DATA LOCATION IN A FOUR-DIMENSIONAL PERIODIC SYSTEM OF DIATOMIC MOLECULES

In: Chemical Information and Computational Challenges… ISBN 978-1-61209-923-1Editor: Mihai V. Putz, pp. 197-212 © 2011 Nova Science Publishers, Inc.

Chapter 7

DATA LOCATION IN A FOUR-DIMENSIONALPERIODIC SYSTEM OF DIATOMIC MOLECULES

Gary W. Burdick1 and Ray Hefferlin2

1Department of Physics, Andrews University, Berrien Springs, Michigan 49104, United States of America

2Physics Department, Southern Adventist University, Collegedale, Tennessee 37315, United States of America

ABSTRACT

The chart of the elements provides a strikingrepresentation of the periodic law. The construction of ananalogous periodic system of diatomic molecules hastantalized several investigators over the span of a century.There have been different approaches to the construction. Inthis chapter, while not denigrating other approaches, weconsider the one which is solidly based on the chart of theelements. In its most general form it is a direct product ofthe element chart (considered as a matrix) with itself; itcan be represented therefore as a four-dimensionalarchitecture. Finding data imprinted in the element chart istrivial; finding digital data in a complex four-dimensional

Corresponding author, Email: [email protected]

In: Chemical Information and Computational Challenges… ISBN 978-1-61209-923-1Editor: Mihai V. Putz, pp. 197-212 © 2011 Nova Science Publishers, Inc.

geometrical structure is not difficult with indexing, but inthis chapter we present a semi-visual method to supplementthe indexing process. We also submit a semi-visualrepresentation of a search for new data based on surfacefitting of fixed-group diatomics. The presentation of thesemi-visual method begins with a very simple case in whichthe element chart is imagined to have just four elements andthen extends to its development in the hyperstructure withmain-group and transition-metal molecules. The addition oflanthanoid and actinoid molecules would present no seriousdifficulty.

1. INTRODUCTION

Data structures that are easily accessible to manipulationin four-dimensional space may soon be available. It is alreadysomewhat possible to “visualize” the hypercube [1,3,4] and thehypersphere [2,4], and more complicated objects [5,6] by theuse of perspectives and rotations of projections into threedimensions. These 3-d projections, now usually seen on flatcomputer screens, may soon be available suspended in three-dimensional space (as they are to the crew of the starshipEnterprise in Star Trek, the New Generation).

Table 1. Geometrical properties of a series of related objectsin spaces with from zero to four dimensions

Space Vertices Edges AreasPoint 0 1 0 0Line 1 2 1 0Square 2 4 4 1Cube 3 8 12 6Hypercube 4 16 32 24

Table 1 shows the arithmetical genealogy of the hypercube.It can be seen that the number of vertices is two to the powerof the space dimensionality. The number of edges and the

Data Location in a Four-Dimensional Periodic System…

number of areas can be determined iteratively from theexpressions, En = 2 En-1 + Vn-1 and An = 2 An-1 + En-1, respectively.Notice that this same iterative expression works for countingeach higher-dimensional objects, such as volumes: Voln = 2 Voln-1

+ An-1. Science fiction [7] includes a humorous story of a 3-dhome gone bad because the hypercube, whose projection was usedto design it, unexpectedly rotated; doors now led tocompletely unexpected destinations and escape was onlypossible thanks to an earthquake. The vectors and matrices ofelectrodynamics have long provided an example of how 4-spaceentities are manipulated in physics, and now string theory hasmade four dimensions seem almost trivial.

2. THE FOUR-DIMENSIONAL PERIODIC SYSTEM OF DIATOMICMOLECULES

One candidate awaiting semi-visual data location in 4-d isthe Kronecker-product (direct-product) periodic system ofdiatomics [8-10]. This system is as an umbrella to severalsystems proposed earlier [11-13], allows for the additivity ofsuch properties as atomic radii and high-energy cross sections[12], and has led to predictions of over a hundred molecularspectroscopic constants [14,15].

The periodic chart of the elements locates each element intwo coordinates — the period (or row, R) and the group (orcolumn, C). Considering this chart as being in a matrix andmultiplying this matrix by itself [8-10] produces theKronecker-product periodic system of diatomic molecules. Thesystem addresses each molecule with the period and groupnumbers of each atom (R1, C1, R2, and C2). Given that theelement chart is not a square, when we “square” it from 2-dinto 4-d, we are immediately faced with the difficult task ofvisualizing an architecture far more complex than that of ahypercube.

While location of data in multidimensional data structurescan be accomplished by indexing, a more intuitive approach mayhave value as a supplement. The purpose of this chapter is to

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Gary W. Burdick and Ray Hefferlin

present such an approach to navigating toward a targetmolecule, and whatever data is associated with it, in the 4-dsystem. In order to accomplish this objective, an extremelyoversimplified example will be explored.

3. A SQUARE AND A HYPERCUBE

Let us suppose that the element chart exists of only fouratoms, as shown in Equation (1.1). The direct product (not ausual matrix product!) has 16 diatomic molecules:

(1.1)

It might be more appropriate to use here the four elementsof ancient times (air, earth, fire and water). Such a“periodic chart” is shown in a well-known cartoon by Harris[8]. One of the “molecules” that emerges from a Kroneckerproduct of such a chart with itself is shown in Figure 1 asbeing discussed by two Greek sages.

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Figure 1. The beginnings of a periodic system of molecules based on the four elements of ancient times. Reprinted by permission of Melissa Hefferlin.

In Equation (1.1) the product can be divided into four 22matrices, each of which has molecules with the same left-handatomic symbol (that is, the same values of R1 and C1). Withineach 22 matrix, the individual molecules are located by meansof R2 and C2. Thus there are four independent variables for thelocation of each molecule. These four independent variablesstrongly suggest the axes of a hypercube (Figure 2).

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Figure 2. A commonly-seen projection of a hypercube in 3-space, with the vertices identified with the molecules seen in Figure 1. It is important to keep in mind that in this and subsequent figures, the molecules correspond to vertices, and do not inhabit boxes. The hypercube has 24 square faces. Each face is square, though more or less severely distorted by the projection. Note that Na2 lies on the bottom surface of the outer cube and that MgNa lies on the bottom surface of the inner cube.

Figure 3. The hypercube system of Figure 2, featuring 15 atomic valence-shell electron molecules. Molecules with other electron populations lie on vertices or lines.

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Appreciation of the subtleties in this hypercube can beginby noting the section on which homonuclear molecules (Li2, Be2,Mg2, Na2) lie. These four points define a square, with Li2 andNa2 on what seems to be the outer cube and Be2 and Mg2 on whatappears to be the inner cube. Moreover, molecules with 15atomic valence-shell electrons also lie on the vertices of asquare, as seen in Figure 3. Note that although these squaresappear distorted in their three-dimensional representations,the distortions are merely artifacts of the projection of thehypercube into three dimensions.

Molecules with fixed left-hand atom symbols (atoms 1) lie ona series of planes from left-front to right-rear in thesefigures. An example is given in Figure 4.

Figure 4. The hypercube system of Figure 2, featuring molecules whosefirst atomic symbol is sodium; they are situated on vertices of the back plane.

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Figure 5. The hypercube system of Figure 2, showing the plane on whose vertices lie molecules with atom 2 is Li.

Figure 6. The hypercube system of Figure 2, showing the plane on whose vertices lie molecules with atom 2 is Mg.

Those with fixed atoms 2 lie on a series of squares, two ofwhich are shown in Figures 5 and 6.

If it is desired to find one particular molecule, say NaMg,in the hypercube, then it is necessary first to identify theplane containing Na as atom 1(Figure 4); then to identify theplane containing Mg as atom 2 (Figure 6); and finally to

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locate the point at which the planes touch (clearly at the topof the figure).

4. THE ELEMENT CHART AND MOLECULAR SYSTEM

4.1. The Chart of the Main-Group and Transition-Metal Elements

Consider the skeleton long-form element chart shown inFigure 7.

Figure 7. A common version of the element chart, but with lanthanoidsand actinoids omitted and with row-seven elements extended through element number 118. Atoms are located not in boxes but on line intersections and ends. A few salient atom symbols are shown on the perimeter of the figure, and it is this perimeter which will be used hereafter.

As was the case for atoms in earlier figures, it isimportant to keep in mind that here (and in subsequentfigures) the atoms (and molecules) lie at grid points andvertices, and do not inhabit boxes. An understanding of theexplanations given above will suggest that the direct productof the grid in Figure 7 with itself would result in anextremely complex network of connecting lines, each line-crossing representing a molecule. This network would beuselessly complex, and so in what follows only the perimeterof Figure 7 will be used. Every point and corner of thisperimeter is provided with an atomic symbol.

4.2. The Kronecker Product Diatomic Molecular System

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The direct product of the simplified element chart withitself (the perimeter of Figure 7) has lines connecting eachone of the atom symbols on the perimeter with themselves andwith each of the others. It is given in Figure 8.

Figure 8. The periodic system of diatomic molecules formed by the direct product of Figure 7 with itself, in the same sense that Figure2 is the direct product of a square with itself. It is important to realize that this is just one of an infinite number of possible projections; it is chosen for its beauty and its ability to display molecular symbols somewhat unambiguously. The center of the system isat a distance of 4 units from the observer in 4-space, where one unitis the distance between molecules along one of the axes. The C2 axis has been rotated 40 degrees toward the C1 axis. Finally, the center of the projection is at a distance of 8 units from the observer in 3-space.

Figure 9. Some “horizontal” sections of Figure 8. Note the similarityof each one, (actually including the “center line”) with Figure 7.

The element chart of Figure 7 is evident scores of times, invarious orientations and with different distortions, in Figure

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8. For example, some of the “horizontal” sections of Figure 8are isolated in Figure 9.

4.3. Molecules in the Kronecker Product System

Figure 10 is the same as Figure 8 except that some molecularsymbols have been supplied. This figure will now be usedrepeatedly in what follows. It was chosen from among thepossible representations on the basis of its elegance and itsseparating the various molecular symbols so as to be seeneasily.

Figure 10. The same as Figure 8 but supplied with molecular symbols.

Further appreciation of the subtleties of the architecturefollows from a look at the outline containing homonuclearmolecules (Figure 11); it also echoes the form of Figure 7.

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Figure 11. The same as Figure 8 except that the profile containing homonuclear molecules is shown.

Isoelectronic molecules are quite a different story because,due to the non-compact nature of the periodic chart, they willlie on multiple disjoint planes.

4.4. Molecules with fixed Left-Hand Atomic Symbol

Molecules with fixed left-hand atom symbols (atoms 1) lie ona series of planes, each representing again the element chart,as shown in Figures 12 to 17. The figures in this and thefollowing section are arranged in a sequence with a logic likethat of Figures 4 to 6. Six “snapshots” are presented; a fewmore are available upon request to the second author ([email protected]).

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Figure 12. The same as Figure 10, showing all molecules having hydrogen as their left-hand atomic symbol.

Figure 13. The same as Figure 10, showing all molecules having francium as their left-hand atomic symbol. As the alkali-metal atoms 1 are selected from progressively higher row numbers in the chart of the elements, the cross sections move (in this projection) slightly back and noticeably to the left.

Figure 14. The same as Figure 10, showing all molecules having beryllium as their left-hand atomic symbol. The difference from Figure 13 is slight but important.

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Figure 15. The same as Figure 10, showing all molecules having calcium as their left-hand atomic symbol.

Figure 16. The same as Figure 10, showing all molecules having heliumas their left-hand atomic symbol.

Figure 17. The same as Figure 10, showing all molecules having element number 118 as their left-hand atomic symbol.

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4.5. Molecules with fixed Right-Hand Atomic Symbol

Molecules with fixed right-hand atom symbols (atoms 2) lieon a series of planes, each representing again the elementchart, as shown in Figures 18 to 23. The figures in thissection are also based on Figure 9. More figures are availablefrom the second author upon request.

Figure 18. The same as Figure 10, showing all molecules having hydrogen as their right-hand atomic symbol. In this projection, the shaded area is as a disadvantageous angle and so the position of H2 has been singled out especially.

Figure 19. The same as Figure 10, showing all molecules having francium as their right-hand atomic symbol.

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Figure 20. The same as Figure 10, showing all molecules having oxygenas their right-hand atomic symbol.

Figure 21. The same as Figure 10, showing all molecules having heliumas their right-hand atomic symbol.

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Figure 22. The same as Figure 10, showing all molecules having neon as their right-hand atomic symbol.

Figure 23. The perfectly predictable section of Figure 10 containing all molecules having 118 as the right-hand atomic symbol.

4.6. Identifying a Target Molecule

Now the time has come to complete one primary objective ofthis chapter, to pinpoint the location of a molecule in thehypersystem. Let that molecule be CO. It is necessary first toidentify the plane containing atom 1 (carbon, the dark planein the Figure); then to identify the plane containing atom 2(oxygen, as in Figure 20); and finally to locate the point atwhich the planes touch, as in Figure 24. The question arises,where is the identical molecule oxygen carbide? The planecontaining all molecules with the left-hand symbol, oxygen, issimilar to that for carbon molecules in Figure 24 but a bittoward the left front. The plane containing all molecules withthe right-hand symbol, carbon, is similar to that for oxygenmolecules in Figure 24 but toward the left rear (note how theHeO corner should move toward HeB). Thus, oxygen carbideshould be slightly to the left of the plane containinghomonuclear molecules (Figure 11).whereas carbon monoxide isslightly to the right of that plane.

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Figure 24. Carbon monoxide is located at the point where the planes containing carbon as atom 1 (darker shading) and oxygen (lighter shading) meet. Even in this two-dimensional presentation of a three dimensional projection of the four dimensional system, the point can be identified.

4.7. Visualization of a Prediction Strategy

A representative prediction strategy begins with analysis ofthe data for some property of fixed-group molecules, forinstance group 2-group 16, which lie on a rectangle defined bymolecules at its four corners, namely BeO, Be116, Ra116, andRaO. This rectangle is shown on the right side of in Figure25.

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Figure 25. At the right, group-2, group-16 molecules, a representative choice for least-squares or neural-network fitting andprediction, are shown. At the left, the identical molecules are shownin group-16, group-2 format.

Table 2. Predictions of vibration frequency ωe (cm-1) andinternuclear separation re (Å) using least-squares and neural-networks, with absolute and percentage standard deviations.

Bold-face numbers indicate agreement within sigma bars

Molecule

Property

Prediction

Standarddeviation

LowerBound

Upperbound

Sigma inpercent

CRCvalue

BePo ωe 575.71 119.24 456.47

694.95 20.71

BaO ωe 559.92 106.26 453.66

666.18 18.98 669.76

BaS ωe 321.59 62.36 259.23

383.95 19.39 379.42a

BaSe ωe 218.79 42.37 176.41

261.16 19.37

CaO ωe 716.62 58.55 658.07

775.18 8.17 732.03b

CaS ωe 438.87 42.65 396.22

481.51 9.72 462.23

MgS ωe 533.42 52.57 480.85

585.99 9.86 528.74a

MgPo ωe 324.39 56.85 267.54

381.25 17.53

SrSe ωe 261.01 46.77 214.24

307.78 17.92

BeSe re 1.858 0.081 1.777 1.940 4.37BeTe re 2.017 0.116 1.901 2.133 5.76CaTe re 2.637 0.110 2.526 2.747 4.18MgTe re 2.501 0.132 2.369 2.633 5.26SrSe re 2.613 0.086 2.527 2.699 3.29

a This datum from Reference 18.b This datum from Reference 17.

The identical conjugate molecules, OBe, 116Be, 116Ra, and116Be, lie on a rectangle at the left side of the figure. Thisobservation is a generalization of the situation for CO and OCin the previous section. Table 2 shows the predictionsobtained by the use of both least-squares and neural-networks,

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where it was required that the error bars of one methodoverlapped those of the other.

The methodology of the least-squares analyses was formulatedfrom [16] and the training data were taken from [17]; thetraining data for the neural-networks analyses were from [18].By looking at isovalent species, such as BaO, BaS, and BaSeone can see impressive evidence of periodicity: vibrationfrequencies decrease, and internuclear separations increase,as (R1,R2) increases. The percentage errors shown are largerthan those for the average of all predicted molecules.

CONCLUSION

We investigated how to navigate in the Kronecker-productperiodic system of diatomic molecules. We have shown how tofind molecules whose chemical symbol contains a chosen left-hand or chosen right-hand atomic symbol; how to pin-point aspecific molecule; how to find ensembles of molecules such asthose are used in least-squares or neural-network fitting andprediction; and how to locate conjugate molecules (withreversed atomic symbols) on opposite sides of the planecontaining homonuclear species. Of necessity, we have selectedone originating element chart, one projection in four-space,and one projection in three dimensions; the logic, however,can be applied when other choices are made.

REFERENCES

[1] Banchoff, T. F. and Strauss, C., The Hypercube:Projections and Slicing (film), Banchoff and Strauss Productions,Providence, 1978.

[2] Banchoff, T. F. and Strauss, C., The Hypersphere:Foliation and Slicing (film), Banchoff and Strauss Productions,Providence, 1979.

[3] http://dogfeathers.com/java/hyprcube.html (last accessedJune 27, 2010.

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[4] Banchoff, T. F., Beyond the Third Dimension: Geometry,Computer Graphics, and Higher Dimensions, Scientific AmericanLibrary, New York, 1990.

[5] Heiserman, D. L., Experiments in Four Dimensions, Tab Books,1983.

[6] Hanson, A. J. and Heng, P. A., Visualizing the FourthDimension Using Geometry and Light, Proceedings of the 2ndconference on Visualization ’91, IEEE Computer SocietyPress, Los Alamitos, CA, 1991, pp. 321-328.

[7] Heinlein, R. A., “And he built a Crooked House,” in TheFantasies of Robert A. Heinlein, Torr, New York, 1999, pp.91-109.

[8] Hefferlin, R.; Zhuvikin, G.V., K.E. Caviness, andDuerksen, P. J., J. Quant. Spectrosc. Radiat. Transfer, 32, 257-268,1984.

[9] Hefferlin, R. J. Chem. Info. Comp. Sci., 34, 313-317, 1994.[10] Hefferlin, R. In Philosophy of Chemistry; Baird, D; Scerri, E;

McIntyre, L; Eds; Springer: Dordrecht, 2006, pp. 221-243.Fig. (7) of the chapter is in error and the correct figurewill be sent upon request.

[11] Hefferlin, R. Comb. Chem. High Through. Screen., 11, 691-706,2008.

[12] Hefferlin, R. Periodic systems of molecules and theirrelation to the systematic analysis of molecular data,Edwin Mellen Press: Lewiston, NY, 1989.

[13] Grimm, H.G., Elektrochem., 31, 474 – 480, 1925.[14] Hefferlin, R. J. Quant. Spectrosc Radiat. Transfer, 111, 71-77,

2010.[15] Hefferlin, R. Internuclear separations using least

squares and neural networks for 46 new main-groupdiatomics, Int. J. Chem. Model., submitted June 3, 2010.

[16] Hefferlin, R., Priddy, K. and Sutton, E., The SystematicPrediction of Diatomic Molecular Properties, Paper 07P56,Congress of Pacific Basin Chemical Societies, Honolulu,Hawaii, December 21, 1984.

[17] Lide, D. R., Ed., Handbook of Physics and Chemistry,Taylor and Francis, New York, 2009.

[18] Huber, K. P. and Herzberg, G., Constants of DiatomicMolecules, Van Nostrand Reihnold, New York, 1979.

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DATA LOCATION IN A FOUR-DIMENSIONAL PERIODIC SYSTEM OF DIATOMIC MOLECULES

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