Density Laplacians

Keywords Hydrogen bond AIM electron density Laplacian of electron density water DNA ... 

First row reference numbers. N is the number of HBs included in the sample, R, and flniax are minimum and maximum O H separations, PBcp( niiii)/V PBCp( ) PBCp( max)/ PBCp(- max) are the density/Laplacian at the BCP corresponding to R and R . Minimum/ maximum values in the sample MinpBcp M PBCP and MinV pecp/ MaxV pecp- Units are e and A... 

Note that from the mathematical standpoint the condition (6.178) implies that Vrp is parallel to the tangent to the surface S. Instead of examining the expression occurring in (6.178), the density Laplacian can be analysed, since substituting into the volume integral in (6.176) for i Vp, the condition (6.178) may be replaced by the requirement... 

Therefore, the effect of the density Laplacian is included implicitly in the kinetic energy density. It is natural that the next step in density gradient correction is the kinetic energy density correction on Jacob s ladder (see Sect. 5.1). Major meta-GGA functionals include the van Voorhis-Scuseria 1998 (VS98) meta-GGA exchange-correlation (van Voorhis and Scuseria 1998), the Perdew-Kurt-Zupan-Blaha (PKZB) meta-GGA exchange-correlation (Perdew et al. 1999), and the Tao-Perdew-Staroverov-Scuseria (TPSS) meta-GGA exchange-correlation (Tao et al. 2003) functionals. 

The term, meta-GGA functional, first appeared in the PKZB meta-GGA exchange-correlation functional (Perdew et al. 1999). The PKZB exchange functional intends to enhance the PBE-GGA functional using the kinetic energy density on the basis of the fundamental conditions extended to the density Laplacian,... 

Table 3 lists the calculated topological properties at the bond critical points. According to the topological theory of atoms in molecules the positive values of the electron density Laplacian at the bond critical point (bcp, where Vp=0) are associated with closed shell interactions (ionic bonds, hydrogen bonds,... 

Another class of recent functionals consists in functionals containing density laplacian terms besides the density and its gradient. This class is also termed as meta-GGAs, and may contain also hybrid functionals (i.e. containing also a pure exchange contribution). Whereas the inclusion of a Laplacian term has been proven to be beneficial for van der Waals or hydrogen bonds 35.36 this is less obvious for other molecular properties of small molecules... 

Noteworthy, BCPs properties usually provide valuable information about the bond itself. For instance, covalent bonds are characterized by high electron density values (pc) at BCPs, while pc is typically very low for non-covalent bonds like hydrogen, agostic, halogen, and van der Waals ones [94—97]. Another important descriptor is the BCP density laplacian value that quantifies the electron accumulation (negative values) or depletion (positive values) at the BCP. Thus, very negative values are observed for covalent bonds ( shared interactions in Bader s vocabulary), while ionic ( closed-shell interactions) ones feature positive values. Many relationships between interaction energies and (p, have been reported in the literature. 

As first examples, we will consider the CI3COX...NH3 complexes (X = F, Cl, and Br) that belong to the R3COX...NH3 family, which was studied by Politzer [56] and recently revisited by us within the framework of conceptual DFT [67] and density Laplacian maps [68]. 

Tognetti V, Joubeit L (2015) Electron density Laplacian and halogen bonds. Theor Chem Acc 134 90... 

Bucinsky L, Kuckova L, Malcek M, KoJisek J, BiskupiiS S et al (2014) Picture change error in quasirelativistic electron/spin density, Laplacian and bond critical points. Chem Phys 438 37 7... 

In a study dedicated to ELF at correlated level, Matito et al. [77] proposed to use as the kernel the sum of the conditional same-spin pair density Laplacians (up-spin and down-spin) ... 

As the 2-particle control functimi can serve, for instance, the same-spin electron pair density, />f"(ri,r2) normalized to the total number of spin electron pairs NaiNg— l)/2. The integral of the same-spin pair density over each micro-cell volume yields the number of same-spin pairs in the micro-cell. As will be shown later, the number of pairs U,- can be approximated by the integral of the pair density Laplacian (in which case 5c = 8/3 and tc(r,) is the Fermi hole curvature). The number of micro-cells with Uf" restricted to the value coo could be determined if the number of pairs formed between the micro-cells would be known as well. 

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