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TRAFO User Manual

1. 2 3 1 Edit Undo This command restores the file in the current window to the way it was before your most recent edit or cursor movement Undo inserts any characters you deleted deletes any characters you inserted replaces any characters you overwrote and moves your cursor back to a prior position This command is available only if an Edit window is currently active and there is something to undo 2 3 2 Edit Cut This command removes the selected text from your document and places the text in the Clipboard You can then choose Edit Paste to paste the cut text into any other document or somewhere else in the same document You can paste the cut text as many times as you want until you choose Edit Cut or Edit Copy This command is available only if an Edit window is currently active and text has been marked for selection 2 3 3 Edit Copy This command leaves the selected text intact and places an exact copy of it in the Clipboard To paste the copied text into any other document choose Edit Paste To copy text from a Help window display the text you want to copy then select Edit Copy The entire contents of the window is copied to the Clipboard This command is only available if an Edit window is currently active and text has been marked for selection 2 3 4 Edit Paste This command inserts the contents of the Clipboard into the current window at the cursor position This command is available only if an Edit window
2. Angew Chem 2004 116 2070 2083 DOI 10 1002 ange 200380015 2 5 2 Diagonalise d11 gt d22 gt d33 Diagonalise a general chemical shift tensor For an application see section 3 1 2 2 5 3 Diagonalise s11 lt s22 lt s33 Diagonalise a general magnetic shielding tensor For an application see section 3 1 1 May 14 2015 3 Applications 3 1 Dealing with NMR Tensors One of the more frequent questions that I receive with respect to Euler angles and NMR tensors is how to figure out the Euler angles relating two NMR tensors most often an electric field gradient tensor and a magnetic shielding tensor obtained from ab initio calculations Because I put some notes on Euler angles and tensor conventions on my home page people seem to think that I am an expert in this matter and versed in math pretty well Alas the opposite is true my math is pathetic and the reason I wrote those notes is to remind myself of the essentials and to make it easier to communicate with others This said the following procedures will lack mathematical elegance but they have worked for me so far Before starting I should advise you that you have to be very careful with such transformations Fre quently the principal axes systems produced by Gaussian or other programs after diagonalization are either right or left handed systems If you are using a left handed PAS Trafo will notify you when you calculate Euler angles Usually you can turn a left ha
3. If you are curious what kind of result we produced you can load ADP_03_unit cry into Trafo select Action Orthogonalize Coordinates to convert fractional coordinates into Cartesian coordinates and save the result as ADP_03_unit_ca rt cry Read this file delete the comment lines replace the first line the unit cell data by the number of atoms in the file 56 insert a second line with a title e g ADP and save the result as ADP_O3_unit_cart xyz note the double quotes they prevent Trafo from adding the extension CRY You can read this XYZ file into Jmol p 2 and the result should look like Fig 2 2 In the next section Generate Block p 16 we shall see how to use this as base for generating packing diagrams References 1 Tenzer L Frazer B L Pepinsky R Acta Crystallogr 1958 11 505 DOI 10 1107 50365110X5800 1389 2 Khan A A Baur W H Acta Crystallogr Sect B 1973 29 2721 DOI 10 1107 5056774087300 7442 14 May 14 2015 Chapter 2 Reference 7 502 7 502 7 546 90 00 90 00 90 00 P 0 000000 0 000000 0 000000 N 0 000000 0 000000 0 500000 0 0 085000 0 146000 0 115000 H 0 014000 0 111000 0 573000 H 0 227000 0 146000 0 122000 U 0 085000 0 146000 0 115000 H 0 014000 0 111000 0 573000 H 0 227000 0 146000 0 122000 D 0 146000 0 085000 0 115000 H 0 111000 0 014000 0 573000 H 0 146000 0 227000 0 122000 0 0 146000 0 085000 0 115000 H 0 111000 0 014000 0 573000 H
4. 0 0 9 Cl 0 0 1 5 0 0 Cl 0 0 10 00 1 5 Figure 2 8 File MoMeDBP_01_crys_cube xyz showing the orientation of the monoclinic crystal axes a b c in the crystal holder frame x y z 18 May 14 2015 Chapter 2 Reference However now there are many atoms close to each other therefore the programs Mercury p 2 or Jmol p 2 will show clusters that are not very helpful one example is shown in Fig 2 9 that has been produced by reading file MoMeDBP_01_crys_cube xyz into Mercury p 2 The reason is that the XYZ file format contains positions only and no information on bonds between the atoms We can cure this if we export the XYZ file from Mercury p 2 into PDB format This file format also contains information on connectivities and we will simply delete the superfluous bonds in this file The resulting file is shown in Fig 2 10 where we have also numbered the H and Cl atoms so that it is easier to recognize the a b c and x y z axes in Fig 2 11 a representation made using Mercury p 2 Jmol p 2 will give the equivalent result Note the increased complexity of file MoMeDBP_01_crys_cube pdb Fig 2 10 compared to MoMeDBP_01_crys_cube cry Fig 2 7 Now Mer cury p 2 or Jmol p 2 will display the relative orientation of the two frames crystal axes and crystal holder in a much clearer way Fig 2 11 Figure 2 9 File MoMeDBP_01_crys_cube xyz displayed by Mercury as cluster HEADER CSD ENTR
5. Tracker Software Products Ltd http www tracker software com more power ful and reliable than the Acrobat Reader The opening of a PDF file at a specific position is achieved via command line parameters Another viewer used by people working with IAT X is the Sumatra PDF Viewer http blog kowalczyk info software sumatrapdf also supported by Trafo Other PDF viewers such as FoxIt or NitroPDF do not support DDE or command line pa rameters as far as I know e bug fix a few superfluous new lines produced in the output files got eaten now New in Version 2 5 3 21 01 2009 e converted project to Microsoft Visual C 2008 Express Edition e added diagonalization functions for single crystal NMR e some minor bug fixes New in Version 2 4 7 30 04 2004 e some minor bug fixes New in Version 2 4 3 e this is the first release with a 32 bit version some internal changes e comment lines are now copied to output file New in Version 2 3 2 e added calculation of dipolar second moments e better error reporting New in Version 2 2 10 e Added calculation of normals a May 14 2015 Chapter 1 Getting Started 1 4 Known Problems None right now 1 5 Acknowledgements This section covers the following topics Credits Credits for important contributions LA Trademarks Trademark acknowledgements a Copyright The obligatory copyright statement Disclaimer The obligatory disclaimer message 1 5 1 Credits e Micr
6. in a way suitable for single crystal NMR rotation patterns The orientation of the rotation progress in the Cartesian frame of reference is specified by Euler angles p 24 where a and B could also be considered as polar angles describing the orientation of the rotation axis the z axis while y marks the rotation progress where the direction of the external magnetic field is parallel to the y axis An additional constant phase angle q for y can be specified this phase angle is not considered in the output The actual value of y is calculated according to y y start i x y incr The sense of rotation specifies whether the field moves as in a single crystal NMR experiment i e the physical counter clockwise rotation of the crystal actually corresponds to a clockwise rotation of the magnetic field By in the interaction frame The dipolar second moment for a heteronuclear spin system is calculated according to 29 May 14 2015 Chapter 2 Reference 4 Ma hetero S S 1 Y 7R 1 3 cos 9 and for a homonuclear spin system M2 homo 31 1 1 Y 7R 1 3 cos 9 Powder Samples In the case of a powder sample no further data are required The dipolar second moment for a heteronuclear spin system is calculated according to Meiro I 1 Dap and for a homonuclear spin system M gt homo ZI 1 Y 27rR Data Source Most nuclear data have originated from Mason s extremely use
7. something else into the clipboard Read the result file 06_csa_in_efg cry delete the empty line and select Action Rotations Calculate Euler Angles This gives the Euler angles for going from the EFG tensor to the shielding tensor 209 8756245150 176 2039520054 90 9096041873 Figure 3 11 Euler angles that orient the shielding tensor in the EFG tensor You are done 34 May 14 2015 Index actions 11 Alchemy 1 bugs 4 Cartesian 2 Cartesian coordinates 21 clear 10 comment line 3 comments 3 coordinates 3 copy 9 copyright 5 cross product 24 crystal system 2 cut 9 data 3 diagonalization 32 dipolar 30 direction cosines 21 disclaimer 5 edit 9 clear 10 copy 9 cut 9 paste 9 select all 10 undo 9 Euler angles 24 26 28 29 exit 8 file 8 new 8 open 8 save 8 file format 1 fractional coordinates 21 handedness 25 introduction 1 left handed 25 menu items 7 new file 8 NMR 30 32 normal vector 23 open file 8 orthogonalization 18 21 23 overview 1 paste 9 polar angles 29 positions 11 17 problems 4 reference 7 revision 4 right handed 25 Rollett 18 21 23 rotations 25 26 28 29 save file 8 second moment 30 select all 10 symmetry operations 11 17 tensors 33 trademarks 5 transformations 28 29 undo 9 vector algebra 23 24 May 14 2015
8. three lines in the box into Trafo and save the file as 01_shielding cry a UE May 14 2015 Chapter 3 Applications 541 6395 2 7673 1 7010 1 2823 541 3822 1 2516 0 5447 0 9831 645 2708 Figure 3 1 Save file as 01 shielding cry Select from the Trafo menu NMR Diagonalize s11 lt s22 lt s33 and specify as output file 02_csa in mf cry This file will now contain the directions of the shielding tensor principal axes in the molecular frame mf Open this file in Trafo 1 0 1 0 1 0 90 0 90 0 90 0 OR 0 0 0 0 0 0 Ti1 0 685342 0 728193 0 006350 T22 0 728140 0 685370 0 008941 T33 0 010862 0 001504 0 999940 Figure 3 2 Content of file 02_csa_in_mf cry Here I have deleted the comment lines that provide further information Compare the result with the eigenvectors reported by Gaussian above the third vector is inverted because the Gaussian setup is a left handed setup If you want you can calculate the Euler angles for the orientation of the shielding tensor in the molecular frame select Action Rotations Calculate Euler angles alpha 172 1167014131 beta 179 3717030882 gamma 305 3860273406 Figure 3 3 Euler angles calculated for the direction cosines in file 02_csa_in_mf cry 3 1 2 The Electric Field Gradient Tensor The EFG tensor may look like this in a Gaussian file Center Electric Field Gradient XX XY ZZ 2 Atom 5494 450187 5494 472
9. which is requested from the user The output is almost always directed to an output file the name of which is also asked for Note that currently the edit window has a limited capacity of about 64 KB which is sufficient for most operations but can be exceeded easily after application of Action Generate Block p 16 there s no such low limit on file oriented operations The format of the input data should conform to the format expected by Alchemy for crystal data files although Trafo is a little more flexible in this respect I have chosen this file format because in the 90 ties when I started the development of Trafo we used the molecular modeling programs Alchemy II and Alchemy 3 to visualize molecules and the results of single crystal NMR data Another advantage of this file format is its simplicity An example is shown in Fig 1 1 Several of the examples shown later in this document use this example as starting point hence you could copy and paste the data in the figure into Trafo and save them for perusal later Crystal data files to be recognized by Alchemy must have the extension CRY All operations are line oriented i e each line of the ASCII type data forms one data record Basically there are three different types of data records each of them with some restrictions as to their format The three data types are e Crystal axis crystal system data a May 14 2015 Chapter 1 Getting Started 7 502 7 502
10. 0 146000 0 227000 0 122000 P 0 500000 0 000000 0 250000 N 0 500000 0 000000 0 250000 D 0 415000 0 146000 0 135000 H 0 486000 0 111000 0 323000 H 0 273000 0 146000 0 128000 U 0 585000 0 146000 0 135000 H 0 514000 0 111000 0 323000 H 0 727000 0 146000 0 128000 D 0 646000 0 085000 0 365000 H 0 611000 0 014000 0 823000 H 0 646000 0 227000 0 372000 0 0 354000 0 085000 0 365000 H 0 389000 0 014000 0 823000 H 0 354000 0 227000 0 372000 P 0 500000 0 500000 0 500000 N 0 500000 0 500000 1 000000 U 0 585000 0 646000 0 615000 H 0 514000 0 611000 1 073000 H 0 727000 0 646000 0 622000 U 0 415000 0 354000 0 615000 H 0 486000 0 389000 1 073000 H 0 273000 0 354000 0 622000 0 0 354000 0 585000 0 385000 H 0 389000 0 514000 0 073000 H 0 354000 0 727000 0 378000 U 0 646000 0 415000 0 385000 H 0 611000 0 486000 0 073000 H 0 646000 0 273000 0 378000 P 1 000000 0 500000 0 750000 N 1 000000 0 500000 0 250000 D 0 915000 0 646000 0 635000 H 0 986000 0 611000 0 177000 H 0 773000 0 646000 0 628000 0 1 085000 0 354000 0 635000 H 1 014000 0 389000 0 177000 H 1 227000 0 354000 0 628000 U 1 146000 0 585000 0 865000 H 1 111000 0 514000 1 323000 H 1 146000 0 727000 0 872000 0 0 854000 0 415000 0 865000 H 0 889000 0 486000 1 323000 H 0 854000 0 273000 0 872000 Figure 2 3 Alchemy crystal data file for ammonium dihydrogen phosphate ADP NH H PO gen erated from the data file shown in Fig 2 1 corresponding to the unit cell content Save th
11. 059 5494 687879 Center Electric Field Gradient XY XZ YZ 2 Atom 0 008592 0 013931 0 009529 Figure 3 4 EFG tensor from Gaussian output Take this to construct the EFG tensor in the standard orientation cf Sect 3 1 1 32 May 14 2015 Chapter 3 Applications 5494 450187 0 008592 0 013931 0 008592 5494 472059 0 009529 0 013931 0 009529 5494 687879 Figure 3 5 EFG tensor in standard frame save as 03 efg cry Select NMR Diagonalize s11 lt s22 lt s33 and specify 04_efg_in mf cry as output open the result 1 0 1 0 1 0 90 0 90 0 90 0 OR 0 0 0 0 0 0 T11 0 059759 0 046104 0 997148 T22 0 303124 0 950924 0 062133 T33 0 951076 0 305972 0 042852 Starting Matrix X Y Z X 5494 4502 0 0086 0 0139 Y 0 0086 5494 4722 0 0095 Z 0 0139 0 0095 5494 6880 Symmetrized tensor in reference frame 5494 4502 0 0086 0 0139 0 0086 5494 4722 0 0095 0 0139 0 0095 5494 6880 Tensor principal components and direction cosines in reference frame T ii x y z 5494 689453 0 059759 0 046104 0 997148 5494 474121 0 303124 0 950924 0 062133 5494 446777 0 951076 0 305972 0 042852 Figure 3 6 Content of file 04_efg in mf cry This tensor is not in the convention that I use in my simulation programs for EFG tensors traceless and Vx
12. 2187 C3 2221 1267 21725 C3 2768 4399 4197 Gey spells Sa ilga 1734 PIE 0484 0614 SIS 32 1916 l Tetraethyl diphosphine disulfide TEPS C8 H20 P2 S2 S N Dutta M M Woolfson Acta Crystallogr 14 178 1961 Output fractional coordinates of TEPS converted to Cartesian coordinates 1 000 1 000 1 000 90 000 90 000 90 000 c3 1 045919 1 667075 1 269696 c3 1 947180 0 354132 1 213294 c3 2 426743 2 115630 1 680507 c3 3 432333 0 077197 1 102878 P3 1 049426 0 072799 0 345577 s3 1 606139 1 465968 1 393590 C3 1 045919 1 667075 1 269696 C3 1 947180 0 354132 1 213294 C3 2 426743 2 115630 1 680507 c3 3 432333 0 077197 1 102878 P3 1 049426 0 072799 0 345577 s3 1 606139 1 465968 1 393590 21 May 14 2015 Chapter 2 Reference References 1 Rollett J S Computing Methods in Crystallography Pergamon Press Oxford 1975 2 4 7 Order of Axes In the dialog box Order of Axes one defines Jorderotaxes A new old axes X ca Cb Ce Y Ca cb L Z Ca Cb ec With the dialog box Order of Axes one assigns which of the crystal a b c axes should be aligned to the Cartesian X Y Z axes The default setting assigns the old a axis to the new X axis b to Y and c to Z This information is required for e Orthogonalization of axes p 17 e Orthogonalization of coordinates p 20 2 4 8 Action Orthogonalize Calculate Normal G
13. 27000 0 146000 0 122000 0 0 146000 0 085000 0 115000 H 0 111000 0 014000 0 573000 H 0 146000 0 227000 0 122000 D 0 146000 0 085000 0 115000 H 0 111000 0 014000 0 573000 H 0 146000 0 227000 0 122000 P 0 500000 0 000000 0 250000 N 0 500000 0 000000 0 250000 0 0 415000 0 146000 0 135000 H 0 486000 0 111000 0 323000 H 0 273000 0 146000 0 128000 0 0 585000 0 146000 0 135000 H 0 514000 0 111000 0 323000 H 0 727000 0 146000 0 128000 0 0 646000 0 085000 0 365000 H 0 611000 0 014000 0 823000 H 0 646000 0 227000 0 372000 0 0 354000 0 085000 0 365000 H 0 389000 0 014000 0 823000 H 0 354000 0 227000 0 372000 Figure 2 1 Alchemy crystal data file for ammonium dihydrogen phosphate ADP NH H POy gen erated from the data in Fig 1 1 corresponding to one half of the unit cell Save the data as ADP_02_unithalf cry if you want to repeat some of the example actions shown later 13 May 14 2015 Chapter 2 Reference Figure 2 2 File ADP_03_unit_cart xyz displayed using Jmol Generate Positions and specify ADP 03_unit cry as output O As number of positions we specify 2 and the first operation as identity operation shall simply copy the first eight positions factor shift source X 10 0 0 x Y 10 00 y Z 10 0 0 z The last operation is the centering operation factor shift source X 10 05 x Y 10 05 y Z 10 05 z The resulting file ADP_O3_unit cry should look like Fig 2 3
14. 3 2 A b 7 7917 5 A c 17 9522 12 B 122 135 4 space group C2 c with respect to this cube frame was deter mined Fig 2 6 This procedure also yields the direction cosines of the crystal axes with respect to the crystal holder Fig 2 7 In the data file shown in Fig 2 7 I have included the cube frame x y z axes but 1 5 A away from the origin OR We can easily convert this to a XYZ file compare Fig 2 4 but Mercury p 2 or Jmol p 2 will not be able to deal with the result properly The reason is that we are specifying atoms OR a x z that are unknown to those programs One way around this is to replace them by sensible atoms e g the XYZ file displayed in Fig 2 8 N Z X Figure 2 6 Orientation of the monoclinic crystal axes a b c in the crystal holder frame x y z 1 00 1 00 1 00 90 00 90 00 90 00 or 0 00000 0 00000 0 00000 a 0 91826 0 31965 0 23380 b 0 18295 0 87884 0 43607 c 0 19058 0 46992 0 86036 2 lda 020 00 y 0 0 1 5 0 0 z 0 0 0 0 1 5 0C 5Mo Me DBP crystal axis system in cube axis system Figure 2 7 File MoMeDBP_01_crys_cube cry showing the orientation of the monoclinic crystal axes a b c in the crystal holder frame x y z 7 OC 5Mo Me DBP crystal axis system in cube axis system C 0 00000 0 00000 0 00000 H 0 91826 0 31965 0 23380 H 0 18295 0 87884 0 43607 H 0 19058 0 46992 0 86036 Cl 1 5 0
15. 7 546 90 00 90 00 90 00 P 0 000 0 000 0 000 N 0 000 0 000 0 500 O 0 085 0 146 0 115 H 0 014 0 111 0 573 H 0 227 0 146 0 122 crystal structure of NH4H2P04 space group I 42d No 122 neutron diffraction data Tenzer L Frazer B L Pepinsky R Acta Cryst 1958 11 505 Figure 1 1 Example of an Alchemy crystal data file here for ammonium dihydrogen phosphate ADP NH H PO 1 You can save the data as ADP_01_fract_coord cry if you want to repeat some of the example actions shown later Comment lines e Regular data lines One disadvantage of the Alchemy crystal data file format is that most other programs do not accept this format Most of them will be able to deal with Alchemy MOL files but not the CRY files However it is not very difficult to convert the CRY files into a format that is recognized by a wider variety of programs the so called XYZ format You can always use Trafo to convert your fractional coordinates into Cartesian coordinates In the resulting file simply replace the first line with the Cartesian crystal axes data by the number of atoms in the file add a second line with a title and keep the rest with the Cartesian coordinates an example of a XYZ file is shown in Fig 2 4 Such XYZ files can be read into Jmol or Mercury for example On this type of data one can perform transformations called Actions in Trafo Links to some of the software
16. Angles xj The Euler angles were determined to be Alpha 247 92451477 deg Beta 30 51503372 deg Gamma 120 05549622 deg e Rl e May 14 2015 Chapter 2 Reference 2 4 11 Invert Axis In the dialog box Axis Inversion one defines xi Your data define a left handed axis system In order to get a right handed sense select an axis to invert or cancel Axis to invert C Xx MY ez Inversion of a coordinate axis changes the handedness of the system Select one axis to invert Usually this dialog is called if the input data define a left handed coordinate system All calculations in Trafo are designed for right handed coordinate systems 2 4 12 Euler Angles The triplet of Euler angles a B y is useful to describe rotations or relative orientations of orthogonal coordinate systems Unfortunately their definition is not unique and in the literature there are as many different conventions as authors The convention employed here is one of the more common ones All rotations are in a counter clockwise fashion right handed mathematically positive sense The Euler angles a 6 y relate two orthogonal coordinate systems having a common origin The transition from one coordinate system to the other is achieved by a series of two dimensional rota tions The rotations are performed about coordinate system axes generated by the previous rotation step the step by step procedure is illus
17. TRAFO Coordinate Transformations USER MANUAL Klaus Eichele May 14 2015 11 May 14 2015 Contents 1 Getting Started 1 UL TORA e or a a EE o E a Be ee 1 L2 LIONE AAA 1 T al Asis eir AR e ad R Pea T RARE wee 2 122 Comment LIDS oi oe ek cee a dc a A a oe 3 ize Eesular Dats Lis 3 6 ooo 2 Ae cee ME Ree ke A ee ES 3 To a ANA 4 L4 Knows Problems 2266406452 4454 g sinipi bd RL RE a e RS ES 5 aa II 5 L51 Credib on si esa dew ese ppa ER A Ep 5 1 5 2 Trademark Acknowledgement 2 R ee ee 5 Ioa Copyright Message socio Behe e iaa ead ee eae eos 5 154 Disclamer of Warianty oe bg a a Pee a eR ae ed 6 2 Reference 7 241 Helpon Menta lems cocoa mioni a a She Cs He We di 7 A RN 8 tel A AA eh ed eee OE ee ek 8 2ra Wile Fe E RR 8 mee Ml pave FEAS 6 ea de A Aaa 8 ee MUS ESE as at a a e Re ee ee on SS 8 20 EIUMeRd o acn cca een ee PEA eee Ee eRe eee eed eee He as 8 oe BCS WG cone eas Ae Sy GA aces as aia E 9 Soe EOS 1 keyed ha ee OPE a heey se eee ek ee eas 9 2 BOALO a a is ee a A a S 9 E A ie Sane ee ke eres ds eee 9 e Bl IESO oe OVA ee eee AA BE ERA eee eae eek 9 C BOO SLAM o wos ok oe A REG eh row PAS A 9 2A HOMO MON a ee ae eee See whee ewe as 10 241 suction Generate Posies ee sot a a A ee en e 10 242 Symmetry Operations ed ee web ee EE EAS Ewe EER R 10 24 3 Action Generale Block cc 62645 bi go ceras Ew EE KS 16 244 Action Onhogonalize Axes lt cerea cato ema ee R ee as 17 ea Kie sock a
18. Y Factor 1 00000000 Shift 0 00000000 CIO IE 7 Factor 1 00000000 Shift 0 00000000 Cx Cy Oz ro Number of sites to be generated In order to generate a unit cell enter the number of molecules or sites or formula units as the number of positions For the first site apply the identity op eration which is the default symmetry operation For the following sites apply the necessary symmetry operations e Symmetry operation required to generate the sites The symmetry operations can be written as combinations of multiplications Factor and summations Shift applied to the input coor dinates where it is possible that the source of coordinates are interchanged The general equation to calculate the new coordinates is New k Factor x Old m Shift with k m x y Z It is easier to apply symmetry operations on fractional coordinates rather than Cartesian coordinates Example Space Group 42d No 122 For this example we shall use the fractional coordinates of ammonium dihydrogen phosphate ADP NH H PO presented in Fig 1 1 as a starting point This example is sufficiently complex to illustrate quite a few points e the application of simple symmetry operations e how to deal with special positions e why it could be necessary to change the source The space group symbol I indicates that the crystal structure of ADP is a body centered lattice there fore there is a so called centering operat
19. Y 0C 5Mo Me DBP crystal axis system in cube axis system CRYST1 1 0000 1 0000 1 0000 90 00 90 00 90 00 SCALE1 1 000000 0 000000 0 000000 0 000000 SCALE2 0 000000 1 000000 0 000000 0 000000 SCALE3 0 000000 0 000000 1 000000 0 000000 HETATM 1 C UNK 1 0 000 0 000 0 000 1 00 0 00 C HETATM 2 Hi UNK 1 0 918 0 320 0 234 1 00 0 00 H HETATM 3 H2 UNK 1 0 183 0 879 0 436 1 00 0 00 H HETATM 4 H3 UNK 1 0 191 0 470 0 860 1 00 0 00 H HETATM 5 Cli UNK 1 1 500 0 000 0 000 1 00 0 00 cl HETATM 6 C12 UNK 1 0 000 1 500 0 000 1 00 0 00 cl HETATM 7 C13 UNK 1 0 000 0 000 1 500 1 00 0 00 cl CONECT 1 2 3 4 5 CONECT 1 6 7 MASTER 0 0 0 0 0 0 0 3 7 0 10 0 END Figure 2 10 File MoMeDBP_01_crys_cube pdb showing the orientation of the monoclinic crystal axes a b c in the crystal holder frame x y z Using Jmol p 2 or Mercury p 2 with file MoMeDBP_01_crys_cube pdb you can verify that the oblique angle H1 C H3 corresponds to 6 122 135 4 whereas the other two angles are 90 The purpose of Action Orthogonalize Axes will be to turn B also into a right angle by generat ing a new crystal axis a that is orthogonal to b and c Read file MoMeDBP_01_crys_cube cry into Trafo select Action Orthogonalize Axes keep the Order of Axes settings at their default specify MoMeDBP_02_crys_cube_orth cry as output file and agree to the normalization of the unit vectors Convert MoMeDBP_02_crys_cube_orth cry into XYZ file format as outlined above use M
20. ains unaffected The rotation matrix to describe this operation is given by cosa sina 0 R a sina cosa 0 0 0 1 Second Rotation The second rotation involves the Euler angle 6 The x 2 y 2 z 2 axis system is rotated about the y 2 axis through an angle D counterclockwise to generate the new coordinate system x 3 y 3 z 3 Analogously to the first Euler rotation this mixes the coordinates along x 2 and z 2 while the coordinate along y 2 remains unaffected This operation also generates a line of nodes parallel to the direction of y 2 The rotation matrix to describe this operation is given by cosB 0 sinfB Ry B 0 1 0 sinf 0 cosfB Third Rotation The last rotation involves the Euler angle y The x 3 y 3 z 3 axis system is rotated about the z 3 axis through an angle y counterclockwise to generate the final coordinate system x y Z Analogously to the first Euler rotation this mixes the coordinates along x 3 and y 3 while the coordinate along z 3 remains un affected The rotation matrix to describe this operation is given by cosy siny 0 Rz y siny cosy 0 0 0 1 The combined effect of these three rotations is given by this transformation matrix cos amp cos Bcos y sina sin y sinacosBcosy cosasiny sinfcosy Rz 7 Ry B Rz cosacosfsiny sinacosy sinacos Bsin y cosacosy sinfsiny cos sin D sina sin D cos D Note This type of rotation about sequentia
21. c by means of the fractional coordinates e g symmetry operations In other cases it is more convenient to use axes at right angles and measure distances in Angstrom units The Orthogonalize Coordinates command from the Actions Orthogonalize menu orthogonalizes the fractional coordinates of an oblique axis system to Cartesian coordinates in the order given in Rollett s procedure In fact it will convert any fractional coordinates into Cartesian coordinates It requires to assign the crystal axes a b c to the coordinate axes X Y Z to define the order of axes p 22 Z Rollett s procedure 1 The axis selected as the c axis is aligned with the coordinate sys tem Z axis 2 The axis selected as the b axis is brought into the YZ plane b 3 The axis selected as a points somewhere along X as required by X a its angles with c and b 20 May 14 2015 Chapter 2 Reference This gives a non trivial transformation matrix L asin f sin y 0 0 L asinfcosy bsina 0 a cos D bcos c cos y cos cos B cos y sin a sin B sin y 1 cos 7 The crystal axes data line p 2 is required Origin data lines p 3 are converted as well Example input the fractional coordinates of TEPS in a triclinic system 8 9800 6 4500 6 1500 113 0 85 200 102 500 C3 1193 3168 3217 C3 2221 1267 ZO C3 2768 4399 4197 C3 3915 1312 1734 P3 1197 0484 0614 3 1832 1916 1257 LIT 3168 3
22. cry as output file Copy this file to ADP_04_unit_block_cart xyz and use an editor that has a higher capacity as Trafo the Windows editor Notepad will do to replace the Cartesian crystal data line by the number of atoms 1512 and insert a second line with a title c f Fig 2 4 Save the file and open it in Jmol p 2 Two different views of the result are shown in Fig 2 5 16 May 14 2015 Chapter 2 Reference 1512 ADP 27 unit cells P 0 000000 N 0 000000 0 0 637670 H 0 105028 0 000000 0 000000 0 000000 3 773000 1 095292 0 867790 0 832722 4 323858 Figure 2 4 Beginning of file ADP_04_unit_block_cart xyz showing the structure of an XYZ file num ber of atoms in the first line a second line with a title followed by the Cartesian coordi nates of the atoms Figure 2 5 Two views of file ADP_04_unit_block_cart xyz using Jmol 2 4 4 Action Orthogonalize Axes The Orthogonalize Axes command from the Actions Orthogonalize menu orthogonalizes the di rection cosines of an oblique axis system in the order given by Rollett s procedure This is an action required when dealing with single crystal NMR Typically in this experiment a crystal is mounted in some holder and X ray diffraction is used to determine the orientation of the crystal axes with respect to the holder If the crystal axes form an oblique system i e at least one angle differs from 90 it is useful to transform them into an ort
23. ctor shift source X 10 05 y Y 10 0 0 x Z 10 025 z Open the resulting file ADP_02_unithalf cry in Trafo Because we have applied the symmetry operations on all atoms including P and N at special positions we have generated too many N and P positions There are four formula units of ADP per unit cell hence our first 8 symmetry operations should generate two formula units 2 NH H gt PO No gt HgH 4P Os the protons at oxygen have only an occupation no of one half Only one quarter of the P and N positions are required You can see that P positions 2 4 are the same as the first and 6 8 the same as the fifth hence can be deleted leaving two P positions as required It is a little more complicated for nitrogen positions 2 6 are the same as 1 and 5 while 3 4 7 8 correspond to 1 and 5 but shifted by one unit of the c cell axis therefore they are also superfluous Clean up the file as outlined and save the result again as ADP_02 unithalf cry The result should look like Fig 2 1 Now we shall generate the second half of the unit cell by applying the centering operation With file ADP_02_unithalf cry in the edit window of Trafo or specifying this file as input file select Action 7 502 7 502 7 546 90 00 90 00 90 00 P 0 000000 0 000000 0 000000 N 0 000000 0 000000 0 500000 0 0 085000 0 146000 0 115000 H 0 014000 0 111000 0 573000 H 0 227000 0 146000 0 122000 0 0 085000 0 146000 0 115000 H 0 014000 0 111000 0 573000 H 0 2
24. d in this dialog box the identity operation is used with a factor of 1 and a shift of zero for all coordinates X Y Z this operation is the default with the source being x y and z respectively This simply copies the input coordinates into the output file To generate this position the default looks like factor shift source X 1 0 0 0 x Y 10 00 y Z 1 0 0 0 Z Click OK O The second position is related to the first by a two fold rotation around the 4 axis the 4 axis generates a two fold axis expressed in Eq 2 1 as x y z To generate this position enter factor shift source X 10 0 0 x Y 10 0 0 y Z 10 00 z The third position generated by y x Z interchanges the sources for x and y factor shift source X 1 0 00 y Y 10 00 x Z 10 00 z The fourth position generated by y x Z is produced by factor shift source X 10 00 y Y 10 0 0 x Z 10 0 0 z The fifth position generated by Y x y Y z is produced by factor shift source X 10 05 x Y 10 00 y Z 10 025 z The sixth position generated by Y x y 1 4 z is produced by factor shift source X 10 05 x Y 10 0 0 y Z 10 025 z O The seventh position generated by y x 4 z is produced by 12 May 14 2015 Chapter 2 Reference factor shift source X 10 05 y Y 10 00 x Z 10 0235 z O The eigth position generated by Y y x z is produced by fa
25. e data as ADP_03_unit cry if you want to repeat some of the example actions shown later 15 May 14 2015 Chapter 2 Reference 2 4 3 Action Generate Block The Generate Block command from the Actions Generate menu allows to create a block cube of new coordinates positions around the existing one Specifically this generates from a unit cell a cube consisting of several unit cells Such operations might be required to construct packing dia grams or as a convenient tool to create the input coordinates for the calculation of dipolar second moments The entire content of the unit cell itself should be created using Generate Positions p 10 In the dialog box Generate Block one defines Generate Block x This option generates crystallographic positions automatically by translating the input coordinates by multiples n of the unit cell constants A A2 a parameter n describes how many unit cells will get created L av Va or a given value of n this command generates a cube of corner length 2n 1 or 2n 1 2n 1 2n 1 1 2n 1 1 new positions around the input data For a value of n equal to 1 4 a cube of 27 positions will get created as indicated in the figure to the left In the newly created output file the original data set will be placed first followed by the new data this is important for later use of the results to calculate dipolar second moments Example In the sec
26. e using Euler rotations Example Given a set of direction cosines specifying unit vectors that define the axes of the reference frame Fig 2 13 the rotation forward with Euler angles 10 6 20 y 30 creates direction cosines Fig 2 14 which specify the coordinates of the reference frame as seen from the derived frame Fig 2 15 Using the option Calculate Euler Angles on this new set of data yields the Euler angles for going from the derived frame to the reference frame 150 6 20 y 170 1 0 1 0 1 0 90 0 90 0 90 0 x 1 0 0 0 0 0 y 0 0 1 0 0 0 z 0 0 0 0 1 0 Figure 2 13 Orientation of the reference frame in its own axis system 01 unit cry 1 0 1 0 1 0 90 0 90 0 90 0 x 0 714610 0 613092 0 336824 y 0 633718 0 771281 0 059391 z 0 296198 0 171010 0 939693 Figure 2 14 Orientation of the reference frame rotated forward by Euler angles a 10 20 y 30 02_unit_forward cry May 14 2015 Chapter 2 Reference Figure 2 15 File 02 unit_forward cry Fig 2 14 after conversion to XYZ and PDB format as ex plained in Sect 2 4 4 displayed using Mercury It shows the orientation of the trans formed reference frame H1 H2 H3 as seen from the derived frame C11 C12 C13 2 4 16 Action Rotations Euler Rotations Backwards Rotates Cartesian coordinates from the new coordinate system derived frame into the old one ref erence frame using Euler rotations E
27. ercury p 2 to convert to PDB format clean up the superfluous bonds and read the result back into Mercury p 2 Measure all H C H angles and verify that they are at 90 Measure also the angle between the crystal holder z axis and the crystal c axis 30 55 You can now use file MoMeDBP_02_crys_cube_orth cry 19 May 14 2015 Chapter 2 Reference Figure 2 11 File MoMeDBP_01_crys_cube pdb Fig 2 10 displayed by Mercury and showing the orien tation of the crystal axes in the crystal holder frame cf Fig 2 6 1 000 1 000 1 000 90 000 90 000 90 000 or 0 000000 0 000000 0 000000 a 0 964246 0 074545 0 254308 b 0 183882 0 879224 0 439491 6 0 190832 0 470540 0 861496 Figure 2 12 File MoMeDBP_02_crys_cube_orth cry showing the orientation of the orthogonalized crystal axes a b c in the crystal holder frame x y z in Trafo to calculate the Euler angles p 24 between both axes systems compare the Euler angle with the angle between z and c axes just determined 30 55 References 1 Rollett J S Computing Methods in Crystallography Pergamon Press Oxford 1975 2 Eichele K Wasylishen R E Maitra K Nelson J H Britten J F Inorg Chem 1997 36 3539 DOF 10 1021 ic970260q 2 4 5 Direction Cosines 2 4 6 Action Orthogonalize Coordinates For many purposes it is useful to refer to the position of an atom in the unit cell of a crystal to the natural axes a b
28. ful book on Multinuclear NMR 1 The current version has been updated according to data from the JUPAC Recommendations 2001 7 The following data differ from those in reference 1 all magnetogyric ratios are according to 7 e the nuclear quadrupole moment values are from the Year 2001 Q Values collected by Pekka Pyykk 3 e the magnetogyric ratios of Sn 119 and Sn 117 could be 3 less than the accepted value 2 i e 9 997559e7 rad s T and 9 552955e7 rad s T instead of 10 021e7 rad s T and 9 589e7 rad sT e Nd 145 apparently has a spin of 7 2 instead of 5 2 4 5 Similarly U 235 has a spin of 7 2 instead of 5 2 4 5 U 233 has been added to the tables 4 5 References 1 Joan Mason Multinuclear NMR Plenum Press New York 1987 2 A Laaksonen and R Wasylishen J Am Chem Soc 1995 117 392 400 DOI 10 1021 ja00106a044 3 P Pyykk Mol Phys 2001 99 1617 1629 DOI 10 1080 00268970110069010 4 Quantities Units and Symbols in Physical Chemistry IUPAC 5 CRC Handbook of Chemistry and Physics 6 A Abragam The Principles of Nuclear Magnetism Clarendon Press Oxford 1961 p 125 Dipolar Broadening in Magnetically Diluted Substances 7 R K Harris E D Becker S M Cabral de Menezes R Goodfellow P Granger Pure Appl Chem 2001 73 1795 1818 DOI 10 1351 pac200173111795 Solid State Nucl Magn Reson 2002 22 458 483 DOI 10 1006 snmr 2002 0063
29. hogonal axes set because one can then determine the Euler angles p 24 that relate the crystal axes to the crystal holder frame This action requires to assign the crystal axes a b c to the coordinate axes X Y Z to define the order of axes p 22 Z Rollett s procedure 1 1 The axis selected as the c axis is aligned with the coordinate sys tem Z axis 2 The axis selected as the b axis is brought into the YZ plane To orthogonalize this axis the normal n to the be plane is determined as the cross product p 23 b x c the new b axis results from the cross product c x n 3 The axis selected as a usually points somewhere along the X di rection The orthogonalized a is the cross product of b x c The input data need to contain three regular data lines corresponding to the direction cosines of the oblique axis system All following data lines are ignored Crystal axes data lines p 2 and origin data lines p 3 are neglected 17 May 14 2015 Chapter 2 Reference See also the topic Orthogonalize Coordinates p 20 for more information Example The example results from a P single crystal NMR experiment on 5 methyldibenzophosphole penta carbonylmolybdenum 0 and has been published 2 A large crystal of this compound has been grown and glued onto a hollow three sided alumina cube measuring 4 mm on each side Using X ray diffraction the orientation of the monoclinic crystal axis system a 31 11
30. ion that generates this centered part Altogether there are 16 symmetry related general positions half of them generated by this centering operation This can be expressed as 0 0 0 22 EZ UNE ee MR x yY Zz x a z y KALS 4 y x 4 z 2 1 This generally adopted setting for ADP at least the researchers that determined the structure of ADP called it generally adopted is obtained by a rotation of 90 about the 4 axis relative to the setting in International Tables for X ray Crystallography 1965 for the space group 142d 1 2 If you use the setting found in the International Tables to generate the positions of ADP from the coordinates in Fig 1 1 you will get a wrong structure To generate all 16 general positions of ADP we will take two steps in the first step we will apply the eight operations shown in the second part of Eq 2 1 and save the result to a file In this file we will need to clean up a few positions related to the special positions of P and N before application of the second step the centering operation If you have not done so already generate your starting file as outlined in the caption of Fig 1 1 Read this file into Trafo and select Action Generate Positions As name for the output file enter ADP_02_unithalf cry May 14 2015 Chapter 2 Reference O For now we want to generate eight positions hence we enter 8 in No of Positions For the first position already querie
31. irst two characters of the line have to be a slash and an asterisk 1 2 3 Regular Data Lines Each line with regular coordinate data consists of four data entries all of them are required The general format will be LABEL x xxxx y yyyy Z ZZZZ where LABEL a label of at least 1 and up to 6 characters describing the atom or data type no spaces or other whitespace characters e g tabs within this name are allowed but underscores etc for use with Alchemy the label should specify a valid Alchemy atom type but Trafo does not care x y z the x y z coordinates as floating point numbers in free format A special type of data line will be recognized by Trafo which is a line describing the origin of the coordinate system OR X XXX y yyy Z ZZZ where Trafo doesn t really care about the coordinates right now but this line shall usually be ne glected the position and number of such lines doesn t matter either The purpose of this origin is to serve as starting point for the display of principal axes systems e g of chemical shift tensors whereby the directions of the principal components were indicated as dummy atoms e g d11 d22 d33 that are 1 A away from the origin May 14 2015 Chapter 1 Getting Started 1 3 Revision History This page summarizes the changes made compared to previous versions of Trafo Experienced users may use this information as a quick update on new program features New in Versi
32. is currently active and there is something to paste 2 3 5 Edit Clear This command removes the selected text but does not copy it to the Clipboard This means you cannot paste the text as you could if you had chosen Edit Cut or Edit Copy Although you cannot paste the cleared text you can undo the Clear command with Edit Undo Clear is useful if you want to delete text but you do not want to overwrite text held in the Clipboard This command is available only if an Edit window is currently active and text has been marked for selection 2 3 6 Edit Select All This command selects the entire contents of the active Edit window You can then use Edit Copy or Edit Cut to copy it to the Clipboard or perform any other editing action This command is available only if an Edit window is currently active 9 May 14 2015 Chapter 2 Reference 2 4 Actions Menu The Actions Menu combines all the commands to perform transformations on input data Currently the following actions can be performed Trafo 01_ADP_Khan_fract_coord cry gt Postions 5 Generate Orthogonalize tructure of NH4H2P04 tu enana enan a T BAA fhia an 2 4 1 Action Generate Positions The Generate Positions command from the Actions Generate menu allows to create new coordi nates positions from the existing one by applying symmetry operations One possible application is to create all the molecules within the unit cell from
33. iven two vectors this routine calculates the normal vector to the plane defined by the two vectors The two vectors are defined by the Cartesian coordinates of their common starting point followed by the Cartesian coordinates of the two end points Note that the direction of this normal vector depends on the order in which the two vectors are defined because the normal vector is calculated via the cross product p 23 of the two vectors Example input 1 00 1 00 1 00 90 0 90 0 90 0 OR 1 00 1 00 0 00 Pi 1 00 3 00 0 00 P2 3 00 0 00 0 00 Output normal vector 0 00000000 0 00000000 4 00000000 normalized 0 00000000 0 00000000 1 00000000 This command considers only the first three coordinate triples A Cartesian crystal axis line p 2 must precede the data May 14 2015 Chapter 2 Reference 2 4 9 Cross Product The Vector or Cross Product of two vectors A and B C AxB generates a vector C with direction perpendicular to the plane of A and B such that A B and C forma right handed system The magnitude of C is defined as C A B sin q where q is the angle between Aand B An alternate definition of the cross product consists in specifying the components of C Cy AyB AzBy Cy AzBr AxBz C gt A By AyBy Ci AjB ALB where i j k are all different and cyclic permutations of the indices x y z 2 4 10 Action Rotations Calculate Euler Angles The Calculate Eule
34. k Sha CES eee AR A E ee 20 2 4 6 Action Orthogonalize Coordinates 1 0 00000000 o 20 eA Her Gr AXE oes eae baal ga ee SSD ER Case ee E 22 2 4 8 Action Orthogonalize Calculate Normal ooo o 22 CAP CROSS PIO lt on hae EA eiaa ee oe OE SE A PSA eae 23 2 4 10 Action Rotations Calculate Euler Angles R o 23 Zl TONES iris A ENE Eee a PENSE ER OEP ES 24 Pte E A iia ds ee ER Ba ee BRE Ra OP ee Pa ES 24 E Rotation T voca ew a d ie ee ae ee a we a E ES 25 2414 Determine Euler Amgles oc ceim poe as FO A Re ee ra R NT 27 2 4 15 Action Rotations Euler Rotations Forward lt o e sesse sranie 27 2 4 16 Action Rotations Euler Rotations Backwards o 28 2417 Action Polar Angles Calculate 2 6 caa RRR ce we es 28 20 INICIEN ordre A ae Re ee e A A ee eS 29 20 1 DipolarSecorid Moment aes esris aiioe cestas a a e 29 222 Diagonallse dll gt ARSS cosa ee R R a A 30 iii May 14 2015 Contents Zour Diagonaliso 6 lt 522 8381 acosa eee ee dd 30 3 Applications 31 D1 Dolne with NMR enS ooe se ee Re E e e RRR eS 31 3 1 1 The Absolute Magnetic Shielding Tensor o ooo ooo caws 31 3 1 2 The Electric Field Gradient Tensor e cec e R T R R cea K i E 32 3 13 Transforming the CSA into the EFG frame o coace cr 34 May 14 2015 1 Getting Started This chapter covers the following topics Introduction An introduction to Trafo Overviews An o
35. l for non commercial use and by any govern ment organization Although the copyright holder retains all rights to this document and the software package you are allowed to copy and distribute verbatim copies of them as you received them in any medium provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty keep intact all the notices that refer to this license and to the absence of any warranty and distribute a copy of this license along with it This package may not be distributed as a part of any commercial package You are expressly not allowed to sell or license this package May 14 2015 Chapter 1 Getting Started 1 5 4 Disclaimer of Warranty Because this software package is licensed free of charge there is no warranty to the extent per mitted by applicable law This package is provided as is without warranty of any kind either expressed or implied including but not limited to the implied warranties of merchantability and fitness for a particular purpose the entire risk as to the quality and performance of the contents of this package is with you should this package prove defective you assume the cost of all nec essary servicing repair or correction in no event unless required by applicable law or agreed to in writing will any copyright holder or any other party who may modify and or redistribute this package as permitted in the lice
36. lly newly generated axes produces the same result as rota tions by the same angles about the fixed original axes if the order of angles is reversed Rz a Ry B Rz y cf Mehring s book appendix 1 May 14 2015 Chapter 2 Reference References 1 M Mehring Principles of High Resolution NMR in Solids 2nd ed Springer Verlag Berlin 1983 2 4 14 Determining Euler Angles Given the relative orientations of two coordinate systems how do we go about determining the Euler angles relating them First we need to decide on the coordinate system to take as the reference coordinate system X Y Z and which one as derived coordinate system x y z Because the Euler transformations allow to switch between coordinate systems easily it does not really matter which one is selected The angle f is simply the angle between the z axes of both co ordinate systems while a is the angle between the X axis of the reference coordinate system and the projection of z into the X Y plane Finally is the angle between y and the line of nodes Computationally we can work your way backwards from the components of the transformation matrix in Eq 2 2 e the zz component gives us cos f e use that to get from components zx and zy e use that a to get y from components xz and yz 2 4 15 Action Rotations Euler Rotations Forward Rotates Cartesian coordinates from the old coordinate system reference frame into the new one derived fram
37. mentioned e Jmol http www jmol org e Mercury http www ccdc cam ac uk mercury References 1 Tenzer L Frazer B L Pepinsky R Acta Crystallogr 1958 11 505 DOI 10 1107 50365110X58001389 1 2 1 Crystal Axis System Data The data line with crystal axes data describes the geometry of the unit cell of a crystal and consists of six floating point numbers a b c a b y where a b c length of the respective crystal axis in Angstrom a BP y angle between pairs of crystal axes a is the angle between b and c B between a and c and y between a and b For Cartesian coordinates this line will read 1 0 1 0 1 0 90 0 90 0 90 0 Note This data line is required for many operations and has always to be the first line in this case For actions where this line is not required it is optional but if present it also has to be the first line 2 May 14 2015 Chapter 1 Getting Started 1 2 2 Comment Lines Comment lines can be added to the data in order to assist later in identifying the data set In Alchemy comment lines can only be added to the end of the data set but in Trafo they may appear anywhere except for the first line in cases where crystal axis system data are required Comment lines are written in the same format as comment lines in the C programming language as the following ex ample illustrates this is a C type comment line In order to qualify as a comment line the f
38. nates p 20 Orthogonalize Calculate normal p 22 Rotations Calculate Euler Angles p 23 Rotations Euler Rotations Forwards p 27 Rotations Euler Rotations Backwards p 28 generate symmetry related sites generate a cube of several unit cells transform an oblique axis set into an orthogonal set transform coordinates from an oblique system into a Cartesian system calculate the normal vector calculate Euler Angles from direction cosines transform from the old system into the new sys tem transform from the new system into the old sys tem 7 May 14 2015 Chapter 2 Reference Polar angles Calculate p convert Cartesian coordinates into polar coordi 28 nates NMR Perform some NMR relevant calculations Dipolar Moment p 29 calculates dipolar second moments for single crystal or powder samples Diagonalise 411 gt d22 gt diagonalise a general chemical shift tensor d33 p 30 Diagonalise s11 lt s22 lt diagonalise a general magnetic shielding tensor s33 p 30 2 2 File Menu The File pop up menu consists of the following items Trafo 01_ADP_Khan_fract_coord cry File Edit Action NMR Help 5494 90 00 96 66 0000 6 6666 0000 6 5666 1466 0 1151 0890 6 5636 1500 6 1256 2 2 1 File New Execution of the command New in the File popup menu clears the edit window If the current content of the edit window has been modified but not saved the user has
39. nded system into a right handed system by inverting the direction cosines of one axis Also eigenvectors are often reported in col umn format but in a Cartesian system they are required row wise Thus the output needs to be transposed The following example will assume that you have calculated an electric field gradient tensor and an absolute magnetic shielding tensor for one atom and that you want to determine the Euler angles orienting the shielding tensor in the EFG tensor 3 1 1 The Absolute Magnetic Shielding Tensor Unless specified explicitly your ab initio software may not report the eigenvectors for example the following excerpt from a Gaussian output may only show the shielding tensor elements with respect to the Gaussian Standard orientation with Gaussian 03 you will get the eigenvectors only if you specify the keyword NMR PrintEigenvectors 2 Al Isotropic 576 0975 Anisotropy 103 7786 XX 541 6395 YX 1 2823 ZX 0 5447 XY 2 7673 YY 541 3822 ZY 0 9831 XZ 1 7010 YZ 1 2516 ZZ 645 2708 Eigenvalues 539 4777 543 5316 645 2832 Eigenvectors 1 0 685346 0 728190 0 006350 2 0 728136 0 685374 0 008940 3 0 010862 0 001504 0 999940 Lets pretend you don t have the eigenvectors You could use Trafo to diagonalize the shielding tensor but Trafo uses a transposed setup XX XY XZ YX YAY en YZ ZX ZY ZZ Therefore for use with Trafo your input should look like this you can copy the
40. nse be liable to you for damages including any general special incidental or consequential damages arising out of the use or inability to use the package including but not limited to loss of data or data being rendered inaccurate or losses sustained by you or third parties even if such holder or other party has been advised of the possibility of such damages In any case liability will be limited to the amount of money that the copyright holder received from you for the use of this program May 14 2015 2 Reference 2 1 Help on Menu Items The menu system of Trafo consists of the following pop up menus Trafo 01_ADP_Khan_fract_coord cry File Edit Action NMR Help az 4997 7 4997 7 5494 90 00 96 60 0 0000 6 6000 0 0000 0 0000 6 6600 0 5008 6 6843 6 1466 6 1151 Category Menu Item Action File File and document management New p 8 clears the content of the Trafo edit window Open p 8 read a file into Trafo s edit window Save as p 8 save the content of the edit window to a new file Exit p 8 exit Trafo Edit Editor window actions Undo p 9 undo last changes Cut p 9 cut selection into clipboard Copy p 9 copy selection into clipboard Paste p 9 insert content of clipboard Clear p 9 clear content Select All p 9 select all content Action Perform transformations Generate Positions p 10 Generate Block p 16 Orthogonalize Axes p 17 Orthogonalize Coordi
41. on 2 5 7 14 05 2015 e new feature the help file contains a lot of examples that illustrate the use of Trafo e new feature the help file is now in PDF format because it is suitable for both online viewing as well as printing To view or print this documentation any PDF viewer should work Also most viewers should be able to deal with the hyperlinks that cross link topics in this help file However opening a PDF file at a specified position is more complicated Therefore not every PDF viewer will work with Trafo to display context sensitive help This matter is aggravated by the fact that every viewer has its own mechanism Currently context sensitive help should work with the following PDF viewers Adobe uses with its Acrobat Reader series of products a mechanism called Dynamic Data Exchange DDE to open a PDF file at a specific position For ages the name of the DDE server has been acroview However with the introduction of Acrobat Reader X this tradi tion has been broken on purpose breaking many applications that rely on this mechanism Trafo should be able to work with older versions of the Acrobat Reader as well as with the acroviewR10 DDE server of Adobe Reader X Because I have been quite dissatisfied with the Adobe products from Reader 5 onwards I am using a different PDF viewer and will not always check whether Acrobat is still working My preferred PDF viewer is the free PDFXChangeEditor the successor of PDFXChangeViewer by
42. option calculates dipolar second moments for single crystal or powder samples using coordinates supplied in a separate file or in the editor window of Trafo The coordinates must be Cartesian coordinates p 20 First it is necessary to specify the spin system in question For that purpose you need to specify the isotope corresponding to the observed nucleus It is assumed that the first atom specified in the input data by a coordinate triple corresponds to this observed nucleus Secondly the isotope of the coupled nucleus is specified The abundance of this isotope is then automatically set to the natural abundance of this isotope but can be modified to reflect the actual level of enrichment if any It is assumed that all coordinates after the first atom the observed nucleus belong to this coupled nucleus The dipolar second moment of a system where only a fraction f of the crystal sites is occupied by nuclear spins is proportional to this fraction f 6 If desired a cutoff distance can be specified Any nucleus that is more remote than the cutoff distance is not included in the calculation This is useful if the convergence of the calculation is to be tested but only one input file is used The type of moment calculation specifies whether the calculation should be performed for a powder or single crystal sample Single Crystals In the case of a single crystal sample the orientation of the magnetic field needs to be specified This is done
43. osoft for providing Visual C 2008 Express Edition for free e Jordan Russell for making Inno Setup available http www jrsoftware org e Jochen Kalmbach for demonstrating how to statically link against the Microsoft CRT and thus get rid of VCREDIST_X86 EXE http blog kalmbach software de chicks for demonstrating in his pdfp PDF tools how to establish Dynamic Data Exchange DDE with Adobe Acrobat Reader http www esnips com web PDFTools e This manual has been produced using the MiKTEX http www miktex org distribution of TATEX in combination with the TeXstudio editor http texstudio sourceforge net Irfan Skiljan s IrfanView http www irfanview com has been used to process bitmapped images and Inkscape www inkscape org for dealing with vector graphics 1 5 2 Trademark Acknowledgement e Microsoft MS is a registered trademark and MS DOS MS Word and MS Windows are trade marks of Microsoft Corporation e Alchemy is a copyrighted program and registered trademark of TRIPOS Associates Inc a sub sidiary of Evans amp Sutherland e Other brand and or product names are used for identification purposes only and are trade marks registered trademarks or copyrights of their respective owners 1 5 3 Copyright Message Copyright 1995 2015 Klaus Eichele All rights reserved This program executable help file and related files may be distributed freely and may be used without fee by any individua
44. r Angles command from the Actions Rotations menu allows to calculate Euler angles p 24 from the direction cosines of the new coordinate system with respect to the old coordi nate system In order to be able to do this the data are checked for validity with procedures pioneered by Michael D Lumsden Halifax N S Canada e check if the unit vectors are indeed unit vectors e check if the unit vectors are orthogonal e check if the unit vectors are right handed if left handed the user has the choice of inverting one axis p 24 selectively If the data pass this test the Euler angles are calculated using specific components of the unit vectors to get the angles The result will be displayed by a dialog box but also be copied to the clip board From there the routines to perform Euler rotations will retrieve them automatically and use them as defaults Note however that any editing tasks copy cut etc may remove these data from the clipboard The input data may be preceded by a line defining the crystal axes p 2 and may have some com ment lines p 3 or lines describing the origin p 3 interspersed they are simply ignored The routine takes the first three regular data lines p 3 as input An example is presented in Fig 2 12 the direction cosines of an orthogonalized crystal axis system in a crystal holder frame Using this file as input this orientation can also be specified using the triple of Euler angles Euler
45. simplify the problem let us start with a two dimensional rotation Suppose the coordinates x y of a point in the two dimensional XY system are known but we are actually in terested in knowing the coordinates of this point in another coordinate system X Y which is related to the XY system by a counter clockwise rotation by an angle 9 As the figure indicates the coordinates of the given point in the new coordinate system will be x xcosp ysing t y xsing ycoso cosp sing sing coso Start Coincidence Z Now transferred to a three dimensional problem the goal will be z 1 to describe the coordinates in a final rotated system x y z which is related to some initial coordinate system X Y Z by the Euler or in matrix notation R p angles The final system is developed in three steps each step y 1 involving a rotation described by one Euler angle At the start both coordinate systems X Y Z and x 1 y 1 z 1 shall be Y coincident X x 1 May 14 2015 Chapter 2 Reference First Rotation The first rotation involves the Euler angle a The x 1 y 1 z 1 axis system is rotated about the Z axis through an angle counterclockwise relative to X Y Z to give the new system x 2 y 2 z 2 It is clear from the figure that this rotation mixes the coordinates along X and Y completely analogous to the two dimensional rotation described above while the coordinate along Z rem
46. the choice of writing the current content of the edit window to a file Once the data are gone they are gone 2 2 2 File Open File Execution of the command Open in the File popup menu retrieves the content of a data file into the edit window If the current content of the edit window has been modified but not saved the user has the choice of writing the current content of the edit window to a file Once the data are gone they are gone 2 2 3 File Save File As Execution of the command Save as in the File popup menu saves the content of the edit window if available into a data file To prevent Trafo from adding the extension cry enclose your file name in double quotes N B It overwrites existing data files without a warning 2 2 4 File Exit Execution of the command Exit in the File popup menu quits Trafo If the current content of the edit window has been modified but not saved the user has the choice of writing the current content of the edit window to a file Once the data are gone they are gone 2 3 Edit Menu The Edit Menu provides various commands which act on the content of the edit window Depending on previous actions not all commands might be available at a given time In their functionality the commands are typical of a Windows application May 14 2015 Chapter 2 Reference The Edit pop up menu consists of the following items Trafo 01_ADP_Khan_fract_coord cry File Edit Action NMR
47. the fractional crystal coordinates of one given molecule and applying the generating symmetry operations on them as detailed in 1 Also after generating one complete unit cell one could create a cube of unit cells by using the Generate Block p 16 command from the Actions Generate menu ER For example consider the situation illustrated in the fig ure The goal is to generate a new molecule which is re lated to the current one by a twofold screw axis operation Such a symmetry operation requires the rotation of atomic positions by 180 about the axis of the screw followed by a translational element The detailed procedure together with some examples is explained in the topic symmetry operations p 10 References 1 International Tables for X ray Crystallography Kynoch Press Birmingham England 1974 Vol 1 2 4 2 Symmetry Operations The Generate Positions command from the Actions Generate menu allows to create new coordi nates positions from the existing ones by applying symmetry operations One application is to generate all positions in the unit cell from the content of the asymmetric unit It calls the dialog box Symmetry Operations A detailed example at the bottom of this topic illustrates the procedure In the dialog box Symmetry Operations one defines 10 May 14 2015 Chapter 2 Reference Symmetry Operations x No of Positions fi Source Factor 1 00000000 Shift 0 00000000 Ex Cy Cz
48. tion Symmetry Operations p 10 we demonstrated how to generate the content of the unit cell given the fractional coordinates of the asymmetric unit using ammonium dihydrogen phosphate ADP as an example If you look at Fig 2 2 the result is not exactly what is expected naively Rather than resulting in a sort of block such generating symmetry operations often generate positions that lie in the neighboring unit cell e g in Fig 2 3 all atoms that have a negative coordinate or a coordi nate greater than one We would need to shift all these positions by integer multiples of the unit cell axes i e add or subtract 1 to bring them into our expected block of 0 lt x y z lt 1 Another solution is to simply add more unit cells around our central unit cell e Read file ADP_03_unit cry Fig 2 3 into Trafo remove the comment lines if present and save the file under the same name Select Action Generate Block specify n 1 and save the result as ADP_04_unit_block cry Note that this creates 26 additional unit cells 27 in total corresponding to 1512 atomic posi tions The file ADP_04_unit_block cry is too big gt 64 KB for Trafo s edit window but you can still perform file related actions E g to visualize this new result clear the Trafo edit window File New and select Action Orthogonalize Coordinates leave the order of axes at their defaults specify ADP_04_unit_block cry as input file and ADP_04_unit_block_cart
49. trated in the topic Rotation Matrices p 25 The convention used here is that x is a rotation about the Z axis of the initial coordinate system About the y axis of this newly generated coordinate system a rotation by B is performed followed by a rotation by y about the new z axis Given the Euler angles the step by step procedure illustrates how to move from one coordinate system to the other However given the two coordinate systems how can one determine the Euler angles relating them This is described in topic Determining Euler Angles p 27 24 May 14 2015 Chapter 2 Reference The usual ranges for a p y are 0 lt lt 360 0 lt 6 lt 180 0 lt y lt 360 References 1 G Arfken Mathematical Methods for Physicists 3rd ed Academic Press New York 1985 2 M E Rose Elementary Theory of Angular Momentum Wiley New York 1957 3 K Schmidt Rohr and H W Spiess Multidimensional Solid State NMR and Polymers Academic Press London 1994 2 4 13 Rotation Matrices Rotations or transformations from one coordinate system into another are conveniently described by the triplet of Euler angles p 24 B y Using the Euler angles this three dimensional problem can be dissected into a sequence of two dimensional rotations whereby in each rotation one axis remains invariant Here all rotations are counter clockwise right handed mathematically positive sense 2D Analogy In order to
50. verview of some of the common tasks and actions Revision History Summary of changes versus previous versions Known Problems Summary of known problems Acknowledgements Credits for important contributions etc 1 1 Introduction Trafo is a program to perform coordinate transformations useful in combination with the analysis of single crystal NMR spectra as well as to assist in some related procedures such as diagonalizations of tensors Although Trafo will often deal with crystallographic data it is not smart in that respect it is a simple tool and not an expert system If you are looking for powerful software to deal with crystallographic questions consider Platon or Mercury both available freely Trafo is part of a simulation package developed during my stay at the Department of Chemistry Dalhousie University Halifax Canada The initial version has been written in C using Borland C 4 52 and has been developed to run on IBM compatible personal computers under the MS Windows environment in the 16 bit subsystem For the 32 bit version Borland C 5 0 has been used initially The current version has been produced using Microsoft Visual C 2008 Express Edition 1 2 Overview The actions selected by the menu commands operate s either directly on the content of the edit window the default if there is anything available e or if the edit window is empty you can select File New p 8 to clear the window on an input file the name of
51. x lt Vyy lt Vz2z2 work out the isotropic part and subtract it from the principal components order the eigenvectors as specified above and save the file as 05_efg_in_ mf_ordered cry 1 0 1 0 1 0 90 0 90 0 90 0 OR 0 0 0 0 0 0 Ti1 0 303124 0 950924 0 062133 T22 0 951076 0 305972 0 042852 T33 0 059759 0 046104 0 997148 Figure 3 7 EFG in molecular frame in the proper convention save as file 05_efg_in_ mf_ordered cry Now you need to calculate the Euler angles for the orientation of the EFG tensor in the molecular frame select Action Rotations Calculate Euler angles alpha 322 3498547908 beta 4 3286051469 gamma 145 4072983840 Figure 3 8 Euler angles calculated for the EFG tensor in file 05_efg in mf ordered cry We will need these angles to transform the CSA into the EFG frame Trafo copies those Euler angles to the clipboard 33 May 14 2015 Chapter 3 Applications 322 3498547908 4 3286051469 145 4072983840 Figure 3 9 Euler angles and they will stay there unless you copy something else 3 1 3 Transforming the CSA into the EFG frame Read the file 02_csa in mf cry select Action Rotations Euler rotations forwards select a name for the output file 06_csa_in_efg cry The EFG Euler angles should already be set properly unless 322 3498547908 4 3286051469 145 4072983840 Figure 3 10 Euler angles of the EFG tensor you put
52. xample Given a set of direction cosines specifying unit vectors the rotation backward creates direc tion cosines which specify the coordinates of the derived frame in the reference frame Using the option Calculate Euler Angles on this new set of data yields the Euler angles for going from the reference frame to the derived frame i e it yields back the Euler angles used for the transformation neglecting round off errors 2 4 17 Action Polar Angles Calculate This command transforms Cartesian coordinates into polar coordinates A Cartesian crystal axis line p 2 must precede the data The output will consist of lines giving the atom label the azimuth angle a the polar angle f and the distance from the origin in Angstroems 28 May 14 2015 Chapter 2 Reference 2 5 NMR Menu 2 5 1 Dipolar Second Moment Dipolar Second Moment Define Dipolar Interaction Observed nucleus Coupled nucleus Natural abundance JHA JHA 100 00000000 Type of Moment Calculation C powder single crystal define rotation axis in next section Define Orientation of Rotation Axis Alpha 0 00000000 deg Gamma star 0 00000000 deg Beta 90 00000000 deg Gamma incr 1 00000000 deg Gamma end 180 00000000 deg Phase angle 0 00000000 deg Sense of rotation clockwise single crystal NMR anticlockwise Cutoff distance 1000 000 A va Ka This program

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