From chuang@iris.bc.ntu.edu.tw Thu Mar 23 01:22:38 1995 Received: from iris.bc.ntu.edu.tw for chuang@iris.bc.ntu.edu.tw by www.ccl.net (8.6.10/930601.1506) id BAA15474; Thu, 23 Mar 1995 01:13:50 -0500 From: Received: by iris.bc.ntu.edu.tw (920330.SGI/920502.SGI) for chemistry@ccl.net id AA08244; Thu, 23 Mar 95 14:15:24 -0800 Date: Thu, 23 Mar 95 14:15:24 -0800 Message-Id: <9503232215.AA08244@iris.bc.ntu.edu.tw> To: DIBUG@comp.bioz.unibas.ch Subject: correlation time estimation Cc: amber@cgl.ucsf.EDU, quanta-l@msi.com, xplor-l@msi.com, str-nmr@net.bio.net, chemistry@ccl.net, molmodel@net.bio.net Dear netters: When doing NOE simulation from structure, it is needed to give the correlation time value. Yah, to my know, it is dependent on the molecular weight, the shape (structure) of the molecule. How to roughtly estimate the correlation time when doing this kind of calculation ??? Any suggestion will be great appreciated. Any suggestion can mail to me directly. Thanks. ccchuang ============================================================ Chyh-Chong Chuang Institude of Biological Chemistry, Academia Sinica,Taipei, Taiwan ------------------------------------------------------------ P.O. BOX 23-106, Phone: (02) 362-0261 ext 2021 Taipei, Taiwan, Fax: (02) 363-5038 R.O.C. E-Mail: chuang@bc.ntu.edu.tw ============================================================ From gl@coil.mdy.univie.ac.at Thu Mar 23 05:37:42 1995 Received: from coil.mdy.univie.ac.at for gl@coil.mdy.univie.ac.at by www.ccl.net (8.6.10/930601.1506) id FAA17792; Thu, 23 Mar 1995 05:35:49 -0500 Received: by coil.mdy.univie.ac.at (931110.SGI/930416.SGI.AUTO) for CHEMISTRY@ccl.net id AA02459; Thu, 23 Mar 95 11:33:53 +0100 From: "Gerald Loeffler" Message-Id: <9503231133.ZM2457@coil.mdy.univie.ac.at> Date: Thu, 23 Mar 1995 11:33:52 +0100 X-Mailer: Z-Mail (3.1.0 22feb94 MediaMail) To: CHEMISTRY@ccl.net Subject: ? X-PLOR mailing list/newsgroup ? Content-Type: text/plain; charset=us-ascii Mime-Version: 1.0 Hi! Do you know of a mailing list or newsgroup dedicated to Axel Bruengers X-PLOR ? Thank you (hope this isn't a FAQ) -- Gerald Loeffler PhD student in Theoretical Biochemistry Email: gl@mdy.univie.ac.at Phone: +43 1 40480 612 Fax: +43 1 4028525 Mail: University of Vienna Institute for Theoretical Chemistry Theoretical Biochemistry Group Waehringerstrasse 17/Parterre A-1090 Wien, Austria #include "fancy_ASCII_picture" #include "funny_statement" #include "standard_disclaimer" From qftramos@usc.es Thu Mar 23 07:07:45 1995 Received: from uscmail.usc.es for qftramos@usc.es by www.ccl.net (8.6.10/930601.1506) id GAA18549; Thu, 23 Mar 1995 06:56:30 -0500 Received: by uscmail.usc.es (5.67b/1.34) id AA01839; Thu, 23 Mar 1995 12:59:29 +0100 Date: Thu, 23 Mar 1995 12:59:29 +0100 From: qftramos@usc.es (Antonio Fernandez Ramos) Message-Id: <199503231159.AA01839@uscmail.usc.es> To: chemistry@ccl.net Subject: least squares fit Dear netters, I am trying to do a non linear least squares fit, using a Marquardt type algoritm, but I need to restrict some parameters to make my function behaves correctly. Is available any program to constrain parameters to be positive? Thanks.  From tevelde@chem.vu.nl Thu Mar 23 07:37:43 1995 Received: from gallium.chem.vu.nl for tevelde@chem.vu.nl by www.ccl.net (8.6.10/930601.1506) id HAA19003; Thu, 23 Mar 1995 07:35:23 -0500 Received: from thallium.chem.vu.nl by gallium.chem.vu.nl (5.64/AUX-3.1.1) id AA02269; Thu, 23 Mar 95 13:35:12 MET Message-Id: <9503231235.AA02269@gallium.chem.vu.nl> Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Thu, 23 Mar 1995 13:35:14 +0100 To: CHEMISTRY@ccl.net From: tevelde@chem.vu.nl (Bert te Velde) Subject: optimization with constraints Dear Antonio (CCL), >I am trying to do a non linear least squares fit, >using a Marquardt type algoritm, but I need to restrict some >parameters to make my function behaves correctly. >Is available any program to constrain parameters to be positive? formulate your problem with transformed parameters, such that p_new = ln(p_old) optimize p_new (without constraints) and obtain your original parameter (p_old) as p_old = exp(p_new) (this is just one of the many transformations that map {-inf,+inf} onto {0,+inf}. Any other transformation that does the same might be used equally well. Sometimes the nature of the problem suggests a particular transformation. The choice of transformation may affect the efficiency of the optimization algorithm) bye ------------------------------------------------------------------------ Bert te Velde SCIENTIFIC COMPUTING & MODELLING phone: +31-20 44 47625 (direct) Vrije Universiteit, Theor. Chemistry 47644 (secr.) De Boelelaan 1083 fax +31-20-44 47643 1081 HV Amsterdam e-mail: tevelde@chem.vu.nl The Netherlands ------------------------------------------------------------------------ From Matthew.Harbowy@tjlus.sprint.com Thu Mar 23 09:52:46 1995 Received: from sprintf.merit.edu for Matthew.Harbowy@tjlus.sprint.com by www.ccl.net (8.6.10/930601.1506) id JAA20730; Thu, 23 Mar 1995 09:38:37 -0500 X400-Received: by mta merit in /PRMD=internet/ADMD=telemail/C=us/; Relayed; Thu, 23 Mar 1995 09:34:39 -0500 X400-Received: by /ADMD=TELEMAIL/C=US/; Relayed; Thu, 23 Mar 1995 09:30:54 -0500 X400-Received: by /PRMD=SMXFL2/ADMD=TELEMAIL/C=US/; Relayed; Thu, 23 Mar 1995 09:36:34 -0500 X400-Received: by /PRMD=LANGATE/ADMD=TELEMAIL/C=GB/; Relayed; Thu, 23 Mar 1995 09:36:00 -0500 Date: Thu, 23 Mar 1995 09:36:00 -0500 X400-Originator: Matthew.Harbowy@tjlus.sprint.com X400-Recipients: non-disclosure:; X400-MTS-Identifier: [/PRMD=LANGATE/ADMD=TELEMAIL/C=GB/;Thu Mar 23 09:36:25 199503] X400-Content-Type: P2-1984 (2) Content-Identifier: BSSE and semiemp From: Matthew.Harbowy@tjlus.sprint.com Message-ID: <"Thu Mar 23 09:36:25 199503*/G=Matthew/S=Harbowy/OU=2812NJLP/O=TMUS.TJL/PRMD=LANGATE/ADMD=TELEMAIL/C=GB/"@MHS> To: chemistry@ccl.net Subject: BSSE and semiempirical methods One of the drawbacks of doing 'corrections', no matter how valid or seemingly necessary, on semiempirical methods such as AM1, is that the parameterization is fitted such that energies and geometries are 'corrected' to compensate for the limitations of the method. This means that, if you would like to do CI, or BSSE removal, or amy other 'very appropriate' 'improvement' technique, you are, in some sense, double-correcting. What that means for transition states and reactive intermediates is anyone's guess: however, doing limited CI to 'correct' the wavefunction to non-closed-shell states seems ok, but doing extensive CI to get 'accurate' energies seems pointless. But, it would seem that BSSE should be present in semiempirical methods. matthew e. harbowy thomas j. lipton, inc. From raman@bioc01.uthscsa.edu Thu Mar 23 10:07:48 1995 Received: from thorin.uthscsa.edu for raman@bioc01.uthscsa.edu by www.ccl.net (8.6.10/930601.1506) id KAA21240; Thu, 23 Mar 1995 10:00:39 -0500 Received: from bioc01.uthscsa.edu by thorin.uthscsa.edu with SMTP; Thu, 23 Mar 1995 9:03:20 -0600 (CST) Received: by bioc01.uthscsa.edu (4.1/SMI-4.1) id AA11848; Thu, 23 Mar 95 09:00:39 CST From: raman@bioc01.uthscsa.edu (C.S.RAMAN) Message-Id: <9503231500.AA11848@bioc01.uthscsa.edu> Subject: Re: CCL:? X-PLOR mailing list/newsgroup ? To: gl@coil.mdy.univie.ac.at (Gerald Loeffler) Date: Thu, 23 Mar 1995 09:00:38 -0600 (CST) Cc: chemistry@ccl.net In-Reply-To: <9503231133.ZM2457@coil.mdy.univie.ac.at> from "Gerald Loeffler" at Mar 23, 95 11:33:52 am X-Mailer: ELM [version 2.4 PL3] Content-Type: text Content-Length: 1685 > Do you know of a mailing list or newsgroup dedicated to Axel Bruengers X-PLOR ? Try xplor-l@msi.com (sending messages to this address will be distributed to the entire list). Cheers -raman -- _/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/ _/ _/ _/ C.S.RAMAN _/ _/ Department of Biochemistry _/ _/ University of Texas Health Science Center _/ _/ 7703 Floyd Curl Drive _/ _/ San Antonio, TX 78284-7760 _/ _/ USA _/ _/ _/ _/ Tel: (210) 567-6623 _/ _/ Fax: (210) 567-6595 _/ _/ E-mail: raman@bioc01.uthscsa.edu _/ _/ _/ _/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/ _/ _/ _/ How can it be that mathematics, a product of human thought _/ _/ independent of experience, is so admirably adapted to the _/ _/ objects of reality? "-Albert Einstein" _/ _/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/ From gavin@vangogh.chem.uab.edu Thu Mar 23 11:07:47 1995 Received: from vangogh.chem.uab.edu for gavin@vangogh.chem.uab.edu by www.ccl.net (8.6.10/930601.1506) id LAA22593; Thu, 23 Mar 1995 11:02:14 -0500 From: Received: by vangogh.chem.uab.edu (AIX 3.2/UCB 5.64/4.03) id AA16378; Thu, 23 Mar 1995 09:56:17 -0600 Date: Thu, 23 Mar 1995 09:56:17 -0600 Message-Id: <9503231556.AA16378@vangogh.chem.uab.edu> To: CHEMISTRY@ccl.net Subject: bond order analysis Dear Netters, What commerical programs can do the ab initio calculations of the bond order ? ANY INFO will be appreciated ! Gavin gavin@vangogh.chem.uab.edu From WSMITH@msnet.mathstat.uoguelph.ca Thu Mar 23 11:07:50 1995 Received: from ccshst06.cs.uoguelph.ca for WSMITH@msnet.mathstat.uoguelph.ca by www.ccl.net (8.6.10/930601.1506) id LAA22583; Thu, 23 Mar 1995 11:01:51 -0500 From: Received: from msnet.nw.uoguelph.ca (msnet.mathstat.uoguelph.ca) by ccshst06.cs.uoguelph.ca with SMTP (1.37.109.14/16.2) id AA088874593; Thu, 23 Mar 1995 11:03:13 -0500 Received: from MSNET/MAILQ by msnet.nw.uoguelph.ca (Mercury 1.11); Thu, 23 Mar 95 11:01:53 GMT-5 Received: from MAILQ by MSNET (Mercury 1.11); Thu, 23 Mar 95 11:01:38 GMT-5 Organization: Dept. of Math & Stats. To: chemistry@ccl.net Date: Thu, 23 Mar 1995 11:01:32 EDT Subject: Re: CCL:least squares fit X-Pmrqc: 1 Priority: normal X-Mailer: Pegasus Mail v3.1 (R1a) Message-Id: <17506D8184C@msnet.nw.uoguelph.ca> Hi, There are many ways (and there is more to it than just this) but the simplest way to get what you want is to just replace your parameter a by another parameter a^2 in your model. BTW, Marquardt-type algorithms are _bound_ to fail to converge in many cases - so beware. The best is to use a _real_ nonlinear optimization code. > Dear netters, >I am trying to do a non linear least squares fit, >using a Marquardt type algoritm, but I need to restrict some >parameters to make my function behaves correctly. >Is available any program to constrain parameters to be positive? > Best Regards, W. R. Smith Professor Dept. of Mathematics and Statistics and School of Engineering University of Guelph FAX: 519-837-0221 Guelph, Ontario Tel: 519-824-4120, ext. 3038 CANADA N1G 2W1 From hans.lischka@itc.univie.ac.at Thu Mar 23 11:14:20 1995 Received: from itc.univie.ac.at for hans.lischka@itc.univie.ac.at by www.ccl.net (8.6.10/930601.1506) id LAA22629; Thu, 23 Mar 1995 11:03:21 -0500 Received: from retsina.itc.univie.ac.at. (retsina.itc.univie.ac.at) by itc.univie.ac.at (5.65c/ITC-Vie(tk)-1.1m) id AA19768; Thu, 23 Mar 1995 17:03:22 +0100 Received: by retsina.itc.univie.ac.at. (5.65c/ITC-Vie(tk)-1.1s) id AA21263; Thu, 23 Mar 1995 17:03:22 +0100 From: hans.lischka@itc.univie.ac.at (Hans Lischka) Message-Id: <199503231603.AA21263@retsina.itc.univie.ac.at.> Subject: MRCI, COLUMBUS query To: chemistry@ccl.net Date: Thu, 23 Mar 1995 17:03:22 +0100 (MEZ) X-Mailer: ELM [version 2.4 PL20] Content-Type: text Content-Length: 1627 Dear Netters, since there have been some enquiries on the COLUMBUS program recently I would like to take this opportunity to inform interested people about an extended version of COLUMBUS which is available on our anonymous ftp server ftp.itc.univie.ac.at in Vienna. Look into the directory pub/quant/COLUMBUS for columbus.new.date.tar.Z (where date stands for a date which is updated from time to time). The procedures for installation are very similar to the original ones (for details see the README.NEW file). Some of the new features are: availability of the analytical MRCI gradient interactive input routines (they make life much easier now!) geometry optimization routines first-order properties polarizabilities via finite field calculation ACPF and MR-AQCC methods and for those who are interested to get experience with parallel computing we have a parallel version of the CI section of the program. It works very well on about 100 processors of the Intel-Paragon (depending on the size of the CI problem which was 4 million in our case), it is being tested on the IBM SP1/SP2 and also works on workstation clusters. Porting to other parallel platforms should not be too difficult as long as the global array tools developed by Robert Harrison (PNL) are available. One has to note that the new features are not as well tested yet as those of the original program! For comments or questions please contact me (address see below) or Ron Shepard at shepard@tcg.anl.gov. Hans Hans Lischka Institute for Theoretical Chemistry and Radiation Chemistry University of Vienna, Austria e-mail: lischka@itc.univie.ac.at From elewars@alchemy.chem.utoronto.ca Thu Mar 23 11:52:50 1995 Received: from alchemy.chem.utoronto.ca for elewars@alchemy.chem.utoronto.ca by www.ccl.net (8.6.10/930601.1506) id LAA23641; Thu, 23 Mar 1995 11:43:47 -0500 Received: (from elewars@localhost) by alchemy.chem.utoronto.ca (8.6.10/8.6.10) id LAA23359 for chemistry@ccl.net; Thu, 23 Mar 1995 11:43:39 -0500 Date: Thu, 23 Mar 1995 11:43:39 -0500 From: "E. Lewars" Message-Id: <199503231643.LAA23359@alchemy.chem.utoronto.ca> To: chemistry@ccl.net Subject: PYRROLE Z-MATRIX Daniel Glossman said he lost the pyrrole Z-matrix suggestions and asked for a re-mailing of them. I lost his email address so I have to send my response to the net again. The internal coord's of atoms in cyclic systems shouldn't be defined sequentially, because of the ratcheting effect (Tim Clark, *A Handbook of Computational Chemistry*). Labelling distances X2N1..., angles A432..., and dihedrals D4321.... will make Z-matrices easier to interpret, till the happy day when they are extinct and everyone uses Cartesians and an interactive model-builder. The input param's are from a MM opt with the Sybil forcefield, using Spartan for visualization. =========== # etc etc [for a Gaussian job] PYRROLE, C2V 0 1 N1 X2 N1 X2N1 C3 X2 C3X2 N1 90. C4 C3 C4C3 X2 A432 N1 0. C5 X2 C3X2 N1 90. C3 180. C6 C5 C4C3 X2 A432 N1 0. H7 C6 H7C6 C5 A765 X2 180. H8 C5 H8C5 C6 A856 X2 180. H9 C3 H8C5 C4 A856 X2 180. H10 C4 H7C6 C3 A765 X2 180. H11 X2 H11X2 C3 90. C4 0. X2N1 1.9970 C3X2 0.7345 C4C3 1.3351 H7C6 1.090 H8C5 1.089 z A432 105.5 /\ A765 125.8 H9 | H8 A856 127.5 \ | / H11X2 3.029 \ | / C3------X2-----C5 / | \ H10--- C4 / | \C6 ---H7 | | x < ----------------------N1 /| / | / H11 y From mjf@iris120.biosym.com Thu Mar 23 12:26:45 1995 Received: from bioc1.biosym.com for mjf@iris120.biosym.com by www.ccl.net (8.6.10/930601.1506) id MAA24313; Thu, 23 Mar 1995 12:18:30 -0500 Received: from iris120.biosym.com by bioc1.biosym.com (5.64/0.0) id AA25612; Thu, 23 Mar 95 09:18:19 -0800 Received: by iris120.biosym.com (940816.SGI.8.6.9/930416.SGI) id JAA14325; Thu, 23 Mar 1995 09:17:24 -0800 Date: Thu, 23 Mar 1995 09:17:24 -0800 From: mjf@biosym.com (Mark J Forster ) Message-Id: <199503231717.JAA14325@iris120.biosym.com> To: chuang@bc.ntu.edu.tw Subject: Correlation time estimation Cc: chemistry@ccl.net Dear Chyh-Chong The most accurate method of correlation time estimation would be measurement of (say) 13C or 15N relaxation for a nucleus with a directly attached proton and with out extensive molecular motion, so a backbone 15N in a protein would be suitable. An H-H noe build up rate between say a Gly HA1-HA2 might also be suitable. In DNA the Cyt H5-H6 vector is of fixed length and is often used for this purpose. If you want a very rough method, that my experience suggests is still within a factor of two, the try MW in KDa = tauc_c in ns of true tau_c values depend upon the temperature, solvent etc and can in practice differ for different atom-atom vectors in the molecule if it exhibits differential mobility. There are also formulas based upon hydrodynamic models in the literature that can predict tau_c for both isotropic and anisotropic reorientation. Hope this helps. From momot@iris5.chem.Arizona.EDU Thu Mar 23 14:22:49 1995 Received: from Arizona.EDU for momot@iris5.chem.Arizona.EDU by www.ccl.net (8.6.10/930601.1506) id OAA27126; Thu, 23 Mar 1995 14:09:29 -0500 Received: from iris5.chem.Arizona.EDU by Arizona.EDU (PMDF V4.3-10 #2381) id <01HOH3TNZBZK91WFEW@Arizona.EDU>; Thu, 23 Mar 1995 12:09:22 -0700 (MST) Received: by iris5.chem.Arizona.EDU (920330.SGI/911001.SGI) for @maggie.telcom.Arizona.EDU:chemistry@ccl.net id AA23168; Thu, 23 Mar 95 12:20:14 -0700 Date: Thu, 23 Mar 1995 12:19:06 -0700 (MST) From: Konstantin Momot Subject: Re: correlation time estimation In-reply-to: <01HOGZ4WTO6A9852DT@Arizona.EDU> Sender: Konstantin Momot To: chemistry@ccl.net, chuang@iris.bc.ntu.edu.tw Reply-to: Konstantin Momot Message-id: MIME-version: 1.0 Content-type: TEXT/PLAIN; CHARSET=US-ASCII Content-transfer-encoding: 7BIT On Wed, 22 Mar 1995 chuang@iris.bc.ntu.edu.tw wrote: > Dear netters: > > When doing NOE simulation from structure, it is needed to give the correlation time value. Yah, to my know, it is dependent on the molecular weight, the shape > (structure) of the molecule. How to roughtly estimate the correlation time when doing this kind of calculation ??? > > Any suggestion will be great appreciated. > > Any suggestion can mail to me directly. Thanks. > > ccchuang > In general, this depends on the particular model of molecular motion you assume. The most consistent discussion of the question is given in Abragam's book, Ch.VIII, Part E (especially pp. 298-300 of the last English edition). If you only consider rotation in a liquid and assume the molecule to be a rigid sphere (very rough estimate, but sufficient for most practical problems !), then you have the correlation time 4pi*R^3*mu/3kT where mu is solvent's viscosity and R is the molecule's radius. Cheers, Konstantin From separk@cs.usc.edu Thu Mar 23 20:52:54 1995 Received: from pollux.usc.edu for separk@cs.usc.edu by www.ccl.net (8.6.10/930601.1506) id UAA03881; Thu, 23 Mar 1995 20:49:15 -0500 Received: (separk@localhost) by pollux.usc.edu (8.6.10/8.6.4) id RAA07045 for CHEMISTRY@ccl.net; Thu, 23 Mar 1995 17:49:12 -0800 Date: Thu, 23 Mar 95 17:49:09 PST From: SEONGBIN PARK To: CHEMISTRY@ccl.net Subject: elementary question (rms deviation) Message-ID: Hello, I am sure this is a very elementary question, but simply I do not know the answer and hope someone else knows the answer. My question is : Why is "standard deviation" called "rms deviation" ? I think they are exactly same. However, only "rms deviation" seems to be used in the molecular structure comparison literature. Thank you very much for your time ! Regards, Seongbin Park (separk@pollux.usc.edu) From chiremv!andromeda!martine@uunet.uu.net Thu Mar 23 16:20:56 1995 Received: from relay3.UU.NET for chiremv!andromeda!martine@uunet.uu.net by www.ccl.net (8.6.10/930601.1506) id QAA29772; Thu, 23 Mar 1995 16:20:54 -0500 Received: from uucp4.UU.NET by relay3.UU.NET with SMTP id QQyiiv11931; Thu, 23 Mar 1995 16:20:45 -0500 Received: from chiremv.UUCP by uucp4.UU.NET with UUCP/RMAIL ; Thu, 23 Mar 1995 16:20:45 -0500 Received: from andromeda by chiron.com (4.1/SMI-4.1) id AA03981; Thu, 23 Mar 95 13:09:10 PST Received: by andromeda (931110.SGI/911001.SGI) for chiremv!uunet!ccl.net!chemistry-request id AA07252; Thu, 23 Mar 95 12:48:47 -0800 Date: Thu, 23 Mar 95 12:48:47 -0800 From: chiremv!andromeda!martine@uunet.uu.net (Eric Martin) Message-Id: <9503232048.AA07252@andromeda> To: uunet!ccl.net!chemistry@uunet.uu.net Subject: Re: CCL:optimization with constraints Dear Antonio & Bert (CCL), >>I am trying to do a non linear least squares fit, >>using a Marquardt type algoritm, but I need to restrict some >>parameters to make my function behaves correctly. >>Is available any program to constrain parameters to be positive? > > formulate your problem with transformed parameters, such that > p_new = ln(p_old) > optimize p_new (without constraints) and obtain your original parameter (p_old) as > p_old = exp(p_new) > > (this is just one of the many transformations that map {-inf,+inf} > onto {0,+inf}. Any other transformation that does the same might be > used equally well. Sometimes the nature of the problem suggests a > particular transformation. The choice of transformation may affect the > efficiency of the optimization algorithm) My understanding is that fitting a transformed function might not only affect the efficiency, it can also affect the answer. The least squares best fit of the transformed problem will have different parameter values (even after the reverse transorm) than the best fit of the untransformed (albeit constrained) problem, since the sum of squared residuals of a different function is being minimized. This can sometimes be rectified by applying weighted least squares. See: "When, Why and How to Use Weighted Least Squares", Robert de Levie, J. Chem. Edu., 63, (1986) p. 10. Best, -Eric Eric Martin martine@chiron.com 4560 Horton St., Emeryville, CA 94608 (510)601-3306, FAX (510)601-3360