Functions electron probability density

The electron probability density function, p(x,y,z) for a molecule depends upon three spatial coordinates (x,y,z) to specify its value at a chosen position. 

Type-A measures are appropriate for measuring the similarity between one or, at most, a few pairs of structures that are characterized by quantum-mechanical descriptions of various sorts. Following the pioneering studies of Carbo et al.io a large number of similarity measures have been described that fall in this class (see, e.g.. Refs. 11—14). Here, a molecule is described by an electron probability density function, and the similarities between pairs of molecules are calculated with measures that describe the overlap of their density functions. Work in this area is exemplified by that of Manaut et al.i" who describe a procedure to align two molecules so as to maximize the similarity... 

The square of the wavefunction (4-9) teUs us how the electron is distributed about the nucleus. In Fig. 4-3b is plotted - (r) as a function of r. We refer to %lr as the electron probability density function. In this case, the probability density is greatest at the nucleus (r = 0) and decays to zero as r becomes infinite. 

A much more attractive and popular methodology to solve the electronic structure problem of complex chemical systems such as zeolites and other microporous materials is based on the DFT. DFT is not based on wave functions, but on electron density function (electron probability density function, electron density, charge density), p(r). This is the main conceptual difference between the approaches discussed earlier and the DFT methodologies. The attractiveness of these methods is partially owing to the fact that a wave function cannot be experimentally measured. It is still debated whether a wave function is just a convenient mathematical model or a true physical entity. On the contrary, the electron density is a directly measurable property that can be obtained from, for example. X-ray diffraction... 

A functional is a function of a function. Electron probability density p is a function p(r) of a point in space located by radius vector r measured from an origin (possibly an atomic mi dens), and the energy E of an electron distribution is a function of its probability density. E /(p). Therefore E is a functional of r denoted E [pfr). ... 

A time-independent wave function is a function of the position in space (r = x,y,z) and the spin degree of freedom, which can be either up or down. The physical interpretation of the wave function is due to Max Born (25, 26), who was the first to interpret the square of its magnitude, > /(r)p, as a probability density function, or probability distribution function. This probability distribution specifies the probability of finding the particle (here, the electron) at any chosen location in space (r) in an infinitesimal volume dV= dx dy dz around r. I lu probability of finding the electron at r is given by )/(r) Id V7, which is required to integrate to unity over all space (normalization condition). A many-electron system, such as a molecule, is described by a many-electron wave function lF(r, r, l .I -.-), which when squared gives the probability den-... 

Because of interelectronic effects this Hamiltonian is not separable. Only when these effects are ignored may the total probability density ip ip be assumed to be a product of one-electron probability densities and the wave function a product of hydrogenic atomic wave functions... 

In addition to D(ri), we have plotted the charge density p(r ) — D(r )/4nri, the electron-electron probability distribution function P(ri2) (defined similarly to D(ri) but with omission of the prefactor 2 ), and the electron-electron density (sometimes referred to as the intracule function). 

First let us recall the more important electron distribution functions and their origin in terms of corresponding density matrices. The electron probability density ( number density in electrons/unit volume) is the best known distribution function others refer to a pair of electrons, or a cluster of n electrons, simultaneously at given points in space. 

The basis set is the set of madiematical functions from which the wave function is constructed. As detailed in Chapter 4, each MO in HF theory is expressed as a linear combination of basis functions, the coefficients for which are determined from the iterative solution of the HF SCF equations (as flow-charted in Figure 4.3). The full HF wave function is expressed as a Slater determinant formed from the individual occupied MOs. In the abstract, the HF limit is achieved by use of an infinite basis set, which necessarily permits an optimal description of the electron probability density. In practice, however, one cannot make use of an infinite basis set. Thus, much work has gone into identifying mathematical functions that allow wave functions to approach the HF limit arbitrarily closely in as efficient a manner as possible. 

The probability of finding the electron in the ground state of the hydrogen atom between radii r and r + dr is given by D(r)Ar, where D(r) is the radial probability density function shown in Figure 4.5. The most probable distance of the electron from the nucleus is found by locating the maximum in D(r) (see Problem 4.12 below). It should come as no surprise to discover that this maximum occurs at the value r = ao, the Bohr radius. 

The radial probability density function for the electron in the ground state of the hydrogen atom takes the form D(r) = Nr2e 2r a°, where N is a constant. 

Consider the radial probability density function, D(r), for the ground state of the hydrogen atom. This function describes the probability per unit length of finding an electron at a radial distance between r and r + dr (see Figure 6.5). 

From the Uncertainty Principle, we no longer speak of the exact position of an electron. Instead, the electron position is defined by a probability density function. If this function is called p (x,y,z), then the electron is most likely found in the region where p has the greatest value. In fact, p dr is the probability of finding the electron in the volume element dr (= dxdydz) surrounding the point (x,y,z). Note that p has the unit of volume-1, and pdr, being a probability, is dimensionless. If we call the electronic wavefunction f, Born asserted that the probability density function p is simply the absolute square of tjr. 

Now we examine the bonding orbital antibonding orbital ax as well as their probability density functions schematic representation of cr is is shown in Fig. 3.1.5(a). In this combination of two Is orbitals, electron density accumulates in the internuclear region. Also, crls has cylindrical symmetry around the internuclear axis. 

Another approach closely related to the ab initio methods that has gained increasing prominence in recent years is the density functional theory (DFT). This method bypasses the determination of the wavefunction electronic probability density p directly and then calculates the energy of the system from p. 

In the latter way of looking at the build-up of molecules, the successive addition of electrons to a positively charged system is reminiscent of the manner in which the atoms of the Periodic Table were considered in Chapter 1. Here again there are certain configurations permitted the electron clouds, and these cloud shapes (or probability density functions) can be described using quantum numbers. Such probability density descriptions are called molecular orbitals in analogy to the much simpler atomic orbitals. Although the initial setup and subsequent mathematical treatment for molecules are much more complicated than for atoms, there arise certain similarities between the two types of orbitals. 

Figure 3. Scheme of the PL quenching model (A) and and the comparison (B) of experimental quenching rate A, constants (left axis) with calculated probability density functions i/(r) of a Is electron at the outer interface. 

The example of the double-slit experiment already shows that the physical information contained in a state function is inherently probabilistic in nature. In the next chapter this feature will be further developed leading to the central concept of this book orbitals. For example, the orbitals of the H atom are wavefunctions (x,y,z) that enable the electron probability density to be known (x,y,z) (x,y,z). The idea of a trajectory (or orbit) is replaced by the idea of a probability distribution. 

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