Isodensity

Figure 7-22. Isodensity plots of the four. T-orbftals of 1,3-butadIene. The nomenclature corresponds to that in Figures 7-21 and 7-22 /correct /.

The MEP at the molecular surface has been used for many QSAR and QSPR applications. Quantum mechanically calculated MEPs are more detailed and accurate at the important areas of the surface than those derived from net atomic charges and are therefore usually preferable [Ij. However, any of the techniques based on MEPs calculated from net atomic charges can be used for full quantum mechanical calculations, and vice versa. The best-known descriptors based on the statistics of the MEP at the molecular surface are those introduced by Murray and Politzer [44]. These were originally formulated for DFT calculations using an isodensity surface. They have also been used very extensively with semi-empirical MO techniques and solvent-accessible surfaces [1, 2]. The charged polar surface area (CPSA) descriptors proposed by Stanton and Jurs [45] are also based on charges derived from semi-empirical MO calculations. 

Non-covalent interactions between molecules often occur at separations where the van der Waals radii of the atoms are just touching and so it is often most useful to examine the electrostatic potential in this region. For this reason, the electrostatic potential is often calculated at the molecular surface (defined in Section 1.5) or the equivalent isodensity surface as shown in Figure 2.18 (colour plate section). Such pictorial representations... 

The original PCM method uses a cavity made of spherical regions around each atom. The isodensity PCM model (IPCM) uses a cavity that is defined by an isosurface of the electron density. This is defined iteratively by running SCF calculations with the cavity until a convergence is reached. The self-consistent isodensity PCM model (SCI-PCM) is similar to IPCM in theory, but different in implementation. SCI-PCM calculations embed the cavity calculation in the SCF procedure to account for coupling between the two parts of the calculation. 

The second set of illustrations show the spin density plotted on the electron density isosurface the spin density provides the shading for the isodensity surface dark areas indicate positive (excess a) spin density and light areas indicate negative (excess P) spin density. For example, in the allyl radical, the spin density is concentrated around the two terminal carbons (and away from the central carbon). In the Be form, it is concentrated around the substituent, and in acetyl radical, it is centered around the C2 carbon atom. 

The Isodensity PCM (IPCM) model defines the cavity as an isodensity surface of the molecule. This isodensity is determined by an iterative process in which an SCF cycle is performed and converged using the current isodensity cavity. The resultant wavefunction is then used to compute an updated isodensity surface, and the cycle is repeated until the cavity shape no longer changes upon completion of the SCF. 

An isodensity surface is a very natural, intuitive shape for the cavity since it corresponds to the reactive shape of the molecule to as great a degree as is possible (rather than being a simpler, pre-defined shape such as a sphere or a set of overlapping spheres). 

MaxPoints option 174 RCFC option 176 Restart option 174 StepSize option 200 isodensity surface 238... 

Over the years, many workers have addressed the problem of choice of cavity and the reaction field. Tomasi s polarized continuum model (PCM) defines the cavity as a series of interlocking spheres. The isodensity PCM (IPCM) defines the cavity as an isodensity surface of the molecule. This isodensity surface is determined iteratively. The self-consistent isodensity polarized continuum model (SQ-PCM) gives a further refinement in that it allows for a full coupling between the cavity shape and the electron density. 

The raw output of a molecular structure calculation is a list of the coefficients of the atomic orbitals in each LCAO (linear combination of atomic orbitals) molecular orbital and the energies of the orbitals. The software commonly calculates dipole moments too. Various graphical representations are used to simplify the interpretation of the coefficients. Thus, a typical graphical representation of a molecular orbital uses stylized shapes (spheres for s-orbitals, for instance) to represent the basis set and then scales their size to indicate the value of the coefficient in the LCAO. Different signs of the wavefunctions are typically represented by different colors. The total electron density at any point (the sum of the squares of the occupied wavefunctions evaluated at that point) is commonly represented by an isodensity surface, a surface of constant total electron density. 

Figure 1. Relief maps of the electron density of (a) SCI2 and (b) H2O in the plane of the nuclei (density and distances from the origin of the coordinate system in au). Isodensity contour lines are shown in the order 0.001,0.002,0.004,0.008 (four outermost contours) 0.02,0.04,0.08 (next three) 0.2,0.4,0.8 (next three). The density is truncated at 2.00 au (innermost contour). These contours are shown in blue, violet, magenta, and green, respectively, on the figure in the table of contents (p 1028).

Figure 5. Three-dimensional isodensity envelopes of (a) SCI2, (b) H2O, and (c) Cl2. The outer envelope has the value of 0.001 au, the van der Waals envelope the inner one is the bond critical point density envelope (pb-envelope).

Here we see clearly the large concentration of density around the oxygen nucleus, and the very small concentration around each hydrogen nucleus. The outer contour is an arbitrary choice because the density of a hypothetical isolated molecule extends to infinity. However, it has been found that the O.OOlau contour corresponds rather well to the size of the molecule in the gas phase, as measured by its van der Waal s radius, and the corresponding isodensity surface in three dimensions usually encloses more than 98% of the total electron population of the molecule (Bader, 1990). Thus this outer contour shows the shape of the molecule in the chosen plane. In a condensed phase the effective size of a molecule is a little smaller. Contour maps of some period 2 and 3 chlorides are shown in Figure 8. We see that the electron densities of the atoms in the LiCl molecule are only very little distorted from the spherical shape of free ions consistent with the large ionic character of this molecule. In... 

FIGURE 10. Three-dimensional isodensity envelopes of the bond critical point density (pb-envelope)... 

In the IPCM calculations, the molecule is contained inside a cavity within the polarizable continuum, the size of which is determined by a suitable computed isodensity surface. The size of this cavity corresponds to the molecular volume allowing a simple, yet effective evaluation of the molecular activation volume, which is not based on semi-empirical models, but also does not allow a direct comparison with experimental data as the second solvation sphere is almost completely absent. The volume difference between the precursor complex Be(H20)4(H20)]2+ and the transition structure [Be(H20)5]2+, viz., —4.5A3, represents the activation volume of the reaction. This value can be compared with the value of —6.1 A3 calculated for the corresponding water exchange reaction around Li+, for which we concluded the operation of a limiting associative mechanism. In the present case, both the nature of [Be(H20)5]2+ and the activation volume clearly indicate the operation of an associative interchange mechanism (156). 

Fig. 7.5 The serinyl group NHCH(CH2OH)C(=0) cut from the glycine mold represented by the intersection of its van derWaals 0.001 au isodensity surface with the -C(C=0) or C-surface at the top left and the NH- or N-surface at the bottom center. These are the complementary sides of the amidic zero-flux surface characteristic of a polypeptide. All properties of the residue are defined and make additive contributions to the molecule constructed from it. The residue has a net charge of -0.006 e.

Fig. 7.6 The 33-residue fragment of the winter flounder antifreeze protein (AFP), constructed by the conjoining of the Aa amino acid residues defined within the glycine mold , pictured in terms of its 0.001 au isodensity envelope, its van der Waals envelope. This is a view showing the ice-binding motif. It is believed that the AFP strand binds... 

Fig. 2.6-1. Calculated valence molecular orbitals of tetrahedral P4 (isodensity surface drawn at a cut off of 0.05 a.u.). Only one representation of the similar degenerate MOs is given.

The main purpose of the method is to define molecular shapes through isodensity surfaces. Tests on a number of small molecules show that this aim is achieved with a great efficiency in computer time. Discrepancies between MEDLA densities and theoretical distributions, averaged over the grid points, are typically below 10% of the total density. While this does not correspond to an adequate accuracy for an X-ray scattering model, the results do provide important information on the shapes of macromolecules. 

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