18.6.1 Promolecular Densities
Densities are stable (see Sect. 18.3.2.2) to such an extent that NCI characteristics
are already contained in the sum of atomic densities, ρ i
at [63, 64]. The resulting
molecular density, also known as promolecular density, ρ
pro , is then given by:
q
pro
¼
X
i
q
at
i
ð18:4Þ
A promolecular density obtained from simple exponential atomic pieces is able
to qualititatively predict low-density, low-reduced-gradient regions similar to
density-functional results. The free atomic densities used in these calculations consist
of one Slater-type function for each electron shell, fit to closely reproduce
spherically-averaged, density-functional atomic densities (see Appendix, Fig. 18.19).
Approximate promolecular densities were constructed by summing exponential
atomic densities for bicyclo[2,2,2]octene, and the homomolecular dimers of
methane and water.
Resultant plots of s
pro versus ρ
pro for these species show the same 2D features seen
in Fig. 18.1. Also, 3D isosurfaces generated from the promolecular density are very
similar to those obtained previously with self-consistent DFT and even MP2 densities
(Fig. 18.5). For all cases considered, results at the self-consistent and promolecular
level are qualitatively equivalent. Only slight quantitative differences are introduced
by density relaxation that, as expected, shift the s versus ρ spikes to more bonding
regimes. Specifically, a large shift towards smaller density values is observed in the
spikes corresponding to non-bonded overlap, introducing less repulsion and greater
stability. Taking this shift into account in the choice of isosurfaces, results at the
self-consistent and promolecular level are qualitatively equivalent for all cases
considered (see Fig. 18.5 bottom). For example, lower cut-offs on the gradient (0.25–
0.35) and higher cut-offs on the density (0.08–0.09 a.u.) were required in order to
generate the isosurface for bicyclo[2,2,2]octene.
18.6.2 Examples
Promolecular densities obviously lack relaxation; however, the promolecular densities are extremely useful in biomolecular systems, such as proteins or DNA.
Because the calculation of the electron density in these systems becomes extremely
computationally expensive, the promolecular density becomes an attractive option:
non-covalent interactions can be analyzed with only the molecular geometry
required as input.
We first consider two model polypeptides: an α-helix consisting of 15 alanine
residues and an anti-parallel β-sheet consisting of 17 glycine residues. Geometries
of the polypeptides were obtained with the MMFF force field using the spartan
506
C. Narth et al.
Densities are stable (see Sect. 18.3.2.2) to such an extent that NCI characteristics
are already contained in the sum of atomic densities, ρ i
at [63, 64]. The resulting
molecular density, also known as promolecular density, ρ
pro , is then given by:
q
pro
¼
X
i
q
at
i
ð18:4Þ
A promolecular density obtained from simple exponential atomic pieces is able
to qualititatively predict low-density, low-reduced-gradient regions similar to
density-functional results. The free atomic densities used in these calculations consist
of one Slater-type function for each electron shell, fit to closely reproduce
spherically-averaged, density-functional atomic densities (see Appendix, Fig. 18.19).
Approximate promolecular densities were constructed by summing exponential
atomic densities for bicyclo[2,2,2]octene, and the homomolecular dimers of
methane and water.
Resultant plots of s
pro versus ρ
pro for these species show the same 2D features seen
in Fig. 18.1. Also, 3D isosurfaces generated from the promolecular density are very
similar to those obtained previously with self-consistent DFT and even MP2 densities
(Fig. 18.5). For all cases considered, results at the self-consistent and promolecular
level are qualitatively equivalent. Only slight quantitative differences are introduced
by density relaxation that, as expected, shift the s versus ρ spikes to more bonding
regimes. Specifically, a large shift towards smaller density values is observed in the
spikes corresponding to non-bonded overlap, introducing less repulsion and greater
stability. Taking this shift into account in the choice of isosurfaces, results at the
self-consistent and promolecular level are qualitatively equivalent for all cases
considered (see Fig. 18.5 bottom). For example, lower cut-offs on the gradient (0.25–
0.35) and higher cut-offs on the density (0.08–0.09 a.u.) were required in order to
generate the isosurface for bicyclo[2,2,2]octene.
18.6.2 Examples
Promolecular densities obviously lack relaxation; however, the promolecular densities are extremely useful in biomolecular systems, such as proteins or DNA.
Because the calculation of the electron density in these systems becomes extremely
computationally expensive, the promolecular density becomes an attractive option:
non-covalent interactions can be analyzed with only the molecular geometry
required as input.
We first consider two model polypeptides: an α-helix consisting of 15 alanine
residues and an anti-parallel β-sheet consisting of 17 glycine residues. Geometries
of the polypeptides were obtained with the MMFF force field using the spartan
506
C. Narth et al.