FT and Least Square methods [Archive]

Zigoteau

Jun23-05, 09:27 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nHi, Zaw,\n\n\n> What is the difference between Least Square method and Fourier methods\n> including Difference Fourier Transforms?\n\n\nYou are talking chalk and cheese here. The Least Squares method and\nFourier transformation are not competing methods for doing the same\nthing, but complementary methods used in different parts of the\ncalculation.\n\nI think you\'re talking about crystallography here. The least-squares\nmethod is used to improve an initial estimate for the positions of\natoms in a crystal. The electron density distribution around a given\natom does vary a bit with its environment, as the bonding to\nneighboring atoms changes, but to a first approximation it can be taken\nto be spherically-symmetrical and independent of the environment, and\nall the textbooks on the subject have tables of scattering factors as a\nfunction of d-spacing for each type of atom. The arrangement of the\nnuclei in space leads to interference between the X-rays scattered from\neach atom, and hence quite complicated variation of the X-ray\nintensities scattered in the different directions. You can measure the\nintensities. With the help of Fourier transformation, a given assumed\ndistribution of nuclear positions will in general give you a different\ndistribution of intensities. You take the difference between each\ncalculated and measured values, square it, and add them all together.\nYou then iteratively change the assumed positions of the nuclei until\nthe sum of the squared differences is minimized, in a method initially\ndeveloped by Gauss.\n\nIn order to use the least-squares method you need a reasonable first\nestimate of the nuclear positions. For molecular weights up to 100 or\nso you can get guesses for the phase of each reflection from the\nHauptmann-Karl procedure. Fourier transforming gives you an initial\nestimate for the electron density distribution, whose prominent peaks\nwill correspond to the positions of the heaviest atoms. Usually you\ndon\'t start to see the hydrogens until you have fairly precise\npositions for the other atoms.\n\nWhen you have the positions of all the atoms sorted out, you can then\nrefine your structure with Debye-Waller factors corresponding to\nthermal vibrations of each atom.\n\nCheers,\n\nZigoteau.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hi, Zaw,

> What is the difference between Least Square method and Fourier methods
> including Difference Fourier Transforms?

You are talking chalk and cheese here. The Least Squares method and
Fourier transformation are not competing methods for doing the same
thing, but complementary methods used in different parts of the
calculation.

I think you're talking about crystallography here. The least-squares
method is used to improve an initial estimate for the positions of
atoms in a crystal. The electron density distribution around a given
atom does vary a bit with its environment, as the bonding to
neighboring atoms changes, but to a first approximation it can be taken
to be spherically-symmetrical and independent of the environment, and
all the textbooks on the subject have tables of scattering factors as a
function of d-spacing for each type of atom. The arrangement of the
nuclei in space leads to interference between the X-rays scattered from
each atom, and hence quite complicated variation of the X-ray
intensities scattered in the different directions. You can measure the
intensities. With the help of Fourier transformation, a given assumed
distribution of nuclear positions will in general give you a different
distribution of intensities. You take the difference between each
calculated and measured values, square it, and add them all together.
You then iteratively change the assumed positions of the nuclei until
the sum of the squared differences is minimized, in a method initially
developed by Gauss.

In order to use the least-squares method you need a reasonable first
estimate of the nuclear positions. For molecular weights up to 100 or
so you can get guesses for the phase of each reflection from the
Hauptmann-Karl procedure. Fourier transforming gives you an initial
estimate for the electron density distribution, whose prominent peaks
will correspond to the positions of the heaviest atoms. Usually you
don't start to see the hydrogens until you have fairly precise
positions for the other atoms.

When you have the positions of all the atoms sorted out, you can then
refine your structure with Debye-Waller factors corresponding to
thermal vibrations of each atom.

Cheers,

Zigoteau.