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 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 parameters Asymptotic 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. 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_cwhere: 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 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)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 = A46BEE1ADr 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/ iD8DBQFQWHfUUxlJ7aRr7hoRAr4VAJ4isN0PLYafsdZgVOYseV+MricBVgCfftQd 4a7EpBF1hud6sM6L0SxBXqE= =ijgy -----END PGP SIGNATURE-----
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