0-kl+k3 + /k2-k1), in-volves taking products over hundred of terms forming the setK. They derive from the fact that a given set k, - k2, -kl + kS,k2-k, can be formed from many sets, ki, k2, k3, chosen fromamong the observed data. The data used in the calculationsdescribed here were subjected to a restriction that facilitatedthese calculations greatly. On the basis that formula 24 is par-ticularly responsive to sets of 14k1 1,1421,1| k in which twoor three 161 are large when the structure factor magnitudesassociated with the invariant of interest are also large, such setsfor a given k1- k2, -k1 + k3, k2 - k3 were restricted to havingat least two of 1k4 |, 1421,1|SiI greater than 1.5. Double pre-cision may be required for certain of the terms in Eq. 24 formany computers in current use.The quantity V4,p (Eq. 3) is a function of determinants that

are formed from a fourth-order one. In the original derivationof the general determinantal joint probability distribution (1),the function V arose from a heuristic argument. According tothe theory, the coefficient of V4,p should be unity. In practice,this has been found to be well satisfied. The calculations re-ported in this paper were performed either with a factor of 0.99or 1.00. An appropriate factor can be determined by requiringthat the average of the values for the cosines of a succession of50-100 invariants be equal to a known theoretical value. Anadjustment in the value of the coefficient of V4,p can be madeso that the average of the computed cosine invariants would bein agreement with the theoretical value. An increase in the

Proc. Natl. Acad. Sci. USA 77 (1980)

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[9]

01 =- (P-ki+k2, I

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Table 1. Analysis of computation of triplet-phase invariants forL-arginine dihydrate by use of Eq. 24

Invariants a for a forSampling No. in 4) set at a for 4)cj, fromrange sample Conditions zero exp Eq. 24

1-150 114 4caic

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Table 3. Phase determination with special invariants for L-arginine dihydratePhase of

From Eq. 24, third CorrectP(4)= )P4= 7r) reflection value of

k- k2 -k1 + k3 k2-k3 (n of 1on) 42-3 42-k3104 104 208 34 0 03010 3010 0020 22 0 0103 103 206 14 r 7r208 206 4014 15 X3010 4014 104 48 rr206 3010 104 6430 10 104 406 42 0 03010 406 1016 28 r r206 3010 1016 33 Xr1016 104 2012 29 r3010 2012 502 50 0 03010 208 502 30 0 02012 3010 102 62 0 0206 2012 406 20 0 03010 609 301 30 t3010 4013 103 23 t

The ratios formed from the probability that these special invariants have a value of zero divided bythe probability that they have a value of 7r, shown in the fourth column, permit the evaluation of thephases associated with k2 - k3 in sequential steps. There are large uncertainties associated with thistype of calculation, but the ratios greatly exceed the uncertainties.* 03,0,10 is specified to be equal to zero as an origin determiner ( 163,0o1o = 3.46).t The last two invariants show that phase information is also available for additional parity groups. Inorder to obtain a phase value for k6,0,9, the phase of 03,0,1 (163,o,1 = 2.77) would have to be specifiedas an additional origin determiner or known otherwise.

in the vicinity of zero degrees. The density increases as the angleincreases. Since the triplet-phase invariants appear in Eq. 24as the arguments of a cosine function, the probability distri-butions are symmetric about 4) = 0, leading to the ambiguityin the signs of the invariants. Determination of the proper signsof the invariants involves enantiomorph or axis direction se-lection (or both) and concerns another part of procedures forphase determination.

Tables 1 and 2 show that the lowest standard deviations fromthe correct answers for the triplet phase invariants are obtainedfrom Eq. 24 when invariants computed to be larger than 600are omitted. The second calculation in Table 1 illustrates whysuch a selection criterion is suggested. A large value for o- was

obtained when the selection was not made. Other similarcomparisons are shown in Table 2. It is also seen that the stan-dard deviations, obtained when expected values from third-order probability distributions replace the calculations of Eq.24, are sometimes significantly larger and sometimes not.Larger standard deviations are obtained if it is assumed that alltriplet-phase invariants are equal to zero, as is required inprocedures for phase determination. Even for this case, how-ever, it is seen from column 4 of Tables 1 and 2 that the selectioncriterion based on Eq. 24 gives some improvement, even if theselected invariants are assumed to be equal to zero. For column4, the errors are always measured systematically from the as-sumed value of zero, whereas in columns 5 and 6 the errors are

Table 4. Phase determination with special invariants for harringtonolidePhase of

From Eq. 24, third CorrectP(4 = 0)/P(4 = r) reflection value of

k- k2 -k1 + k3 k2- k3 (n of 1on) Ok2-3 Ok2-k101 10 1 00 2 17 r r0173 0173 00 6 14 0 0002 006 00 8 16 r r00 2 035 03 7 15 -7r/2* -r/200 2 035 03 3 23 -ir/2 -7r/200 8 03 5 03 3 11 -7r/2 -r/200 2 006 00 8 16 r r012 035 043 23 7rt 7r012 035 023 25 r 7r002 012 014 13 r/2 r/200 8 0175 0173 1000 2 0175 0173 1203 5 0173 0142 23012 0173 0181 16The phase determination proceeds by use of these special invariants in a sequence of steps facilitated

by the calculations in the fourth column. Phases associated with k2 - k3 are individually evaluated.*00,3,5 is specified to be equal to zero as an origin determiner (I 60,3,51 = 3.94).t0O,1,2 is specified to be equal to -r/2 (I6ol.21 = 3.10).Illustrations of additional phase relationships available.

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randomly distributed about the estimated or computed valuesof the triplet invariants.Special calculationsThere are special triplet invariants in the higher space groupswhose evaluation can immediately give the value of a semin-variant phase. For example, the first line of Table 3 refers to thetriplet invariant 0-1(4 + 4T(4 + 0208, and the evaluation of thisinvariant with Eq. 24 yields immediately the value of Om.8 Thisis so because it is known that for an appropriate origin specifi-cation for space group P212121, the phase value of 0-IC4 mustequal to 0 or ir and the sum of two of them must therefore al-ways be zero. Other special invariants involving two-dimen-sional reflections are shown. With the specification of the valuesof appropriate phases to determine the origin and, when re-quired, enantiomorph or axis direction (or both), they permitthe sequential development of phase values. This is illustratedin Tables 3 and 4, in which k2-k3, given in the fifth column,is obtained from knowledge of 4kj-k2 and 0-k,+kI and thestrong indication that the value of the triplet-phase invariantis equal to zero as shown in the fourth column. The values ofcOkk-k2 and '-kj+k3 are known in the initial steps from spacegroup considerations and after that from previous determina-tions and phase specifications. Some of the latter are noted inthe footnotes to Tables 3 and 4.The fourth column of Tables 3 and 4 gives the ratio of the

probability that the value of the invariant of interest has a valueof zero to the probability that it has a value of -r, as computedfrom Eq. 24. The two largest values computed for the invariantsknown to have a value of ir for arginine were 107 and 102. Forharringtonolide, the three largest values for such invariants were108, 105, and 102. These values could be reduced by decreasingfurther the factor of 0.99 that multiplied V4,p, in Eq. 24. Therewould, of course, also be some corresponding decreases in theratios shown in Tables 3 and 4. It is not necessary to perform thisadditional calculation and, although the results shown in thefourth column have a large uncertainty, it is seen from the sizeof the exponents that the values of the triplet invariants arecomputed with very high reliability. Examination of the se-quence of steps of Tables 3 and 4 in which, for example, thesame phase is determined in more than one way, reveals a highdegree of internal consistency.

It is seen from Tables 3 and 4 that the values of certainseminvariant phases can be obtained by direct calculation ofspecial triplet invariants by use of Eq. 24. Such phase infor-mation has immediate application in procedures for phasedetermination.

DISCUSSIONThe illustrative examples computed with Eq. 24 indicate thatthis formula can provide useful phase information for non-centrosymmetric crystals. The information is of varied sorts.The computations, which are based on fourth-order determi-nantal probability distributions, provide a means for selectingsets of invariants that are distributed more closely about thevalue of zero than the sets that are normally used. The selectionof the latter is based on third-order determinantal probabilitydistributions. The information obtained from the calculationsof special invariants, as illustrated in Tables 3 and 4, can be usedboth as a part of direct phase determination and also as a meansfor selecting among alternative solutions after a phase deter-mination has been carried out.The information obtained from Eq. 24 deteriorates in de-

finitiveness and reliability as structures increase in complexity.

For example, numerous calculations performed with data froma crystal of antamanide (15), which has 368 nonhydrogen atomswith roughly the same atomic number, in a unit cell of spacegroup P212121 resulted in little information that was not alreadyavailable from the third-order theory.The function V4,p plays an important role in Eq. 24. This

function is a feature of the new determinantal joint probabilitydistributions (1) and does not arise in the usual way of derivingjoint probability distributions for structure factors-for ex-ample, by use of characteristic functions. V4,p helps to take intoproper account the effect of structure factors of large magni-tude.The form of Eq. 24 shows that it does not have the problem

of asymptotic convergence, a property of the usual form forjoint probability distributions of structure factors. Asymptoticconvergence can lead to inaccuracies, particularly whenmagnitudes of structure factors are large and the number ofatoms in the unit cell of a structure of interest is moderate.

Future implicationsThe decrease in definitiveness and reliability of calculationswith Eq. 24 as the complexity of a structure increases impliesthat the fourth-order theory developed here will probably notmaterially aid in the solution of the difficult problems foundamong structures with 100 or more atoms of roughly equalatomic number that crystallize in noncentrosymmetric spacegroups. This is so despite the fact that Eq. 24 involves the entireset of measured data, an amount that normally greatly over-determines the structure problem.One possible implication of this circumstance is that the step

from Eq. 22 to Eq. 24 that involves the assumption of the in-dependence of individual contributors is the source of thelimitation of Eq. 24. In this regard, it is worth noting that thehigher order determinants contain the appropriate interrela-tionships among the structure factors and the correspondingjoint probability distributions are available (1). This suggeststhat the development of formulas from the higher order de-terminantal joint probability distributions might be fruitful ifpracticable computing techniques can be developed to facilitatetheir application.

The computer programs required for the test examples were pre-pared by Mr. S. A. Brenner. His help with the programs and calcula-tions is greatly appreciated.

1. Karle, J. (1978) Proc. Natl. Acad. Sci. USA 75, 2545-2548.2. Karle, J. (1979) Proc. Natl. Acad. Sci. USA 76, 2089-2093.3. Hauptman, H. & Karle, J. (1957) Acta Crystallogr. 10, 267-

270.4. Vaughan, P. A. (1958) Acta Crystallogr. 11, 111-1 15.5. Karle, J. & Hauptman, H. (1958) Acta Crystallogr. 11, 264-

269.6. Karle, J. (1970) Acta Crystallogr. Sect. B 26, 1614-1617.7. Tsoucaris, G. (1970) Acta Crystallogr. Sect. A 26, 492-499.8. Hauptman, H. (1972) Z. Kristallogr. 135, 1-17.9. Messager, J. C. & Tsoucaris, G. (1972) Acta Crystallogr. Sect. A

28,482-484.10. Giacovazzo, C. (1976) Acta Crystallogr. Sect. A 32,967-976.11. Giacovazzo, C. (1977) Acta Crystallogr. Sect. A 33,527-531.12. Karle, I. L. & Karle, J. (1964) Acta Crystallogr. 17, 835-841.13. Flippen-Anderson, J. L. & Duesler, E. (1979) Acta Crystallogr.

Sect. B 35, 1253-1254.14. Karle, J. & Gilardi, R. D. (1973) Acta Crystallogr. Sect. A 29,

401-407.15. Karle, I. L., Wieland, T., Schermer, D. & Ottenheym, H. C. J.

(1979) Proc. Natl. Acad. Sci. USA 76, 1532-1536.

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