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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
100 0.217171 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*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_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 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
>> factors" at all, since at modest resolutions all we really need
>> is Z (the atomic number) and the B factor. But, then again, I
>> suppose it doesn't hurt either.
>>
>>
>> So, what does this have to do with series termination? Series
>> termination arises in the inverse Fourier transform (making a map
>> from structure factors). Technically, the "tails" of a Gaussian
>> never reach zero, so any sort of "resolution cutoff" always
>> introduces some error into the electron density calculation.
>> That is, if you create an arbitrary electron-density map, convert
>> it into structure factors and then "fft" it back, you do _not_
>> get the same map that you started with! How much do they differ?
>> Depends on the RMS value of the high-angle structure factors that
>> have been cut off (Parseval's theorem). The "infinitely sharp
>> spike" problem exacerbates this, because the B=0 structure
>> factors do not tend toward zero as fast as a Gaussian with the
>> "atomic width" would.
>>
>> So, for a given resolution, when does the B factor get "too
>> sharp"? Well, for "protein" atoms, the following B factors will
>> introduce an rms error in the electron density map equal to about
>> 5% of the peak height of the atoms when the data are cut to the
>> following resolution: d B 1.0 <5 1.5 8 2.0 27 2.5 45 3.0 65
>> 3.5 86 4.0 >99
>>
>> smaller B factors than this will introduce more than 5% error at
>> each of these resolutions. Now, of course, one is often not
>> nearly as concerned with the average error in the map as you are
>> with the error at a particular point of interest, but the above
>> numbers can serve as a rough guide. If you want to see the
>> series-termination error at a particular point in the map, you
>> will have to calculate the "true" map of your model (using a
>> program like SFALL), and then run the map back and forth through
>> the Fourier transform and resolution cutoff (such as with SFALL
>> and FFT). You can then use MAPMAN or Chimera to probe the
>> electron density at the point of interest.
>>
>> But, to answer the OP's question, I would not recommend trying to
>> do fancy map interpretation to identify a mystery atom. Instead,
>> just refine the occupancy of the mystery atom and see where that
>> goes. Perhaps jiggling the rest of the molecule with "kick maps"
>> to see how stable the occupancy is. Since refinement only does
>> forward-FFTs, it is formally insensitive to series termination
>> errors. It is only map calculation where series termination can
>> become a problem.
>>
>> Thanks to Garib for clearing up that last point for me!
>>
>> -James Holton MAD Scientist
>>
>>
>>
>> On 9/15/2012 3:12 AM, Tim Gruene wrote:
>>
> 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:
>>>>>
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>>>>>>
>>>>>> 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<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
>
>>>
>>
- --
- --
Dr Tim Gruene
Institut fuer anorganische Chemie
Tammannstr. 4
D-37077 Goettingen
GPG Key ID = A46BEE1A
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