Electron Densities for Molecular Quantum Similarity

Clearly, the electron density will depend on the method that obtains it. Nonetheless, even the simple Hartree-Fock (HF) electron density performs well to obtain first-order electron densities. The electron density in this MO LCAO framework is given by 

this usual notation may be found in Szabo and Ostlund. The Xv(f) d X (r) are the basis functions and Dv is the density matrix, also called the charge density-bond order matrix. It follows that the MQSM in this framework is given by 

Clearly, the Hirshfeld promolecular electron density is not likely to simplify the integrals in Eq. [39]. The essential difference between the Hirshfeld and ASA promolecular densities is that in the ITirshfeld method, the isolated atom electron densities pa(r) are obtained in the same basis set as the one in the ab initio calculation of the true molecular electron density, whereas in the ASA approach, the isolated atom densities are obtained in the way as described below. In the ASA method, we use a slightly different promolecular atomic shell approximation (PASA) electron density, where the number of electrons Pa attached to each atom a is introduced. The total promolecular electron density for an N-atom molecule is given by 

In most applications, this number is set equal to the atomic number of the element involved, Z. In some applications, is set to the number of electrons attributed to the atom according to the results of some type of population analysis. The isolated atom electron densities are optimal linear combinations of s-type Gaussians. That is. 

The tVi are the expansion coefficients for the M s-type Gaussians, and we can see immediately the link between Eq. [42] and the wave function quadrature. So, for the calculation of ASA-based promolecular electron densities, we first need to develop a scheme for the fitting of the atomic densities. The exponents of the Gaussians may be chosen from, e.g., a well-tempered series.The coefficients may then be fitted against the true atomic ab initio electron density. Once these exponents and coefficients are set, these Gaussian exponents and coefficients are universally applicable. Promolecular densities p (r) can then be obtained quickly from Eq. [41]. 

---

Earlier density extension results were proven only for parts of artificial molecular electron densities, where the complete molecule was assumed to be confined to a finite, bounded region of the three-dimensional space [21], a condition that violates quantum mechanics. However, the new Holographic Electron Density Fragment Theorem quoted here proves the unique extension property of parts of quantum-mechanically correct, boundaryless electron densities of molecules. This new theorem is of special importance with respect to transferability, establishing that for complete, boundaryless molecular electron densities no actual fragment density of sharp boundaries is perfectly transferable. This result has implications on using averaged electron densities for similarity analysis [162]. 

Although one can extend this discussion to even higher order electron densities, p(ri) and p(ri,r2) are the most commonly used quantities in molecular quantum similarity. It is worth noting that often no order is mentioned in publications when considering electron density. It is then commonly accepted that one then refers to the first-order density. In the remainder of this chapter, we assume that the first-order density is used, except when an order is mentioned explicitly. An extremely important property of p(ri) is its positive definite nature, which may seem like a simple consequence of its probability nature, but this point can hardly be overemphasized for its application in quantum similarity, as will be shown. 

Having established that the electron density is the basic molecular descriptor, and that a theoretical justification exists for its selection, the theory of molecular quantum similarity can now be developed. 

Two reasons exist for the good performance of ASA and PASA densities in studying molecular quantum similarity. First, because the highest electron density is concentrated near the nucleus, it is easy to understand that if we use functions that give a fair or good representation of that area, the contribution from that area of the molecule in the total similarity will be approximated well. Second, these subvalence regions are usually reasonably transferable... 

As discussed earlier, we cannot only derive first-order electron densities, but also we can extend them to higher order electron densities. We have used the second-order electron density p(ri,r2) in lieu of the first-order electron density on several occasions in molecular quantum similarity, because the second-order electron density is in fact the lowest order density where electron correlation becomes apparent. It has been used extensively by Ponec et al. "- in the study of similarity in pericyclic reactions where the second-order electron density offers important advantages over the first-order electron density. In another contribution, Ponec et al. went to the third-order electron density. Again, most of the discussion relating to molecular quantum similarity indices and molecular alignment is also applicable to higher order electron densities, replacing where necessary the first-order electron density by, for example, the second-order electron density. 

It is beyond the scope of this chapter to discuss the range of structure-based methods that chemists can use for molecular afignment. This field of research has been, and continues to be, very active. One algorithm, called TGSA, will be presented here in some detail, however, because of its popularity in molecular quantum similarity studies. Structure-based techniques differ from the aforementioned techniques in several respects. First, they not attempt to maximize the MQSM for a pair of molecules. Second, they do not make a specific reference to molecular quantum similarity as such, they are aimed at a wider range of applications. Third, they are not based on electron density in a formal way, but instead they take a more familiar approach based on chemical topology. Consequently, they apply well-known concepts such as chemical bonds and try to overlap the most similar and largest common structure elements in both molecules. 

A field that has attracted special attention on different occasions is that of the similarity between two atoms located in two different molecules. Eor example, we can imagine calculating the similarity between two carbonyl carbon atoms, one in molecule A and one in molecule B, or calculating the similarity between a carbon atom in a molecule and the isolated atom, or between two different carbon atoms in the same molecule. From the perspective of molecular quantum similarity, calculating the similarity measure will require atomic electron densities within the molecule. Therefore, before turning to the similarity calculation, two methods for obtaining such densities will be briefly presented. 

Boon et al. also studied several chiral molecules, which included again two amino acids (Ala and Leu) and CHFClBr, a prototype of chiral molecules. Ab initio total molecular electron densities yielded both Euclidean distances and Carbo indices between the enantiomers of these molecules. Molecular superposition was performed with, on the one hand, a manual alignment based on chemical intuition and the QSSA method, on the other hand. When analyzing the tables of the work by Boon et al. and comparing the results to the work by Mezey et al., similar values for the Euclidean distances between the two enantiomers appear for Ala and Leu, which once again illustrates the power of the ASA promolecular densities to yield quantum similarity measures in good agreement with those obtained from ab initio calculations. 

In more recent years, additional progress and new computational methodologies in macromolecular quantum chemistry have placed further emphasis on studies in transferability. Motivated by studies on molecular similarity [69-115] and electron density representations of molecular shapes [116-130], the transferability, adjustability, and additivity of local density fragments have been analyzed within the framework of an Additive Fuzzy Density Fragmentation (AFDF) approach [114, 131, 132], This AFDF approach, motivated by the early charge assignment approach of Mulliken [1, 2], is the basis of the first technique for the computation of ab initio quality electron densities of macromolecules such as proteins [133-141],... 

Mestres, J., M. Sola, M. Duran, and R. Carbo. 1996. On the Calculation of Ab Initio Quantum Molecular Similarities for Large Systems Fitting the Electron Density. J. Comp. Chem. 15, 1113. 

Apparently, the concept of similarity plays an important role in the chemistry of functional groups. Motivated by the recent revival of interest in molecular similarity [7-39], we shall present a systematic approach towards a quantum chemical description of functional groups. There are two main components of the approach described in this report. The first component is shape-similarity, based on the topological shape groups and topological similarity measures of molecular electron densities[2,19-34], whereas the second component is the Density Domain approach to chemical bonding [4]. The topological Density Domain is a natural basis for a quantum... 

---