Electrons probability distribution

CT bond (Section 2 3) A connection between two atoms in which the electron probability distribution has rotational symmetry along the mtemuclear axis A cross section per pendicular to the mtemuclear axis is a circle... 

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). 

Like the Coulombic forces, the van der Waals interactions decrease less rapidly with increasing distance than the repulsive forces. They include interactions that arise from the dipole moments induced by nearby charges and permanent dipoles, as well as interactions between instantaneous dipole moments, referred to as dispersion forces (Israelachvili 1992). Instantaneous dipole moments can be thought of as arising from the motions of the electrons. Even though the electron probability distribution of a spherical atom has its center of gravity at the nuclear position, at any very short instance the electron positions will generally not be centered on the nucleus. 

The evaluation of elements such as the M n,fin s is a very difficult task, which is performed with different levels of accuracy. It is sufficient here to mention again the so called sudden approximation (to some extent similar to the Koopmans theorem assumption we have discussed for binding energies). The basic idea of this approximation is that the photoemission of one-electron is so sudden with respect to relaxation times of the passive electron probability distribution as to be considered instantaneous. It is worth noting that this approximation stresses the one-electron character of the photoemission event (as in Koopmans theorem assumption). 

The electrochemical behavior of a molecule is directly related to the MO energy levels, while experimental EPR parameters correlate with the electron probability distribution. 

Taking the squared absolute value of Eq. (6.12) and using the definition of the parabolic coordinates given in Eq. (6.4), we can write an expression for the electron probability distribution in spherical coordinates,1... 

We can illustrate this by comparing the energies of the Is, 2s and 2 p orbitals for a helium atom, which has two electrons. The first electron goes into the Is orbital. Thus the atom He+ has an electronic probability distribution which is given by putting Z = 2 into Equation 6.24 above ... 

The system electron density p(r) and hence the one-electron probability distribution p(r) = p(r)/N, that is, the density per electron or the shape factor of p, are determined by the first-order density matrix y in the AO representation, also called the charge and bond order (CBO) matrix,... 

Another way of representing the electron probability distribution for the Is orbital is to calculate the probability at points along a line drawn outward in any direction from the nucleus. The result is shown in Fig. 12.16(b), where R2 (the square of the radial part—the part that depends on r—of the Is orbital) is plotted versus r. Note that the probability of finding the electron at... 

Representation of the 2p orbitals, (a) The electron probability distribution for a 2p orbital. Generated from a program by Robert Allendoerfer on Project SERAPHIM disk PC 2402 reprinted with permission, (b) The boundary surface representations of all three 2p orbitals. Note that the signs inside the surface indicate the phases (signs) of the orbital in that region of space. 

A cross section of the electron probability distribution for a Ip orbital. 

The orbital properties of greatest interest are size, shape (described by the electron probability distribution), and energy. These properties for the hydrogen MOs are represented in Fig. 14.26. From Fig. 14.26 we can note several important points ... 

The electron probability distribution in the bonding MO of the HF molecule. Note the greater electron density close to the fluorine atom. 

The same stockholder division principle applies to the associated -electron probability distribution (/t-electron shape function) ... 

This definition depends on the possibility to define a unique single configurational energy. As we have seen above, that might often not be possible if we do not want to fall back on the unsatisfactory UHF approach. We can as an alternative use the statistical definition of electron correlation. The electronic motion is uncorrelated if the probability to find electron one in a small volume element around xi and electron two around X2 is the product of the one-electron probability distributions ... 

The ground state (Is) of the hydrogen atom a. plotted against r. b. plotted against r. c. plotted against r. d. Representation of the electron probability distribution. 

Two representations of the hydrogen 1s, 2s, and 3s orbitals, (a) The electron probability distribution, (b) The surface that contains 90% of the total electron probability (the size of the orbital, by definition). 

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