Bond critical points, electron density analysis

Becke and Roussel (BR) functional, 185 Bell, Evans, Polanyi (BEP) principle, 364 Bell correction, tunnelling, 391 Bending energy, in force fields, 11 BLYP function, 188 Boltzman probability function, 374 Bond critical points, electron density analysis, 225... 

Vector quantities, such as a magnetic field or the gradient of electron density, can be plotted as a series of arrows. Another technique is to create an animation showing how the path is followed by a hypothetical test particle. A third technique is to show flow lines, which are the path of steepest descent starting from one point. The flow lines from the bond critical points are used to partition regions of the molecule in the AIM population analysis scheme. 

Bond critical points represent extremes of electronic density. For this reason, these points are located in space where the gradient vector V p vanishes. Then the two gradient paths, each of which starts at the bond critical point and ends at a nucleus, will be the atomic interaction line. When all the forces on all the nuclei vanish, the atomic interaction line represents a bond path. In practice, this line connects two nuclei which can consequently be called bonded [5]. In terms of topological analysis of the electron density, these critical points and paths of maximum electron density (atomic interaction lines) yield a molecular graph, which is a good representation of the bonding interactions. 

A topological analysis of the electron density in the framework of AIM theory, performed for the systems in Figure 6.2, has completely confirmed their formulation as dihydrogen-bonded complexes. In accord with the AIM criteria, the pc and V pc parameters at the bond critical points found in the H- - -H directions are typical of dihydrogen bonds 0.042 and 0.057 au for complex LiH HF and 0.046 and 0.048 au for complex NaH- - -HF, respectively. The presence of the bond critical points can be well illustrated by the molecular graph in Figure 6.3, obtained for the HCCH H-Li complex by Grabowski and co-workers [8]. 

For comparison, the authors have probed a complex formed by the same proton-donor molecule and molecular hydrogen. In this very weak complex, HCCH- - (H2), the H- - (H2) distance has been calculated as 2.606 A (i.e., significantly larger than the sum of the van der Waals radii of H). It is extremely interesting that a topological analysis of the electron density also leads to the appearance of the bond critical point in the H- - (H2) direction. However, the Pc and V pc values are very small (0.0033 and 0.0115 au, respectively) compared with those in the HCCH- - -HLi complex (0.0112 and 0.0254 au, respectively). The most important conclusion of this comparison is There is no evident borderline between the dihydrogen-bonded complexes and the van der Waals systems. 

NH4-CH4]+ complex in the gas phase [36]. Topological analysis of the electron density performed in the framework of AIM theory shows the bond critical points on the H- H directions with pc values of 0.013 an. It is interesting that the electron density in this complex is larger than that obtained for the BH4 - CH4 dihydrogen-bonded system (pc = 0.007 an), the CH4 molecule of which acts as a proton donor. In accordance with the electronic density, the H- H distances in the BH4 - H4C complex were remarkably longer than 2.4 A (2.797, 2.929,... 

AIM topological analysis of the electron density performed for two complexes and for isolated components is shown in Table 6.15. The bond critical points found in the H H directions are characterized by the small electronic density with Pc = 0.002 and 0.009 au in the CILj- HF and SilLj- HF systems, respectively. The Laplacian, V pc, is also small but takes positive values in accordance with the AIM criteria for dihydrogen bonding. 

Here, pb is the bond critical point (saddle point in three dimensions, a minimum on the path of the maximum electron density). In Eq. (44), and A.2 are the principal curvatures perpendicular to the bond path. The parameters A and B in Eq. (45) determined using various basis sets are given in Bader et al. [83JA(105)5061]. Convenient parameters in the quantitative analysis of a conjugation effect are the relative 7r-character tj (in %) of the CC formal double or single bonds determined with reference to the bond of ethylene (90MI2) ... 

A fourth type of approach relies on analyzing the overall electron density, rather than the density contributed by individual orbitals. A particularly popular analysis of this type is the atoms in molecules analysis (AIM).6 Variation of the density through space can be shown to map onto the bonding pattern. The presence of a bond between two atoms is revealed by the presence of a saddle point, or bond critical point, in the density near the bond midpoint. 

The atoms-in-molecules (AIM) analysis of electron density, using ab initio calculations, was considered in Section 5.5.4. A comparison of AIM analysis by DFT with that by ab initio calculations by Boyd et al. showed that results from DFT and ab initio methods were similar, but gradient-corrected methods were somewhat better than the SVWN method, using QCISD ab initio calculations as a standard. DFT shifts the CN, CO, and CF bond critical points of HCN, CO, and CH3F toward the carbon and increases the electron density in the bonding regions, compared to QCISD calculations [107]. 

The electron density analysis within the AIM methodology shows the presence of bond critical points due to the formation of the HB and, in some homochiral dimers, also between oxygen atoms of different molecules (Fig. 3.13). The presence of the latter bcps is associated to those complexes where the homochiral complex is more stable than the heterochiral one. 

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