Re: [ccp4bb] Series termination effect calculation.
Hi James,
using dynamic N-Gaussian approximation to form-factor tables as described
here (pages 27-29):
http://cci.lbl.gov/publications/download/iucrcompcomm_jan2004.pdf
and used in Phenix since 2004, avoids both: singularity at B=0 and
inaccurate density values (compared to the raw forma-factor tables) for
B-0.
Attached is the script that proves this point. To run, simply
phenix.python run.py.
Pavel
On Sun, Sep 16, 2012 at 11:32 PM, James Holton jmhol...@lbl.gov wrote:
Yes, the constant term in the 5-Gaussian structure factor tables does
become annoying when you try to plot electron density in real space, but
only if you try to make the B factor zero. If the B factors are ~12 (like
they are in 1m1n), then the electron density 2.0 A from an Fe atom is not
-0.2 e-/A^3, it is 0.025 e-/A^3. This is only 1% of the electron density at
the center of a nitrogen atom with the same B factor.
But if you do set the B factor to zero, then the electron density at the
center of any atom (using the 5-Gaussian model) is infinity. To put it in
gnuplot-ish, the structure factor of Fe (in reciprocal space) can be
plotted with this function:
Fe_sf(s)=Fe_a1*exp(-Fe_b1*s*s)**+Fe_a2*exp(-Fe_b2*s*s)+Fe_a3***
exp(-Fe_b3*s*s)+Fe_a4*exp(-Fe_**b4*s*s)+Fe_c
where:
Fe_c = 1.036900;
Fe_a1 = 11.769500; Fe_a2 = 7.357300; Fe_a3 = 3.522200; Fe_a4 = 2.304500;
Fe_b1 = 4.761100; Fe_b2 = 0.307200; Fe_b3 = 15.353500; Fe_b4 = 76.880501;
and s is sin(theta)/lambda
applying a B factor is then just multiplication by exp(-B*s*s)
Since the terms are all Gaussians, the inverse Fourier transform can
actually be done analytically, giving the real-space version, or the
expression for electron density vs distance from the nucleus (r):
Fe_ff(r,B) = \
+Fe_a1*(4*pi/(Fe_b1+B))**1.5***safexp(-4*pi**2/(Fe_b1+B)*r*r) \
+Fe_a2*(4*pi/(Fe_b2+B))**1.5***safexp(-4*pi**2/(Fe_b2+B)*r*r) \
+Fe_a3*(4*pi/(Fe_b3+B))**1.5***safexp(-4*pi**2/(Fe_b3+B)*r*r) \
+Fe_a4*(4*pi/(Fe_b4+B))**1.5***safexp(-4*pi**2/(Fe_b4+B)*r*r) \
+Fe_c *(4*pi/(B))**1.5*safexp(-4*pi2/(B)*r*r);
Where here applying a B factor requires folding it into each Gaussian
term. Notice how the Fe_c term blows up as B-0? This is where most of the
series-termination effects come from. If you want the above equations for
other atoms, you can get them from here:
http://bl831.als.lbl.gov/~**jamesh/pickup/all_atomsf.**gnuplothttp://bl831.als.lbl.gov/~jamesh/pickup/all_atomsf.gnuplot
http://bl831.als.lbl.gov/~**jamesh/pickup/all_atomff.**gnuplothttp://bl831.als.lbl.gov/~jamesh/pickup/all_atomff.gnuplot
This infinitely sharp spike problem seems to have led some people to
conclude that a zero B factor is non-physical, but nothing could be further
from the truth! The scattering from mono-atomic gasses is an excellent
example of how one can observe the B=0 structure factor. In fact, gas
scattering is how the quantum mechanical self-consistent field calculations
of electron clouds around atoms was experimentally verified. Does this
mean that there really is an infinitely sharp spike in the middle of
every atom? Of course not. But there is a very sharp spike.
So, the problem of infinite density at the nucleus is really just an
artifact of the 5-Gaussian formalism. Strictly speaking, the 5-Gaussian
structure factor representation you find in ${CLIBD}/atomsf.lib (or Table
6.1.1.4 in the International Tables volume C) is nothing more than a curve
fit to the true values listed in ITC volume C tables 6.1.1.1 (neutral
atoms) and 6.1.1.3 (ions). These latter tables are the Fourier transform
of the true electron density distribution around a particular atom/ion
obtained from quantum mechanical self-consistent field calculations (like
those of Cromer, Mann and many others).
The important thing to realize is that the fit was done in _reciprocal_
space, and if you look carefully at tables 6.1.1.1 and 6.1.1.3, you can see
that even at REALLY high angle (sin(theta)/lambda = 6, or 0.083 A
resolution) there is still significant elastic scattering from the heavier
atoms. The purpose of the constant term in the 5-Gaussian representation
is to try and capture this high-angle tail, and for the really heavy
atoms this can be more than 5 electron equivalents. In real space, this is
equivalent to saying that about 5 electrons are located within at least
~0.03 A of the nucleus. That's a very short distance, but it is also not
zero. This is because the first few shells of electrons around things like
a Uranium nucleus actually are very small and dense. How, then, can we
have any hope of modelling heavy atoms properly without using a map grid
sampling of 0.01A ? Easy! The B factors are never zero.
Even for a truly infinitely sharp peak (aka a single electron), it doesn't
take much of a B factor to spread it out to a reasonable size. For
example, applying a B factor of 9 to a point charge will give it a
full-width-half max (FWHM) of
Re: [ccp4bb] Series termination effect calculation.
-BEGIN PGP SIGNED MESSAGE-
Hash: SHA1
tg@slartibartfast:~/tmp$ phenix.python run.py
0.001 627.413-4.01639e+06 303880
0.1 275.984 275.247 435.678
0.5 92.2049 92.206 93.6615
1 47.8941 47.8936 47.9421
10 3.54414 3.54415 3.5439
1000.217171 0.21717 0.21714
weird numbers. A proper description would have 6e/A^3 for a C at
x=(0,0,0) with B=0. How are these numbers 'not inaccurate'?
Cheers,
Tim
On 09/19/2012 06:47 PM, Pavel Afonine wrote:
Hi James,
using dynamic N-Gaussian approximation to form-factor tables as
described here (pages 27-29):
http://cci.lbl.gov/publications/download/iucrcompcomm_jan2004.pdf
and used in Phenix since 2004, avoids both: singularity at B=0 and
inaccurate density values (compared to the raw forma-factor tables)
for B-0.
Attached is the script that proves this point. To run, simply
phenix.python run.py.
Pavel
On Sun, Sep 16, 2012 at 11:32 PM, James Holton jmhol...@lbl.gov
wrote:
Yes, the constant term in the 5-Gaussian structure factor
tables does become annoying when you try to plot electron density
in real space, but only if you try to make the B factor zero. If
the B factors are ~12 (like they are in 1m1n), then the electron
density 2.0 A from an Fe atom is not -0.2 e-/A^3, it is 0.025
e-/A^3. This is only 1% of the electron density at the center of
a nitrogen atom with the same B factor.
But if you do set the B factor to zero, then the electron density
at the center of any atom (using the 5-Gaussian model) is
infinity. To put it in gnuplot-ish, the structure factor of Fe
(in reciprocal space) can be plotted with this function:
Fe_sf(s)=Fe_a1*exp(-Fe_b1*s*s)**+Fe_a2*exp(-Fe_b2*s*s)+Fe_a3***
exp(-Fe_b3*s*s)+Fe_a4*exp(-Fe_**b4*s*s)+Fe_c
where: Fe_c = 1.036900; Fe_a1 = 11.769500; Fe_a2 = 7.357300;
Fe_a3 = 3.522200; Fe_a4 = 2.304500; Fe_b1 = 4.761100; Fe_b2 =
0.307200; Fe_b3 = 15.353500; Fe_b4 = 76.880501; and s is
sin(theta)/lambda
applying a B factor is then just multiplication by exp(-B*s*s)
Since the terms are all Gaussians, the inverse Fourier transform
can actually be done analytically, giving the real-space version,
or the expression for electron density vs distance from the
nucleus (r):
Fe_ff(r,B) = \
+Fe_a1*(4*pi/(Fe_b1+B))**1.5***safexp(-4*pi**2/(Fe_b1+B)*r*r) \
+Fe_a2*(4*pi/(Fe_b2+B))**1.5***safexp(-4*pi**2/(Fe_b2+B)*r*r) \
+Fe_a3*(4*pi/(Fe_b3+B))**1.5***safexp(-4*pi**2/(Fe_b3+B)*r*r) \
+Fe_a4*(4*pi/(Fe_b4+B))**1.5***safexp(-4*pi**2/(Fe_b4+B)*r*r) \
+Fe_c *(4*pi/(B))**1.5*safexp(-4*pi2/(B)*r*r);
Where here applying a B factor requires folding it into each
Gaussian term. Notice how the Fe_c term blows up as B-0? This
is where most of the series-termination effects come from. If you
want the above equations for other atoms, you can get them from
here:
http://bl831.als.lbl.gov/~**jamesh/pickup/all_atomsf.**gnuplothttp://bl831.als.lbl.gov/~jamesh/pickup/all_atomsf.gnuplot
http://bl831.als.lbl.gov/~**jamesh/pickup/all_atomff.**gnuplothttp://bl831.als.lbl.gov/~jamesh/pickup/all_atomff.gnuplot
This infinitely sharp spike problem seems to have led some
people to conclude that a zero B factor is non-physical, but
nothing could be further from the truth! The scattering from
mono-atomic gasses is an excellent example of how one can observe
the B=0 structure factor. In fact, gas scattering is how the
quantum mechanical self-consistent field calculations of electron
clouds around atoms was experimentally verified. Does this mean
that there really is an infinitely sharp spike in the middle
of every atom? Of course not. But there is a very sharp
spike.
So, the problem of infinite density at the nucleus is really
just an artifact of the 5-Gaussian formalism. Strictly speaking,
the 5-Gaussian structure factor representation you find in
${CLIBD}/atomsf.lib (or Table 6.1.1.4 in the International Tables
volume C) is nothing more than a curve fit to the true values
listed in ITC volume C tables 6.1.1.1 (neutral atoms) and 6.1.1.3
(ions). These latter tables are the Fourier transform of the
true electron density distribution around a particular
atom/ion obtained from quantum mechanical self-consistent field
calculations (like those of Cromer, Mann and many others).
The important thing to realize is that the fit was done in
_reciprocal_ space, and if you look carefully at tables 6.1.1.1
and 6.1.1.3, you can see that even at REALLY high angle
(sin(theta)/lambda = 6, or 0.083 A resolution) there is still
significant elastic scattering from the heavier atoms. The
purpose of the constant term in the 5-Gaussian representation
is to try and capture this high-angle tail, and for the really
heavy atoms this can be more than 5 electron equivalents.
Re: [ccp4bb] Series termination effect calculation.
-BEGIN PGP SIGNED MESSAGE-
Hash: SHA1
Sorry, 6e^-/A^3 (or -6e/A^3 for charge density people) this should
have said.
On 09/20/2012 11:39 AM, Tim Gruene wrote:
tg@slartibartfast:~/tmp$ phenix.python run.py 0.001 627.413
-4.01639e+06 303880 0.1 275.984 275.247
435.678 0.5 92.2049 92.206 93.6615 1
47.8941 47.8936 47.9421 10 3.54414
3.54415 3.5439 1000.217171 0.21717
0.21714
weird numbers. A proper description would have 6e/A^3 for a C at
x=(0,0,0) with B=0. How are these numbers 'not inaccurate'?
Cheers, Tim
On 09/19/2012 06:47 PM, Pavel Afonine wrote:
Hi James,
using dynamic N-Gaussian approximation to form-factor tables as
described here (pages 27-29):
http://cci.lbl.gov/publications/download/iucrcompcomm_jan2004.pdf
and used in Phenix since 2004, avoids both: singularity at B=0
and inaccurate density values (compared to the raw forma-factor
tables) for B-0.
Attached is the script that proves this point. To run, simply
phenix.python run.py.
Pavel
On Sun, Sep 16, 2012 at 11:32 PM, James Holton
jmhol...@lbl.gov wrote:
Yes, the constant term in the 5-Gaussian structure factor
tables does become annoying when you try to plot electron
density in real space, but only if you try to make the B factor
zero. If the B factors are ~12 (like they are in 1m1n), then
the electron density 2.0 A from an Fe atom is not -0.2 e-/A^3,
it is 0.025 e-/A^3. This is only 1% of the electron density at
the center of a nitrogen atom with the same B factor.
But if you do set the B factor to zero, then the electron
density at the center of any atom (using the 5-Gaussian model)
is infinity. To put it in gnuplot-ish, the structure factor of
Fe (in reciprocal space) can be plotted with this function:
Fe_sf(s)=Fe_a1*exp(-Fe_b1*s*s)**+Fe_a2*exp(-Fe_b2*s*s)+Fe_a3***
exp(-Fe_b3*s*s)+Fe_a4*exp(-Fe_**b4*s*s)+Fe_c
where: Fe_c = 1.036900; Fe_a1 = 11.769500; Fe_a2 = 7.357300;
Fe_a3 = 3.522200; Fe_a4 = 2.304500; Fe_b1 = 4.761100; Fe_b2 =
0.307200; Fe_b3 = 15.353500; Fe_b4 = 76.880501; and s is
sin(theta)/lambda
applying a B factor is then just multiplication by exp(-B*s*s)
Since the terms are all Gaussians, the inverse Fourier
transform can actually be done analytically, giving the
real-space version, or the expression for electron density vs
distance from the nucleus (r):
Fe_ff(r,B) = \
+Fe_a1*(4*pi/(Fe_b1+B))**1.5***safexp(-4*pi**2/(Fe_b1+B)*r*r) \
+Fe_a2*(4*pi/(Fe_b2+B))**1.5***safexp(-4*pi**2/(Fe_b2+B)*r*r)
\ +Fe_a3*(4*pi/(Fe_b3+B))**1.5***safexp(-4*pi**2/(Fe_b3+B)*r*r)
\ +Fe_a4*(4*pi/(Fe_b4+B))**1.5***safexp(-4*pi**2/(Fe_b4+B)*r*r)
\ +Fe_c *(4*pi/(B))**1.5*safexp(-4*pi2/(B)*r*r);
Where here applying a B factor requires folding it into each
Gaussian term. Notice how the Fe_c term blows up as B-0?
This is where most of the series-termination effects come from.
If you want the above equations for other atoms, you can get
them from here:
http://bl831.als.lbl.gov/~**jamesh/pickup/all_atomsf.**gnuplothttp://bl831.als.lbl.gov/~jamesh/pickup/all_atomsf.gnuplot
http://bl831.als.lbl.gov/~**jamesh/pickup/all_atomff.**gnuplothttp://bl831.als.lbl.gov/~jamesh/pickup/all_atomff.gnuplot
This infinitely sharp spike problem seems to have led some
people to conclude that a zero B factor is non-physical, but
nothing could be further from the truth! The scattering from
mono-atomic gasses is an excellent example of how one can
observe the B=0 structure factor. In fact, gas scattering is
how the quantum mechanical self-consistent field calculations
of electron clouds around atoms was experimentally verified.
Does this mean that there really is an infinitely sharp spike
in the middle of every atom? Of course not. But there is a
very sharp spike.
So, the problem of infinite density at the nucleus is really
just an artifact of the 5-Gaussian formalism. Strictly
speaking, the 5-Gaussian structure factor representation you
find in ${CLIBD}/atomsf.lib (or Table 6.1.1.4 in the
International Tables volume C) is nothing more than a curve fit
to the true values listed in ITC volume C tables 6.1.1.1
(neutral atoms) and 6.1.1.3 (ions). These latter tables are
the Fourier transform of the true electron density
distribution around a particular atom/ion obtained from quantum
mechanical self-consistent field calculations (like those of
Cromer, Mann and many others).
The important thing to realize is that the fit was done in
_reciprocal_ space, and if you look carefully at tables
6.1.1.1 and 6.1.1.3, you can see that even at REALLY high
angle (sin(theta)/lambda = 6, or 0.083 A resolution) there is
still significant elastic scattering from the heavier atoms.
The purpose of the constant term in the 5-Gaussian
representation is to try and capture this high-angle tail,
and for the really heavy atoms this can be
Re: [ccp4bb] Series termination effect calculation.
Tim, I don't follow your argument: why should the density be 6A^-3 at
the centre of a C atom?
-- Ian
On 20 September 2012 10:39, Tim Gruene t...@shelx.uni-ac.gwdg.de wrote:
-BEGIN PGP SIGNED MESSAGE-
Hash: SHA1
tg@slartibartfast:~/tmp$ phenix.python run.py
0.001 627.413-4.01639e+06 303880
0.1 275.984 275.247 435.678
0.5 92.2049 92.206 93.6615
1 47.8941 47.8936 47.9421
10 3.54414 3.54415 3.5439
1000.217171 0.21717 0.21714
weird numbers. A proper description would have 6e/A^3 for a C at
x=(0,0,0) with B=0. How are these numbers 'not inaccurate'?
Cheers,
Tim
On 09/19/2012 06:47 PM, Pavel Afonine wrote:
Hi James,
using dynamic N-Gaussian approximation to form-factor tables as
described here (pages 27-29):
http://cci.lbl.gov/publications/download/iucrcompcomm_jan2004.pdf
and used in Phenix since 2004, avoids both: singularity at B=0 and
inaccurate density values (compared to the raw forma-factor tables)
for B-0.
Attached is the script that proves this point. To run, simply
phenix.python run.py.
Pavel
On Sun, Sep 16, 2012 at 11:32 PM, James Holton jmhol...@lbl.gov
wrote:
Yes, the constant term in the 5-Gaussian structure factor
tables does become annoying when you try to plot electron density
in real space, but only if you try to make the B factor zero. If
the B factors are ~12 (like they are in 1m1n), then the electron
density 2.0 A from an Fe atom is not -0.2 e-/A^3, it is 0.025
e-/A^3. This is only 1% of the electron density at the center of
a nitrogen atom with the same B factor.
But if you do set the B factor to zero, then the electron density
at the center of any atom (using the 5-Gaussian model) is
infinity. To put it in gnuplot-ish, the structure factor of Fe
(in reciprocal space) can be plotted with this function:
Fe_sf(s)=Fe_a1*exp(-Fe_b1*s*s)**+Fe_a2*exp(-Fe_b2*s*s)+Fe_a3***
exp(-Fe_b3*s*s)+Fe_a4*exp(-Fe_**b4*s*s)+Fe_c
where: Fe_c = 1.036900; Fe_a1 = 11.769500; Fe_a2 = 7.357300;
Fe_a3 = 3.522200; Fe_a4 = 2.304500; Fe_b1 = 4.761100; Fe_b2 =
0.307200; Fe_b3 = 15.353500; Fe_b4 = 76.880501; and s is
sin(theta)/lambda
applying a B factor is then just multiplication by exp(-B*s*s)
Since the terms are all Gaussians, the inverse Fourier transform
can actually be done analytically, giving the real-space version,
or the expression for electron density vs distance from the
nucleus (r):
Fe_ff(r,B) = \
+Fe_a1*(4*pi/(Fe_b1+B))**1.5***safexp(-4*pi**2/(Fe_b1+B)*r*r) \
+Fe_a2*(4*pi/(Fe_b2+B))**1.5***safexp(-4*pi**2/(Fe_b2+B)*r*r) \
+Fe_a3*(4*pi/(Fe_b3+B))**1.5***safexp(-4*pi**2/(Fe_b3+B)*r*r) \
+Fe_a4*(4*pi/(Fe_b4+B))**1.5***safexp(-4*pi**2/(Fe_b4+B)*r*r) \
+Fe_c *(4*pi/(B))**1.5*safexp(-4*pi2/(B)*r*r);
Where here applying a B factor requires folding it into each
Gaussian term. Notice how the Fe_c term blows up as B-0? This
is where most of the series-termination effects come from. If you
want the above equations for other atoms, you can get them from
here:
http://bl831.als.lbl.gov/~**jamesh/pickup/all_atomsf.**gnuplothttp://bl831.als.lbl.gov/~jamesh/pickup/all_atomsf.gnuplot
http://bl831.als.lbl.gov/~**jamesh/pickup/all_atomff.**gnuplothttp://bl831.als.lbl.gov/~jamesh/pickup/all_atomff.gnuplot
This infinitely sharp spike problem seems to have led some
people to conclude that a zero B factor is non-physical, but
nothing could be further from the truth! The scattering from
mono-atomic gasses is an excellent example of how one can observe
the B=0 structure factor. In fact, gas scattering is how the
quantum mechanical self-consistent field calculations of electron
clouds around atoms was experimentally verified. Does this mean
that there really is an infinitely sharp spike in the middle
of every atom? Of course not. But there is a very sharp
spike.
So, the problem of infinite density at the nucleus is really
just an artifact of the 5-Gaussian formalism. Strictly speaking,
the 5-Gaussian structure factor representation you find in
${CLIBD}/atomsf.lib (or Table 6.1.1.4 in the International Tables
volume C) is nothing more than a curve fit to the true values
listed in ITC volume C tables 6.1.1.1 (neutral atoms) and 6.1.1.3
(ions). These latter tables are the Fourier transform of the
true electron density distribution around a particular
atom/ion obtained from quantum mechanical self-consistent field
calculations (like those of Cromer, Mann and many others).
The important thing to realize is that the fit was done in
_reciprocal_ space, and if you look carefully at tables 6.1.1.1
and 6.1.1.3, you can see that even at REALLY high angle
(sin(theta)/lambda = 6, or 0.083 A resolution) there is still
significant elastic scattering from the heavier atoms. The
purpose of the constant
Re: [ccp4bb] Series termination effect calculation.
Hi Tim,
I'm not sure I understand your argument either. Anyway, I hope this Ralf's
paper (and references therein) will make it more clear:
http://cci.lbl.gov/~rwgk/my_papers/CCN_2011_01_electron_density.pdf
All the best,
Pavel
On Thu, Sep 20, 2012 at 5:19 AM, Ian Tickle ianj...@gmail.com wrote:
Tim, I don't follow your argument: why should the density be 6A^-3 at
the centre of a C atom?
-- Ian
On 20 September 2012 10:39, Tim Gruene t...@shelx.uni-ac.gwdg.de wrote:
-BEGIN PGP SIGNED MESSAGE-
Hash: SHA1
tg@slartibartfast:~/tmp$ phenix.python run.py
0.001 627.413-4.01639e+06 303880
0.1 275.984 275.247 435.678
0.5 92.2049 92.206 93.6615
1 47.8941 47.8936 47.9421
10 3.54414 3.54415 3.5439
1000.217171 0.21717 0.21714
weird numbers. A proper description would have 6e/A^3 for a C at
x=(0,0,0) with B=0. How are these numbers 'not inaccurate'?
Cheers,
Tim
On 09/19/2012 06:47 PM, Pavel Afonine wrote:
Hi James,
using dynamic N-Gaussian approximation to form-factor tables as
described here (pages 27-29):
http://cci.lbl.gov/publications/download/iucrcompcomm_jan2004.pdf
and used in Phenix since 2004, avoids both: singularity at B=0 and
inaccurate density values (compared to the raw forma-factor tables)
for B-0.
Attached is the script that proves this point. To run, simply
phenix.python run.py.
Pavel
On Sun, Sep 16, 2012 at 11:32 PM, James Holton jmhol...@lbl.gov
wrote:
Yes, the constant term in the 5-Gaussian structure factor
tables does become annoying when you try to plot electron density
in real space, but only if you try to make the B factor zero. If
the B factors are ~12 (like they are in 1m1n), then the electron
density 2.0 A from an Fe atom is not -0.2 e-/A^3, it is 0.025
e-/A^3. This is only 1% of the electron density at the center of
a nitrogen atom with the same B factor.
But if you do set the B factor to zero, then the electron density
at the center of any atom (using the 5-Gaussian model) is
infinity. To put it in gnuplot-ish, the structure factor of Fe
(in reciprocal space) can be plotted with this function:
Fe_sf(s)=Fe_a1*exp(-Fe_b1*s*s)**+Fe_a2*exp(-Fe_b2*s*s)+Fe_a3***
exp(-Fe_b3*s*s)+Fe_a4*exp(-Fe_**b4*s*s)+Fe_c
where: Fe_c = 1.036900; Fe_a1 = 11.769500; Fe_a2 = 7.357300;
Fe_a3 = 3.522200; Fe_a4 = 2.304500; Fe_b1 = 4.761100; Fe_b2 =
0.307200; Fe_b3 = 15.353500; Fe_b4 = 76.880501; and s is
sin(theta)/lambda
applying a B factor is then just multiplication by exp(-B*s*s)
Since the terms are all Gaussians, the inverse Fourier transform
can actually be done analytically, giving the real-space version,
or the expression for electron density vs distance from the
nucleus (r):
Fe_ff(r,B) = \
+Fe_a1*(4*pi/(Fe_b1+B))**1.5***safexp(-4*pi**2/(Fe_b1+B)*r*r) \
+Fe_a2*(4*pi/(Fe_b2+B))**1.5***safexp(-4*pi**2/(Fe_b2+B)*r*r) \
+Fe_a3*(4*pi/(Fe_b3+B))**1.5***safexp(-4*pi**2/(Fe_b3+B)*r*r) \
+Fe_a4*(4*pi/(Fe_b4+B))**1.5***safexp(-4*pi**2/(Fe_b4+B)*r*r) \
+Fe_c *(4*pi/(B))**1.5*safexp(-4*pi2/(B)*r*r);
Where here applying a B factor requires folding it into each
Gaussian term. Notice how the Fe_c term blows up as B-0? This
is where most of the series-termination effects come from. If you
want the above equations for other atoms, you can get them from
here:
http://bl831.als.lbl.gov/~**jamesh/pickup/all_atomsf.**gnuplot
http://bl831.als.lbl.gov/~jamesh/pickup/all_atomsf.gnuplot
http://bl831.als.lbl.gov/~**jamesh/pickup/all_atomff.**gnuplot
http://bl831.als.lbl.gov/~jamesh/pickup/all_atomff.gnuplot
This infinitely sharp spike problem seems to have led some
people to conclude that a zero B factor is non-physical, but
nothing could be further from the truth! The scattering from
mono-atomic gasses is an excellent example of how one can observe
the B=0 structure factor. In fact, gas scattering is how the
quantum mechanical self-consistent field calculations of electron
clouds around atoms was experimentally verified. Does this mean
that there really is an infinitely sharp spike in the middle
of every atom? Of course not. But there is a very sharp
spike.
So, the problem of infinite density at the nucleus is really
just an artifact of the 5-Gaussian formalism. Strictly speaking,
the 5-Gaussian structure factor representation you find in
${CLIBD}/atomsf.lib (or Table 6.1.1.4 in the International Tables
volume C) is nothing more than a curve fit to the true values
listed in ITC volume C tables 6.1.1.1 (neutral atoms) and 6.1.1.3
(ions). These latter tables are the Fourier transform of the
true electron density distribution around a particular
atom/ion obtained from quantum mechanical
Re: [ccp4bb] Series termination effect calculation.
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Because C has 6 electrons and without thermal vibrations (T=0/B=0) I
thought you'd catch all six of them with a box of 1A side length.
Is this too simple thinking?
Tim
On 09/20/2012 02:19 PM, Ian Tickle wrote:
Tim, I don't follow your argument: why should the density be 6A^-3
at the centre of a C atom?
-- Ian
On 20 September 2012 10:39, Tim Gruene t...@shelx.uni-ac.gwdg.de
wrote: tg@slartibartfast:~/tmp$ phenix.python run.py 0.001
627.413-4.01639e+06 303880 0.1 275.984
275.247 435.678 0.5 92.2049 92.206
93.6615 1 47.8941 47.8936 47.9421 10
3.54414 3.54415 3.5439 1000.217171
0.21717 0.21714
weird numbers. A proper description would have 6e/A^3 for a C at
x=(0,0,0) with B=0. How are these numbers 'not inaccurate'?
Cheers, Tim
On 09/19/2012 06:47 PM, Pavel Afonine wrote:
Hi James,
using dynamic N-Gaussian approximation to form-factor tables
as described here (pages 27-29):
http://cci.lbl.gov/publications/download/iucrcompcomm_jan2004.pdf
and used in Phenix since 2004, avoids both: singularity at B=0 and
inaccurate density values (compared to the raw forma-factor
tables) for B-0.
Attached is the script that proves this point. To run,
simply phenix.python run.py.
Pavel
On Sun, Sep 16, 2012 at 11:32 PM, James Holton
jmhol...@lbl.gov wrote:
Yes, the constant term in the 5-Gaussian structure
factor tables does become annoying when you try to plot
electron density in real space, but only if you try to make
the B factor zero. If the B factors are ~12 (like they are
in 1m1n), then the electron density 2.0 A from an Fe atom
is not -0.2 e-/A^3, it is 0.025 e-/A^3. This is only 1% of
the electron density at the center of a nitrogen atom with
the same B factor.
But if you do set the B factor to zero, then the electron
density at the center of any atom (using the 5-Gaussian
model) is infinity. To put it in gnuplot-ish, the
structure factor of Fe (in reciprocal space) can be plotted
with this function:
Fe_sf(s)=Fe_a1*exp(-Fe_b1*s*s)**+Fe_a2*exp(-Fe_b2*s*s)+Fe_a3***
exp(-Fe_b3*s*s)+Fe_a4*exp(-Fe_**b4*s*s)+Fe_c
where: Fe_c = 1.036900; Fe_a1 = 11.769500; Fe_a2 =
7.357300; Fe_a3 = 3.522200; Fe_a4 = 2.304500; Fe_b1 =
4.761100; Fe_b2 = 0.307200; Fe_b3 = 15.353500; Fe_b4 =
76.880501; and s is sin(theta)/lambda
applying a B factor is then just multiplication by
exp(-B*s*s)
Since the terms are all Gaussians, the inverse Fourier
transform can actually be done analytically, giving the
real-space version, or the expression for electron density
vs distance from the nucleus (r):
Fe_ff(r,B) = \
+Fe_a1*(4*pi/(Fe_b1+B))**1.5***safexp(-4*pi**2/(Fe_b1+B)*r*r)
\
+Fe_a2*(4*pi/(Fe_b2+B))**1.5***safexp(-4*pi**2/(Fe_b2+B)*r*r)
\
+Fe_a3*(4*pi/(Fe_b3+B))**1.5***safexp(-4*pi**2/(Fe_b3+B)*r*r)
\
+Fe_a4*(4*pi/(Fe_b4+B))**1.5***safexp(-4*pi**2/(Fe_b4+B)*r*r)
\ +Fe_c *(4*pi/(B))**1.5*safexp(-4*pi2/(B)*r*r);
Where here applying a B factor requires folding it into
each Gaussian term. Notice how the Fe_c term blows up as
B-0? This is where most of the series-termination effects
come from. If you want the above equations for other atoms,
you can get them from here:
http://bl831.als.lbl.gov/~**jamesh/pickup/all_atomsf.**gnuplothttp://bl831.als.lbl.gov/~jamesh/pickup/all_atomsf.gnuplot
http://bl831.als.lbl.gov/~**jamesh/pickup/all_atomff.**gnuplothttp://bl831.als.lbl.gov/~jamesh/pickup/all_atomff.gnuplot
This infinitely sharp spike problem seems to have led
some people to conclude that a zero B factor is
non-physical, but nothing could be further from the truth!
The scattering from mono-atomic gasses is an excellent
example of how one can observe the B=0 structure factor.
In fact, gas scattering is how the quantum mechanical
self-consistent field calculations of electron clouds
around atoms was experimentally verified. Does this mean
that there really is an infinitely sharp spike in the
middle of every atom? Of course not. But there is a
very sharp spike.
So, the problem of infinite density at the nucleus is
really just an artifact of the 5-Gaussian formalism.
Strictly speaking, the 5-Gaussian structure factor
representation you find in ${CLIBD}/atomsf.lib (or Table
6.1.1.4 in the International Tables volume C) is nothing
more than a curve fit to the true values listed in ITC
volume C tables 6.1.1.1 (neutral atoms) and 6.1.1.3 (ions).
These latter tables are the Fourier transform of the true
electron density distribution around a particular atom/ion
obtained from quantum mechanical self-consistent field
calculations (like those of Cromer, Mann and many others).
The important thing to realize is that the fit was done in
_reciprocal_ space, and if you look carefully at tables
6.1.1.1 and 6.1.1.3, you can see that even at REALLY high
angle
Re: [ccp4bb] Series termination effect calculation.
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P.S.: run.py reads
format_value(%.6g,ug_n.electron_density(0, b_iso))
so I thought the output of the first line states the calculated
electron density a position 0 (0,0,0) for a Carbon atom (top lines) at
the given b_iso values.
Cheers,
Tim
On 09/20/2012 02:47 PM, Tim Gruene wrote:
Because C has 6 electrons and without thermal vibrations (T=0/B=0)
I thought you'd catch all six of them with a box of 1A side
length.
Is this too simple thinking?
Tim
On 09/20/2012 02:19 PM, Ian Tickle wrote:
Tim, I don't follow your argument: why should the density be
6A^-3 at the centre of a C atom?
-- Ian
On 20 September 2012 10:39, Tim Gruene t...@shelx.uni-ac.gwdg.de
wrote: tg@slartibartfast:~/tmp$ phenix.python run.py 0.001
627.413-4.01639e+06 303880 0.1 275.984
275.247 435.678 0.5 92.2049 92.206
93.6615 1 47.8941 47.8936 47.9421 10
3.54414 3.54415 3.5439 1000.217171
0.21717 0.21714
weird numbers. A proper description would have 6e/A^3 for a C at
x=(0,0,0) with B=0. How are these numbers 'not inaccurate'?
Cheers, Tim
On 09/19/2012 06:47 PM, Pavel Afonine wrote:
Hi James,
using dynamic N-Gaussian approximation to form-factor
tables as described here (pages 27-29):
http://cci.lbl.gov/publications/download/iucrcompcomm_jan2004.pdf
and used in Phenix since 2004, avoids both: singularity at B=0 and
inaccurate density values (compared to the raw
forma-factor tables) for B-0.
Attached is the script that proves this point. To run,
simply phenix.python run.py.
Pavel
On Sun, Sep 16, 2012 at 11:32 PM, James Holton
jmhol...@lbl.gov wrote:
Yes, the constant term in the 5-Gaussian structure
factor tables does become annoying when you try to plot
electron density in real space, but only if you try to
make the B factor zero. If the B factors are ~12 (like
they are in 1m1n), then the electron density 2.0 A from
an Fe atom is not -0.2 e-/A^3, it is 0.025 e-/A^3. This
is only 1% of the electron density at the center of a
nitrogen atom with the same B factor.
But if you do set the B factor to zero, then the
electron density at the center of any atom (using the
5-Gaussian model) is infinity. To put it in gnuplot-ish,
the structure factor of Fe (in reciprocal space) can be
plotted with this function:
Fe_sf(s)=Fe_a1*exp(-Fe_b1*s*s)**+Fe_a2*exp(-Fe_b2*s*s)+Fe_a3***
exp(-Fe_b3*s*s)+Fe_a4*exp(-Fe_**b4*s*s)+Fe_c
where: Fe_c = 1.036900; Fe_a1 = 11.769500; Fe_a2 =
7.357300; Fe_a3 = 3.522200; Fe_a4 = 2.304500; Fe_b1 =
4.761100; Fe_b2 = 0.307200; Fe_b3 = 15.353500; Fe_b4 =
76.880501; and s is sin(theta)/lambda
applying a B factor is then just multiplication by
exp(-B*s*s)
Since the terms are all Gaussians, the inverse Fourier
transform can actually be done analytically, giving the
real-space version, or the expression for electron
density vs distance from the nucleus (r):
Fe_ff(r,B) = \
+Fe_a1*(4*pi/(Fe_b1+B))**1.5***safexp(-4*pi**2/(Fe_b1+B)*r*r)
\
+Fe_a2*(4*pi/(Fe_b2+B))**1.5***safexp(-4*pi**2/(Fe_b2+B)*r*r)
\
+Fe_a3*(4*pi/(Fe_b3+B))**1.5***safexp(-4*pi**2/(Fe_b3+B)*r*r)
\
+Fe_a4*(4*pi/(Fe_b4+B))**1.5***safexp(-4*pi**2/(Fe_b4+B)*r*r)
\ +Fe_c *(4*pi/(B))**1.5*safexp(-4*pi2/(B)*r*r);
Where here applying a B factor requires folding it into
each Gaussian term. Notice how the Fe_c term blows up
as B-0? This is where most of the series-termination
effects come from. If you want the above equations for
other atoms, you can get them from here:
http://bl831.als.lbl.gov/~**jamesh/pickup/all_atomsf.**gnuplothttp://bl831.als.lbl.gov/~jamesh/pickup/all_atomsf.gnuplot
http://bl831.als.lbl.gov/~**jamesh/pickup/all_atomff.**gnuplothttp://bl831.als.lbl.gov/~jamesh/pickup/all_atomff.gnuplot
This infinitely sharp spike problem seems to have led
some people to conclude that a zero B factor is
non-physical, but nothing could be further from the
truth! The scattering from mono-atomic gasses is an
excellent example of how one can observe the B=0
structure factor. In fact, gas scattering is how the
quantum mechanical self-consistent field calculations of
electron clouds around atoms was experimentally verified.
Does this mean that there really is an infinitely sharp
spike in the middle of every atom? Of course not. But
there is a very sharp spike.
So, the problem of infinite density at the nucleus is
really just an artifact of the 5-Gaussian formalism.
Strictly speaking, the 5-Gaussian structure factor
representation you find in ${CLIBD}/atomsf.lib (or Table
6.1.1.4 in the International Tables volume C) is nothing
more than a curve fit to the true values listed in ITC
volume C tables 6.1.1.1 (neutral atoms) and 6.1.1.3
(ions). These latter tables are the Fourier transform of
the true electron density distribution around a
Re: [ccp4bb] Series termination effect calculation.
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Dear James,
Mann only fitted their data to sin\lambda/theta 1.5, and up to there
the fit is pretty good. 30 years ago the computational means must have
been very different and what takes 5s now would have taken minutes or
hours then.
I am going to do the re-fitting, compare with a couple of test
structures where we have observed problems which might well be related
to this discussion. Fortunately with shelxl the testing is a piece of
cake and does not even require a change of the source code.
The results should be published within the year. Maybe you get an
updated atomsf.lib then with ccp4 6.1.4 ;-)
Best wishes,
Tim
On 09/19/2012 02:16 AM, James Holton wrote:
That's really interesting! Since the fits then and now were both
least-squares, I wonder how Cromer Mann could have gotten it so
far off? Looking at the residuals, I see that although that of
nitrogen oscillates badly, even the worst outlier is still within
0.01 electrons of the Hartree-Fock values. Perhaps 1% of an
electron was their convergence limit?
Either way, I think it would be valuable to have a re-fit of the
Table 6.1.1.1/3 values without the c term. Then we can go all
the way to B=0 without worrying about singularities.
For example, I attach here a plot of the electron density at the
center of a nitrogen atom vs B factor (in real space). The red
curve is the result of a 20-Gaussian fit to the data for nitrogen
in table 6.1.1.1 all the way out to sin(theta)/lambda = 6 (although
7 Gaussians is more than enough). This true curve approaches
1000 e-/A^3 as B approaches zero, but the 5-Gaussian model using
the Cromer-Mann coefficients form 6.1.1.4 (blue curve) starts to
deviate when B becomes less than one, and actually goes negative
for B 0.1. A simpler model (without the c term, but re-fit) is
the green line. Very much like what Tim suggested.
Not exactly a problem for typical macromolecular refinement, but
still... I wonder what would happen if I edited my
${CLIBD}/atomsf.lib ?
-James Holton MAD Scientist
On 9/18/2012 6:32 AM, Tim Gruene wrote: Hello Oliver,
when you fit the values from ICA Tab 6.1.1.1 with gnuplot, the
values of C and N become much more comparable. c(C) = 0.017 and
especially c(N) = 0.025 0!!! for C: Final set of parameters
Asymptotic Standard Error ===
==
a1 = 0.604126 +/- 0.02326 (3.85%) a2
= 2.63343 +/- 0.03321 (1.261%) a3 =
1.52123 +/- 0.03528 (2.319%) a4 = 1.2211
+/- 0.02225 (1.822%) b1 = 0.185807 +/-
0.00629 (3.385%) b2 = 14.6332 +/- 0.1355
(0.9263%) b3 = 41.6948 +/- 0.5345
(1.282%) b4 = 0.717984 +/- 0.01251
(1.743%) c = 0.0171359+/- 0.002045
(11.93%)
for N: Final set of parametersAsymptotic Standard
Error ===
==
a1 = 0.723788 +/- 0.04334 (5.988%) a2
= 3.24589 +/- 0.04074 (1.255%) a3 =
1.90049 +/- 0.04422 (2.327%) a4 =
1.10071 +/- 0.0413 (3.752%) b1 =
0.157345 +/- 0.007552 (4.8%) b2 = 10.106
+/- 0.1041 (1.03%) b3 = 30.0211 +/-
0.3946 (1.314%) b4 = 0.567116 +/-
0.01914 (3.376%) c = 0.0252303+/-
0.003284 (13.01%)
In 1967, Mann only calculated to sin \theta/lambda = 0, ... 1.5,
and their tabulated values do indeed fit decently within that
range, but not out to 6A.
I thought this was notworthy, and I am curious which values for
these constants refinement programs use nowadays. Maybe George,
Garib, Pavel, and Gerard may want to comment?
Cheers, Tim
On 09/18/2012 10:11 AM, Oliver Einsle wrote:
Hi there,
I was just pointed to this thread and should comment on the
discussion, as actually made the plots for this paper. James
has clarified the issue much better than I could have, and
indeed the calculations will fail for larger Bragg angles if
you do not assume a reasonable B-factor (I used B=10 for the
plots).
Doug Rees has pointed out at the time that for large theta
the c-term of the Cromer/Mann approximation becomes dominant,
and this is where chaos comes in, as the Cromer/Mann
parameters are only derived from a fit to the actual
HF-calculation. They are numbers without physical meaning,
which becomes particularly obvious if you compare the
parameters for C and N:
C: 2.3100 20.8439 1.0200 10.2075 1.5886 0.5687
0.8650 51.6512 0.2156 N: 12.2126 0.0057 3.1322 9.8933
2.0125 28.9975 1.1663 0.5826 -11.5290
The scattering factors for these are reasonably similar, but
the c-values are entirely different. The B-factor dampens
Re: [ccp4bb] Series termination effect calculation.
Hi there, I was just pointed to this thread and should comment on the discussion, as actually made the plots for this paper. James has clarified the issue much better than I could have, and indeed the calculations will fail for larger Bragg angles if you do not assume a reasonable B-factor (I used B=10 for the plots). Doug Rees has pointed out at the time that for large theta the c-term of the Cromer/Mann approximation becomes dominant, and this is where chaos comes in, as the Cromer/Mann parameters are only derived from a fit to the actual HF-calculation. They are numbers without physical meaning, which becomes particularly obvious if you compare the parameters for C and N: C: 2.3100 20.8439 1.0200 10.2075 1.5886 0.5687 0.8650 51.6512 0.2156 N: 12.2126 0.0057 3.1322 9.8933 2.0125 28.9975 1.1663 0.5826 -11.5290 The scattering factors for these are reasonably similar, but the c-values are entirely different. The B-factor dampens this out and this is an essential point. For clarity: I made the plots using Waterloo Maple with the following code: restart; SF :=Matrix(17,9,readdata(scatter.dat,float,9)); biso := 10; e:= 1; AFF := (e)-(SF[e,1]*exp(-SF[e,2]*s^2)+SF[e,3]*exp(-SF[e,4]*s^2) +SF[e,5]*exp(-SF[e,6]*s^2)+SF[e,7]*exp(-SF[e,8]*s^2) +SF[e,9])*exp(-biso*s^2/4); H:= AFF(1); C:= AFF(2); N:= AFF(3); Ox := AFF(4); S:= AFF(5); Fe := AFF(6); Fe2 := AFF(7); Fe3 := AFF(8); Cu := AFF(9); Cu1 := AFF(10); Cu2 := AFF(11); Mo := AFF(12); Mo4 := AFF(13); Mo5 := AFF(14); Mo6 := AFF(15); // Plot scattering factors plot([C,N,Fe,S], s=0..1); // Figure 1: rho0 := (r) - Int((4*Pi*s^2)*Fe2*sin(2*Pi*s*r)/(2*Pi*s*r), s=0..1/dmax); dmax := 1.0; plot (rho0, -5..5); // Figure 1 (inset): Electron Density Profile rho := (r,f) -(Int((4*Pi*s^2)*f*sin(2*Pi*s*r)/(2*Pi*s*r),s=0..1/dmax)); cofactor:= 9*rho(3.3,S) + 6*rho(2.0,Fe2) + 1*rho(3.49,Mo6) + 1*rho(3.51,Fe3); plot(cofactor, dmax=0.5..3.5); The file scatter.dat is simply a collection of some form factors, courtesy of atomsf.lib (see attachment). Cheers, Oliver. Am 9/17/12 11:24 AM schrieb Tim Gruene unter t...@shelx.uni-ac.gwdg.de: -BEGIN PGP SIGNED MESSAGE- Hash: SHA1 Dear James et al., so to summarise, the answer to Niu's question is that he must add a factor of e^(-Bs^2) to the formula of Cromer/Mann and then adjust the value of B until it matches the inset. Given that you claim rho=0.025e/A^3 (I assume for 1/dmax approx. 0) for B=12 and the inset shows a value of about 0.6, a somewhat higher B-value should work. Cheers, Tim On 09/17/2012 08:32 AM, James Holton wrote: Yes, the constant term in the 5-Gaussian structure factor tables does become annoying when you try to plot electron density in real space, but only if you try to make the B factor zero. If the B factors are ~12 (like they are in 1m1n), then the electron density 2.0 A from an Fe atom is not -0.2 e-/A^3, it is 0.025 e-/A^3. This is only 1% of the electron density at the center of a nitrogen atom with the same B factor. But if you do set the B factor to zero, then the electron density at the center of any atom (using the 5-Gaussian model) is infinity. To put it in gnuplot-ish, the structure factor of Fe (in reciprocal space) can be plotted with this function: Fe_sf(s)=Fe_a1*exp(-Fe_b1*s*s)+Fe_a2*exp(-Fe_b2*s*s)+Fe_a3*exp(-Fe_b3*s*s )+Fe_a4*exp(-Fe_b4*s*s)+Fe_c where: Fe_c = 1.036900; Fe_a1 = 11.769500; Fe_a2 = 7.357300; Fe_a3 = 3.522200; Fe_a4 = 2.304500; Fe_b1 = 4.761100; Fe_b2 = 0.307200; Fe_b3 = 15.353500; Fe_b4 = 76.880501; and s is sin(theta)/lambda applying a B factor is then just multiplication by exp(-B*s*s) Since the terms are all Gaussians, the inverse Fourier transform can actually be done analytically, giving the real-space version, or the expression for electron density vs distance from the nucleus (r): Fe_ff(r,B) = \ +Fe_a1*(4*pi/(Fe_b1+B))**1.5*safexp(-4*pi**2/(Fe_b1+B)*r*r) \ +Fe_a2*(4*pi/(Fe_b2+B))**1.5*safexp(-4*pi**2/(Fe_b2+B)*r*r) \ +Fe_a3*(4*pi/(Fe_b3+B))**1.5*safexp(-4*pi**2/(Fe_b3+B)*r*r) \ +Fe_a4*(4*pi/(Fe_b4+B))**1.5*safexp(-4*pi**2/(Fe_b4+B)*r*r) \ +Fe_c *(4*pi/(B))**1.5*safexp(-4*pi**2/(B)*r*r); Where here applying a B factor requires folding it into each Gaussian term. Notice how the Fe_c term blows up as B-0? This is where most of the series-termination effects come from. If you want the above equations for other atoms, you can get them from here: http://bl831.als.lbl.gov/~jamesh/pickup/all_atomsf.gnuplot http://bl831.als.lbl.gov/~jamesh/pickup/all_atomff.gnuplot This infinitely sharp spike problem seems to have led some people to conclude that a zero B factor is non-physical, but nothing could be further from the truth! The scattering from mono-atomic gasses is an excellent example of how one can observe the B=0 structure factor. In fact, gas scattering is how the quantum
[ccp4bb] AW: [ccp4bb] Series termination effect calculation.
Good point! With 9 parameters there must be a zillion combinations that produce a decent fit. But even if they are chosen to be similar they have no physical meaning, right? Cheers, Oliver. -Ursprüngliche Nachricht- Von: Tim Gruene [mailto:t...@shelx.uni-ac.gwdg.de] Gesendet: Dienstag, 18. September 2012 15:32 An: Oliver Einsle Cc: CCP4BB@JISCMAIL.AC.UK Betreff: Re: [ccp4bb] Series termination effect calculation. -BEGIN PGP SIGNED MESSAGE- Hash: SHA1 Hello Oliver, when you fit the values from ICA Tab 6.1.1.1 with gnuplot, the values of C and N become much more comparable. c(C) = 0.017 and especially c(N) = 0.025 0!!! for C: Final set of parametersAsymptotic Standard Error ===== a1 = 0.604126 +/- 0.02326 (3.85%) a2 = 2.63343 +/- 0.03321 (1.261%) a3 = 1.52123 +/- 0.03528 (2.319%) a4 = 1.2211 +/- 0.02225 (1.822%) b1 = 0.185807 +/- 0.00629 (3.385%) b2 = 14.6332 +/- 0.1355 (0.9263%) b3 = 41.6948 +/- 0.5345 (1.282%) b4 = 0.717984 +/- 0.01251 (1.743%) c = 0.0171359+/- 0.002045 (11.93%) for N: Final set of parametersAsymptotic Standard Error ===== a1 = 0.723788 +/- 0.04334 (5.988%) a2 = 3.24589 +/- 0.04074 (1.255%) a3 = 1.90049 +/- 0.04422 (2.327%) a4 = 1.10071 +/- 0.0413 (3.752%) b1 = 0.157345 +/- 0.007552 (4.8%) b2 = 10.106 +/- 0.1041 (1.03%) b3 = 30.0211 +/- 0.3946 (1.314%) b4 = 0.567116 +/- 0.01914 (3.376%) c = 0.0252303+/- 0.003284 (13.01%) In 1967, Mann only calculated to sin \theta/lambda = 0, ... 1.5, and their tabulated values do indeed fit decently within that range, but not out to 6A. I thought this was notworthy, and I am curious which values for these constants refinement programs use nowadays. Maybe George, Garib, Pavel, and Gerard may want to comment? Cheers, Tim On 09/18/2012 10:11 AM, Oliver Einsle wrote: Hi there, I was just pointed to this thread and should comment on the discussion, as actually made the plots for this paper. James has clarified the issue much better than I could have, and indeed the calculations will fail for larger Bragg angles if you do not assume a reasonable B-factor (I used B=10 for the plots). Doug Rees has pointed out at the time that for large theta the c-term of the Cromer/Mann approximation becomes dominant, and this is where chaos comes in, as the Cromer/Mann parameters are only derived from a fit to the actual HF-calculation. They are numbers without physical meaning, which becomes particularly obvious if you compare the parameters for C and N: C: 2.3100 20.8439 1.0200 10.2075 1.5886 0.5687 0.8650 51.6512 0.2156 N: 12.2126 0.0057 3.1322 9.8933 2.0125 28.9975 1.1663 0.5826 -11.5290 The scattering factors for these are reasonably similar, but the c-values are entirely different. The B-factor dampens this out and this is an essential point. For clarity: I made the plots using Waterloo Maple with the following code: restart; SF :=Matrix(17,9,readdata(scatter.dat,float,9)); biso := 10; e:= 1; AFF := (e)-(SF[e,1]*exp(-SF[e,2]*s^2)+SF[e,3]*exp(-SF[e,4]*s^2) +SF[e,5]*exp(-SF[e,6]*s^2)+SF[e,7]*exp(-SF[e,8]*s^2) +SF[e,9])*exp(-biso*s^2/4); H:= AFF(1); C:= AFF(2); N:= AFF(3); Ox := AFF(4); S:= AFF(5); Fe := AFF(6); Fe2 := AFF(7); Fe3 := AFF(8); Cu := AFF(9); Cu1 := AFF(10); Cu2 := AFF(11); Mo := AFF(12); Mo4 := AFF(13); Mo5 := AFF(14); Mo6 := AFF(15); // Plot scattering factors plot([C,N,Fe,S], s=0..1); // Figure 1: rho0 := (r) - Int((4*Pi*s^2)*Fe2*sin(2*Pi*s*r)/(2*Pi*s*r), s=0..1/dmax); dmax := 1.0; plot (rho0, -5..5); // Figure 1 (inset): Electron Density Profile rho := (r,f) -(Int((4*Pi*s^2)*f*sin(2*Pi*s*r)/(2*Pi*s*r),s=0..1/dmax)); cofactor:= 9*rho(3.3,S) + 6*rho(2.0,Fe2) + 1*rho(3.49,Mo6) + 1*rho(3.51,Fe3); plot(cofactor, dmax=0.5..3.5); The file scatter.dat is simply a collection of some form factors, courtesy of atomsf.lib (see attachment). Cheers, Oliver. Am 9/17/12 11:24 AM schrieb Tim Gruene unter t...@shelx.uni-ac.gwdg.de: Dear James et al., so to summarise, the answer to Niu's question is that he must add a factor of e^(-Bs^2) to the formula of Cromer/Mann and then adjust the value of B until it matches the inset. Given that you claim rho=0.025e/A^3 (I assume for 1/dmax approx. 0) for B=12 and the inset shows a value of about 0.6, a somewhat higher B-value should work
Re: [ccp4bb] Series termination effect calculation.
That's really interesting! Since the fits then and now were both
least-squares, I wonder how Cromer Mann could have gotten it so far
off? Looking at the residuals, I see that although that of nitrogen
oscillates badly, even the worst outlier is still within 0.01 electrons
of the Hartree-Fock values. Perhaps 1% of an electron was their
convergence limit?
Either way, I think it would be valuable to have a re-fit of the Table
6.1.1.1/3 values without the c term. Then we can go all the way to
B=0 without worrying about singularities.
For example, I attach here a plot of the electron density at the center
of a nitrogen atom vs B factor (in real space). The red curve is the
result of a 20-Gaussian fit to the data for nitrogen in table 6.1.1.1
all the way out to sin(theta)/lambda = 6 (although 7 Gaussians is more
than enough). This true curve approaches 1000 e-/A^3 as B approaches
zero, but the 5-Gaussian model using the Cromer-Mann coefficients form
6.1.1.4 (blue curve) starts to deviate when B becomes less than one, and
actually goes negative for B 0.1. A simpler model (without the c
term, but re-fit) is the green line. Very much like what Tim suggested.
Not exactly a problem for typical macromolecular refinement, but
still... I wonder what would happen if I edited my ${CLIBD}/atomsf.lib ?
-James Holton
MAD Scientist
On 9/18/2012 6:32 AM, Tim Gruene wrote:
-BEGIN PGP SIGNED MESSAGE-
Hash: SHA1
Hello Oliver,
when you fit the values from ICA Tab 6.1.1.1 with gnuplot, the values
of C and N become much more comparable. c(C) = 0.017 and especially
c(N) = 0.025 0!!!
for C:
Final set of parametersAsymptotic Standard Error
=====
a1 = 0.604126 +/- 0.02326 (3.85%)
a2 = 2.63343 +/- 0.03321 (1.261%)
a3 = 1.52123 +/- 0.03528 (2.319%)
a4 = 1.2211 +/- 0.02225 (1.822%)
b1 = 0.185807 +/- 0.00629 (3.385%)
b2 = 14.6332 +/- 0.1355 (0.9263%)
b3 = 41.6948 +/- 0.5345 (1.282%)
b4 = 0.717984 +/- 0.01251 (1.743%)
c = 0.0171359+/- 0.002045 (11.93%)
for N:
Final set of parametersAsymptotic Standard Error
=====
a1 = 0.723788 +/- 0.04334 (5.988%)
a2 = 3.24589 +/- 0.04074 (1.255%)
a3 = 1.90049 +/- 0.04422 (2.327%)
a4 = 1.10071 +/- 0.0413 (3.752%)
b1 = 0.157345 +/- 0.007552 (4.8%)
b2 = 10.106 +/- 0.1041 (1.03%)
b3 = 30.0211 +/- 0.3946 (1.314%)
b4 = 0.567116 +/- 0.01914 (3.376%)
c = 0.0252303+/- 0.003284 (13.01%)
In 1967, Mann only calculated to sin \theta/lambda = 0, ... 1.5, and
their tabulated values do indeed fit decently within that range, but
not out to 6A.
I thought this was notworthy, and I am curious which values for these
constants refinement programs use nowadays. Maybe George, Garib,
Pavel, and Gerard may want to comment?
Cheers,
Tim
On 09/18/2012 10:11 AM, Oliver Einsle wrote:
Hi there,
I was just pointed to this thread and should comment on the
discussion, as actually made the plots for this paper. James has
clarified the issue much better than I could have, and indeed the
calculations will fail for larger Bragg angles if you do not assume
a reasonable B-factor (I used B=10 for the plots).
Doug Rees has pointed out at the time that for large theta the
c-term of the Cromer/Mann approximation becomes dominant, and this
is where chaos comes in, as the Cromer/Mann parameters are only
derived from a fit to the actual HF-calculation. They are numbers
without physical meaning, which becomes particularly obvious if you
compare the parameters for C and N:
C: 2.3100 20.8439 1.0200 10.2075 1.5886 0.5687 0.8650
51.6512 0.2156 N: 12.2126 0.0057 3.1322 9.8933 2.0125
28.9975 1.1663 0.5826 -11.5290
The scattering factors for these are reasonably similar, but the
c-values are entirely different. The B-factor dampens this out and
this is an essential point.
For clarity: I made the plots using Waterloo Maple with the
following code:
restart; SF :=Matrix(17,9,readdata(scatter.dat,float,9));
biso := 10; e:= 1; AFF :=
(e)-(SF[e,1]*exp(-SF[e,2]*s^2)+SF[e,3]*exp(-SF[e,4]*s^2)
+SF[e,5]*exp(-SF[e,6]*s^2)+SF[e,7]*exp(-SF[e,8]*s^2)
+SF[e,9])*exp(-biso*s^2/4);
H:= AFF(1); C:= AFF(2); N:= AFF(3); Ox :=
AFF(4); S:= AFF(5); Fe := AFF(6); Fe2 := AFF(7); Fe3 :=
AFF(8); Cu := AFF(9); Cu1 := AFF(10); Cu2 := AFF(11); Mo
:= AFF(12); Mo4 := AFF(13); Mo5 := AFF(14); Mo6 := AFF(15);
// Plot scattering factors
Re: [ccp4bb] Series termination effect calculation.
Yes, the constant term in the 5-Gaussian structure factor tables does
become annoying when you try to plot electron density in real space, but
only if you try to make the B factor zero. If the B factors are ~12
(like they are in 1m1n), then the electron density 2.0 A from an Fe atom
is not -0.2 e-/A^3, it is 0.025 e-/A^3. This is only 1% of the electron
density at the center of a nitrogen atom with the same B factor.
But if you do set the B factor to zero, then the electron density at the
center of any atom (using the 5-Gaussian model) is infinity. To put it
in gnuplot-ish, the structure factor of Fe (in reciprocal space) can be
plotted with this function:
Fe_sf(s)=Fe_a1*exp(-Fe_b1*s*s)+Fe_a2*exp(-Fe_b2*s*s)+Fe_a3*exp(-Fe_b3*s*s)+Fe_a4*exp(-Fe_b4*s*s)+Fe_c
where:
Fe_c = 1.036900;
Fe_a1 = 11.769500; Fe_a2 = 7.357300; Fe_a3 = 3.522200; Fe_a4 = 2.304500;
Fe_b1 = 4.761100; Fe_b2 = 0.307200; Fe_b3 = 15.353500; Fe_b4 = 76.880501;
and s is sin(theta)/lambda
applying a B factor is then just multiplication by exp(-B*s*s)
Since the terms are all Gaussians, the inverse Fourier transform can
actually be done analytically, giving the real-space version, or the
expression for electron density vs distance from the nucleus (r):
Fe_ff(r,B) = \
+Fe_a1*(4*pi/(Fe_b1+B))**1.5*safexp(-4*pi**2/(Fe_b1+B)*r*r) \
+Fe_a2*(4*pi/(Fe_b2+B))**1.5*safexp(-4*pi**2/(Fe_b2+B)*r*r) \
+Fe_a3*(4*pi/(Fe_b3+B))**1.5*safexp(-4*pi**2/(Fe_b3+B)*r*r) \
+Fe_a4*(4*pi/(Fe_b4+B))**1.5*safexp(-4*pi**2/(Fe_b4+B)*r*r) \
+Fe_c *(4*pi/(B))**1.5*safexp(-4*pi**2/(B)*r*r);
Where here applying a B factor requires folding it into each Gaussian
term. Notice how the Fe_c term blows up as B-0? This is where most of
the series-termination effects come from. If you want the above
equations for other atoms, you can get them from here:
http://bl831.als.lbl.gov/~jamesh/pickup/all_atomsf.gnuplot
http://bl831.als.lbl.gov/~jamesh/pickup/all_atomff.gnuplot
This infinitely sharp spike problem seems to have led some people to
conclude that a zero B factor is non-physical, but nothing could be
further from the truth! The scattering from mono-atomic gasses is an
excellent example of how one can observe the B=0 structure factor. In
fact, gas scattering is how the quantum mechanical self-consistent field
calculations of electron clouds around atoms was experimentally
verified. Does this mean that there really is an infinitely sharp
spike in the middle of every atom? Of course not. But there is a
very sharp spike.
So, the problem of infinite density at the nucleus is really just an
artifact of the 5-Gaussian formalism. Strictly speaking, the
5-Gaussian structure factor representation you find in
${CLIBD}/atomsf.lib (or Table 6.1.1.4 in the International Tables volume
C) is nothing more than a curve fit to the true values listed in ITC
volume C tables 6.1.1.1 (neutral atoms) and 6.1.1.3 (ions). These
latter tables are the Fourier transform of the true electron density
distribution around a particular atom/ion obtained from quantum
mechanical self-consistent field calculations (like those of Cromer,
Mann and many others).
The important thing to realize is that the fit was done in _reciprocal_
space, and if you look carefully at tables 6.1.1.1 and 6.1.1.3, you can
see that even at REALLY high angle (sin(theta)/lambda = 6, or 0.083 A
resolution) there is still significant elastic scattering from the
heavier atoms. The purpose of the constant term in the 5-Gaussian
representation is to try and capture this high-angle tail, and for the
really heavy atoms this can be more than 5 electron equivalents. In
real space, this is equivalent to saying that about 5 electrons are
located within at least ~0.03 A of the nucleus. That's a very short
distance, but it is also not zero. This is because the first few shells
of electrons around things like a Uranium nucleus actually are very
small and dense. How, then, can we have any hope of modelling heavy
atoms properly without using a map grid sampling of 0.01A ? Easy! The
B factors are never zero.
Even for a truly infinitely sharp peak (aka a single electron), it
doesn't take much of a B factor to spread it out to a reasonable size.
For example, applying a B factor of 9 to a point charge will give it a
full-width-half max (FWHM) of 0.8 A, the same as the diameter of a
carbon atom. A carbon atom with B=12 has FWHM = 1.1 A, the same as a
point charge with B=16. Carbon at B=80 and a point with B=93 both
have FWHM = 2.6 A. As the B factor becomes larger and larger, it tends
to dominate the atomic shape (looks like a single Gaussian). This is
why it is so hard to assign atom types from density alone. In fact,
with B=80, a Uranium atom at 1/100th occupancy is essentially
indistinguishable from a hydrogen atom. That is, even a modest B factor
pretty much washes out any sharp features the atoms might have.
Sometimes I wonder why we bother with form
Re: [ccp4bb] Series termination effect calculation.
Le Lundi 17 Septembre 2012 08:32 CEST, James Holton jmhol...@lbl.gov a écrit Hello May I add a few words after the thorough comments by James. I lmay be easier to consider series termination in real space as follows. The effect of series termination in 3D on rho(r) is of convoluting the exact rho(r) with the approximation of a delta function resulting from the limit in resolution. This approximation in 3D is given exactly by the function G[X] = 3*[Sin(X) - X*Cos(X)]/X^3, where X = 2*Pi*r/d (r in Angstrom and d the resolution, also in Angstrom). This is the function appearing in the rotation function (for exactly the same reason of truncating the resolution). If you consider that the iron atom is punctual (i.e. its Fourier transform would be merely constant), then the approximation resulting from series termination is just given by G[X] (apart for a scaling factor). And if you convolute the exact and ideal rho(r) with G[X], you will obtain the exact form of rho[r] affected by series termination. Note that, considering the Gaussian approximation of the structure factors, this would amount to convolute gaussians with G[X] (see James comments). I join a figure corresponding to the simplification of a punctual iron atom. I only put on this figure the curves corresponding to the limits in resolution, 1.3, 2 an 2.5 Angstrom because at a resolution of 1 Angstrom, the iron atom is definitely not punctual. I used the same color codes as in Fig. 1 of the paper. One can see that the ripples on my approximate figure are essentially the same as in Fig. 1 of the paper. Of course, it cannot reproduce the features of rho[r] for r--0 since the iron aton is definitely not punctual. Practical comment. It is quite useful to consider the following rule of thumb: the first minimum of G[X] appears at a distance equal to 0.92*d (d = resolution) and the first maximum at 1.45*d. Therefore, if one suspects that series terminaiton effects might cause a spurious through, or peak, it may be enough to recalculate the e.d. map at different resolutions to check whether these features are moving or not. Philippe Dumas PS: it is instructive to make a comparison with the Airy function in astronomy. Airy calculated this function to take into account the distorsion brought by the limlited optical resolution of a telescope to a punctual image of a star. Nothing else than our problem, with an iron atom replacing a star... Plus ça change, plus c'est la même chose. Yes, the constant term in the 5-Gaussian structure factor tables does become annoying when you try to plot electron density in real space, but only if you try to make the B factor zero. If the B factors are ~12 (like they are in 1m1n), then the electron density 2.0 A from an Fe atom is not -0.2 e-/A^3, it is 0.025 e-/A^3. This is only 1% of the electron density at the center of a nitrogen atom with the same B factor. But if you do set the B factor to zero, then the electron density at the center of any atom (using the 5-Gaussian model) is infinity. To put it in gnuplot-ish, the structure factor of Fe (in reciprocal space) can be plotted with this function: Fe_sf(s)=Fe_a1*exp(-Fe_b1*s*s)+Fe_a2*exp(-Fe_b2*s*s)+Fe_a3*exp(-Fe_b3*s*s)+Fe_a4*exp(-Fe_b4*s*s)+Fe_c where: Fe_c = 1.036900; Fe_a1 = 11.769500; Fe_a2 = 7.357300; Fe_a3 = 3.522200; Fe_a4 = 2.304500; Fe_b1 = 4.761100; Fe_b2 = 0.307200; Fe_b3 = 15.353500; Fe_b4 = 76.880501; and s is sin(theta)/lambda applying a B factor is then just multiplication by exp(-B*s*s) Since the terms are all Gaussians, the inverse Fourier transform can actually be done analytically, giving the real-space version, or the expression for electron density vs distance from the nucleus (r): Fe_ff(r,B) = \ +Fe_a1*(4*pi/(Fe_b1+B))**1.5*safexp(-4*pi**2/(Fe_b1+B)*r*r) \ +Fe_a2*(4*pi/(Fe_b2+B))**1.5*safexp(-4*pi**2/(Fe_b2+B)*r*r) \ +Fe_a3*(4*pi/(Fe_b3+B))**1.5*safexp(-4*pi**2/(Fe_b3+B)*r*r) \ +Fe_a4*(4*pi/(Fe_b4+B))**1.5*safexp(-4*pi**2/(Fe_b4+B)*r*r) \ +Fe_c *(4*pi/(B))**1.5*safexp(-4*pi**2/(B)*r*r); Where here applying a B factor requires folding it into each Gaussian term. Notice how the Fe_c term blows up as B-0? This is where most of the series-termination effects come from. If you want the above equations for other atoms, you can get them from here: http://bl831.als.lbl.gov/~jamesh/pickup/all_atomsf.gnuplot http://bl831.als.lbl.gov/~jamesh/pickup/all_atomff.gnuplot This infinitely sharp spike problem seems to have led some people to conclude that a zero B factor is non-physical, but nothing could be further from the truth! The scattering from mono-atomic gasses is an excellent example of how one can observe the B=0 structure factor. In fact, gas scattering is how the quantum mechanical self-consistent field calculations of electron clouds around atoms was experimentally verified. Does this mean that there really is an infinitely sharp spike in the
Re: [ccp4bb] Series termination effect calculation.
-BEGIN PGP SIGNED MESSAGE-
Hash: SHA1
Dear James et al.,
so to summarise, the answer to Niu's question is that he must add a
factor of e^(-Bs^2) to the formula of Cromer/Mann and then adjust the
value of B until it matches the inset. Given that you claim
rho=0.025e/A^3 (I assume for 1/dmax approx. 0) for B=12 and the inset
shows a value of about 0.6, a somewhat higher B-value should work.
Cheers,
Tim
On 09/17/2012 08:32 AM, James Holton wrote:
Yes, the constant term in the 5-Gaussian structure factor tables does
become annoying when you try to plot electron density in real space, but
only if you try to make the B factor zero. If the B factors are ~12
(like they are in 1m1n), then the electron density 2.0 A from an Fe atom
is not -0.2 e-/A^3, it is 0.025 e-/A^3. This is only 1% of the electron
density at the center of a nitrogen atom with the same B factor.
But if you do set the B factor to zero, then the electron density at the
center of any atom (using the 5-Gaussian model) is infinity. To put it
in gnuplot-ish, the structure factor of Fe (in reciprocal space) can be
plotted with this function:
Fe_sf(s)=Fe_a1*exp(-Fe_b1*s*s)+Fe_a2*exp(-Fe_b2*s*s)+Fe_a3*exp(-Fe_b3*s*s)+Fe_a4*exp(-Fe_b4*s*s)+Fe_c
where:
Fe_c = 1.036900;
Fe_a1 = 11.769500; Fe_a2 = 7.357300; Fe_a3 = 3.522200; Fe_a4 = 2.304500;
Fe_b1 = 4.761100; Fe_b2 = 0.307200; Fe_b3 = 15.353500; Fe_b4 = 76.880501;
and s is sin(theta)/lambda
applying a B factor is then just multiplication by exp(-B*s*s)
Since the terms are all Gaussians, the inverse Fourier transform can
actually be done analytically, giving the real-space version, or the
expression for electron density vs distance from the nucleus (r):
Fe_ff(r,B) = \
+Fe_a1*(4*pi/(Fe_b1+B))**1.5*safexp(-4*pi**2/(Fe_b1+B)*r*r) \
+Fe_a2*(4*pi/(Fe_b2+B))**1.5*safexp(-4*pi**2/(Fe_b2+B)*r*r) \
+Fe_a3*(4*pi/(Fe_b3+B))**1.5*safexp(-4*pi**2/(Fe_b3+B)*r*r) \
+Fe_a4*(4*pi/(Fe_b4+B))**1.5*safexp(-4*pi**2/(Fe_b4+B)*r*r) \
+Fe_c *(4*pi/(B))**1.5*safexp(-4*pi**2/(B)*r*r);
Where here applying a B factor requires folding it into each Gaussian
term. Notice how the Fe_c term blows up as B-0? This is where most of
the series-termination effects come from. If you want the above
equations for other atoms, you can get them from here:
http://bl831.als.lbl.gov/~jamesh/pickup/all_atomsf.gnuplot
http://bl831.als.lbl.gov/~jamesh/pickup/all_atomff.gnuplot
This infinitely sharp spike problem seems to have led some people to
conclude that a zero B factor is non-physical, but nothing could be
further from the truth! The scattering from mono-atomic gasses is an
excellent example of how one can observe the B=0 structure factor. In
fact, gas scattering is how the quantum mechanical self-consistent field
calculations of electron clouds around atoms was experimentally
verified. Does this mean that there really is an infinitely sharp
spike in the middle of every atom? Of course not. But there is a
very sharp spike.
So, the problem of infinite density at the nucleus is really just an
artifact of the 5-Gaussian formalism. Strictly speaking, the
5-Gaussian structure factor representation you find in
${CLIBD}/atomsf.lib (or Table 6.1.1.4 in the International Tables volume
C) is nothing more than a curve fit to the true values listed in ITC
volume C tables 6.1.1.1 (neutral atoms) and 6.1.1.3 (ions). These
latter tables are the Fourier transform of the true electron density
distribution around a particular atom/ion obtained from quantum
mechanical self-consistent field calculations (like those of Cromer,
Mann and many others).
The important thing to realize is that the fit was done in _reciprocal_
space, and if you look carefully at tables 6.1.1.1 and 6.1.1.3, you can
see that even at REALLY high angle (sin(theta)/lambda = 6, or 0.083 A
resolution) there is still significant elastic scattering from the
heavier atoms. The purpose of the constant term in the 5-Gaussian
representation is to try and capture this high-angle tail, and for the
really heavy atoms this can be more than 5 electron equivalents. In
real space, this is equivalent to saying that about 5 electrons are
located within at least ~0.03 A of the nucleus. That's a very short
distance, but it is also not zero. This is because the first few shells
of electrons around things like a Uranium nucleus actually are very
small and dense. How, then, can we have any hope of modelling heavy
atoms properly without using a map grid sampling of 0.01A ? Easy! The
B factors are never zero.
Even for a truly infinitely sharp peak (aka a single electron), it
doesn't take much of a B factor to spread it out to a reasonable size.
For example, applying a B factor of 9 to a point charge will give it a
full-width-half max (FWHM) of 0.8 A, the same as the diameter of a
carbon atom. A carbon atom with B=12 has FWHM = 1.1 A, the same as a
point charge with B=16. Carbon at B=80
Re: [ccp4bb] Series termination effect calculation.
-BEGIN PGP SIGNED MESSAGE- Hash: SHA1 Dear Ian, provided that f(s) is given by the formula in the Cromer/Mann article, which I believe we have agreed on, the inset of Fig.1 of the Science article we are talking about is claimed to be the graph of the function g, which I added as pdf to this email for better readability. Irrespective of what has been plotted in any other article meantioned throughout this thread, this claim is incorrect, given a_i, b_i, c 0. I am sure you can figure this out yourself. My argument was not involving mathematical programs but only one-dimensional calculus. Cheers, Tim On 09/14/2012 04:46 PM, Ian Tickle wrote: On 14 September 2012 15:15, Tim Gruene t...@shelx.uni-ac.gwdg.de wrote: -BEGIN PGP SIGNED MESSAGE- Hash: SHA1 Hello Ian, your article describes f(s) as sum of four Gaussians, which is not the same f(s) from Cromer's and Mann's paper and the one used both by Niu and me. Here, f(s) contains a constant, as I pointed out to in my response, which makes the integral oscillate between plus and minus infinity as the upper integral border (called 1/dmax in the article Niu refers to) goes to infinity). Maybe you can shed some light on why your article uses a different f(s) than Cromer/Mann. This explanation might be the answer to Nius question, I reckon, and feed my curiosity, too. Tim Niu, oops yes a small slip in the paper there, it should have read 4 Gaussians + constant term: this is clear from the ITC reference given and the $CLIBD/atomsf.lib table referred to. In practice it's actually rendered as a sum of 5 Gaussians after you multiply the f(s) and atomic Biso factor terms, so unless Biso = 0 (very unphysical!) there is actually no constant term. My integral for rho(r) certainly doesn't oscillate between plus and minus infinity as d_min - zero. If yours does then I suspect that either the Biso term was forgotten or if not then a bug in the integration routine (e.g. can it handle properly the point at r = 0 where the standard formula for the density gives 0/0?). I used QUADPACK (http://people.sc.fsu.edu/~jburkardt/f_src/quadpack/quadpack.html) which seems pretty good at taking care of such singularities (assuming of course that the integral does actually converge). Cheers -- Ian - -- - -- Dr Tim Gruene Institut fuer anorganische Chemie Tammannstr. 4 D-37077 Goettingen GPG Key ID = A46BEE1A -BEGIN PGP SIGNATURE- Version: GnuPG v1.4.12 (GNU/Linux) Comment: Using GnuPG with Mozilla - http://enigmail.mozdev.org/ iD8DBQFQVFSBUxlJ7aRr7hoRAoPYAKDNQu84ozIz5Mn/qmRKiLxXPw/zPgCgwd75 KUHsKzaSdi9mL5kzZBeOqUI= =mbnY -END PGP SIGNATURE- integral-crop.pdf Description: Adobe PDF document
Re: [ccp4bb] Series termination effect calculation.
-BEGIN PGP SIGNED MESSAGE-
Hash: SHA1
Dear Niu,
as far as I can tell, all your parameters are correct and the
scattering term for f(s) you use is also correct. f(s) furthermore
matches very closely those tabulated in the Intl. Tables C Tab. 6.1.1.1.
My reproduction of the mentioned formula with r=2.0 using MAXIMA also
shows quite a different graph.
The graph does not make much sense:
as d_max - 0 1/d_max - infinity and the integrand goes to infinity,
because f(s) contains a constant positive term. Hence the integral
should oscilatingly approach infinity and not stabilise as the upper
integral limit approaches infinity.
Best wishes,
Tim
On 09/13/2012 11:09 PM, Niu Tou wrote:
Dear Colleagues,
I am trying to repeat a series termination effect calculation
displayed as figure 2 in a publihsed paper
(http://www.ncbi.nlm.nih.gov/pubmed/12215645). Formula (1) was used
to implement this calculation. Since f(s) is not defined in detail
in this paper, I used formula and parameters listed in another
paper (http://scripts.iucr.org/cgi-bin/paper?a05896) to calculate
it.
However, the result I got is not consistent with figure 2 of the
first paper. I am not sure if the formulas I used are right or not.
Or if there is any problem in the MatLab code, which I list below:
###
clear all;clc;format compact;format long;
% matrix of a, b, c coefficients:
% rows: Fe, S, Fe1, Mo
% columns: A1; B1; A2; B2; A3; B3; A4; B4; C
fM = ...
[11.9185 4.87394 7.04848 0.34023 3.34326 15.9330 2.27228 79.0339
1.40818;...
7.18742 1.43280 5.88671 0.02865 5.15858 22.1101 1.64403 55.4651
-3.87732;...
11.9185 4.87394 7.04848 0.34023 3.34326 15.9330 2.27228 79.0339
1.40818;...
19.3885 0.97877 11.8308 10.0885 3.75919 31.9738 1.46772 117.932
5.55047];
%%% store radius data:
% distance from: origin
% columns: Fe, S, Fe, Mo
R_el = [2.0 3.3 3.5 3.5];
RHO_t = zeros(4,400);
for numel = 1:4
EL = numel;
RHO = zeros(1,400);
dmax = zeros(1,400);
for iter = 1:400
dmax(iter) = iter/100; % in angstroms
% numerical integration
int_fun = @(s) 4*pi*(s.^2).* ...
(fM(EL,1).*exp(-fM(EL,2).*(s.^2)*0.25) + ...
fM(EL,3).*exp(-fM(EL,4).*(s.^2)*0.25) + ...
fM(EL,5).*exp(-fM(EL,6).*(s.^2)*0.25) + ...
fM(EL,7).*exp(-fM(EL,8).*(s.^2)*0.25) + fM(EL,9)).* ...
sin(2*pi*s*R_el(EL))./(2*pi*s*R_el(EL));
RHO(iter) = quad(int_fun,0,1/dmax(iter));
clc;display(iter);display(numel);
end
RHO_t(numel,:) = RHO;
end
RHO_t(1,:)= 6*RHO_t(1,:);
RHO_t(2,:)= 9*RHO_t(2,:);
figure;
axis([0.5 3.5 -10 10]); hold on;
plot(dmax,RHO_t(1,:),...
dmax,RHO_t(2,:),...
dmax,RHO_t(3,:),...
dmax,RHO_t(4,:),...
dmax,sum(RHO_t,1));
title('Electron Density Profile');
legend('Fe','S','Fe1','Mo','Sum');
xlabel('d_m_a_x'); ylabel('Rho(r)');
set(gca,'XDir','reverse');
##
Any suggestions will be appreciated. Thanks!
Niu
- --
- --
Dr Tim Gruene
Institut fuer anorganische Chemie
Tammannstr. 4
D-37077 Goettingen
GPG Key ID = A46BEE1A
-BEGIN PGP SIGNATURE-
Version: GnuPG v1.4.12 (GNU/Linux)
Comment: Using GnuPG with Mozilla - http://enigmail.mozdev.org/
iD8DBQFQUx2lUxlJ7aRr7hoRAi6IAJ9cU5etz4nb8Y7ti9zSjjL5P3ptJgCgtsl4
RM8Zn+l70MLPXoO1MpGKYUI=
=Lq9Z
-END PGP SIGNATURE-
Re: [ccp4bb] Series termination effect calculation.
On 14 September 2012 13:05, Tim Gruene t...@shelx.uni-ac.gwdg.de wrote: -BEGIN PGP SIGNED MESSAGE- Hash: SHA1 Dear Niu, as far as I can tell, all your parameters are correct and the scattering term for f(s) you use is also correct. f(s) furthermore matches very closely those tabulated in the Intl. Tables C Tab. 6.1.1.1. My reproduction of the mentioned formula with r=2.0 using MAXIMA also shows quite a different graph. The graph does not make much sense: as d_max - 0 1/d_max - infinity and the integrand goes to infinity, d_min surely, i.e. minimum d-spacing? because f(s) contains a constant positive term. Hence the integral should oscilatingly approach infinity and not stabilise as the upper integral limit approaches infinity. Tim, exactly so, in fact like this: http://journals.iucr.org/d/issues/2012/04/00/dz5235/dz5235.pdf See eqn 2 and Fig 11(b). Note that although rho(r) itself does indeed tend to zero as d_min - 0 as expected, the volume integral of rho(r) (i.e. the calculated number of electrons) does not (unless d_min - 0) ! Cheers -- Ian
Re: [ccp4bb] Series termination effect calculation.
-BEGIN PGP SIGNED MESSAGE- Hash: SHA1 Hello Ian, your article describes f(s) as sum of four Gaussians, which is not the same f(s) from Cromer's and Mann's paper and the one used both by Niu and me. Here, f(s) contains a constant, as I pointed out to in my response, which makes the integral oscillate between plus and minus infinity as the upper integral border (called 1/dmax in the article Niu refers to) goes to infinity). Maybe you can shed some light on why your article uses a different f(s) than Cromer/Mann. This explanation might be the answer to Nius question, I reckon, and feed my curiosity, too. Cheers, Tim On 09/14/2012 02:39 PM, Ian Tickle wrote: On 14 September 2012 13:05, Tim Gruene t...@shelx.uni-ac.gwdg.de wrote: -BEGIN PGP SIGNED MESSAGE- Hash: SHA1 Dear Niu, as far as I can tell, all your parameters are correct and the scattering term for f(s) you use is also correct. f(s) furthermore matches very closely those tabulated in the Intl. Tables C Tab. 6.1.1.1. My reproduction of the mentioned formula with r=2.0 using MAXIMA also shows quite a different graph. The graph does not make much sense: as d_max - 0 1/d_max - infinity and the integrand goes to infinity, d_min surely, i.e. minimum d-spacing? because f(s) contains a constant positive term. Hence the integral should oscilatingly approach infinity and not stabilise as the upper integral limit approaches infinity. Tim, exactly so, in fact like this: http://journals.iucr.org/d/issues/2012/04/00/dz5235/dz5235.pdf See eqn 2 and Fig 11(b). Note that although rho(r) itself does indeed tend to zero as d_min - 0 as expected, the volume integral of rho(r) (i.e. the calculated number of electrons) does not (unless d_min - 0) ! Cheers -- Ian - -- - -- Dr Tim Gruene Institut fuer anorganische Chemie Tammannstr. 4 D-37077 Goettingen GPG Key ID = A46BEE1A -BEGIN PGP SIGNATURE- Version: GnuPG v1.4.12 (GNU/Linux) Comment: Using GnuPG with Mozilla - http://enigmail.mozdev.org/ iD8DBQFQUzwQUxlJ7aRr7hoRAuQWAJ40ufwGUZrPmt2cLpXdccR2MMrboACgiOaG CTfi2MtYiBpy5oxY1kcyFJ4= =fIwH -END PGP SIGNATURE-
[ccp4bb] Series termination effect calculation.
Dear Colleagues,
I am trying to repeat a series termination effect calculation displayed as
figure 2 in a publihsed paper
(http://www.ncbi.nlm.nih.gov/pubmed/12215645). Formula
(1) was used to implement this calculation. Since f(s) is not defined in
detail in this paper, I used formula and parameters listed in another
paper (http://scripts.iucr.org/cgi-bin/paper?a05896) to calculate it.
However, the result I got is not consistent with figure 2 of the first
paper. I am not sure if the formulas I used are right or not. Or if there
is any problem in the MatLab code, which I list below:
###
clear all;clc;format compact;format long;
% matrix of a, b, c coefficients:
% rows: Fe, S, Fe1, Mo
% columns: A1; B1; A2; B2; A3; B3; A4; B4; C
fM = ...
[11.9185 4.87394 7.04848 0.34023 3.34326 15.9330 2.27228 79.0339 1.40818;...
7.18742 1.43280 5.88671 0.02865 5.15858 22.1101 1.64403 55.4651
-3.87732;...
11.9185 4.87394 7.04848 0.34023 3.34326 15.9330 2.27228 79.0339 1.40818;...
19.3885 0.97877 11.8308 10.0885 3.75919 31.9738 1.46772 117.932 5.55047];
%%% store radius data:
% distance from: origin
% columns: Fe, S, Fe, Mo
R_el = [2.0 3.3 3.5 3.5];
RHO_t = zeros(4,400);
for numel = 1:4
EL = numel;
RHO = zeros(1,400);
dmax = zeros(1,400);
for iter = 1:400
dmax(iter) = iter/100; % in angstroms
% numerical integration
int_fun = @(s) 4*pi*(s.^2).* ...
(fM(EL,1).*exp(-fM(EL,2).*(s.^2)*0.25) + ...
fM(EL,3).*exp(-fM(EL,4).*(s.^2)*0.25) + ...
fM(EL,5).*exp(-fM(EL,6).*(s.^2)*0.25) + ...
fM(EL,7).*exp(-fM(EL,8).*(s.^2)*0.25) + fM(EL,9)).* ...
sin(2*pi*s*R_el(EL))./(2*pi*s*R_el(EL));
RHO(iter) = quad(int_fun,0,1/dmax(iter));
clc;display(iter);display(numel);
end
RHO_t(numel,:) = RHO;
end
RHO_t(1,:)= 6*RHO_t(1,:);
RHO_t(2,:)= 9*RHO_t(2,:);
figure;
axis([0.5 3.5 -10 10]); hold on;
plot(dmax,RHO_t(1,:),...
dmax,RHO_t(2,:),...
dmax,RHO_t(3,:),...
dmax,RHO_t(4,:),...
dmax,sum(RHO_t,1));
title('Electron Density Profile');
legend('Fe','S','Fe1','Mo','Sum');
xlabel('d_m_a_x'); ylabel('Rho(r)');
set(gca,'XDir','reverse');
##
Any suggestions will be appreciated. Thanks!
Niu
Re: [ccp4bb] Series termination effect calculation.
Hi,
pointers listed here may be of help:
1) CCP4 Newsletterhttp://www.ccp4.ac.uk/newsletters/newsletter42/content.html
On the Fourier series truncation peaks at subatomic resolution
Anne Bochow, Alexandre Urzhumtsev
2) https://www.phenix-online.org/presentations/latest/pavel_maps.pdf
3) Central Ligand in the FeMo-Cofactor Nitrogenase MoFe-Protein at 1.16 Å
Resolution: A.
Oliver Einsle, et al. Science, 1696 (2002) 297
4) Page 267 Figure 4:
On the possibility of the observation of valence electron density for
individual
bonds in proteins in conventional difference maps
P. V. Afonine, V. Y. Lunin, N. Muzet and A. Urzhumtsev
Acta Cryst. (2004). D60, 260-274
Pavel
On Thu, Sep 13, 2012 at 2:09 PM, Niu Tou niutou2...@gmail.com wrote:
Dear Colleagues,
I am trying to repeat a series termination effect calculation displayed as
figure 2 in a publihsed paper (http://www.ncbi.nlm.nih.gov/pubmed/12215645).
Formula
(1) was used to implement this calculation. Since f(s) is not defined in
detail in this paper, I used formula and parameters listed in another
paper (http://scripts.iucr.org/cgi-bin/paper?a05896) to calculate it.
However, the result I got is not consistent with figure 2 of the first
paper. I am not sure if the formulas I used are right or not. Or if there
is any problem in the MatLab code, which I list below:
###
clear all;clc;format compact;format long;
% matrix of a, b, c coefficients:
% rows: Fe, S, Fe1, Mo
% columns: A1; B1; A2; B2; A3; B3; A4; B4; C
fM = ...
[11.9185 4.87394 7.04848 0.34023 3.34326 15.9330 2.27228 79.0339
1.40818;...
7.18742 1.43280 5.88671 0.02865 5.15858 22.1101 1.64403 55.4651
-3.87732;...
11.9185 4.87394 7.04848 0.34023 3.34326 15.9330 2.27228 79.0339
1.40818;...
19.3885 0.97877 11.8308 10.0885 3.75919 31.9738 1.46772 117.932 5.55047];
%%% store radius data:
% distance from: origin
% columns: Fe, S, Fe, Mo
R_el = [2.0 3.3 3.5 3.5];
RHO_t = zeros(4,400);
for numel = 1:4
EL = numel;
RHO = zeros(1,400);
dmax = zeros(1,400);
for iter = 1:400
dmax(iter) = iter/100; % in angstroms
% numerical integration
int_fun = @(s) 4*pi*(s.^2).* ...
(fM(EL,1).*exp(-fM(EL,2).*(s.^2)*0.25) + ...
fM(EL,3).*exp(-fM(EL,4).*(s.^2)*0.25) + ...
fM(EL,5).*exp(-fM(EL,6).*(s.^2)*0.25) + ...
fM(EL,7).*exp(-fM(EL,8).*(s.^2)*0.25) + fM(EL,9)).* ...
sin(2*pi*s*R_el(EL))./(2*pi*s*R_el(EL));
RHO(iter) = quad(int_fun,0,1/dmax(iter));
clc;display(iter);display(numel);
end
RHO_t(numel,:) = RHO;
end
RHO_t(1,:)= 6*RHO_t(1,:);
RHO_t(2,:)= 9*RHO_t(2,:);
figure;
axis([0.5 3.5 -10 10]); hold on;
plot(dmax,RHO_t(1,:),...
dmax,RHO_t(2,:),...
dmax,RHO_t(3,:),...
dmax,RHO_t(4,:),...
dmax,sum(RHO_t,1));
title('Electron Density Profile');
legend('Fe','S','Fe1','Mo','Sum');
xlabel('d_m_a_x'); ylabel('Rho(r)');
set(gca,'XDir','reverse');
##
Any suggestions will be appreciated. Thanks!
Niu
