Equidensity

In addition to the critical points an important feature appearing in one-electron densities are the closed contour curves corresponding to a given value of the density [20]. These closed equidensity curves envelop each one of the nuclei up to the bond critical point located along the bond path joining two nuclei. Other closed equidensity contours envelop both nuclei, etc. This is shown schematically in Fig. 2 for the LiH molecule. 

Let us consider now the application of local-scaling transformations to sets of single-particle functions or orbitals. As it was shown in Sect. 2.1, a set of plane waves gives rise to the transformed orbitals described by Eq. (2). In particular, the application of this transformation to one-dimensional plane-waves leads to Harriman s equidensity orbitals [27], which are given by ... 

Note that these orbitals satisfy the equidensity condition iA (a, b, c) b, c) = p(a, b, c)/N hence, a single Slater determinant constructed from these orbitals yields the density ... 

In polar coordinates, we have the following explicit form for these equidensity orbitals [27] ... 

Any equidensity path y(p) = C is closed and continuous and the whole set of these paths covers the entire space of when C ranges over the domain of... 

Figure 1. Intramolecular vibrational density redistribution IVR of Na3

These locally-scaled plane-waves are precisely the equidensity orbitals of Har-riman [25]. When the generating orbitals of Eq. (20) are substituted into Eqs. (9) and (10) (for the latter we assume spherical symmetry, viz., 1 = 0 for all i) we obtain the following constant term for the local-scaling plane-wave modulating function... 

Figure 3. Isosurfaces of electron densities calculated by a MEM analysis using from a preliminary structure model that does not contain H20 molecules. Balls indicate atoms used in the structure model. Equidensity level 1.2 A-3.

Figure 4. Structure model (left) and isosurfaces of electron densities determined for the sodium cobalt oxide hydrate by MEM analysis (right). Equidensity level 1.0 A 3.

One may use different types of molecular cavities and surfaces definitions (e.g. equipotential surfaces, equidensity surfaces, van der Waals surfaces). Among them there is a subset that shares a common trait they consider that a molecule may be represented as a set of rigid interlocking spheres. There are three such surfaces a)... 

The particular choice of the equidensity orthmormal orbitals defining the Slater determinant that yields a prescribed electrcm density p(r) has been proposed by Harriman [18] on the basis of the pioneering works by Macke [19] and Gilbert [20]. Alternative constructions and extensions have also been suggested [21, 22]. in die density functional theory such A-electron wavefiinctions are mvolved in the formal density constrained search of Levy [17]. 

Let us consider, for simplicity, the one-dimensional case. The Harriman [18] equidensity orbitals for isolated subsystems A and S, e.g., very distant reactants in the combined M =A -8 system, that give rise to the prescribed separated subsystem densities p (x) and p (x), normalized to = f p (x)dx and... 

These orbitals, although no longer of the equidensity type, do preserve the subsystem densities ... 

The 0-0 axis is practically collinear to the C-H bond for the closed shell state and perpendicular for the open shell one. Also, the 0-0 distance is much larger for the latter than for the former. The C-H bond for the open shell state is long and hence weakened. The spin equidensity surface at 0.02 e /bohr is evocative of the incipient formation of a 1 electron second C3-O2 bond and of a 1 electron bond between Ci and Oi. 

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