Electron-nucleus attractions

The small differences m stability between branched and unbranched alkanes result from an interplay between attractive and repulsive forces within a molecule (intramo lecular forces) These forces are nucleus-nucleus repulsions electron-electron repul sions and nucleus-electron attractions the same set of fundamental forces we met when... 

The derivative of the core operator h is a one-electron operator similar to the nucleus-electron attraction required for the energy itself (eq. (3.55)). The two-electron part yields zero, and the V n term is independent of the electronic wave function. The remaining terms in eqs. (10.89), (10.90) and (10.95) all involve derivatives of the basis functions. When these are Gaussian functions (as is usually the case) the derivative can be written in terms of two other Gaussian functions, having one lower and one higher angular momentum. 

The only term in this expression that can be derived directly from the charge distribution is the Coulombic energy. It consists of nucleus-nucleus repulsion, nucleus-electron attraction, and electron-electron repulsion terms. For a medium of unit dielectric constant,... 

The electrostatic interaction energy between two spherical atoms or ions located at A and B is the sum of the internuclear repulsions, the nucleus-electron attractions, and the electron-electron repulsions (Su and Coppens 1995) ... 

FIGURE 7.1 A covalent H-H bond is the net result of attractive and repulsive electrostatic forces. The nucleus-electron attractions (blue arrows) are greater than the nucleus-nucleus and electron-electron repulsions (red arrows), resulting in a net attractive force that holds the atoms together to form an H2 molecule. 

The ground state electronic energy of the real molecule is the sum of the electron kinetic energies, the nucleus-electron attraction potential energies, and the electron-electron repulsion potential energies ... 

The nucleus-electron attraction integrals simplify as well ... 

Gaseous atoms and ions are specified because they are far from any other chemical species (Section 12.10), and thus few extraneous interactions interfere with the nucleus-electron attraction. Note that this is not the energy for the more familiar reaction of solid sodium to produce sodium ions in a lattice or in solution. 

The overall force between two atoms Is the result of electron-electron repulsion, nucleus-nucleus repulsion, and nucleus-electron attraction. The arrows in this diagram show the net force acting on two fluorine atoms as they move toward each other. 

Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions. 

For orbitals in the same shell but in different subshells, a combination of nucleus-electron attraction and electron-electron repulsion influences the orbital energies. 

Once is calculated (for the given configuration of the nuclei), the total energy E is obtained as a sum of nucleus—nucleus repulsions, nucleus—electron attractions (not critical from the point of view of the present discussion, since they are easy to compute) and two-electron repulsion contributions which can again be divided into coulombic, J, and exchange terms, K ... 

The small differences in stability between branched and unbranched alkanes result from an interplay between attractive and repulsive forces within a molecule (intramolecular forces). These forces are nucleus-nucleus repulsions, electron-electron repulsions, and nucleus-electron attractions, the same set of fundamental forces we met when talking about chemical bonding (see Section 1.12) and van der Waals forces between molecules (see Section 2.14). When the energy associated with these interactions is calculated for all of the nuclei and electrons within a molecule, it is found that the attractive forces increase more than the repulsive forces as the structure becomes more compact. Sometimes, though, two atoms in a molecule are held too closely together. WeTl explore the consequences of that in Chapter 3. 

Second, suppose the effect of electron repulsion is simply a reduction in nucleus-electron attraction. In that case we will have Z0e2/4peo jq (Z0 is some number smaller than Z) and another term like that for... 

The ground state electronic energy of our real molecule is the sum of the electron kinetic energies, the nucleus-electron attraction potential energies, and the electron-electron repulsion potential energies (more precisely, the sum of the quantum-mechanical average values or expectation values, each denoted (value)) and each is a functional of the ground-state electron density ... 

Despite its great success in accounting for the spectral lines of the H atom, the Bohr model failed to predict the spectrum of any other atom, even that of helium, the next simplest element. In essence, the Bohr model predicts spectral lines for the H atom and other one-electron species, such as He" (Z = 2), Li (Z = 3), and Be (Z = 4). But, it fails for atoms with more than one electron because in these systems, electron-electron repulsions and additional nucleus-electron attractions are present as well. Nevertheless, we still use the terms ground state and excited state and retain one of Bohr s central ideas in our current model the energy of an atom occurs in discrete levels. 

The simplest independent-electron model results from a complete neglect of electron-electron interactions. This is the bare-nucleus model for which the Hamiltonian h is merely a sum of kinetic energy and nucleus-electron attraction terms. The bare-nucleus model has a number of unique features which make its use in atomic and molecular studies attractive (see, for example. Refs. 7 and 17-22). The most widely used independent-electron model is the Hartree-Fock model. In this model, as is well known, the... 

---