Proc. Nati. Acad. Sci. USAVol. 88, pp. 10099-10103, November 1991Applied Mathematics

Direct calculation of atomic coordinates from diffraction intensities:Space group P1

(least squares/extended convergence/global minima/refinement space)

JEROME KARLELaboratory for the Structure of Matter, Naval Research Laboratory, Washington, DC 20375-5000

Contributed by Jerome Karle, August 14, 1991

ABSTRACT A method is presented that can extend therange of convergence for the least-squares minimization tech-nique in nonlinear systems. The problem of determining di-rectly the positions of atomic coordinates in a crystal from themeasured intensities of x-ray scattering is used to illustrate themethod. The minimization is facilitated by altering the mini-mization function in a way that reduces the number of falseminima and, in addition, by altering the character of the falseminima from time to time by changing the particular deck ofdata that is used and the values of a variety of parameters thatmay occur in the defining equations. With a structure consist-ing of 30 equal atoms in space group P1 and the use of exactdata with a Cray XMP/216 computer, convergence to a globalminimum was generally obtained, from a random positioningof the 90 coordinates, in a few minutes. Convergence within anhour for a 40-atom problem was much more sporadic, showinga considerable decrease in success with the increase in com-plexity. The possibility of extending the method to othermathematical systems is apparent.

With the intensity equations appropriate to crystal structuredetermination as an example, a methodology is illustrated forovercoming the entrapment ofleast-squares functions causedby false minima, thereby facilitating the determination ofglobal minima. The basic concepts contained in this approachare as follows.

(i) Modification of the character of the defining equationsso as to reduce the population of false minima in the least-squares function while preserving the location of globalminima.

(ii) Alteration, from time to time, of the shape of theminimization function so that the remaining false minima maychange in their number and location, while the location ofglobal minima does not change.The motivation for item i is clear. For the same number of

defining equations, for example, a space that has many fewerfalse minima could be expected, in general, to be moreamenable to the application of minimization techniques. Themotivation for item ii derives from the possibility that thealterations of the shape of the minimization function, coupledwith the driving pressure of the least-squares technique,would be useful in guiding the variables to their values at aglobal minimum.A variety of cases can arise. There may be an infinite

number ofacceptable global minima for a problem of interest.This is the case for the examples discussed in this article.Serious interference with the process can arise if there isexcessive occurrence of deep but false minima having valuesthat are comparable to those for the minima or minimum ofinterest. In many cases, it should be possible to determinewhether such false minima are obtained by the least-squares

process, by testing the minima against data that were not usedto discover them.

The Crystal Structure Problem

Crystal structures are ideally composed of atomic arrange-ments that repeat periodically in three dimensions. A repet-itive unit is contained in a volume called the unit cell that maybe described in terms of a conveniently chosen set of refer-ence axes. If the atomic arrangement in a unit cell is known,the atomic arrangement in the entire crystal is known. Herewe are concerned with the problem of locating the equilib-rium positions of atoms whose surrounding electron densitydistributions are approximated as spherical. In this presen-tation, the atoms are to be located solely by use of theintensities of scattering. Alternatively, this problem has beenlargely solved by the intermediate use of the so-called phasesof the diffracted waves.There is motivation for pursuing the approach followed

here despite the considerable success in structure determi-nation already achieved by the use of phases. The methodconsidered here, which may be described as the use ofdynamic alteration of the minimization function, may haveapplicability to other problems.The electron density distribution in a crystal represents the

structure of a crystal since the maxima of this distributionlocate the centers of the atoms. Consistent with the three-dimensional periodicity of a crystal, the electron densitydistribution p(r) can be described by a Fourier series,

oc

p(r) = V-' I Fhexp(-27rih-r),h-x0

[Il

where V is the volume of the unit cell, r is a vector whosecomponents locate a point in the unit cell, and h is a vectorwhose integral components label imaginary planes that cutthrough the crystal. The components of h are the integers h,k, and 1, whose values are inversely proportional to theintercepts of the corresponding planes on the axes chosen todefine the unit cell. The coefficients Fh, called structurefactors, are real or complex numbers that represent charac-teristics of the x-ray scattering that can be associated with theplanes h.The structure factors may be written

Fh = lFhlexp(ioh), [2]

where the angle oh is the phase associated with Fh. Theincident x-ray wave is scattered by the atoms in a unit cell insuch a way that it appears to be reflected by segments ofcrystal planes in a unit cell labeled by h with a maximumscattered amplitude of jFhI and at a point in the scattered wavecycle given by the phase Oh. In this case, reflection occursonly for a specific value of the angle of incidence.

10099

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If the coefficients Fh of the Fourier series were directlyavailable from a diffraction experiment, crystal structurescould be readily calculated from Eq. 1. The intensities of thediffracted beams are proportional to the JFh12 and thus, in anordinary diffraction experiment, only the magnitudes |Fhb areobtained but the associated phases Oh are not. Phase valuescan be obtained from the intensities by use of formulas frominequality theory enhanced by probabilistic measures (1).Such approaches are not considered here.By performing the Fourier inversion of Eq. 1 and replacing

the integral expression for the Fourier coefficient by a sum,since p(r) can be represented as a sum of N discrete, ratheraccurately known atomic electron distributions per unit cell,we obtain

N

JFhbexp(i4h) = E fihexp(27irthrj), [3]

where fjh is the atomic scattering factor, the amplitude ofelastic scattering, of the jth atom in a unit cell containing Natoms, and rj is the position of the jth atom. By multiplyingboth sides of Eq. 3 by their complex conjugates, we obtain

/N 2 /N 2

IFh12= fhcos 27rh-rj + fhsin 2lrhr.[4)

Since the atomic electron distributions are known to goodapproximation, the fih values can be computed and, in fact,are found in tables. The JFhI2 values are obtained fromexperiment. Inherent in the use of Eq. 4 is the assumptionthat in deriving the FhbI2 values from the measured intensitiesof scattering, a number of corrections have been made,including proper scaling and the elimination of positionaldisorder. The positions rj are the only unknown quantities inEq. 4, clearly a system of simultaneous equations since thereare many independent IFbh2 values measured. Our objectiveis to solve Eq. 4 for the unknown quantities rp.There is another form for Eq. 4 that is convenient to use in

which the atomic scattering factors are essentially constant,corresponding to the scattering from atoms that are essen-tially points. The transformation that is used yields so-callednormalized structure factor magnitudes IEhl from the JFhI, asfollows:

lEhbI = lFhbI >E 4jh), [5]

where e' is a factor dependent upon the nature of thecomponents ofh and the space group (2). An additional usefulassumption can be made, namely that

fjh-Zjfhb [6]

wherefb is regarded as a universal shape function for atomicscattering factors and Zj is the atomic number ofthejth atom.This assumption is quite accurate when a structure containsonly atoms having atomic numbers that are close in value-e.g., an organic compound consisting of C, N, and 0 inaddition to H, which scatters relatively weakly.With the use of Eqs. 5 and 6, Eq. 4 becomes

lEhl2 [

N 2

Y. Zjcos 2rh-rjJ=l

+ E Zisin 21rh-ri)] /( z i). [7]

Eq. 4 or 7, as preferred, can form the basis for the proceduresto be described. If we were to base the minimization on the

observed normalized structure factor magnitudes squared,IEhobs, the function to be minimized would be

E (jEh s -_ EhbIac)2 [8]

where IEhlcc is obtained from computation of the right sideof Eq. 7. Instead of using Eq. 7 as a set of simultaneousdefining equations, attempts were made to modify the defin-ing equations with the objective of improving the shapes ofthe minimization functions. The latter can be written moregenerally as

[9]

where f implies the usual "function of."

Modification of Defining Equations

Consistent with item i in the Introduction, some simplenumerical calculations were carried out in one dimension inthe attempt to discover defining equations that would pro-duce improved minimization functions. The calculations re-

b

0.6Coordinate

1.0

FIG. 1. Effect of altering the least-squares function in a one-dimensional space. The ordinate is the residual, which is a functionof the fractional coordinate (the repeat value for the coordinatescaled to unity) given by the abscissa. The defining equations are Ih= exp(as2)jEhj2 (Eq. 10) where a = 0 for curve a, a = 1 for curve b,and a = 3 for curve c. The calculation involves 12 intensity data (h= 1, . , 12). It is seen that when a = 0, the refinement spacecontains many false minima, but when a = 3 there are none. The twominima in curve c are characteristic of the problem and both areacceptable solutions. A similar one-dimensional calculation with Eq.11 with v = 0.6, 61 = E2 = 1 and a = 0, 1, 2, . . . yielded the followingresults: except for scale, the curve for a = 0 was essentiallyindistinguishable from curve a, the curve fora = 1 was smoother thancurve b and essentially indistinguishable from that obtained when a= 2 in Eq. 10 (not shown), and the curve for a = 2 was essentiallyindistinguishable from curve c.

10100 Applied Mathematics: Karle

Ya lf(lEhlo'bs) -f(lEhlc'aic)]2,h

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quire the use of two atoms, one with fixed coordinate and thesecond with a variable coordinate.

Fig. 1 shows the results of some of the experiments. It isa plot of the values of the minimization functions (residuals)versus the values of the variable coordinate. In all thecalculations, 12 data were used (h - 1, 2, . . . , 12) and thecoordinates ofthe atoms were, in fractional coordinates, 0.10and 0.37. The coordinate 0.10 was held fixed in the calcula-tion ofthe minimization function to establish an origin but notan enantiomorph. Since no enantiomorph is specified, thereis also a minimum at 0.10 + (0.10 - 0.37) = -0.17, which,because of periodicity, is equivalent to 0.83. Note that thepatterns are symmetric about the points 0.10 and 0.60.

Fig. 1 is based on the use of Eqs. 7 and 9, where

fl(IEhI2) = exp(-acs2)IEhI2. [10]

a is given selected values, s = sin O/A, 20 is the angle betweenthe incident and reflected beams, A is the wavelength of theincident radiation, and s can also be expressed in terms of hor, more generally, in terms of h and appropriate referenceaxes for three dimensions (3). The minimization functions inFig. 1 differ from each other by the values of a. It is apparentthat for a = 0 and 1, there are false minima that would preventa least-squares process from reaching a global minimum. Fora = 3, essentially any arbitrary starting point for the unknowncoordinate would be brought to a global minimum by aleast-squares process. For this simple example, a change inthe minimization function afforded a general solution to theglobal minimum problem. There is no reason, however, toassume that such an alteration would lead to a generalsolution in a multidimensional space. Nevertheless, an im-provement in the minimum function may well ensue.Two types of minimization functions that have been con-

sidered and have proven to be useful with fairly complexproblems are Eqs. 10 and 11.

f2(IEhI2)

= exp(-as2)(e1 + IEhI2) ln[e2 + exp(-as2)(El + IEhI2)v], [11]

where a, El, E2, and v are parameters whose values can beselected. Calculations similar to those for Fig. 1 that makeuse of the minimization function 11 have also been per-formed. As indicated in the legend for Fig. 1, parameters forEq. 11 could be readily found that gave similar curves. Thelarger values for a in both Eq. 10 and Eq. 11 gave curves inwhich the only minima were the global minima.

Eq. 11 has also been associated with a weighting functionfor each of the defining equations of the form

sin 2irhE sin 2irkE sin 2vle

2fhE 2 frkE 27rle [12]

where the components of h are given by h (h, k, l) and Eis a variable whose values are to be selected. There areevidently a number of parameters whose values can be variedin the course of a minimization calculation.

Rolling

Item ii in the Introduction concerns the alteration, from timeto time, of the shape of the minimization function. Theprocess for achieving this is called "rolling." The meaning ofrolling and how it is carried out will now be illustrated.

Let us first consider rolling of the data. For this purpose,we select a set of data ("Store"), a subset of the data("Deck") that varies in content from time to time during thecalculation, and a smaller subset ("Roll") that also varies incontent in the course of the minimization. For the crystal

structure problem, the data have been chosen from the ranges,,, < s < sl,, where smin is the lowest value of s for whichdiffraction takes place and Smut is some selected value of sthat gives a desired number of data. The Deck and the Rollare selected from the Store. The number of data in the Deckplus the Roll may be less than or equal to the number in theStore.A typical selection of Store, Deck, and Roll (SDR) for a

30-atom problem in space group P1 is 250, 171, and 41,respectively. If Eq. 11 were to be used to form the definingequations and rolling would be limited solely to the data,fixed values would be required for the remaining parametersin Eq. 11.

In space group P1, the procedure may be initiated by anarbitrary selection of the coordinates for all the atoms andthey may all be treated as variables without the necessity ofspecifying a particular origin or enantiomorph. The solutionobtained will automatically contain such information. Theprocedure involves the following steps.

1. The initial set of coordinates is placed into the mini-mization procedure with the use ofa Deck formed from thosedata with the lowest s values in the set comprising Store.

2. A maximum of eight repetitions of the least-squarescalculation is applied for a particular Deck. After each suchcalculation, the changes in all the parameters are divided bysuccessive powers of 2 and those changes giving the lowestresidual are selected. Division by powers of 2 is discontinuedas soon as the residual increases over that of a previous step.This process prevents the calculation from skipping past thelocal minimum and diverging out of control.

3. There are fewer than 8 repetitions of the least-squarescalculation if (a) in carrying out successive divisions bypowers of 2, all changes in the residuals are <0.005 or (b) thedifference between the resulting residuals in two successiverepetitions of the least squares calculation is <10-5.

4. Once the repetitions of the least-squares calculation,applied for a particular Deck, are stopped, the minimizationproceeds by rolling the Deck. This is achieved by use of theStore. The particular manner used for rolling the Deck is asfollows. The sets of data in Store and Deck are arranged inincreasing values of s and numbered in sequence so that thelowest number is attached to the datum having the lowest svalue. On rolling, those data associated with the highestnumbers in the list are removed from the Store in an amountcorresponding to the number of elements in the Roll andplaced at the bottom of the list in the Deck. The list in theDeck is now renumbered with those data at the bottom giventhe lowest numbers and the remaining ones in the Deck allshifted upward numerically, as appropriate. Similarly and atthe same time the proper number of the topmost data in thelisting for the Deck, as indicated by the Roll, are transferredto the bottom of the Store with a renumbering of the listingcomparable to that for the Deck. This results in an altered setof data for the Deck of the correct number, and the least-squares minimization can continue. The same process, in-volving a shifting of the data, can be repeated as often asnecessary.

5. The minimization process has probably reached asuccessful conclusion when the residuals are low and remainso with the addition of additional data that were not used inthe minimization process. A final least-squares calculationwith the original (unmodified) function, as well as a plot oftheresults, may also be appropriate.The parameters a, e, v, El, and E2 in Eqs. 11 and 12 afford

additional opportunities for carrying out rolling procedures.A way to vary the parameters is to have a table of severalvalues for each of the parameters a, e, v, El, and E2. Individualvalues can be chosen for the parameters from the table by useof a random number selector for each of the quantities.Typically, rolling on the data may occur several times before

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switching to a new set of values for the parameters a, E, v, El,and E2. It is also possible to vary the number ofdata in the setsthat form the Store, Deck, and Roll.

Criteria are required to halt a calculation. For the examplespresented in this article, the criteria were simple to applysince exact data were used. The calculations were haltedwhen an excellent fit to the data used in the calculations wasobtained-i.e., a very low residual was computed for theunmodified function (e.g., 10-6). With experimental data,extremely low residuals are generally not likely to be ob-tained, even with a correct answer. Realistic estimates ofresiduals for halting calculations will depend on the problemat hand and will have to be decided accordingly. It may alsobe important to examine the results in terms of physicalmeaning. In the case of the type of problem studied here, thefinal arrangements of the atoms would be expected to obeycertain structural criteria, as well as to produce a good fit tothe experimental data.

Applications

Some results are presented here that show the degree ofsuccess obtained to date. Indications concerning the validityofthe method are apparent, but the calculations do not permitan assessment of the relative merits of the very large numberof alternative procedures available for carrying out themethod or of its ultimate limitations.

Calculations have been performed that illustrate the natureof the procedures. They concern 30- and 40-equal-atomproblems in space group P1 for which exact data have beenused. The source of atomic coordinates for the 30- and40-atom test problems was the structure of zervamycin IIAanalog (4). The first 30 or 40 sets of atomic coordinates listed

Table 1. Application of Eq. 10 to a 30-atom problem in spacegroup P1

Time, sec

Randomstarting set

of coordinates

12345678910

(Sum)

CY = 8;NR*;

Ar = 0.5%

328486462453190220460161435139

3330

CY = 8;NR = 2;Ar = 0.5%

320603177141294275308242241846

3447

The least-squares calculations illustrate the use of rolling of dataand, in one case, the changing of a parameter in the course of acalculation. The Store consisted of 250 data of lowest values for s,171 data were used in the least-squares calculations, and 41 data werechanged each time the data were rolled. For the second column, awas held at the value of 50. When the value of the parameter a waschanged for the calculations whose results are given in the thirdcolumn, the values were chosen at random from 40, 50, and 60. Theresults for 10 random starting sets of coordinates, labeled 1-10, aregiven. All 10 initial random starts converged to a correct resultwithout exception in the times listed. Note that CY is the maximumnumber of cycles (repetitions of the least-squares calculation as newvalues of the variables are obtained), NR is the number of times thedata are rolled before selected parameters are changed, and Ar is thechange in the residual between cycles below which the data arerolled, even if the maximum number ofpermitted cycles has not beenreached.*Not applicable.

in table 1 of ref. 4 were used in the calculations and weretreated as if they all belonged to the same type of atom.Actually, they belonged to C, N, and 0 atoms. The calcu-lations were made from random starts for the positions of theatoms and all the atoms were allowed to move freely,permitting the system to converge to an arbitrary origin andenantiomorph. A Cray XMP/216 computer was used.

Table 1 illustrates the application of Eq. 10 to a 30-atomproblem in P1 and Table 2 illustrates the application of Eqs.11 and 12 to the same problem. There are three coordinatesper atom and the problem was treated as a 90-variableproblem. All tests in this article concern equal-atom struc-tures. The calculations involve rolling on the data and thechanging of various ofthe parameters a, E, v, E1, and 62 as thecomputations proceed. The calculations presented in Tables1 and 2 were made from a succession of arbitrarily differentstarting coordinates. Each proceeded readily to a globalminimum. A comparison of Tables 1 and 2 suggests thepossibility that the use of Eq. 11 or of Eq. 11 combined withEq. 12 has advantages over the use of Eq. 10. If there is anadvantage, it may derive from the opportunity to changemore parameters in Eqs. 11 and 12 than in Eq. 10.Numerous least-squares calculations were made with the

30-atom problem in space group P1 with several hundredinitial sets of random values for the atomic coordinates andthe use of Eq. 10 with 171 data associated with the lowestvalues of s. Without the application of rolling, none of themconverged to a global minimum. As many as 200 cycles ofleast-squares refinement were permitted for 70 of the initialsets. With the 30-atom problem in space group P1 and the useof Eq. 10 with rolling, numerous Store, Deck, Roll setsranging from 170, 141, 19 to 350, 271, 79 were tested withinitial random coordinate sets. Of 30 tests only 3 did notconverge to a global minimum within the limit of 1000 sec.Some additional calculations for 30- and 40-atom problems

are listed in Tables 3 and 4, respectively. The trials concernthe use of Eq. 11 with Eq. 12 under various conditions. Eachvariation of the conditions involved single trials for the

Table 2. Application of Eqs. 11 and 12 to the same 30-atomproblem in space group P1 as considered in Table 1

Time, sec

Randomstarting sets

of coordinates

12345678910

CY = 8;NR = 2;Ar = 0.5%

78173356242268397131129316238

CY = 8;NR = 3;Ar = 0.5%

237331292247194179234133214300

(Sum) 2328 2361As before, the Store consisted of 250 data of lowest values for s,

171 data were used in the least-squares calculations, and 41 data werechanged each time the data were rolled. When the parameters werechanged, they were selected independently and at random from thefollowing table:

a 40 50 60E 0.02 0.04 0.06v 0.30 0.50 0.80

E1, 82 1.00 1.20 1.40The results for 10 random starting sets of coordinates, labeled 1-10,are given. All 10 initial random starts converged to a correct resultwithout exception in the times listed. CY, NR, and Ar have the samemeaning as in Table 1.

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Proc. Natl. Acad. Sci. USA 88 (1991) 10103

Table 3. Application of various modifications of Eq. 11, withadditional use of Eq. 12, to a 30-atom problem in space group P1

ConvergenceStore Deck Roll Conditions time, sec

300 243 57 a = 75; v = 0.05; 290, 371*E1 = 62 = 1

250 171 41 a = 40; E = 0.06; 205v = 0.5; 6E = E2 = 1

250 171 41 a = 50; E = 0.04 2%v = 0.5; E1 = 62 = 1

250 171 41 Random change of a and E; 175V = 0.5; E1 = E2 = 1

250 171 41 Random change of 227all parameters

The calculations were started with random choices for values ofthe coordinates. It is seen that a global minimum was attained inabout the same times shown in Table 2. In these calculations CY =8, NR = 2, and Ar = 0.5%. When the parameters were changed, theywere selected independently and at random from the following table:

a 40 50 60 70 80E 0.02 0.04 0.06 0.07 0.08v 0.03 0.05 0.08 1.20 1.50

E1, 62 1.0 1.2 1.4 1.6 1.8All calculations involved Eqs. 11 and 12 except the first, whichomitted Eq. 12.*Values for two separate trials.

30-atom problem except for the first row of Table 3, wherethere were two. All trials for the 30-atom problem readilyconverged to a global minimum. This was not the case for the40-atom problem. For every individual trial shown in Table4, there were a number ofrandom starts that did not convergeto a global minimum within 1 hr, which was the limit at whicha calculation was stopped. Probably no more than 1 in 10random starting sets of coordinates would converge to aglobal minimum with the techniques and procedures used sofar.

Concluding Remarks

The virtue of modifying the refinement function and applyingrolling procedures is apparent from the results of the calcu-lations. The convergence to a global minimum is not onlyfacilitated, it is extremely improbable that it would beachieved without a rolling procedure. The method employsan excess of data compared to unknown quantities.Only one type of problem has been considered, i.e., the

location of atomic coordinates. This problem has specialfeatures that were taken advantage of-for example, low-frequency terms that were given a disproportionately heavierweight by use of the exponential function exp(-as2) and

Table 4. Application of various modifications of Eq. 11, withadditional use of Eq. 12, to a 40-atom problem in P1

ConvergenceStore Deck Roll Conditions time, sec

300 247 53 a = 50; E = 0.04 2993v = 0.5; E1 = E2 = 1

300 247 53 Random change* of a and E; 988v = 0.5; E1 = 62 = 1

300 247 53 Random change 25%of all parameterst

300 247 53 Random change 1716of all parameterst

330 210 57 Rolling on the 3 sets 1009360 230 71 of data and random390 270 87 change of all parameterstThe calculations were started with random choices for the values

of the coordinates. Note that the fifth entry involved variation ofStore, Deck, and Roll as the calculation proceeded. With a 1-hr limiton the calculations, most of the selections for the starting coordinatesdid not reach a global minimum, unlike the circumstance for the30-atom problem. In these calculations CY = 8, NR = 2, and Ar =0.5%.*See Table 3 legend for parameter values.tOnly columns 1, 3, and 5 of parameter values (Table 3 legend) wereused.

additionally by Eq. 12. It also appears that working in spacegroup P1 without the specification of origin or enantiomorphoffers more opportunities to obtain a global minimum.There are several matters that require investigation before

practical application to the crystal structure problem could beconsidered. These matters concern the effect of incorporat-ing space-group symmetry into the calculations, the treat-ment of unequal atoms, and the improvement of the successrate for the more complex structures. The applicability of themethod of this article to other mathematical systems may wellinvolve other types of function modification.

Further progress may come from general developments inthe character of the functions used for the defining equations.The method offers a myriad of possibilities for further study.

My thanks to Mr. Stephen Brenner for writing the appropriateprograms and making the computations. This research was sup-ported in part by Public Health Service Grant GM30902 and in partby the Office of Naval Research.

1. Karle, J. (1986) Angew. Chem. Int. Ed. Engl. 25, 614-629.2. Stewart, J. M., Karle, J., Iwasaki, H. & Ito, T. (1977) Acta

Crystallogr. Sect. A 33, 519.3. Vand, V. (1%5) in Computing Methods in Crystallography, ed.

Rollett, J. S. (Pergamon, Oxford), pp. 214-215.4. Karle, I. L., Sukumar, M. & Balaram, P. (1986) Proc. Natl.

Acad. Sci. USA 83, 9284-9288.

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