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 > 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: >>>>>> >>>>>>> -----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<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 > > -----BEGIN PGP SIGNATURE----- > Version: GnuPG v1.4.12 (GNU/Linux) > Comment: Using GnuPG with Mozilla - http://enigmail.mozdev.org/ > > iD8DBQFQWuRdUxlJ7aRr7hoRAix4AJ9u/dT5RQuuL9wY0+2BTQre9TdsywCeJ5Jg > uWoFQGip/L4ZTbGQvaMAIHQ= > =Gdhk > -----END PGP SIGNATURE-----
