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The Calculation of X-ray Scattering Factors for Spherically Averaged Molecules
Example: Methane, CH4
All molecular gas-phases become crystalline when subjected to sufficient pressure.
It seems that the lowest pressure phases may be characterized as close-packed struc-
hIres wherein each molecule is freely rotating about an associated point. As pressure
is increased the motion is constrained and a lowering of symmetry occurs. In order
to model the freely rotating molecule, the molecular electron-density wave functions
were computed using the Gaussian92 program for energy-optimized model molecules.
The electron-density was then spherically averaged in order to mimic the rotations.
The Fourier transform of this spherically averaged electron density provides the scat-
tering factor for the rotating molecule. Displacement parameters obtained from the
structure refinements provide estimates of translational motion.
A genera.l outline of the procedure is a.s follows:
1) Find the minimum energy structure of CH4 using the Gaussia.n code.
2) Calcul~te the molecular orbital coefficients for this configuration.
:3) Calculate the spherically averaged electron density for the molecule.
4) Transform the electron density into scattering factors.
The Minimum Energy Structure of CH4
The structure of CH4 was calculated using the Gaussian 92 for Windows code
assuming tetrahedral symmetry with a high level 6-:31G** basis set and MP2 M~lIer-
Plesset configuration interactions. The length of the CH bond was varied (optimized
at 1.085.5 A) until a minimum energy (-40.3646259 a.u.) was obtained. The optimized
bond length compares well with a reported CD bond length of 1.084 Afor solid heavy
methane at :34.5 K (Press, 1972).
Molecular Orbital Coefficients
A single point calculation was then carried out at the minimum energy configu-
ration in order to obtain the molecular orbital coefficients. A brief review of the form
of the basis sets is now given. See Szabo and Ostlund (1982) for a more complete
description.
Consider an expansion of a molecular orbital, 'l/Ji,
K
'l/Ji= L C/1i<P/1'
/1=1
where C/1i are the expansion coefficients and <P/1are previously prescribed linear com-
binations (or contractions) of Gaussian functions (primitives). The contraction has
the formL
<p/1(r - RA) = L dp/19p(ap/1' r - RA),
p=1
where ap/1 and dp/1 are the contraction exponents and coefficients, L is the length of
the contraction, and RA is the position of the center of the atom A. The normalized
Gaussian primitives are of the forms
91s( a, r) = (8a3/,rr3)1/4e-ar2
92Px( a, r) = (128a5/,rr3)1/4xe-ar2
93dxx (a, r) = (2048a7/97r3)1/4xye-ar2
93dxy(a,r) = (2048a7/7r3)1/4xye-ar2.
The contraction coefficients (d's) and exponents (a's) are obtained by a least-squares
fit which, for the Is shell, minimizes the integral
and for the higher order sand p shells, minimizes the integral
2
This means that the contractions for higher order sand p shells have the same radial
behavior because they share the same exponents and thus are treated as sp functions.
This is done in order to save computing time. <jJSFand <jJGGFrepresent Slater functions
and contracted Gaussian functions, respectively.
In this study I used a 6-31 G** basis set. The 6-31 G part of the acronym means
that for the carbon atom the inner shell functions are a contraction of 6 primitive
Gaussians and the outer valence shell functions are split into a contraction of 3 prim-
itives and 1 primitive Gaussian. The first *, 6-31 G*, means that an un contracted
d- type polarization function is also included for carbon. Hydrogen does not have
an inner shell, its functions are simply a contraction of 3 primitive Gaussians and 1
primitive Gaussian. The second *, 6-31 G**, means that there is also an un contracted
p-type polarization function included for hydrogen. An example of the basis sets is
given below. The first column are the exponents (a's) and in the second column are
the contraction coefficients (d's). For the sp functions, the first column contains the
exponents and the second and third columns contain the contraction coefficients for
the s and the p functions, respectively.
6-31 G * Basis set for carbon that consists of 4 contractions.
S 6 1. 000.3047524880D+04
0.4573695180D+030.1039486850D+030.2921015530D+020.9286662960D+010.3163926960D+01
SP 3 1. 000.7868272350D+01 -0.1193324200D+000.1881288540D+01 -0.1608541520D+000.5442492580D+00 0.1143456440D+01
SP 1 1. 000.1687144782D+00 0.1000000000D+01 0.1000000000D+01
D 1 1. 000.8000000000D+00 0.1000000000D+01
0.1834737130D-020.1403732280D-010.6884262220D-010.2321844430D+000.4679413480D+000.3623119850D+00
0.6899906660D-010.3164239610D+000.7443082910D+00
3
6-31 G** Basis set for hydrogen that consists of 3 contractions.
S 3 1. 000.18731136960+020.28253943650+010.64012169230+00
S 1 1. 000.16127775880+00
P 1 1. 000.11000000000+01
0.33494604340-010.23472695350+000.81375732620+00
0.10000000000+01
0.10000000000+01
Molecular orbital coefficients are calculated by the Gaussian program and output
to a file in a form like the following.
4
(Ai)--O (Ai)--O (T2)--O (T2)--O (T2)--OEIGENVALUES -- -11.20680 -0.94379 -0.54434 -0.54434 -0.54434
1 1 C is 0.99569 -0.19504 0.00000 0.00000 0.000002 2S 0.02809 0.36481 0.00000 0.00000 0.000003 2PX 0.00000 0.00000 0.16673 0.08239 0.387064 2PY 0.00000 0.00000 0.38776 -0.11790 -0.141945 2PZ 0.00000 0.00000 0.07903 0.40462 -0.120176 3S -0.01324 0.34649 0.00000 0.00000 0.000007 3PX 0.00000 0.00000 0.08255 0.04079 0.191648 3PY 0.00000 0.00000 0.19198 -0.05837 -0.070289 3PZ 0.00000 0.00000 0.03913 0.20033 -0.0595010 XX -0.00178 -0.00116 0.00000 0.00000 0.0000011 YY -0.00178 -0.00116 0.00000 0.00000 0.0000012 ZZ -0.00178 -0.00116 0.00000 0.00000 0.0000013 XY 0.00000 0.00000 0.00550 0.02817 -0.0083714 XZ 0.00000 0.00000 0.02700 -0.00821 -0.0098815 YZ 0.00000 0.00000 0.01161 0.00574 0.0269516 2 H is -0.00003 0.13789 0.25095 0.14621 0.04949
17 2S 0.00242 0.04459 0.22038 0.12840 0.0434618 PX 0.00011 -0.00769 -0.00688 -0.00425 0.0044219 PY 0.00011 -0.00769 -0.00328 -0.00751 -0.0042120 PZ 0.00011 -0.00769 -0.00831 0.00100 -0.0038521 3 H is -0.00003 0.13789 -0.18834 o .17434 -0.14470
22 2S 0.00242 0.04459 -0.16539 0.15310 -0.1270723 PX -0.00011 0.00769 -0.00448 0.00801 0.00077
24 PY -0.00011 0.00769 -0.00088 0.00475 -0.00785
25 PZ 0.00011 -0.00769 0.00849 -0.00007 0.00357
26 4 H is -0.00003 0.13789 0.05624 -0.23961 -0.1619427 2S 0.00242 0.04459 0.04939 -0.21042 -0.1422128 PX -0.00011 0.00769 0.00487 -0.00782 0.0001229 PY 0.00011 -0.00769 0.00417 0.00724 0.0038830 PZ -0.00011 0.00769 0.00344 -0.00257 -0.0081531 5 H is -0.00003 0.13789 -0.11886 -0.08094 0.2571532 2S 0.00242 0.04459 -0.10438 -0.07108 0.2258233 PX 0.00011 -0.00769 0.00726 0.00444 -0.0035334 PY -0.00011 0.00769 0.00177 -0.00502 0.0075235 PZ -0.00011 0.00769 -0.00326 0.00350 0.00788
CH4 coefficients for occupied molecular orbitals
Spherically Averaged Electron Density
If we have an N electron closed-shell molecule described by a single determinant
wave function with each molecular orbital'I/Ja containing two electrons, then the total
5
charge density at the point r is
N/2
p(r) = 2 L l7j1a(r)12,a
where p(r) dr is the probability of finding an electron in the volume element dr at r.
Note that the integral over all space is the total number of electrons,
N/2 N/2
Jp(r) dr = 2 L J l7j1a(r)12= 2 L 1 = N.
a a
Inserting our expansion of a molecular orbital into the density expression gives
N/2
p(r) = 2 L <p~(r)7j1a(r)a
N/2
= 2 L L C:a<p:(r) L C!'a<p!,(r)
a v
N/2
= L [2 L C:aC!'a] <p:(r)<p!,(r)
v!' a
v!'
= <P*P<P
where the density matrix is defined as
N/2
Pv!' = 2 L C:aC!'aa
and <Pis a vector of components, <pv(r). This is the equation that I used to calculate
the electron density.
Initially, in order to compute the spherically averaged electron density, calcula-
tions were undertaken over a 90 x 180 angular grid (16,200 points for each value
of r) that covered a complete sphere around the molecule at a series of fixed radii,
0.000 ::::; 1'.::::; 8.000 bohr, at increments of 0.001 bohr. The spherically averaged
electron density at a given value of l' was then computed as
6
where tl<jJ = 1r189 and tl() = 21r1179. A table of 8001 r versus < p(r) > data points
was constructed.
While this method is theoretically sound, in practice it has its problems. The
solution converges very slowly and takes an amazingly long time to compute. When
summed over all space the total number of electrons exceeds the true value. This
was not really a problem with methane but it was quite apparent with H2, probably
resulting from its apparent lack of spherical symmetry.
What I ended up doing was to obtain < p(r) >, (0 ::::;r" ::::;5A), at intervals of
o.01A. At each value of r the average electron density was computed from a 10 X 10
grid using a 10 step Gaussian quadrature method both in <jJand in (). The results
were much more satisfying.
Figure 1 is a plot of the electron density minus the spherically averaged electron
density along the CH vector. The difference demonstrates the build up of electron
density along the CH vector due to covalent bonding effects. Note that the peak is
located near r = 0.58 A. This is where the electron density from the hydrogen atoms
is concentrated, even though the hydrogen nuclei are at 1.0855 A from the carbon.
Scattering Factors
The scattering factor for an atom or a molecule is difficult to calculate unless the
electron density demonstrates spherical symmetry. However, with the assumption
of spherical electron density, < p(r) >, and with the carbon at the origin, then the
scattering factor of methane, f( 8), is the Fourier transform of p( r),
f(8) =('XJU(r)sin(21rrr*) dro 21rrr* ~ U(r)
sin(21rrr*)tlr,
21rrr*r
where r* = lid = 2sin()l).. = 28 and U(r') = 41rr2 < p(r) > is the radial distribution
function for methane, as plotted in Figure 2. Note that the first peak represent the
electrons from the 18 orbital of C while the second peak is due primarily to electrons
in the higher order shells. The hydrogen nucleii are located near the mid-point of
the second peak. The scattering factors for 201 values of 8, 0.00 ::::;8 ::::;2.00, at 0.01
7
increments were computed and tabulated. The scattering factor curve is presented in
Figure 3 and compared to that of carbon.
The last step in the process is to fit the curve to the standard equation that gives
an analytical approximation to the scattering factor,
4
f«~) = Laiexp(-biS2) +c,i=l
o ~ S ~ 2A-1
where ai and bi are the 8 parameters that are varied and c is obtained from the
constraint
c = f (0) - L ai.
The fitting procedure was accomplished with a minimization program written by
M.B. Boisen Jr. and L.W. Johnson (Boisen and Gibbs, 1993) that is based on a
quasi-Newton method that uses a model trust region (with a double dogleg) and the
BFGS update (Dennis and Schnabel, 198:3). The values of the fitted parameters are
a1.54:3827
-0.5141276.1808722.573929
b0.5808713.995349
13.79743436.110,548
c0.215499
The quantity that was minimized was the error in the fit, defined by Doyle and Turner
(1968) as1/ 2
[
~201 82
]E =
;~~)L.J~O~
i= .0185,
where 8 = f( S)actual - f( S )fitted. The maximum deviation between actual and fitted
values of f( s) is 0.005.
8
References
Boisen, M.B., Jr. and Gibbs, G.V. (1993) A modeling of the structure and compress-
ibility of quartz with a molecular potential and its transferability to cristobalite
and coesite. Physics and Chemistry of Minerals, 20, 123-135.
Dennis, J.E., Jr. and Schnabel, R.B. (1983) Numerical Methods for Unconstrained
Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, pp 378.
Doyle, P.A. and Turner, P.S. (1968) Relativistic Hartree-Fock X-ray and electron
scattering factors. Acta Crystallographica, A24, 390-397.
Press, W. (1972) Structure and phase transitions of solid heavy methane (CD4). The
Journal of Chemical Physics, 56, 2Ej97-2609.
Szabo, A. and Ostlund, H.S. (1982) Modern Quantum Chemistry: Introduction to
Advanced Electronic Structure Theory. MacMillan Publishing Co., Inc, New
York.
9
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