Zeitschrift fur Kristallographie, Bd. 129, S. 124-141 (1969)

Analysis of density distribution of the outerelectrons in hexamine*

By J. L. AMOROS and MARISA Canut-AmorosSchool of Technology, Southern Illinois University, Carbondale, Illinois

(Received May 31, 1968)

AuszugMit Hilfe der ublichen Methoden zur Strukturbestimmung kann die wahre

Elektronendichte-Verteilung in einem KLristall mit Molekiilstruktur nichterhalten werden. Mit der Elektronenschalen-Selektionsmethodo (SES) lassensich hingegen die Beitriige der inneren und der aufleren Elektronen zur Inten-sitat eines Interferenzstrahls getrennt behandeln. Die Methode wurde zur

Untersuchung der Struktur des Hexamethylentetramins angewandt. Isotropeund anisotrope Gaufl-Funktionen zur Beschreibung von Form und Warme-schwingung der C-, N- und O-Atome wurden nach der Methode der kleinstenQuadrate verfeinert. Die drei-dimensionale Elektronendichte-Verteilung in deniiuBeren Elektronenschalen der Atome wurden aus Rontgenstrahl-Interferenz-daten erhalten und mit Daten, die sich mit Hilfe einer Faltungsmethode ergabon,verglichen.

AbstractConventional methods used in x-ray crystallography do not permit a true

density distribution of a molecular crystal to be obtained. The selected-electron-shell method allows treating independently the contribution of the inner andouter electrons of the different atoms to the total x-ray scattering. This methodis applied to the case of hexamine. Isotropic and anisotropic Gaussian functionsdescribing the form and thermal factors of carbon, nitrogen and hydrogen are

refined by the least-squares method. The corresponding three-dimensionalelectron-density maps of the outer electrons of the molecule of hexamine are

obtained from x-ray diffraction data and compared with the calculated ones

using a convolution method.

IntroductionX-ray diffraction is a unique tool that allows the observation of the

electron density of a crystal. After the work of Bragg, in which heshowed that the electron density of a crystal could be obtained from

* Partially supported by the Solid State Sciences Division of the Air ForceOffice of Scientific Research, Grants AF-AFOSR-596-66 and AF-AFOSR-832-67.

Density distribution of the outer electrons in hexamine 125

x-ray diffraction data through Fourier summation, the methodreceived widespread interest and the work of J. M. Robertson (1959)in organic compounds proved the correctness of the approach. Lateron the introduction of high-speed computers showed that it was

possible to determine by an iterative process, not only the atomiccoordinates of the atoms in the structure, but also thermal para-meters describing the individual thermal vibrations of rigid-bodymolecules in a crystal. It is not by chance, then, that crystal-structureanalysts paid much attention to the new trend, forgetting somehowthe study of the electron density of the crystals. The reasons behindthis behavior are well-known: the electron density, as obtained byFourier summation, is not accurate enough to allow the study of theelectron density of bonded atoms, due to many inaccuracies involvedin the original method. First, the atomic scattering factors as tabulatedin the International tables for crystallography, for instance, corre-

spond to free atoms, assumed to be spherically symmetric. Second,the thermal vibrations are assumed to be correctly represented bya probability function that can be described by an ellipsoid. Third,and most important of all, there is no easy way to overcome theinherent limitation due to the selection of a given wavelength in theexperiment.

In order to overcome the serious limitations normally imposed incrystal-structure analysis, several authors have pointed out that theatomic scattering function of a bonded atom is an anisotropic func-tion (McWeeney, 1951, 1952, 1953, 1954). Very recently, also,serious doubts have been arisen (Coppens, 1968) about the legitimacyof using ellipsoids to describe thermal motions of atoms. For instance,comparison of x-ray and neutron ,,thermal ellipsoids" showed an

increase in "thermal vibrations" along directions corresponding tobonding in the molecule, suggesting that probably the "thermalellipsoids" ought to be revised in light of the existence of anisotropyof electron density in bonded atoms, specially in organic crystals.Difference Fourier synthesis have shown, time and again, that residualelectron density in covalently bonded atoms occurs along the bondingdirection.

This shows the necessity for revising the whole question of theelectron density in crystals as obtained from x-ray diffraction tech-nique and to look for a new simple solution of the problem. In a pre-vious paper, Amoros and Canut-Amoros (1968 a) have shown thatit is possible to separate the contributions to x-ray diffraction of the

126 J. L. Amobos and Mabisa Canut-Amobos

inner and outer electrons in molecular crystals. The method is basedin the analytical representation of the atomic scattering factor,treating separately such contributions by associating them with theaccepted shells of the atom; the method can be called appropriatelythe selected electron-shell (SES) method. In this paper, we shallapply the SES method to the analysis of the electron density ofhexamethylenetetramine (HMT).

The selected electron-shell methodThe method assumes that the electron density of an atom can be

given analytically by Gaussian polynomials. Ewald (1940) was thefirst to suggest that the scattering-density distribution of an atomcould be represented by a single Gaussian function. To represent theelectron density, more than a Gaussian function is needed. This can

be done in a variety of ways. However, to give physical meaning toa Gaussian polynomial representation, each term of the Gaussianpolynomial must be associated with each main electron shell of theatom. In this way the assumption is made that the electron densityof a given electron shell, namely the K, L, M,... shell of the Bohratomic model, can be represented by a Gaussian distribution (Amoros,1968).

With this basic assumption, the atomic scattering amplitude isalso given by a Gaussian polynomial because the transform of a Gaus-sian function of width g is a Gaussian of width \\ng. In fact, it ispossible to determine the Gaussian components Gs, i.e., the numberof electrons in the shell s, and gs, the width of the Gaussian of theatomic scattering amplitude from a semi-logarithmic plot of thescattering amplitude of the atoms versus \r* |2 (Amoros and Canut-Amoros, 1968 a). As the Gaussian functions that describe the scatteringpower of the individual shells have different width, each one contribu-tes to certain regions of the reciprocal space. In the region near theorigin of the diffraction space, all the shells contribute, but only thescattering of the innermost electrons contribute to the scattering ofplanes with high scattering angle. Therefore it is possible to separatethe region in reciprocal space where the contribution of the K shellis present alone (see our previous paper).

In real crystals, atoms are subject to thermal vibration. Forsimplicity let us assume that the atom vibrates isotropically in an

harmonic potential field. The smearing function is given by a Gaussianprobability p(r) which is temperature dependent. Then the atomic

Density distribution of the outer electrons in hexamine 127

scattering amplitude is modified by a temperature factor, which isalso given by a Gaussian function. The only consequence of an har-monic vibration is to vary the width of the Gaussian function thatdescribes the shell scattering. Once the Gaussian components are

explicity introduced in the expression of the atomic scattering am-

plitude, the restriction of rigid-body vibrations for the atom is no

longer necessary. Each atomic shell can vibrate with different ampli-tude. The analytical approximation of the atomic scattering factoris therefore very convenient because it allows one to determine thetemperature factors of the different shells independently. In the con-

ventional method used in x-ray crystallography, rigid atoms are

assumed. Several hints exists, however, that this is not the case,and this method will certainly be useful to reach conclusive evidenceon this matter.

If we deal with a structure that has been well-refined by the con-

ventional method it is reasonable to assume that the coordinates ofthe atoms are well determined. We may then accept these as constant

parameters. But even in the case where conventional thermal analysisis assumed to be satisfactory, we are forced to determine thenew B'-s as they now contain the temperature and the Gaussianparameters of a rigid shell. These new parameters contain thecontribution of the average electron density of the given shellof the atom at the temperature at which the observations havebeen made.

We know that, in molecular structures, a bond depends on theoverlapping of charge in the outer shells of the two atoms which are

bonded together (Slater, 1965). This overlapping is expected to bea maximum when the maximum charge densities of the outer shellsof the two atoms coincide. The anisotropy in the electron density ofthe atoms of a molecule depends, then, on the contribution of theouter electrons alone. The distribution of the inner electrons is expect-ed to be spherically symmetric. Then any anisotropy observed in thedistribution of the inner electrons must come from the distortionproduced by the anisotropy of the thermal motion alone. This givesa good way to determine with high accuracy the thermal motion ofthe atomic nuclei and the inner electrons tightly bonded to them.In molecular crystals with C, N, 0 and H, the region outside a limitingsphere of r* about 1.0 A-1 contains the contribution of the inner elec-trons alone. In this case, it is possible to deduce the best values forB'jK using only data of high angles.

128 J. L. Amoros and Marisa Canut-Amoros

Once the best values of the Gaussian parameters of the shelldescribing the inner electrons have been determined, it is just a matterof routine to calculate the contribution of these electrons to all theobserved x-ray data. A difference Fourier synthesis can then be cal-culated using (F0—Fk) as Fourier coefficients for those reflectionswith r* < 1.0 A-1. This new series has important properties: First,it is free from series-termination effect, because the Fouriercoefficients are zero outside the region of r* 1.0 A-1. Second, itcontains the information of the distribution of the most importantelectrons, the outer electrons, in the bonding system of a molecule.It contains also the contribution of the hydrogen atoms if they are

present in the structure. The (F0—Fk) Fourier summation is funda-mental in the analysis of the electron distribution in a molecule andit is the basic equation in the SES method.

Selected electron-shell refinement of hexamethylenetetramineSeveral reasons exist to select hexamethylenetetramine to test the

SES method. Preliminary work in a previous paper (Amoros andCanut-Amoros, 1968a) showed that it was possible to obtain two-dimensional maps of the electron density of the inner and outerelectrons in complicated organic substances, and therefore it seemedconvenient to try the method to study three-dimensional cases.

Hexamethylenetetramine is a fairly simple cubic substance witha well known crystal structure, for which very good x-ray diffractiondata have been provided by Becka and Cruickshank (1963) forthree different temperatures, namely 298°, 100°, and 34°K. Themolecule contains only carbon, nitrogen and hydrogen atoms, andtherefore the substance is well adapted to the application of theSES method. In our study we shall assume that the atomic coordinatesas given by Becka and Cruickshank are correct and that the ob-served scattering amplitudes reported by the authors are good enoughin view of the small discrepancy factors achieved in their refining ofthe crystal structure. Becka and Cruickshank (1963) pointed outthe greater accuracy of their data from crystal 5. Therefore thecorresponding set of data for this crystal has been used in the presentGaussian refinement.

If only a two-term Gaussian approximation is used in describingthe atomic-scattering amplitudes associated with the K and L shellsof carbon and nitrogen atoms at rest, one obtains Gck = 1-8, Gcl =4.2and Ctn/c = 1.8 and G^l = 5.2 electrons rather than 2 and 4 for

Density distribution of the outer electrons in hexamine 129

carbon, and 2 and 5 for nitrogen (Amoros and Canut-Amc-ros, 1968)respectively. McWeeny (1951) calculated the isolated-atom-factorcomponents corresponding to the inner shell and valence shell ofthe carbon atom, for which he reported 2 and 4 electrons, respec-tively.

No attempt was made to refine the Gs parameters by imposingspecial conditions such as Gqk + Gcl = 6 electrons. In one refine-ment, an attempt was made to refine the gs parameters by fixing theGs to 1.8 electrons for GK and NZ, and 4.2 and 5.2 for GL and NA.In another refinement, the Gs parameters were fixed to 2 for CK andNA", and 4 and 5 GL and NA, respectively, and 1 for H. The resultingB values favored this choice.

All the refinements were done by the least-squares method withthe ORFLS program of Busing, Martin and Levi (1962), using theneutron-scattering modification, introducing the electron charges Gsof the corresponding shell instead of the neutron-scattering factor,and allowing the refinement of the "temperature factors" B'js that,with this method, include not only the true Debye-Waller tempera-ture factors Bj, but also the form factors of the s shell. With thisprocedure it is then assumed that both factors are correctly representedby Gaussian functions. In a similar way for the anisotropic case,the (3'ijs substitute for the well-known temperature factor /Sy.

The first step is to determine the Gaussian parameters g'jK de-scribing the K electrons of carbon and nitrogen atoms in hexameth-ylenetetramine at a given temperature. In the previous paper we

have shown that for r* > 0.9 A-1 the only contribution to the scat-tering amplitude is that of the inner electrons of carbon and nitrogen,while the contribution of the hydrogen atoms is so small that it can

be disregarded at this moment. Only observed reflections withr* > 0.9 A-1 have been used in the first step of the refinement.

In the next step, the Gaussian parameters of the outer electronsof carbon and nitrogen, and of one of the hydrogen atoms are refinedby least-squares method using the information available inside thewhole limiting sphere. For this refinement, the values of B'jK or

p'ijK already obtained for the K shell are kept constant and onlythe isotropic parameters B'jL or the anisotropic f3'ijL are allowedto vary.

A preliminary isotropic Gaussian refinement of the atomic scat-

tering amplitude of hexamethylenetetramine was done. The corre-

sponding B'js parameters obtained are given in Table 1. The increase

Z. KristaUogr. Bd. 129, 1-4 9

130 J. L. Amoros and Marisa Canut-Amoros

Table 1. Isotropic refined B' parameters of the carbon, nitrogen and hydrogenatoms in hexamethylenetetramine at three temperatures

AtomB'

34°K 100°K

CarbonNitrogenHydrogenCarbonNitrogenR

KK

LL

2 electrons2I45

1.382 k?1.063

12.719.914.74.9o/o

2.158 A21.753

14.522.116.95

6.20/0

of the R value with temperature indicates the increasing importanceof anisotropic thermal motion with increasing temperature. Thereforean anisotropic refinement was in order. The f}'ijs parameters deducedby the least-squares method are given in Table 2. To deduce thesevalues the point symmetry of the atomic locus was taken into account.The final R values are given also in Table 2 for the different tempera-tures of the experimental values. The results are slightly better thanthose given by Becka and Crtjickshank using rigid-body atoms andthat are included in parenthesis in the same table.

Table 2. Anisotropic refined fl- parameters of the carbon, nitrogen and hydrogenatoms in hexamethylenetetramine crystal at three temperatures

Charge '11 '38 '12 '13

CKNifHCLNi

CKMHCLN£

CKNKHCLNi

.0058

.0056

.0746

.1064

.0780

.0088

.0090

.1130

.0967

.0900

.0166

.0216

.1184

.1224

.0953

.0080

.0056

.0746

.1046

.0780

.0128

.0090

.1130

.1074

.0900

.0289

.0216

.1184

.1326

.0953

.0080

.0056

.0793

.1046

.0780

.0128

.0090

.1026

.1074

.0900

.0289

.0216

.1024

.1326

.0953

0

-

0.000400

-

.0013

0

-

.000600

-

.0010

.0047

.0011

00.0004

.01900

.0013

0.0006.0523

0.0010

).0047.0025

3.0011

0

Density distribution of the outer electrons in hexamine 131

Table 3. Structure factors for hexamethylenetetramine at 34 °K

r IM IM IfkI hkl r

iIM IM IM

16 8 0 5.75 4.75 4.75 7 6 5 I.52X" 6.4814 6 2 2.22 6.82 5.81 5.81 10 3 1 1.52 5.7112 8 4 2.17 9.76 8.9O 8.9O 10 2 2 1.50 12.8914 5 1 2.16 2.93 3.00 3.00 9 4 3 1.49 9.1214 2 2 2.07 7.77 7.19 7-19 10 1 1 1•46 6.3714 1 1 2.04 3.19 3.07 3.07 7 7 2 1.46 6.589 8 7 2.02 5.08 4.80 4.80 8 6 0 1.45 5-208 8 8 2.00 11.30 11.00 11.00 7 7 0 1.43 6.35

13 4 1 1.97 5.95 5.75 5.75 8 5 3 1.43 8.7110 6 6 I.90 9.72 8.85 8.85 8 4 4 1.42 20.3312 5 1 1.89 6.42 5.86 5.86 9 3 2 1.40 5.311163 1.86 4.68 4.63 4.63 7 6 3 1.40 6.9412 4 0 1.83 13.09 13.61 13.61 7 5 4 1.37 7.8511 6 1 1.82 4.42 4.77 4.77 9 2 1 1.34 5.8312 3 1 1.80 6.19 5.99 5.99 655 1.34 5.9510 7 1 1.77 4.48 4.86 4.86 7 6 1 1.34 6.669 8 1 1.75 6.66 7.10 7.10 8 4 2 1.33 5.27

11 5 0 1.75 5.27 5-38 5.38 8 3 3 1.31 8.43965 1.72 4.52 4.19 4.19 9 1 0 1.31 11.05

10 6 2 1.71 10.31 11.03 11.03 7 5 2 1.28 7.0611 4 1 1.70 5.63 5.25 5.25 6 6 2 1.26 16.7611 3 2 I.67 6.03 5.75 5-75 7 4 3 1.25 8.139 7 2 1.67 4.44 4.72 4.72 8 2 2 1.23 4.827 7 6 1.67 5.95 5.68 5.68 6 6 0 1.23 5.35

113 0 I.65 4.72 4.98 4.98 6 5 3 1.21 7.029 7 0 1.65 7.20 7.13 7.13 8 2 0 I.19 5-128 8 0 1.64 15.37 16.85 16.85 6 4 4 1.19 5.27

10 5 1 1.62 4.20 4.58 4.58 8 1 1 1.18 11.0111 2 1 1.62 6.21 5.81 5.81 8 0 0 1.16 25.589 5 4 1.60 9.12 8.52 8.52 7 3 2 1.14 8.92

10 3 3 1.57 5.31 5.95 5.95 6 5 l 1.14 6.748 7 1 1.55 6.74 7.01 7.00 6 4 2 1.08 5.478 5 5 1.55 8.57 8.68 8.67 6 3 3 1.06 8.73

5.945.60

13.688.876.I97.044.266.278.87

20.885-617.1'.8.595.815.707.024,698.72

10.857.00

17.078.234.305.276.Hi4.765.27

11.0826. 199.017.505.509.02

5.9*5.60

13.688.87(>. 197.034.266.238.87

20.885.607-158.583.785.637.004.678.67

10.806.98

17.028.114.305.166.824.775.25

10.9225.918.777.285.388.64

hk 1 r IM iFcl |FKl7 2 1 1.06X"1 9.36 9.10 8.835 5 2 1.06 7.46 7.60 7.036 4 0 1.04 4.60 4.61 4.467 1 0 1.02 9.82 10.57 10.175 50 1.02 14.48 14.17 13.224 4 4 1.00 4.30 4.10 4.466 3 1 0.98 9.44 9.64 8.956 2 2 O.96 21.65 22.32 21.245 4 1 O.94 15.17 14.62 13.356 2 0 0.91 5.5I 6.00 5.986 1 1 0.89 8.76 8.70 8.395 3 2 0.89 9.24 9.52 8.636 0 0 0.87 6.27 6.43 6.474 4 2 0.87 6.23 6.27 5-955 3 0 0.84 15.27 14.91 13.244 3 3 0.84 3.77 3.73 2.424 4 0 0.82 38.27 37.42 32.185 2 1 0.79 9-76 9.69 8.364 3 1 0.74 16.16 15.90 13.094 2 2 0.71 8.19 7.68 6.523 3 2 0.68 15.67 15.91 10.884 2 0 0.65 9.32 8.94 7.193 3 0 0.61 18.54 18.92 12.633 2 1 0.54 17.13 16.69 10.602 2 2 0.50 49.57 51.18 26.523 1 0 0.46 2.97 2.75 1.012 2 0 0.41 14.57 14.56 7.622 1 1 O.35 24.19 25.61 10.632 0 0 0.29 19.43 19.63 7.421 1 0 0.21 46.00 45.99 16.330 0 0 0.00 152.00 40.00

Table 4. Structure factors for hexamethylenetetramine at 100°K

h k 1

12 4 o12 3 19 8 1

10 6 29 7 08 8 0

11 2 110 5 1

9 5 49 6 1

10 3 38 7 18 5 5

10 3 110 2 29 4 3

10 2 010 1 17 7 27 7 08 5 38 4 49 3 27 6 37 5 4

i.83 r11.891.741.711.6 51.63i. 621.62l

-

591.571.571.541.5*1.51l. 501.491.471.461.431.431.411

.

401.401.37

IM IfkI Fd IM6.933.49--.226.664.71

10.074.212.615-052.713.824. 175.804.058.476.472.754.324.603.105.80

14.224.0 35-345-57

7.373.474.436.414.49

10.273.992.875.842.813.814.366.014.088.98o. 292.884.145.013.426.31

14.333.7*5.076. DO

4.446.414.49

10.273.992.875.832.803.814.366.054. OB8.986.312.884.145.013.396.35

14.383.765.105. »6

7 6 19 2 18 4 29 1 08 3 37 5 26 6 27 4 36 6 06 5 38 2 06 4 48 1 18 0 07 3 26 5 17 3 06 4 25 5 27 2 16 3 36 4 07 i 05 5 06 3 1

1.34*-'1.341.321.311.311.271.261 .241.221.211.191.191.171.15

1.001.061 .001.041 .01'1 .02

5.533.803.47

6.145.28

13.055.424.365.183.723.498.51

20.556.995.*91.904.755.187.2 I7.123.747.91

12.958. 18

5.513.803.348.546.205.36

12.585-534.315.323.903.328.90

20.367.635.593.414.485.757.688.133.907.94

12.387.97

5.543.793.318.526.155-38

12.615-364.155-333.883.358.81

20.187.465.423.204

.

JO5.1*7.457.663.85T.72

11.627.50

6 2 25 4 16 2 06 1 15 3 26 0 04 4 25 3 04 3 34 4 05 2 14 3 14 2 23 3 24 2 03 3 04 1 13 2 12 2 23 1 02 2 0

0.96 r10.940.91

0.870.870.840.8'.0.820.790.740.710.68O.650.610.610. 540.500.460.410.350.290.20(1.00

18.8013.444.997.567.004

.

075.88

13.343.16

30.568.42

14.04

15.977.73

17.921.66

15.8844.802.49

14.7125.71

18.6213.125.297.807.714.756.21

13.323.18

31.937.95

13.716.83

15.328.20

16.061.70

16.4745.712.79

13.6724.1519-0846.00

152.00

17.8111.935.147.417.264.615,40

11.832.20

28.377.03

11.695.70

10.316.40

10.980.52

10.3325.27

1.107.44

10.137.50

16.4840.00

Table 5. Structure factors for hexamethylenetetramine at 298 °K

h k 1

8 4 47 6 19 1 07 5 26 6 26 6 06 5 38 2 08 1 17 4 18 0 07 3 26 5 17 3 06 4 2

1.40 X"1.321

.

201.261.241.211.191.171.161.161.141

.

121.121 .081.07

M4.2 42. 1 43. V)2.324.6 43-052.462.D84.361.838.474.242.341.962.75

IM IM IM IM IM IM4.292.563.502.534.832

.

<) 42.532.084.331.728.874.060.001.982.68

4.292. 563.492

.

524.812.932.512. 084.291 .698.774.010.00i.9>2.66

6 3 37 2 17 1 05 5 06 3 16 2 25 4 16 2 06 1 15 3 24 4 25 3 04 3 34 4 05 2 1

1.05 A"1.051.011.010.970.950.920.900.880.880.850.830.830.810.78

4.994.323.056.374. 569.968. 102.6 54.^94.60

9.372.30

21.814.60

6.574.48

10.438.072.874.954.954.239.13

/2.1321.46

4.944.011.016.294.249.957.552.854.394.533.988.231.54

18.633.90

4 2 23 3 24 2 03 3 03 2 12 2 23 1 02 2 02 T 12 0 0

0.73 i0.70O.670.640.600.530.490.450. 400.350.290.200,1(0

10.323.79

10.716.09

10.8114.5435.393.40

14.4421.6118.1850.77

10.303.78

10.795.85

10.8014.50

3.3614.4921.82

8.353.287.654.816.969.23

1.597.339-137.13

16.4740.00

Tables 3 to 5 give the observed structure factors as given byBecka and Cruickshank and the calculated structure factors ob-tained with the anisotropic Gaussian parameters just described. The

9*

132 J. L. Amoros and Marisa Canttt-Amoros

contribution Fk of the inner electrons of the carbon and nitrogenatoms to the total scattering amplitudes is also given in the corre-

sponding tables. The reflections have been arranged in terms of de-creasing modulus of reciprocal vector in order to show the region inwhich contribution of the different electron shells of carbon andnitrogen atoms and the contribution of the hydrogen atoms are

appreciable. It can be seen than, for r* > 0.9 A-1, the contributionof the outer electrons and hydrogen electrons is negligible. Even inthe case of the reflection 800, where the contribution of all the atomsis almost in phase, the contribution of the outer electrons and hydrogenatoms is less than l°/0 of the total structure factor. This shows thatthe method outlined here has a sound physical basis. The analysisof the results also shows that the use of a difference Fourier synthesisincluding all the (F0—Fc) is not the best way to determine theposition of the hydrogen atoms in one structure, as they do not con-

tribute to the scattering of the whole diffraction space. A selectivedifference Fourier synthesis using only the reflections up to the regionwhere the contribution of the hydrogen atoms is effective should bemore reasonable.

Effective atomic scattering amplitudes of carbonand nitrogen in hexamethylenetetramine

The SES method allows us to determine the variation of theeffective scattering amplitudes of the atomic electron shells as a

function of temperature, and to study its consequences. Table 6

gives the Gaussian parameters g's of carbon, nitrogen and hy-drogen atoms calculated by using the refined B's values of Table 1.Figure 1 represents the isotropic atomic-scattering amplitudes of car-

bon and nitrogen as deduced from the present Gaussian least-squares

Table 6. Isotropic Gaussian parameters of the carbon, nitrogen and hydrogen atomsin hexamethylenetetramine crystal at three temperatures

Shell 6?s charge34 °K 100°K

KKLL

2 electrons2451

.186 A

.164

.739

.620

.65

.234 A

.210

.748

.652

.73

Density distribution of the outer electrons in hexamine 133

0 7 2 0 12

Fig. 1. Gaussian components of the isotropic atomic scattering factor for carbon(left part) and nitrogen atoms in hexamethylenetetramine at different tempera-

tures

refinement. The most important temperature variation is observedin the scattering amplitude of the inner electrons. However, theeffective scattering of the outer electrons is only slightly affected byan increase in temperature. Due to this fact the contribution to thetotal scattering of the L electrons at different temperatures is ob-

134 J. L. Amobos and Maeisa Canut-Amobos

served always in the same region of diffraction space, namely up toabout r* < 0.9 A~i.

The atomic electron densities calculated by Fourier inversion ofthe spherically symmetric atomic-scattering amplitudes and its radialdistributions are given in Fig. 2. They correspond to carbon andnitrogen in hexamethylenetetramine at the three temperatures ofobservation. It is readily seen that the most important contributionto the atomic electron density is due to the inner electrons, the outerelectrons contributing very little.

In an analogous way, it can be seen that the temperature variationof the electron density of the inner electrons is by far more pronouncedthan that of the outer electrons. An important consequence is that theouter-electrons density maps are expected to show slight variationwith temperature and therefore are potentially important in the studyof the electron-density distribution of the most important electronsin a molecule—the outer electrons.

The temperature variation of the radial distribution of the electrondensity is again very important for the inner electrons, while theradial distribution of the outer electrons is very little affected bythermal motion. This confirms the view that the normal procedure of

- r[&]Fig.2(i). Gaussian components of electron density of carbon (upper part) and

nitrogen atoms in hexamethylenetetramine at different temperatures

Density distribution of the outer electrons in hexamine 135

calculating bonds by overlap of the radial distribution at rest isa sound approximation to reality.

0 7 2 0 7 2

Fig.2(ii). Gaussian components of radial distribution of the electron densityof carbon and nitrogen atoms in hexamethylenetetramine at different tem-

peratures

136 J. L. Amoeos and Mabisa Canut-Amoeos

Density distribution of the outer electrons in hexamethylenetetramineThree-dimensional electron-density maps of hexamethylenetetra-

mine at 34°, 100°, and 298 °K have been computed by using bothconventional and SES methods. As it is well known, the conventionalmethod uses as Fourier coefficients the observed F0(hkl) values. Thisyields an electron-density map where obviously no analysis of thecontribution of the different electron shells of the atoms is possible.Such maps cannot be used when questions such as distribution ofbonding electrons are asked. Series-termination effects are normallyvery disturbing in the conventional maps arising mainly from theexperimental cutoff of the scattering contribution of the inner elec-trons and by the pronounced contribution of the inner electrons to thetotal electron density, as seen in Fig. 2. Even at room temperature,where the increase in thermal agitation makes the contribution of theouter electrons relatively more important than the inner electronscompared with lower temperatures, the electron-density maps were

unsuitable for the study of the bonding and outer electrons in hexa-methylenetetramine. For this reason those maps are not reproducedhere.

The non-rigid atomic model allows the independent calculationof the electron density of the inner and outer electrons of the atomsin hexamethylenetetramine, because their contributions can be ob-tained independently. However if only Fourier series are used, theK maps cannot be used because the series-termination effect is even

more pronounced than in the conventional method. The practicalway of calculating K maps is by convolution integrals. This methodwill be discussed elsewhere.

The SES method has its best application in the computation ofFourier maps corresponding to the electron-density distribution ofthe outer electrons. In this case a the Fourier series (F0

—

Fx) can becomputed using ordinary techniques. In order to show the potentialityof the method, two kinds of maps have been calculated, namely thesection of the molecule through (110) and a series of sections parallelto (001) at increments of z = 0.04. The actual computations were

done using a Fortran program of Zalkix (1962) with an IBM 7044.The contour maps have been drawn with a digital incrementalCalcomp 565 plotter with off-line magnetic unit Calcomp 470. TheIBM 7040 program for contouring was written for us by C. Notestjne.In order to compare results, similar maps have been computed by theconvolution method using the Gaussian parameters derived from

Density distribution of the outer electrons in hexamine 137

(i) 00 (Hi)

Fig. 3. Outer-electrons density maps of hexamethylenetetramine at threedifferent temperatures. Sections parallel to (001) at z = 0 passing through four

carbon atoms. Contour lines at increments of 0.2 e • A-3

d) (H) an)Fig. 4. Outer-electrons density maps of hexamethylenetetramine at three differ-ent temperatures. Sections parallel to (001) at z = 0.12, passing approximately

through two nitrogen atoms. Contour lines at increments of 0.2 e

A-3

(i) 00 OH)

Fig. 5. Outer-electrons density maps of hexamethylenetetramine at threedifferent temperatures. Sections parallel to (001) at z = 0.36, passing approxi-mately through two hydrogen atoms. Contour lines at increments of 0.2 e • A-3

138 J. L. Amoros and Marisa Canut-Amoros

least-squares analysis and a Fortran program developed in our

Laboratory (Amoros and Casper, 1968). A set of maps correspondingto special positions are reproduced in Figs. 3 to 5 at 34° (i), 100° (ii)and 298°K (Hi) respectively.

Sections (110) are very interesting because they contain informa-tion about C—N bonding and the effect of hydrogen atoms on theelectron density of hexamethylenetetramine molecule. Sections parallelto (001) are interesting because they give information about othersignificant interactions between atoms in the molecule. The mapsshow little variation with temperature, in accordance with the theo-retical part of the method. In all the cases the electron density outsidethe region of the molecule drops to practically zero and it is alwayssignificantly less than 0.2 electrons A-3. The assumption that inhexamethylenetetramine, van der Waals forces are the main cohesiveforces in the crystal, is therefore a correct one.

Some thermal effect is, of course, observed in the maps. This effectis particularly interesting when the peak electron densities of theouter electrons of the atoms are examined. For instance, the peakelectron density of carbon has the following values at 298°K:1.72e-A-3; at 100°K: 2.20 e • A"3 and at 34°K: 2.2 e • A"3. Ina similar way, the peak electron density of the nitrogen atom is2.60 e • A"3, 3.0 e • A"3 and 3.4 e A-3 at 298°, 100° and 34 °K re-

spectively. Peak electron-density values decrease with increasingtemperature as expected from theory.

The SES maps are interesting also because they allow the locationof the hydrogen atoms, at least the direction in which the hydrogenatoms are located. The sections (110) at low temperature show twohills in the electron cloud of the carbon atoms that correspond to thedeformation of the electron-density function due to the presence ofthe hydrogen atom. A comparison with convolution maps calculatedwith and without hydrogen atoms show very clearly the effect (Fig. 6.)At higher temperatures the hills are spread by the thermal motion,nevertheless the distortion of the electron cloud due to the hydrogenatoms is still clearly visible. Therefore SES maps might be very usefulin determining the hydrogen atoms by x-ray diffraction technique.The electron density oberved shows that the general distribution can

be explained by the overlap of the outer electron shells of the differentatoms forming the molecule. This can easily be proved by com-

paring the observed SES map and the convolution map. At this stagethen, we may assume, that the anisotropy of the electron density

Density distribution of the outer electrons in hexamine 139

due to the bonding is not important in hexamethylenetetramine.The positive electron density observed in the central region of themolecule in the sections at the three temperatures is due to the

(iv) (v)Fig. 6. Outer-electrons density maps of hexamethylenetetramine at threedifferent temperatures. Section through (110), contour lines at increments of0.2 e

A"3, (i) at 34°K, (ii) at 100°K, (Hi) at 298°K, (iv) convolution map withhydrogen atoms, (v) conv olution map without hydrogen atoms, both at 34 °K

overlap of the outer electrons alone. Values of 0.7, 0.5 and 0.9 elec-trons • A"3 are observed in this region at 298°, 100° and 34 °K re-

spectively. In the region between carbon atoms in the section 2 = 0

140 J. L. Amobos and Mabisa Canut-Amoeos

parallel to (001) the electron density observed is 0.9, 0.7 and 0.6 elec-trons • A~3 at 298°, 100° and 34 °K respectively. In hexamethylene-tetramine a bond between carbon and nitrogen involves a minimumdensity of about 1.65 electrons in the line of bonding. In any otherdirection where no bonding is expected, the electron density is ofthe order of 0.65 e • A-3 or less.

ConclusionsThe method outlined here may prove advantageous in the study

of bond electrons in molecular crystals. It is also applicable to othertypes of substances, as it will be reported elsewhere.

The importance of the SES method is that it gives an informationthat is almost independent of theory and therefore has significancefor the analysis of theoretical expectation confronted with experi-mental fact. The SES method is an experimental method and, as inany experimental approach, the results obtained depend largely on

the accuracy of the experimental data. For instance, all observedF(hkl) values have to be used in the calculation of SES maps and thevalues have to be properly corrected. Compton and thermal scatteringshould be corrected for. In case of intensities affected from extinctionwe know that the experimental data are not reliable. In hexameth-ylenetetramine the best value reported by Becka and Cruickshankfor P(110) at room temperature was used. For lower temperatures no

P(110) is reported, and the calculated value was used in the finalSES maps. F(222) is also affected by extinction. According to Beckaand Cruickshank the extinction of 222 is less important at the lowertemperatures. The best data reported by Becka and Cruickshankwere used in the calculations. This means that the SES maps are

good only as far as the observed data are good.The question remains on the validity of the one Gaussian approxi-

mation of the inner-shell scattering factor. From the experimentalpoint of view this is the only approximation that makes sense becausewe do not have experimental values outside the limiting sphere.The criterion of a good result is that the (F0

—

Fc) Fourier seriesmust be zero or with unimportant random differences. Thereforethis kind of synthesis gives a good way to check the experimentalsoundness of the refinement.

It is necessary for more experimental work in these directions inorder to get full advantage of the method outlined in this paper.

Density distribution of the outer electrons in hexamine 141

AcknowledgmentsWe wish to thank the Data Processing and Computing Center.

Southern Illinois University for the facilities given to our research,Also we want to thank Mrs. Janice McIntyre for taking care of thepreparation of the computations involved in this work.

ReferencesJose Luis and Marisa Amoros (1968), Molecular crystals: their transforms

and diffuse scattering. Wiley Monographs in Crystallography (John Wileyand Sons, Inc., New York).

J. L. Amoros and Marisa Canut-Amoros (1968a), On the effect of the electronshell structure of the atoms in x-ray diffraction. Z. Kristallogr. 127, 5—20.

J. L. Amoros and Marisa Canut-Amoros (19686), Thermal motion of the core

and outer electrons in molecular crystals. Abstract, winter meeting ACA,page 20.

J. L. Amoros and T. Casper (1968), Computation, free of series-terminationeffect, of the components of the election density of a molecular crystal.Abstract winter meeting ACA, page 21.

L. N. Becka and D. W. J. Cruickshank (1963), The crystal structure ofhexamethylenetetramine I. X-ray studies at 298, 100 and 34°K. Proc. Roy.Soc. [London] A 273, 435-454.

W. R. Busing, K. O. Martin and H. A. Levy (1962). ORFLS, a FORTRANcrystallographic least-squares program. Oak Ridge National Laboratory,ORNL-TM-305.

P. Coppens (1968), The significance of discrepancies between x-ray and neutronthermal parameters. Abstract winter meeting ACA, page 37.

P..P. Ewald (1940), X-ray diffraction by finite and imperfect crystal lattices.Proc. Physic. Soc. [London] A 52, 164—174.

International tables for x-ray crystallography (1962) Volume 3. Kynoch Press,Birmingham.

R. McWeeny (1951), X-ray scattering by aggregates of bonded atoms. I. Analy-tical approximations in single-atom scattering. Acta Crystallogr. 4, 513—519.

R. McWeeny (1952), X-ray scattering by aggregates of bonded atoms. II. Theeffect of the bonds: with an application to H2. Acta Crystallogr. 5, 463—468.

R. McWeeny (1953), X-ray scattering by aggregates of bonded atoms. III. Thebond scattering factor: simple methods of approximation in the general case.

Acta Crystallogr. 6, 631—637.R. McWeeny (1954), X-ray scattering by aggregates of bonded atoms. IV. Ap-

plications to the carbon atom. Acta Crystallogr. 7, 180—186.J. M. Robertson (1953), Organic crystals and molecules. Cornell University

Press.J. C. Slater (1965), Symmetry and energy bands in crystals. McGraw-Hill,

New York.A. Zalkin (1962), A FORTRAN Fourier program. Lawrence Radiation Labo-

ratory, California.