Electron density, definition

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. 

At a physical level. Equation 35 represents a mixing of all of the possible electronic states of the molecule, all of which have some probability of being attained according to the laws of quantum mechanics. Full Cl is the most complete non-relativistic treatment of the molecular system possible, within the limitations imposed by the chosen basis set. It represents the possible quantum states of the system while modelling the electron density in accordance with the definition (and constraints) of the basis set in use. For this reason, it appears in the rightmost column of the following methods chart ... 

We know a number of things about the electron density P(r). First of all, if we integrate it over space, we get the number of electrons, m. This follows simply from the definition... 

Whenever one compound uses its electrons to attack another compound, we call the attacker a nucleophile, and we call the compound being attacked an electrophile. It is very simple to tell the difference between an electrophile and a nucleophile. You just look at the arrows and see which compound is attacking the other. A nucleophile will always use a region of high electron density (either a lone pair or a bond) to attack the electrophile (which, by definition, has a region of low electron density that can be attacked). These are important terms, so let s make sure we know how to identify nucleophiles and electrophiles. 

According to wave mechanics, the electron density in an atom decreases asymptotically towards zero with increasing distance from the atomic center. An atom therefore has no definite size. When two atoms approach each other, interaction forces between them become more and more effective. 

The contribution of the frontier orbitals would be maximized in certain special donor-acceptor reactions. The stabilization energy is represented by Eqs. (3.25) and (3.26). Even in a less extreme case, the frontier orbital contribution maybe much more than in the expression of the superdelocalizability. If we adopt the approximation of Eq. (6.3), the intramolecular comparison of reactivity can be made only by the numerator value. In this way, it is understood that the frontier electron density, /r, is qualified to be an intramolecular reactivity index. The finding of the parallelism between fr and the experimental results has thus become the origin of the frontier-electron theory. The definition of fr is hence as follows ... 

The definition of the radius of an ion in a crystal as the distance along the bond to the point of minimum electron density is identical with the definition of the radius of an atom in a crystal or molecule that we discuss in the analysis of electron density distributions in Chapter 6. The radius defined in this way does not depend on any assumption about whether the bond is ionic or covalent and is therefore applicable to any atom in a molecule or crystal independently of the covalent or ionic nature of the bond, but it is not constant from one molecule or crystal to another. The almost perfectly circular form of the contours in Figure... 

Clearly the form of a deformation density depends crucially on the definition of the reference state used in its calculation. A deformation density is therefore meaningful only in terms of its reference state, which must be taken into account in its interpretation. As we will see shortly, the theory of AIM provides information on bonding directly from the total molecular electron density, thereby avoiding a reference density and its associated problems. But first we discuss experimentally obtained electron densities. 

Calibration to absolute intensity means that the scattered intensity is normalized with respect to both the photon flux in the primary beam and the irradiated volume V. Thereafter the scattering intensity is either expressed in terms of electron density or in terms of a scattering length density. Both definitions are related to each other by Compton s classical electron radius. 

This definition leads to an electron density measured in electrons per nm3 (cf. Sect. 7.10.1). If we are aiming at a treatment in terms of scattering cross-section we define /o = Zre, instead. 

The formal definition of the electronic chemical hardness is that it is the derivative of the electronic chemical potential (i.e., the internal energy) with respect to the number of valence electrons (Atkins, 1991). The electronic chemical potential itself is the change in total energy of a molecule with a change of the number of valence electrons. Since the elastic moduli depend on valence electron densities, it might be expected that they would also depend on chemical hardness densities (energy/volume). This is indeed the case. 

This expression indicates that there is a decreased probability (indicated by the term —2(/>A(/>B) of finding the electrons in the region between the two nuclei. In fact, there is a nodal plane between the positive and negative (with respect to algebraic sign) of the two regions of the molecular orbital. As a simple definition, we can describe a covalent bond as the increased probability of finding electrons between two nuclei or an increase in electron density between the two nuclei when compared to the probability or density that would exist simply because of the presence of two atoms. 

The three quantum numbers may be said to control the size (n), shape (/), and orientation (m) of the orbital tfw Most important for orbital visualization are the angular shapes labeled by the azimuthal quantum number / s-type (spherical, / = 0), p-type ( dumbbell, / = 1), d-type ( cloverleaf, / = 2), and so forth. The shapes and orientations of basic s-type, p-type, and d-type hydrogenic orbitals are conventionally visualized as shown in Figs. 1.1 and 1.2. Figure 1.1 depicts a surface of each orbital, corresponding to a chosen electron density near the outer fringes of the orbital. However, a wave-like object intrinsically lacks any definite boundary, and surface plots obviously cannot depict the interesting variations of orbital amplitude under the surface. Such variations are better represented by radial or contour... 

These definitions apply to any atomic system, molecule or crystal. Fig. 7.3 a illustrates their application to the charge distribution of the guanine-cytosine base-pair. Fig. 7.3 b shows the molecular structure defined by the bond paths and the associated CPs that clearly and uniquely define the three hydrogen bonds that link the two bases. Fig. 7.3 c shows the atomic boundaries and bond paths overlaid on the electron density in the plane of the nuclei. All properties of the atoms can be determined, enabling one, for example, to determine separately the energy of formation of each of the three hydrogen bonds. 

The modem definition from IUPAC says, A covalent bond is a region of relatively high electron density between nuclei which arises (at least partially) from sharing of electrons, and gives rise to an attractive force and characteristic inter-nuclear distance . 

The article is organized as follows in Section 2, a general discussion concerning the definition of electrostatic potentials in the frame of DFT is presented. In Section 3, the solvation energy is reformulated from a model based on isoelectronic processes at nucleus. The variational formulation of the insertion energy naturally leads to an energy functional, which is expressed in terms of the variation of the electron density with respect to... 

Now, as in the case of the energy, up to this point, we have worked with the nonsmooth expression for the electronic density. However, in order to incorporate the second-order effects associated with the charge transfer processes, one can make use of a smooth quadratic interpolation. That is, with the two definitions given in Equations 2.23 and 2.24, the electronic density change Ap(r) due to the electron transfer AN, when the external potential v(r) is kept fixed, may be approximated through a second-order Taylor series expansion of the electronic density as a function of the number of electrons,... 

If population analysis is not synonymous with the concept of an AIM, it becomes necessary to introduce a proper set of requirements before one can speak of an AIM. An AIM is a quantum object and as such has an electron density of its own. This atomic electron density must obviously be positive definite and the sum of these atomic densities must equal the molecular density. Each atomic density pA(r) can be obtained from the molecular density p(r) in the following way ... 

From this wave function, one sees how even in the early beginning of molecular quantum mechanics, atomic orbitals were used to construct molecular wave functions. This explains why one of the first AIM definitions relied on atomic orbitals. Nowadays, molecular ab initio calculations are usually carried out using basis sets consisting of basis functions that mimic atomic orbitals. Expanding the electron density in the set of natural orbitals and introducing the basis function expansion leads to [15]... 

In fact, one can even use a positive definite operator, the electron density of yet a third molecule X acting as template, to obtain [21,22]... 

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