The
Calculation
of X-ray
Scattering
Factors
Example:
All molecular gas-phases
Methane,
become crystalline
It seems that the lowest pressure
the motion is constrained
to model the freely rotating
were computed
tering factor for the rotating
structure
refinements
provide estimates
the molecular
:3) Calculate
the spherically
4) Transform
Plesset configuration
at 1.085.5
A) until
of translational
motion.
model molecules.
obtained
from the
averaged electron density for the molecule.
symmetry
factors.
Energy Structure
of CH4
using the Gaussian
92 for Windows code
with a high level 6-:31 G** basis set and MP2 M~lIer-
interactions.
The length of the CH bond was varied (optimized
a minimum energy (-40.3646259
at :34.5 K (Press,
wave functions
of CH4 using the Gaussia.n code.
bond length compares well with a reported
methane
In order
in order to mimic the rotations.
parameters
of CH4 was calculated
tetrahedral
electron-density
Displacement
the electron density into scattering
The structure
occurs.
orbital coefficients for this configuration.
The Minimum
assuming
averaged
As pressure
is a.s follows:
energy structure
2) Calcul~te
point.
struc-
averaged electron density provides the scat-
molecule.
A genera.l outline of the procedure
1) Find the minimum
as close-packed
program for energy-optimized
of this spherically
Molecules
to sufficient pressure.
about an associated
molecule, the molecular
was then spherically
The Fourier transform
when subjected
and a lowering of symmetry
using the Gaussian92
The electron-density
Averaged
CH4
phases may be characterized
hIres wherein each molecule is freely rotating
is increased
for Spherically
1972).
a.u.) was obtained.
CD bond length of 1.084
The optimized
A for
solid heavy
Molecular
A single point calculation
Orbital Coefficients
was then carried out at the minimum
ration in order to obtain the molecular
of the basis sets is now given.
orbital coefficients.
See Szabo and Ostlund
energy configu-
A brief review of the form
(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
binations
(or contractions)
coefficients
and <P/1are previously
of Gaussian
functions
prescribed
(primitives).
linear
com-
The contraction
has
the form
L
<p/1(r - RA)
L
=
dp/19p( ap/1' r - RA),
p=1
where
ap/1 and dp/1 are the contraction
the contraction,
Gaussian
exponents
and RA is the position
primitives
and coefficients,
L is the length
of the center of the atom A. The normalized
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
fit which, for the Is shell, minimizes
and for the higher order sand
of
(a's) are obtained
the integral
p shells, minimizes
2
the integral
by a least-squares
This means that the contractions
for higher order sand
behavior because they share the same exponents
This is done in order to save computing
and contracted
Gaussian
functions,
p shells have the same radial
and thus are treated
as sp functions.
time. <jJSFand <jJGGF
represent
Slater 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
Gaussians
atom the inner shell functions
and the outer valence shell functions
itives and 1 primitive
d- type polarization
Gaussian.
function
an inner shell, its functions
primitive
Gaussian.
p-type polarization
exponents
of 6 primitive
are split into a contraction
of 3 prim-
The first *, 6-31 G*, means that an un contracted
is also included
for carbon.
are simply a contraction
Hydrogen
of 3 primitive
does not have
Gaussians
and 1
The second *, 6-31 G**, means that there is also an un contracted
function
included
for hydrogen.
given below. The first column are the exponents
the contraction
are a contraction
coefficients
An example of the basis sets is
(a's) and in the second column are
(d's). For the sp functions,
the first column contains
and the second and third columns contain the contraction
the s and the p functions,
respectively.
6-31 G * Basis set for carbon that consists of 4 contractions.
S
6 1. 00
0.3047524880D+04 0.1834737130D-02
0.4573695180D+03
0.1403732280D-01
0.1039486850D+03
0.6884262220D-01
0.2921015530D+02
0.2321844430D+00
0.9286662960D+01
0.4679413480D+00
0.3163926960D+01
0.3623119850D+00
SP
3 1. 00
0.7868272350D+01
-0.1193324200D+00
0.1881288540D+01
-0.1608541520D+00
0.5442492580D+00
0.1143456440D+01
SP
1 1. 00
0.1687144782D+00
0.1000000000D+01
D
1 1. 00
0.8000000000D+00
0.1000000000D+01
0.6899906660D-01
0.3164239610D+00
0.7443082910D+00
0.1000000000D+01
3
the
coefficients for
6-31 G** Basis set for hydrogen
3 1. 00
0.18731136960+02
0.28253943650+01
0.64012169230+00
S
1 1. 00
0.16127775880+00
P
1 1. 00
0.11000000000+01
that consists of 3 contractions.
S
0.33494604340-01
0.23472695350+00
0.81375732620+00
0.10000000000+01
0.10000000000+01
Molecular orbital coefficients are calculated
to a file in a form like the following.
4
by the Gaussian
program
and output
CH4 coefficients for occupied molecular
(Ai)--O
EIGENVALUES --
1 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16 2
17
18
19
20
21 3
22
23
24
25
26 4
27
28
29
30
31 5
32
33
34
35
C
H
H
H
is
2S
2PX
2PY
2PZ
3S
3PX
3PY
3PZ
XX
YY
ZZ
XY
XZ
YZ
is
0.99569
0.02809
0.00000
0.00000
0.00000
-0.01324
0.00000
0.00000
0.00000
-0.00178
-0.00178
-0.00178
0.00000
0.00000
0.00000
2S
PX
PY
PZ
-0.00003
0.00242
0.00011
0.00011
0.00011
is
-0.00003
2S
PX
PY
PZ
0.00242
-0.00011
-0.00011
0.00011
-0.00003
0.00242
-0.00011
0.00011
-0.00011
-0.00003
0.00242
0.00011
-0.00011
-0.00011
is
2S
PX
PY
PZ
H
-11.20680
is
2S
PX
PY
PZ
Spherically
orbitals
(Ai)--O (T2)--O
-0.94379 -0.54434
0.00000
-0.19504
0.36481 0.00000
0.00000 0.16673
0.00000 0.38776
0.00000 0.07903
0.34649 0.00000
0.00000 0.08255
0.00000 0.19198
0.00000 0.03913
-0.00116 0.00000
0.00000
-0.00116
0.00000
-0.00116
0.00000 0.00550
0.00000 0.02700
0.00000 0.01161
0.13789
0.04459
-0.00769
-0.00769
-0.00769
0.13789
0.04459
0.00769
0.00769
-0.00769
0.13789
0.04459
0.00769
-0.00769
0.00769
0.13789
0.04459
-0.00769
0.00769
0.00769
with each molecular
-0.18834
-0.14194
0.40462 -0.12017
0.00000 0.00000
0.04079 0.19164
-0.05837
-0.07028
0.20033 -0.05950
0.00000 0.00000
0.00000 0.00000
0.00000 0.00000
0.02817 -0.00837
-0.00821 -0.00988
0.00574 0.02695
0.14621
0.12840
-0.00425
-0.00751
0.00100
0.15310
0.00801
0.00475
-0.00007
-0.23961
-0.21042
-0.00782
0.00724
-0.00257
-0.08094
-0.07108
0.00444
-0.00502
0.00350
0.04949
0.04346
0.00442
-0.00421
-0.00385
-0.14470
-0.12707
0.00077
-0.00785
0.00357
-0.16194
-0.14221
0.00012
0.00388
-0.00815
0.25715
0.22582
-0.00353
0.00752
0.00788
Density
molecule described
orbital'I/Ja containing
5
-0.11790
o .17434
-0.16539
-0.00448
-0.00088
0.00849
0.05624
0.04939
0.00487
0.00417
0.00344
-0.11886
-0.10438
0.00726
0.00177
-0.00326
Averaged Electron
If we have an N electron closed-shell
wave function
0.25095
0.22038
-0.00688
-0.00328
-0.00831
(T2)--O (T2)--O
-0.54434 -0.54434
0.00000 0.00000
0.00000 0.00000
0.08239 0.38706
by a single determinant
two electrons,
then the total
charge density at the point r is
N/2
L
=2
p(r)
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
J
Inserting
our expansion
p(r) dr
=
N/2
L J
2
l7j1a(r)12= 2
a
of a molecular
L
1
=
N.
a
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
a
C!'a<p!,(r)
v
N/2
=
L L
[2
C:aC!'a] <p:(r)<p!,(r)
a
v!'
v!'
= <P*P<P
where the density matrix is defined as
N/2
Pv!' = 2
L
C:aC!'a
a
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
tions were undertaken
over a 90 x 180 angular
of r) that covered a complete
0.000
electron
::::;
1'.::::; 8.000 bohr,
density
the spherically
sphere around
at increments
averaged
6
density, calcula-
grid (16,200 points for each value
the molecule at a series of fixed radii,
of 0.001 bohr.
at a given value of l' was then
electron
computed
The spherically
as
averaged
tl<jJ = 1r 189 and tl()
where
=
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.
solution converges very slowly and takes an amazingly
long time to compute.
summed
exceeds the true value.
over all space the total number
was not really a problem
resulting
with methane
from its apparent
What
of electrons
but it was quite apparent
a 10 step
This
lack of spherical symmetry.
I ended up doing was to obtain
using
When
with H2, probably
< p(r) >, (0 ::::;r" ::::;5A), at intervals
o.01A. At each value of r the average electron density was computed
grid
The
Gaussian
quadrature
method
both
in <jJand
from a 10
in (). The
of
X 10
results
were much more satisfying.
Figure 1 is a plot of the electron density minus the spherically
density
along the CH vector.
density
along the CH vector due to covalent bonding
located near r
= 0.58
is concentrated,
The difference demonstrates
the hydrogen
density
of spherical
scattering
effects.
nuclei are at 1.0855
Scattering
electron
the build up of electron
Note that the peak is
A. This is where the electron density from the hydrogen atoms
even though
The scattering
averaged electron
A from
the carbon.
Factors
factor for an atom or a molecule is difficult to calculate
demonstrates
electron
symmetry.
However, with the assumption
density, < p(r) >, and with the carbon at the origin, then the
factor of methane,
f(8) =
spherical
unless the
f( 8), is the Fourier transform
('XJU(r)sin(21rrr*)
dr
o
21rrr*
~r U(r)
of p( r),
sin(21rrr*)
tlr,
21rrr*
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
electrons from the 18 orbital of C while the second peak is due primarily
in the higher order shells.
the second
peak.
The hydrogen
The scattering
factors
nucleii are located
for 201 values
7
the
to electrons
near the mid-point
of
of 8, 0.00 ::::;8 ::::;2.00, at 0.01
increments
were computed
Figure 3 and compared
and tabulated.
The scattering
factor curve is presented
to that of carbon.
The last step in the process is to fit the curve to the standard
an analytical
in
approximation
to the scattering
equation
that gives
factor,
4
f«~)
= Laiexp(-biS2)
o ~ S ~ 2A-1
+c,
i=l
where ai and bi are the 8 parameters
that
are varied and c is obtained
from the
constraint
c
The fitting
procedure
M.B. Boisen Jr.
was accomplished
and L.W. Johnson
quasi-Newton
method
BFGS update
(Dennis and Schnabel,
with a minimization
(Boisen and Gibbs,
program
written
by
1993) that is based on a
that uses a model trust region (with a double dogleg) and the
a
1.54:3827
-0.514127
6.180872
2.573929
The quantity
L ai.
= f (0) -
that was minimized
198:3). The values of the fitted parameters
b
0.580871
3.995349
13.797434
36.110,548
are
c
0.215499
was the error in the fit, defined by Doyle and Turner
(1968) as
~201
E
=
;~~) [
1/ 2
82
L.J~O~
i
]
where 8 = f( S)actual - f( S )fitted. The maximum
values of f( s) is 0.005.
8
= .0185,
deviation
between
actual
and fitted
References
Boisen, M.B., Jr. and Gibbs, G.V. (1993) A modeling of the structure
ibility of quartz with a molecular
and coesite.
Physics and Chemistry
Dennis, J.E., Jr. and Schnabel,
Optimization
and Nonlinear
Doyle, P.A. and Turner,
scattering
factors.
of Chemical
Szabo, A. and Ostlund,
Advanced
and its transferability
of Minerals,
Equations.
Prentice-Hall,
P.S. (1968) Relativistic
Electronic
to cristobalite
20, 123-135.
R.B. (1983) Numerical
Acta Crystallographica,
Press, W. (1972) Structure
Journal
potential
and compress-
Methods
for Unconstrained
Englewood
Hartree-Fock
Cliffs, pp 378.
X-ray
and electron
A24, 390-397.
and phase transitions
of solid heavy methane
(CD4). The
Physics, 56, 2Ej97-2609.
H.S. (1982) Modern
Structure
Theory.
York.
9
Quantum
MacMillan
Chemistry:
Publishing
Introduction
to
Co., Inc, New
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