Matta, Boyd. The quantum theory of atoms in molecules

322 12 Fragment Transferability Studied Theoretically and Experimentally with QTAIM

Fig. 12.4 Left: Huber four-circle di ractometer at beamline D3 (Hasylab) equipped with helium cryostat and Bruker Smart 1K CCD area detector. The vacuum chamber wall at the cold head of the cryostat is made of Kapton film. Right: background resulting from di erent vacuum cylinder materials – beryllium (above) and Kapton film (below) [14] Copyright 2003 and Reproduction with Permission from International Union of Crystallography (IUCr).

problem of the vacuum chamber was recently solved by replacing beryllium with a Kapton film as cylinder-wall material of the vacuum chamber (Fig. 12.4) [14].

This results in a much lower and practically unstructured background and makes the combination of a closed-cycle cryostat with a modern area detector feasible. In a comparative study of four high-order data sets for strychnine single crystals at 100, 25, and 15 K, it was shown that I/s ratios of high-order reflections improved most favorably when synchrotron radiation and a temperature of 15 K were used [14].

On the basis of current technical developments, experimental advances in highspeed evaluation of electron densities can be expected – data collection periods of hours or even minutes are within reach. This was demonstrated by a 12-h sequence of di raction experiments recently conducted at the protein crystallography beamline X10 SA of the Swiss light source (SLS) at the Paul Scherrer Institute (Villigen, Switzerland) [15]. It resulted in a total of 400,000 reflections of four high-resolution X-ray data sets for electron density determination. Because of the brilliant primary beam properties, intensities could be observed even for tiny crystals and in very high regions of reciprocal space.

One particular example of this experimental sequence is illustrated in Fig. 12.5, which shows the static map in the purine plane of adenosine after aspherical atom refinement. It was generated from a quick 1 h test dataset of more than 22,000 reflections. The covalent bonding features are properly resolved, even though this data set was acquired in one of the shortest measurements ever conducted in experimental electron-density work.

12.3 Studying Transferability with QTAIM 323

Fig. 12.5 Static deformation density map in the purine plane of adenosine, based, as far as we are aware, on the fastest data set ever acquired for electron-density work (exposure time < 1 h) [15] Copyright 2005 and Reproduction with Permission from American Chemistry Society.

The conclusion reached from all the experimental conditions discussed above is that if beam conditions are stable synchrotron radiation is the first choice for high-resolution electron-density data collection. This is especially true if smaller crystals are to be used; further advantages are reduced absorption and extinction. Cooling to low temperatures of approximately 10–20 K is superior to the 100 K cooling normally used, because many significant high-order reflections can be measured. If, however, crystals are large enough and di ract properly, MoKa radiation can be su cient, as was demonstrated by the data collection of vitamin B12 at 100 K.

The 12-h sequence of di raction experiments described above suggests the possibility of establishing high-throughput techniques in electron density research, thereby making electronic information of entire classes of chemically or biologically related compounds available at an increased pace. Together with the invariom data base approach that will be detailed in the next sections, high-speed evaluation of electron densities could become a routine task to be conducted in a time comparable with those currently needed for conventional X-ray analyses.

12.3

Studying Transferability with QTAIM – Atomic and Bond Topological Properties of Amino Acids and Oligopeptides

A key feature of Bader’s theory of atoms in molecules is the partitioning of a molecular structure into sub-molecular regions, functional groups, or single atoms. The partitioning procedure makes use of the zero-flux surface (ZFS) in the electron-density gradient vector field ‘r(r). Surfaces of this type establish atomic basins around nuclear attractors of the corresponding trajectories of ‘r(r) and

32412 Fragment Transferability Studied Theoretically and Experimentally with QTAIM

Table 12.1 Summary of available experimental charge-density studies on amino acids (from [19]).

uniquely define atomic volumes. Together with identification of critical points on bond paths, rings, and cages, tools are at hand for quantitative evaluation of bonding, atomic, or functional group properties. Transferability of the abovementioned quantitative data is essential for application of database approaches to modeling of the electron density of larger systems. On the electronic level, it can be expected that density and derived properties of a functional group composed of atomic fragments should have high transferability when compared for di erent but chemically related molecules.

The biologically important class of the twenty genetically encoded amino acids was one of the first in which this transferability was systematically examined experimentally and theoretically. Bader and Matta have published complete topological data on all twenty amino acids based on theoretical calculations [16–18] and experimental studies on sixteen of the twenty amino acids have been performed by di erent groups, as detailed in Table 12.1. This class of compounds is thus the first for which a complete set of theoretical electron-density data is available and for which the corresponding experimental studies are approaching completeness.

Quantitative results for bond topological properties are summarized in Fig. 12.6 and Table 12.2 [19]. Figure 12.6a shows r(rbcp) values and Laplacians for the five main chain bonds common to all amino acids (CO, CO(H), CN, Ca aC0 (¼ CCO) and Ca aCb (¼ CCR)) from the sixteen experimental studies. With the

32612 Fragment Transferability Studied Theoretically and Experimentally with QTAIM

Table 12.2 Mean values and absolute and relative standard deviations (s and s=r) for the five main chain bonds. First line – experiment (multipole); second–fifth line – calculations at experimental geometry (HF/6-311þþG**; HF/6-311þþG(3df,3pd); B3LYP/6-311þþG**; B3LYP/6-311þþG(3df,3pd)); sixth line – theoretical values (Matta and Bader, RHF/6-311þþG**) [19].

a n is the number of contributing entries.

12.6b. The positive Laplacians of the CO bonds describe partial ionic character. As a result of this the r(rbcp) values of the CN bond are higher than for the zwitterionic molecules, in accordance with chemical understanding. In the experimental data there is a small di erence between the C0 aO bonds which originates in the weak interactions described.

12.3 Studying Transferability with QTAIM 327

Averages and statistical standard deviations (s values) of the quantities illustrated in Fig. 12.6 are listed in Table 12.2. For the experimental averages of the five bonds considered (first lines in Table 12.2) the s values are, approximately, within 0.07–0.11 e A˚ 3 and 2–5 e A˚ 5 for r(rbcp) and ‘2r(rbcp), respectively. For the 24 entries of Matta and Bader’s theoretical calculations (all obtained with the same basis set RHF/6-311þþG , last lines for each bond in Table 12.2) the s values are much smaller, and here intermolecular interactions are not considered. It was, nevertheless, shown that results of ab initio calculations with di erent basis sets can vary in the same range as experimental data.

It can be concluded that despite a variety of experimental and refinement conditions and di erent b-substituents and crystal environments, the overall results for experiment and theory are very consistent in their respective ranges. With regard to Bader’s concept of transferability of submolecular properties these data are useful for predicting the bond properties of larger systems of biological interest, which often cannot be determined experimentally.

Between single amino acids and macromolecules, for example proteins, are the oligopeptides, which contain the building blocks of proteins. We would like to compare the transferability of atomic volumes and charges in the peptide bond region for several oligopeptides. For this purpose we make use of experimental data for five dipeptides, one hexapeptide, and four tripeptides. The latter are part of a systematic study of tripeptides of the type l-ala–XXX–l-ala, were the central amino acid residue XXX was varied to examine whether this central residue affects transferability in the peptide bond region.

The results obtained so far are summarized in Fig. 12.7. The averages of comparable quantities show that the internal consistency for volumes is <1 A˚ 3. The atomic volumes at the other Ca and C0 atoms are equal within twice the statistical error, except if Ca belongs to a Gly residue. The average volumes of the (non-Gly) Ca atoms are more than 1 A˚ 3 smaller than those of the Gly Ca atoms, in which the second hydrogen atom enables the carbon to expand. A nearest-neighbor effect is also seen for the N atoms. In proline the nitrogen atom is part of the five membered ring and bonded to a third carbon atom instead of hydrogen, which reduces the volume by more than 1 A˚ 3, by analogy with the above quoted volume expansion for the glycine Ca.

The QTAIM charges (for averages see also Fig. 12.7) agree within the given atom types by 0.07–0.16 e, which is a surprisingly small spread. The Ca atoms carry a small positive charge, the hydrogen atoms of the peptide NaH are moderately positively charged and the C0 atoms carry a high positive charge, whereas strong negative charges close to 1 e are observed on the N and O atoms. These experimental results indicate that polarization of Bader atoms is much higher than obtained, for example, from theoretical orbital models (NBO or Mulliken charges) [35] or than used in force field parameterization. For example, the amber [36] force field uses charges of 0.5 for oxygen, þ0.5 for carbon, 0.57 for nitrogen, and þ0.37 e for hydrogen atoms in the peptide bond. The definition and determination of atomic charges have been subjects of controversial discussions in recent years [35] and charges derived from di erent methods may di er significantly.

328 12 Fragment Transferability Studied Theoretically and Experimentally with QTAIM

Fig. 12.7 Average atomic charges (e) and volumes (A˚ 3) of the atoms in the peptide bond region. A refers to five dipeptides and one hexapeptide [33]; B refers to the four tripeptides AAA.12H2O [34], APA.H2O, AYA.H2O, and AYA.C2H5OH (to be published); av is the average over all entries. n is the number of contributing entries.

QTAIM charges are based on well-defined atomic segments of the electronic charge density and can be derived from an experiment [37, 38].

The positive charges on the Ca, C0, and H atoms total approximately þ1.7 e and the negative charges on N and O amount to approximately 2e, so for each peptide bond region an excess of 0.3 e must be compensated by the side-chains or, for glycine, another hydrogen atom.

A preliminary conclusion of this study of the peptide bond is that very reproducible atomic properties for the contributing atoms can be derived if the chemical environment is comparable. A significant experimentally detectable e ect of next-nearest neighbors on the electron density of Ca-type atoms was not observed. The results therefore show the validity of the nearest/next-nearest neighbor approximation and encourage the use of database approaches for electron-density modeling of macromolecules.

12.4

Invariom Modeling

Conventional interpretation of atomic-resolution X-ray data is based on the promolecule model (superimposition of spherically symmetric, isolated atomic den-

330 12 Fragment Transferability Studied Theoretically and Experimentally with QTAIM

The method described here di ers in important aspects from the experimental approach suggested by Pichon-Pesme et al. [45–47] for ultra-high resolution protein crystallography. Because our approach does not rely on experimental data, measurement errors and bias arising as a result of inadequate modeling of the thermally smeared density are excluded. We also provide a consistent definition for transferable sites in terms of invarioms that can be generated, in principle, for all chemically relevant bonding situations at any level of theory.

Some relevant details of a similar database approach by Volkov et al. [48] must be mentioned. In the work of Volkov et al. populations are averaged over model compounds that were chosen from a large possible range, so that deviation from electrical neutrality is di erent from results using our invariom database, in which only one defined model compound is used for each invariom. Rules for setting up a model compound will be given below. Inclusion of a notation scheme in the invariom concept also enables automation of least-squares refinement of experimental X-ray data.

12.4.1

Invariom Notation, Choice of Model Compounds, and Practical Considerations

In invariom notation the element symbol of the atom of interest, in capitals, begins the name, then formal bond order and nearest neighbors follow in lower case ordered by their position in the periodic table (heavier atoms first). The order of the ligands in the invariom name is determined by their decreasing bond order. For mesomerism or delocalized systems a bond order of 1.5 is specified. For these, the next-nearest neighbors must be taken into account in the name. Next-nearest neighbors are also used for invariom names of hydrogen and hypervalent atoms. This is achieved by writing the next-nearest neighbors according to the previous rules in brackets after the nearest neighbors (examples are given in Fig. 12.8). For chiral invarioms, R or S is used as a prefix (CIP rules), separated by a dash. Because, usually, next-nearest neighbors only are considered, chiral invarioms occur a lot less frequently than chiral atoms. Finally a þ or a indicates a charge when the model compound used is an ion, for example the acetic acid anion. The sign is best placed at the very end of the name.

The invariom approach di erentiates between chemical environments by distinguishing between single and mesomeric bonds, which enables rules to be set up for generation of model compounds. For single-bond systems the model compounds simply include the nearest neighbors saturated by hydrogen atoms. This is also valid for double/triple bonded systems. Charged groups, for example RaNH3þ, require a charged model compound. For hydrogen atoms, next-nearest neighbors must be included to achieve proper electroneutrality during summation over invariom fragments at the application stage. Next-nearest neighbors are also included for delocalized systems and hypervalent atoms. When extended mesomeric or delocalized systems occur in a structure, the whole mesomeric fraction of the molecule must be included in the model compound in a suitable

12.4 Invariom Modeling 331

manner. A good example is the model compound of a carbon invariom assigned to any atom of a benzene ring. The obvious advantage of calculating the complete ring is that use of a radical – here the propenyl-radical taking into account nearest neighbors only – can be avoided.

For practical application of invariom modeling analysis of the IAM geometry is the starting point. For this purpose the empirical relationship below [49] is used to identify covalent bonds, where d is the distance between the atoms, in A˚ ngstroms, rc is the covalent radius, and jDðENÞj the di erence between the Allred– Rochow electronegativities [50] of two atoms in a structure.

When d 0:85 is smaller than the sum of the covalent radii of the two atoms multiplied by an electronegativity di erence a bond is found. To di erentiate between single/mesomeric/double, and triple bonds the same empirical relationship, Eq. (1), is used to assign a bond distinguishing term w (Eq. 2):

The distinction between a singe/mesomeric/double bond is arbitrary and not very distinctive for some delocalized systems, but experience with a large number of structures and optimized model compounds led to the establishment of reliable values for sensible distinction where to include next-nearest neighbors and where it is unnecessary.

Software [51] has been developed that enables the automation of the modeling process. It automatically analyses molecular geometry, assigns invariom names to each atom in a structure, uses invariom notation to find and transfer database density values and writes input files for the respective aspherical-atom refinement program xd [52].

12.4.2

Support for Pseudoatom Fragments from QTAIM

We were interested whether or not it is possible to support the invariom approach by means of QTAIM. Integration over the atomic basin as defined by the zero flux surface of r(r) yields well-defined atomic charges, and we will focus on theoretically calculated AIM charges of several chemically related example molecules to investigate the limits of transferability.

The level of transferability needed for the X-ray scattering model is limited to the possible accuracy and resolution of the experimental data. Although perfect transferability is not reached in theory, current experimental limitations enable approximations to be made. One factor reducing transferability is hydrogen bonding; di erent conformations [16] can also lead to slightly di erent charges.