SECOND EXAMPLE (sigi)

In the second example (provided as the file sigi.ins) a small organic structure is refined in the space group P-1. Only the features that are different from the ags4 refinement will be discussed in detail. The structure consists of a five-membered lactone [-C7-C11-C8-C4(O1)-O3-] with a -CH₂-OH group [-C5-O2] attached to C7 and a =C(CH₃)(NH₂) unit [=C9(C10)N6] double-bonded to C8.

Of particular interest here is the placing and refinement of the 11 hydrogen atoms via HFIX instructions. The two -CH₂- groups (C5 and C11) and one tertiary CH (C7) can be placed geometrically by standard methods; the algorithms have been improved relative to those used in SHELX-76, and the hydrogen atoms are now idealized before each refinement cycle (and after the last). Since N6 is attached to a conjugated system, it is reasonable to assume that the -NH₂ group is coplanar with the C8=C9(C10)-N6 unit, which enables these two hydrogens to be placed as ethylenic hydrogens, which requires HFIX (or AFIX) 9n; the program takes into account that they are bonded to nitrogen in setting the default bond lengths. All these hydrogens are to be refined using a 'riding model' (HFIX or AFIX m3) for x, y and z.

The -OH and -CH₃ groups are trickier, in the latter case because C9 is sp²-hybridized, so the potential barrier to rotation is low and there is no fully staggered conformation available as the obvious choice. Since the data are reasonable, the initial torsion angles for these two groups can be found by means of difference electron density syntheses calculated around the circles which represent the loci of all possible hydrogen atom positions. The torsion angles are then refined during the least-squares refinement. Note that in subsequent cycles (and jobs) these groups will be re-idealized geometrically with RETENTION of the current torsion angle; the circular Fourier calculation is performed only once. Two 'free variables' (2 and 3 - yes, they still exist!) have been assigned to refine common isotropic displacement parameters for the 'rigid' and 'rotating' hydrogens respectively. If these had not been specified, the default action would have been to hold the hydrogen U values at 1.2 times the equivalent isotropic U of the atoms to which they are attached (1.5 for the -OH and methyl groups).

The 'sigi.ins' file (which is provided as a test job) is as follows. Note that for instructions with both numerical parameters and atom names such as HFIX and MPLA, it does not matter whether numbers or atoms come first, but the order of the numerical parameters themselves (and in some cases the order of the atoms) is important.

 
TITL SIGI in P-1
CELL 0.71073 6.652 7.758 8.147 73.09 75.99 68.40
ZERR 2 .002 .002 .002 .03 .03 .03
SFAC C H N O
UNIT 14 22 2 6           ! no LATT and SYMM needed for space group P-1
 
L.S. 4
EXTI 0.001               ! refine an isotropic extinction parameter
WGHT .060 0.15           ! (suggested by program in last job);  WGHT
OMIT 2 8 0               ! and OMIT are also based on previous output
 
BOND $H                  ! include H in bond lengths / angles table
CONF                     ! all torsion angles except involving hydrogen
FMAP 2                   ! F_o-F_c Fourier
PLAN -20                 ! printer plots and full analysis of peak list
 
HFIX 147 31 O2           ! initial location of -OH and -CH₃ hydrogens from
HFIX 137 31 C10          ! circular Fourier, then refine torsion, U(H)=fv(3)
 
HFIX 93 21 N6            ! -NH₂ in plane, xyz ride on N, U(H)=fv(2)
HFIX 23 21 C5 C11        ! two -CH₂- groups, xyz ride on C, U(H)=fv(2)
HFIX 13 21 C7            ! tertiary CH, xyz ride on C, U(H)=fv(2)
 
EQIV $1 X-1, Y, Z        ! define symmetry operation and tabulate H-bond
RTAB H..O H2 O1_$1       ! distance and angle to symmetry equivalent of O1
RTAB XHY O2 H2 O1_$1      ! 'H..O' and 'XHY' are table headings
 
RTAB H..O H6A O1         ! include intramolecular H-bond in tables
RTAB XHY N6 H6A O1
 
EQIV $2 X+1, Y, Z-1      ! include a further intermolecular H-bond in the
RTAB H..O H6B O2_$2      ! same tables; involves symmetry equivalent of O2
RTAB XHY N6 H6B O2_$2
                                    ! l.s. planes through 5-ring and through
MPLA 5 C7 C11 C8 C4 O3 O1 N6 C9 C10 ! CNC=CCC moiety, then find deviations
MPLA 6 C10 N6 C9 C8 C11 C4 O1 O3 C7 ! of last 4 and 3 named atoms resp. too
 
FVAR 1 .06 .07           ! overall scale and free variables for U(H)
 
REM name sfac# x y z sof(+10 to fix it) U11 U22 U33 U23 U13 U12 follow
 
O1      4   0.30280   0.17175   0.68006  11.00000   0.02309   0.04802 =
        0.02540  -0.00301  -0.00597  -0.01547
O2      4  -0.56871   0.23631   0.96089  11.00000   0.02632   0.04923 =
        0.02191  -0.00958   0.00050  -0.02065
O3      4  -0.02274   0.28312   0.83591  11.00000   0.02678   0.04990 =
        0.01752  -0.00941  -0.00047  -0.02109
C4      1   0.10358   0.23458   0.68664  11.00000   0.02228   0.02952 =
        0.01954  -0.00265  -0.00173  -0.01474
C5      1  -0.33881   0.18268   0.94464  11.00000   0.02618   0.03480 =
        0.01926  -0.00311  -0.00414  -0.01624
N6      3   0.26405   0.17085   0.33925  11.00000   0.03003   0.04232 =
        0.02620  -0.01312   0.00048  -0.01086
C7      1  -0.25299   0.33872   0.82228  11.00000   0.02437   0.03111 =
        0.01918  -0.00828  -0.00051  -0.01299
C8      1  -0.03073   0.27219   0.55976  11.00000   0.02166   0.02647 =
        0.01918  -0.00365  -0.00321  -0.01184
C9      1   0.05119   0.24371   0.39501  11.00000   0.02616   0.02399 =
        0.02250  -0.00536  -0.00311  -0.01185
C10     1  -0.10011   0.29447   0.26687  11.00000   0.03877   0.04903 =
        0.02076  -0.01022  -0.00611  -0.01800
C11     1  -0.26553   0.36133   0.63125  11.00000   0.02313   0.03520 =
        0.01862  -0.00372  -0.00330  -0.01185
 
HKLF 4     ! read intensity data from 'sigi.hkl'; terminates '.ins' file

The data reduction reports 1904 reflections read with -7 >= h >= 7, -8 >= k >= 9 and -9 >= l >= 9. Note that these are the limiting index values; in fact only about 1.5 times the unique volume of reciprocal space was measured. The maximum 2-theta was 50.00, and there were no systematic absence violations, 34 (not seriously) inconsistent equivalents, and 1297 unique data, of which 1 was suppressed (by OMIT). R(int) was 0.0196 and R(sigma) 0.0151.

It will be seen that the program uses different default distances to hydrogen for different bonding situations (these may be overridden by the user if desired, of course). These defaults depend on the temperature (set using TEMP) in order to allow for librational effects. The list of default X-H distances is followed by the (squashed) circular difference electron syntheses to determine the C-OH and C-CH₃ initial torsion angles:

 
Default effective X-H distances for T =   20.0 C
 
AFIX m =    1     2     3     4   4[N]  3[N]  15[B]  8[O]   9   9[N]   16
d(X-H) =  0.98  0.97  0.96  0.93  0.86  0.89  1.10  0.82  0.93  0.86  0.93
 
 
Difference electron density (eA^-3x100) at 15 degree intervals for AFIX 147
group attached to O2.  The center of the range is eclipsed (cis) to C7 and
rotation is clockwise looking down C5 to O2
  -2  0  1  0  0  0 -1 -5 -8 -9 -6 -2  2  5  9 16 29 42 48 39 23  9  0 -2
 
 
Difference electron density (eA^-3x100) at 15 degree intervals for AFIX 137
group attached to C10.  The center of the range is eclipsed (cis) to N6 and
rotation is clockwise looking down C9 to C10
  34 37 39 41 38 30 20 15 19 28 39 47 50 43 29 15 12 19 29 35 33 27 25 29
 
After local symmetry averaging:   21  28  36  41  40  33  24  20

It can be seen that the hydroxyl hydrogen is very clearly defined, but that the methyl group is rotating fairly freely (low potential barrier). After three-fold averaging, however, there is a single difference electron density maximum. The (squashed) least-squares refinement output follows:

 
Least-squares cycle 1  Maximum vector length =511 Memory required =1771/135569
 
wR2 =  0.1138 before cycle   1 for   1296 data and  105 /  105 parameters
 
GooF = S =     1.134;     Restrained GooF =      1.134  for      0 restraints
 
Weight = 1/[sigma^2(Fo^2)+(0.0600*P)^2+0.15*P] where P=(Max(Fo^2,0)+2*Fc^2)/3
 
    N      value        esd    shift/esd  parameter
 
    1     0.97914     0.00386    -5.406    OSF
    2     0.03486     0.00263    -9.959   FVAR  2
    3     0.07515     0.00396     1.048   FVAR  3
    4     0.02334     0.00951     2.349   EXTI
 
Mean shift/esd =   0.911    Maximum =  -9.959 for FVAR  2
 
Max. shift = 0.038 A for H10C      Max. dU =-0.026 for H5A
 
     .......... etc (cycles 2 and 3 omitted) .........
 
 
Least-squares cycle 4  Maximum vector length =511 Memory required =1771/135569
 
wR2 =  0.1044 before cycle   4 for   1296 data and  105 /  105 parameters
 
GooF = S =     1.025;     Restrained GooF =      1.025  for      0 restraints
 
Weight = 1/[sigma^2(Fo^2)+(0.0600*P)^2+0.15*P] where P=(Max(Fo^2,0)+2*Fc^2)/3
 
    N      value        esd    shift/esd  parameter
 
    1     0.97903     0.00361    -0.001    OSF
    2     0.03607     0.00178     0.022   FVAR  2
    3     0.07346     0.00379    -0.009   FVAR  3
    4     0.02502     0.01089    -0.004   EXTI
 
Mean shift/esd =   0.006    Maximum =  -0.182 for tors H10A
 
Max. shift = 0.003 A for H10B      Max. dU = 0.000 for H5A
 
 
Largest correlation matrix elements
 
    0.509 U12 O2 / U22 O2                   0.506 U12 O3 / U11 O3
    0.508 U12 O2 / U11 O2                   0.500 U12 O3 / U22 O3
 
 
 
Idealized hydrogen atom generation before cycle   5
 
Name     x       y       z    AFIX  d(X-H)  shift  Bonded   Conformation
                                                    to      determined by
H2   -0.6017  0.2095  0.8833  147   0.820   0.000   O2        C5  H2
H5A  -0.2721  0.0676  0.9001   23   0.970   0.000   C5        O2  C7
H5B  -0.2964  0.1554  1.0576   23   0.970   0.000   C5        O2  C7
H6A   0.3572  0.1389  0.4085   93   0.860   0.000   N6        C9  C8
H6B   0.3073  0.1559  0.2347   93   0.860   0.000   N6        C9  C8
H7   -0.3331  0.4598  0.8575   13   0.980   0.000   C7        O3  C5  C11
H10A -0.2044  0.4191  0.2694  137   0.960   0.000   C10       C9  H10A
H10B -0.1761  0.2034  0.2962  137   0.960   0.000   C10       C9  H10A
H10C -0.0176  0.2950  0.1525  137   0.960   0.000   C10       C9  H10A
H11A -0.3575  0.2948  0.6198   23   0.970   0.000   C11       C8  C7
H11B -0.3198  0.4943  0.5737   23   0.970   0.000   C11       C8  C7

The final structure factor calculation, analysis of variance etc. produces the following edited output:

 
Final Structure Factor Calculation for  SIGI in P-1
 
Total number of l.s. parameters = 105    Maximum vector length =  511
 
wR2 =  0.1044 before cycle   5 for   1296 data and    0 /  105 parameters
 
GooF = S =     1.025;     Restrained GooF =      1.025  for      0 restraints
 
Weight = 1/[sigma^2(Fo^2)+(0.0600*P)^2+0.15*P] where P=(Max(Fo^2,0)+2*Fc^2)/3
 
R1 =  0.0365 for   1189 Fo > 4.sigma(Fo)  and  0.0399 for all   1297 data
wR2 =  0.1060,  GooF = S =   1.042,  Restrained GooF =    1.042  for all data
 
 
Principal mean square atomic displacements U
 
  0.0504   0.0254   0.0188   O1
  0.0491   0.0229   0.0190   O2
  0.0513   0.0194   0.0165   O3
  0.0326   0.0208   0.0159   C4
  0.0375   0.0204   0.0190   C5
  0.0440   0.0320   0.0214   N6
  0.0329   0.0201   0.0185   C7
  0.0276   0.0190   0.0181   C8
  0.0288   0.0220   0.0191   C9
  0.0494   0.0353   0.0181   C10
  0.0353   0.0215   0.0183   C11
 
 
Analysis of variance for reflections employed in refinement
K = Mean[Fo^2] / Mean[Fc^2]  for group
 
Fc/Fc(max)     0.000 0.009 0.017 0.027 0.038 0.049 0.065 0.084 0.110 0.156 1.0
 
Number in group    135.  125.  130.  139.  119.  133.  130.  128.  131.  126.
 
           GooF   1.110 1.006 1.082 1.046 1.093 1.014 0.923 0.996 1.027 0.930
 
            K     1.521 1.121 0.966 1.023 1.008 0.990 0.998 0.998 1.008 1.010
 
 
Resolution(A)  0.84  0.88  0.90  0.95  0.99  1.06  1.14  1.25  1.44  1.79  inf
 
Number in group    136.  127.  128.  128.  136.  124.  128.  130.  130.  129.
 
           GooF   1.007 0.890 0.865 0.867 0.864 0.921 0.874 1.095 1.256 1.432
 
            K     1.024 1.013 1.017 0.990 0.991 0.989 1.013 0.995 1.037 1.004
 
            R1    0.062 0.049 0.051 0.046 0.034 0.034 0.031 0.039 0.039 0.037
 
 
Recommended weighting scheme:  WGHT   0.0548   0.1468

The analysis of variance should be examined carefully for indications of systematic errors. If the Goodness of Fit is significantly higher than unity and the scale factor K is appreciably lower than unity in the extreme right columns in terms of both F_c and resolution, then an extinction parameter should be refined (the program prints a warning in such a case). This does not show here because an extinction parameter is already being refined. The scale factor is a little high for the weakest reflections in this example; this may well be a statistical artifact and may be ignored (selecting the groups on F_c will tend to make F_o² greater than F_c² for this range). The increase in the GooF at low resolution (the 1.79 to infinity range) is caused in part by systematic errors in the model such as the use of scattering factors based on spherical atoms which ignore bonding effects, and is normal for purely light-atom structures (this interpretation is confirmed by the fact that difference electron density peaks are found in the middle of bonds). In extreme cases the lowest or highest resolution ranges can be conveniently suppressed by means of the SHEL instruction; this is normal practice in macromolecular refinements.

The weighting scheme suggested by the program is designed to produce a flat analysis of variance in terms of F_c, but makes no attempt to fit the resolution dependence of the Goodness of Fit. It is also written to the end of the .res file, so that it is easy to update it before the next job. In the early stages of refinement it is better to retain the default scheme of WGHT 0.1; the updated parameters should not be incorporated in the next '.ins' file until all atoms have been found and at least the heavier atoms refined anisotropically.

The list of most disagreeable reflections and tables of bond lengths and angles (BOND $H - omitted here) and torsion angles (CONF) are followed by the RTAB and MPLA tables:

 
Selected torsion angles
 
 -175.08 ( 0.12)  C7 - O3 - C4 - O1
    5.72 ( 0.15)  C7 - O3 - C4 - C8
  109.70 ( 0.12)  C4 - O3 - C7 - C5
  -11.64 ( 0.15)  C4 - O3 - C7 - C11
  171.12 ( 0.10)  O2 - C5 - C7 - O3
  -72.04 ( 0.15)  O2 - C5 - C7 - C11
   -1.47 ( 0.24)  O1 - C4 - C8 - C9
  177.61 ( 0.12)  O3 - C4 - C8 - C9
 -176.27 ( 0.14)  O1 - C4 - C8 - C11
    2.81 ( 0.16)  O3 - C4 - C8 - C11
    3.09 ( 0.22)  C4 - C8 - C9 - N6
  176.93 ( 0.13)  C11 - C8 - C9 - N6
 -177.23 ( 0.13)  C4 - C8 - C9 - C10
   -3.38 ( 0.22)  C11 - C8 - C9 - C10
  176.04 ( 0.13)  C9 - C8 - C11 - C7
   -9.39 ( 0.14)  C4 - C8 - C11 - C7
   12.36 ( 0.14)  O3 - C7 - C11 - C8
 -104.74 ( 0.13)  C5 - C7 - C11 - C8
 
 
Distance H..O
 
     2.041 (0.003)  H2 - O1_$1
     2.225 (0.002)  H6A - O1
     2.172 (0.002)  H6B - O2_$2
 
 
Angle XHY
 
  174.03 (2.37)  O2 - H2 - O1_$1
  129.29 (0.05)  N6 - H6A - O1
  155.07 (0.05)  N6 - H6B - O2_$2
 
 
 
Least-squares planes (x,y,z in crystal coordinates) and deviations from them
(* indicates atom used to define plane)
 
 2.344 (0.004) x + 7.411 (0.004) y - 0.015 (0.005) z = 1.978 (0.004)
 
*   -0.074 (0.001)  C7
*    0.068 (0.001)  C11
*   -0.042 (0.001)  C8
*   -0.006 (0.001)  C4
*    0.054 (0.001)  O3
    -0.006 (0.002)  O1
    -0.098 (0.003)  N6
    -0.056 (0.002)  C9
    -0.031 (0.003)  C10
 
Rms deviation of fitted atoms =   0.055
 
 
 2.544 (0.004) x + 7.349 (0.004) y - 0.166 (0.004) z = 1.863 (0.003)
 
Angle to previous plane (with approximate esd) =  2.45 ( 0.07 )
 
*    0.005 (0.001)  C10
*    0.008 (0.001)  N6
*   -0.005 (0.001)  C9
*   -0.034 (0.001)  C8
*    0.013 (0.001)  C11
*    0.012 (0.001)  C4
     0.057 (0.002)  O1
     0.021 (0.002)  O3
    -0.154 (0.002)  C7
 
Rms deviation of fitted atoms =   0.016

All esd's printed by the program are calculated rigorously from the full covariance matrix, except for the angle between two least-squares planes, which involves some approximations. The contributions to the esds in bond lengths, angles and torsion angles also take the errors in the unit-cell parameters (as input on the ZERR instruction) rigorously into account; an approximate treatment is used to obtain the (rather small) contributions of the cell errors to the esds involving least-squares planes.

The free torsional motion of H2 is virtually at right angles to the fairly linear hydrogen bond, so the O-H..O angle has a large esd. On the other hand the 'riding model' constraint applied to the N-H bonds effectively prevents the estimation of a meaningful esd in the two N-H..O angles, hence the unrealistically small values for these two esds.

There follows the difference electron density synthesis and line printer 'plot' of the structure and peaks. The highest and lowest features are 0.28 and -0.17 eA^-3 respectively, and the rms difference electron density is 0.04. These values confirm that the treatment of the hydrogen atoms was adequate, and are indeed typical for routine structure analysis of small organic molecules. This output is too voluminous to give here, and indeed users of the Siemens SHELXTL molecular graphics program XP will almost always suppress it by use of the default option of a positive number on the PLAN instruction, and employ interactive graphics instead for analysis of the peak list.

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