Matta, Boyd. The quantum theory of atoms in molecules

12.4 Invariom Modeling 333

Fig. 12.10 Series of amides with low fragment transferability. From left to right: formamide, acetamide, N-methylformamide, and N-methylacetamide.

If one wishes to reconstruct the electron density based on the sum of atomic QTAIM fragments as successfully attempted earlier [56], the carbon atom attached to the OH group cannot be regarded as transferable and the assumption of carbon transferability would introduce a systematic error in a scattering model, whereas the oxygen and hydrogen atoms have very similar or identical charges.

We have also calculated atomic QTAIM charges of a number of amides (Fig. 12.10 and Table 12.4). For these molecules transferability is only found for the oxygen atoms that are partly double-bonded to the carbon atom of the peptide bond. The nitrogen atoms of the amino group are similar in formamide and acetamide, and in N-methylformamide and N-methylacetamide, but di er in the two groups. The situation is analogous with the carbon atom – charges di er between substituted and nonsubstituted atoms, but are in agreement when the neighbors are similar. We conclude that, to a good approximation, atoms have similar QTAIM charges when their nearest neighbored atoms are identical [57]; this approach is termed the nearest-neighbor approximation (NNA).

We can also observe ‘‘compensatory transferability’’ in the charges, to use the term introduced by Bader [53]. When two possible mesomeric Lewis formulae can be written for a molecule, charge compensation can be assumed to be more pronounced than for a molecule consisting of single-bonded atoms only.

These observations illustrate the reasons for the choice of model compounds used to predict the electron density for invariom modeling as mentioned above.

Table 12.4 Atomic charges for a series of amides.

33412 Fragment Transferability Studied Theoretically and Experimentally with QTAIM

12.5

Applications of Aspherical Invariom Scattering Factors

12.5.1

Molecular Geometry and Anisotropic Displacement Properties

Invariom modeling improves the accuracy of molecular geometry from X-ray single crystal di raction, especially for hydrogen atoms, for which bond distances become comparable with results from neutron di raction or theoretical calculations. Modeling with aspherical scattering factors has been shown to be more appropriate than application of the IAM [58]. Asphericity shifts [59] are bond-length aberrations because of the use of the IAM. They occur, for example, in longer CaO bonds, because of spherical averaging over the oxygen valence density in the IAM at low resolution; CbC bonds can also be a ected. Asphericity shifts disappear in invariom model refinements, although such di erences are often within the error range of the least squares.

Improvement of molecular geometry for dl-serine by using invarioms is shown in Table 12.5 by comparison of bond distances of the 298 K X-ray invariom data with room-temperature neutron study results for di erent resolution [60]. In essence, invariom modeling provides an improved scattering factor model and, as a result, the least-squares fit of aspherical scatterers to experimental structure factors usually leads to substantial improvement of the crystallographic R-factors and the goodness of fit, and to a reduction of the remaining residual electron density.

Table 12.5 Neutron (N) bond lengths of dl-serine compared with X-ray (X) invariom and the IAM model at room temperature.

12.5 Applications of Aspherical Invariom Scattering Factors 335

Fig. 12.11 Mean value for bonds between C, N, and O for the Hirshfeld test plotted against resolution. Above 10 10 4 A˚ 2 (dotted line) the test is not regarded as fulfilled, as suggested by Hirshfeld [58] Copyright 2005 and Reproduction with Permission from IUCr.

Another result of the modeling process is the increased physical significance of the anisotropic displacement parameters that describe thermal motion (and disorder) in a structure, as can be proven by the results of the Hirshfeld test [5]. Figure 12.11 shows the e ect of the inclusion of the aspherical density in the anisotropic temperature parameters with regard to data resolution for dl-serine. Temperature data for carbon, nitrogen and oxygen are not regarded as including bonding e ects if the di erence of the mean-square displacement amplitudes (DMSDA) is smaller than 0.001 A˚ 2. A mean DMSDA value for the six nonhydrogen bonds was used in Fig. 12.11 in the study of dl-serine [58] and this value was plotted against resolution for invariom scattering factors obtained with the basis set B3LYP/6-311þþG(3df,3pd). Whereas for the IAM at 100 K deconvolution of electron density and thermal e ects was not achieved and the test was not fulfilled at a resolution of 0.55 A˚ 1 in sin y=l (or d ¼ 0:9 A˚ ), by using the invariom model the Hirshfeld test was fulfilled for all three temperatures (20 K, 100 K, 298 K) investigated. It was concluded that this, or higher, resolution is recommended for invariom modeling.

12.5.2

Using the Enhanced Multipole Model Anomalous Dispersion Signal

Absolute configuration of light atom structures is of crucial importance in the pharmaceutical industry. Although the phenomenon of chirality was discovered many years ago, the relevance of the absolute structure of a drug applied to the

336 12 Fragment Transferability Studied Theoretically and Experimentally with QTAIM

human organism was not recognized until the Contergan/Thalidomide scandal in the early sixties. These events made clear that absolute structure can be of outmost importance and that it should be carefully analyzed for every drug candidate before it is registered as a drug, even though it was later found that Thalidomide racemizes in vivo. The proportion of single-enantiomer drugs among new drugs introduced into the market is rapidly growing, and reached @70% in 2002 [61]. There are stringent legal requirements to determine the absolute configuration of drug molecules. A related aspect is the possibility of extending or sidestepping patents by invoking chirality. Single-crystal X-ray structure analysis enables determination of the absolute structure from the intensity di erence between Friedel pairs, because of anomalous dispersion. The objective of such an analysis is, usually, to assign absolute configuration to chiral molecules in a crystal structure and considerable e ort has been devoted to this research topic. Introduction of the x parameter by Flack [62] enabled unambiguous assignment of enantiomorph polarity and the x parameter is widely used and implemented in many least-squares refinement software, for example crystals [63], shelxl [64] or gfmlx [65]. Absolute determination of structure requires that the anomalous dispersion signal is su ciently pronounced. We have recently shown that the multipole model provides additional inversion distinguishing power [66]. Invariom modeling with fixed density terms provides a means of exploiting this to improve absolute structure determination with CuKa, and it has been found that high-res- olution MoKa datasets can provide almost similar information when evaluated with the invariom approach.

12.5.3

Modeling the Electron Density of Oligopeptide and Protein Molecules

For the purpose of structural refinement of amino acid, oligopeptide and protein molecules the naturally occurring amino acids were analyzed in terms of their invariom fragments, and a database with 73 entries that covers this class of compound was generated from 37 model compounds. This invariom database has been validated on 42 experimental small-molecule crystal structures (from IUCr journals), of di erent quality and resolution, covering not only the naturally occurring amino acids, but some of their derivatives, their protonated/unprotonated states, and most common solvents (details of the structures studied have been published elsewhere [67]).

Figure 12.12a shows the crystallographic R-factor for IAM and invariom models for these trial structures. R(F) is always equal to or better for the invariom model, when compared with the IAM. The di erence RðFIAMÞ RðFinvÞ depends on the resolution and temperature of an experiment. Improvements are more apparent at lower temperatures, because of better deconvolution of thermal and electronic e ects. Because low-resolution datasets do not contain as much information on the aspherical part of r(r), improvement of the R-factor is smaller than when more extended datasets are used. Figure 12.12b compares the positive residual electron density Dr(r) for the IAM and invariom models. Here, the most interesting feature can be observed for a high resolution structure containing sulfur,

12.5 Applications of Aspherical Invariom Scattering Factors 337

Fig. 12.12 (a) Comparison of the crystallographic R(F) factor for the IAM and invariom models. (b) Comparison of the residual density Dr(r) for the IAM and invariom models [67] Copyright 2006 and Reproduction with Permission from IUCr.

where the residual density for the invariom model is substantially reduced and becomes comparable with that of other structures that do not contain sulfur.

In disordered structures analogous behavior is seen for Dr(r) and reduction of the R-factor. For such structures the residual density is similar or even increased

338 12 Fragment Transferability Studied Theoretically and Experimentally with QTAIM

by use of the invariom model. One can conclude that the remaining, unmodeled, density is the reason for only small improvements of the R-factor for disordered structures.

An important conclusion for protein refinement can therefore be drawn from disordered structures – it is the modeling of disorder and the completeness of the structural model, rather than the aspherical electron density contribution, that limit the fit of calculated and experimental structure factors and, therefore, the quality of the results. As already remarked, invariom modeling requires data resolution of d a0:9 A˚ or sin y=l b0:55 A˚ 1. This resolution is also recommended for protein data.

12.6 Conclusion

From systematic study of amino acids and oligopeptides by experimental electron density analysis using Bader’s QTAIM an understanding of fragment transferability emerged. Use of appropriate model compounds now enables reproduction of molecular electron density r(r) from fragments; the result is a good approximation to the total density when the invariom pseudoatoms density description is employed. A pseudoatomic fragmentation of a molecular electron density retaining the local chemical environment of an atom enables convenient extension of the IAM scattering model. Such invariom modeling improves the inversion distinguishing power in absolute structure determination and yields ‘‘charge-density quality’’ geometries from low-resolution standard X-ray datasets. Anisotropic displacement parameters become physically meaningful and all properties that can be derived from a model electron density can be rapidly calculated for larger molecules so that a convenient approach to electron densities of macromolecules (proteins, polynucleotides) is feasible.

To apply invariom modeling for standard small-molecule structures no further calculations nor extra experimental procedures are necessary, making it a rapid, readily accessible, and useful tool for standard crystallographic work. In this way high-throughput techniques, which have been established in a variety of fields of X-ray di raction-based structure research, can also be applied in experimental electron-density work. This would be of special importance in the biological sciences. In medical chemistry, where it is a fundamental challenge to understand drug–target recognition processes, knowledge of electron-density distribution is a valuable completion of structure information and serves as a basis for a better understanding of such interactions than consideration of steric properties only. Since rapid screening of a large number of chemical compounds is indispensable in structure-guided drug discovery, the generation of electronic information of entire classes of chemically or pharmacologically related compounds would be highly desirable. This can become a routine task to be performed in time periods comparable to those currently needed for conventional X-ray analyses.