Isovalue

Besides the expressions for a surface derived from the van der Waals surface (see also the CPK model in Section 2.11.2.4), another model has been established to generate molecular surfaces. It is based on the molecular distribution of electronic density. The definition of a Limiting value of the electronic density, the so-called isovalue, results in a boundary layer (isoplane) [187]. Each point on this surface has an identical electronic density value. A typical standard value is about 0.002 au (atomic unit) to represent electronic density surfaces. 

Isovalue-based surfaces are also often used for the representation of molecular orbitals. 

Besides molecular orbitals, other molecular properties, such as electrostatic potentials or spin density, can be represented by isovalue surfaces. Normally, these scalar properties are mapped onto different surfaces see above). This type of high-dimensional visualization permits fast and easy identification of the relevant molecular regions. 

Figure 2-125. Different isovalue-based surfaces of phenylalanine a) isoelectronic density b) molecular orbitals (HOMO-LUMO) c) isopotential surface and d) isosurface of the electron cryo-microscopic volume of the ribosome of Escherichia coii.

Atomic volumes play an important role in relating physicochemical properties to biological effects. Most atoms in molecules are not entirely bounded by interatomic surfaces and an atomic volume is defined as a measure of the space enclosed by the intersection of the atom s zero-flux surfaces with some outer envelope of the density. The envelope with a value of 0.001 au is generally chosen as this has been shown to yield molecular sizes in good agreement with experimentally assigned van der Waals radii [16, 17]. A related property is the van der Waals surface area, which QTAIM determines by integrating an atom s exposed contribution to a molecule s isovalued surface. 

Figure 1. Symmetry-unique SC orbitalsfor the gas-phase Diels-Alder reaction along the CASSCF(6,6) IRC at IRC -0.6 amu bohr (leftmost column), TS (IRC = 0) and IRC +0.6 amu bohr (rightmost column). Three-dimensional isovalue surfaces, corresponding to / = 0.08, were drawn from virtual reality modelling language (VRML) files produced by MOLDEN [il].

The evolution of the shapes of the SC orbitals with the progress of the cyclohexadiene ring-opening is illustrated by Fig. 5. The symmetry-unique SC orbitals tj/i-ti/s are shown as three-dimensional isovalue surfaces at the cyclohexadiene end of the IRC segment (leftmost column of orbitals), at the TS (central column of orbitals) and at the hexatriene end of the IRC segment (rightmost column of orbitals). The reflections of t /i, tj/2 and /3 in the symmetry plane which is retained throughout this IRC interval, result in /6, /5 and tj/4, respectively. 

Among the quantities which have proven of value as graphical models are the molecular orbitals, the electron density, the spin density (for radicals and other molecules with unpaired electrons), the electrostatic potential and the local ionization potential. These may all be expressed as three-dimensional functions of the coordinates. One way to display them on a two-dimensional video screen (or on a printed page) is to define a surface of constant value, a so-called isovalue surface or, more simply, isosurface. ... 

The isochromes, lines where the transmitted intensity is cancelled by interference, provide isovalue curves for the difference in the principal stresses. Along the die axis, the maximum order of the fiinges in the inflow area will be denoted Ke and the extreme order in the outflow area denoted Ko. The order of the fidnges at the wall before the outflow area will be denoted Kw. 

Fig. 6. Refractive indices and extinction coefficients (both given at a wavelength of 550 ran) of several metals and semiconductor materials, as found in the literature. Some isovalue-curves of nk product are shown (most optical constants values are extracted from Palik, 1985, and from J. A. Woollam WVASE software, 2009).

Fig. 30.4 o and redox active molecular orbitals (RAMOs) of the 1V54B and 1V540 models of the Cua site in the gas phase. AU isovalue surfaces are set at 0.03 (e/A ). Molecular structures are shown in thin lines... 

Let it be the collection of all isovalued curves (or surfaces) of v in Y. Note that different isovalued curves never intersect. Since Yis convex and u is continuous, V = u[y] is an interval which contains dense countable rational numbers. Their corresponding isovalued curves are thus countable and dense in ). 

Figure 1. Isovalue surfaces of A2FI (p) for (a) tolane, 1, in the absence of an external field, (b) tolane thiolate, 2, in the absence of an external field, (c) tolane thiolate in an external field oriented in cooperation with the electron donation of sulfur (forward bias), and (d) tolane thiolate in an external field oriented in opposition to the electron donation of sulfur(reversebias). A2II (p) = -0.025 au everywhere on these surfaces, and A2n (p)< -0.025 au everywhere within these envelopes. The orientations of the molecules, and the momentum-space axes are as described in the "GettingOriented" section.

Figure 8.3 Spin density distribution for OLYP, spin density difference distributions with respect to OLYP and the seven highest occupied MOs that determine the spin densities. An isovalue of 0.003 was chosen. A blue surface indicates an excess of a-electron density, while a yellow surface...

Fig. 11 Unpaired electron density for the Ag (top) and B3u (bottom) states of the 11-acene (isovalue 0.005 a.u.) of the n-MR-AQCC/RAS(6)/ CAS (4,4)/AUX(6)/6-31G calculation with individual atomic populations computed from a Mulliken analysis...

Is an accepted convention to call maps of the isovalue contours plotting of... 

Fig. 10 RdPCA 3D-free energy representation of the dihydroxylated compound IV each of the nine iso-surfaces corresponds to points of the 3D-space (PCI, PC2, PC3) with a constant free energy isovalue (in kcal mor ) the ring symbolizes the projection on the previous (PCI, PC2) 2D-representation energy origin is the same as in Fig. 9B charge set 1 in vacuum (1 ns).

Isovalue surfaces of L(r) = 0 for CO (left) and BH3 (right) showing regions of valence-shell charge concentration (VSCC, inside surfaces) and depletion (VSCD, holes) that define the reactive surface and direction of interaction between a Lewis base and a Lewis acid. Adapted from ref [42]. Copyright 1994 Oxford University Press USA. 

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