One-electron species

The energy of any one-electron species in its nth state (n = principal quantum number) is given by E = —BZVn2, where Z is the charge on the nucleus and B is 2.180 X 10 18 J. Find the ionization energy of the Li2+ ion in its first excited state in kilojoules per mole. 

As we have seen in earlier sections, wave functions can be used to perform useful calculations to determine values for dynamical variables. Table 2.2 shows the normalized wave functions in which the nuclear charge is shown as Z (Z = 1 for hydrogen) for one electron species (H, He+, etc.). One of the results that can be obtained by making use of wave functions is that it is possible to determine the shapes of the surfaces that encompass the region where the electron can be found some fraction (perhaps 95%) of the time. Such drawings result in the orbital contours that are shown in Figures 2.3, 2.4, and 2.5. 

Figure 3.1 Diatomic potential-energy curves for H2, Li2 and their cations. (The one-electron species H2+ is calculated at UHF/6-311++G" level others at B3LYP/6-311++G level.)...

Calculate the energy (in kj/mol) required to remove the electron in the ground state for each of the following one-electron species, using the Bohr model. 

This equation has been solved exactly only for one-electron species such as the hydrogen atom and the ions He and Li. Simplifying assumptions are necessary to solve the equation for more complex atoms and molecules. Chemists and physicists have used their inmition and ingenuity (and modern computers), however, to apply this equation to more complex systems. 

Fig. 3. Redox/protonation cycle of tetraheme cytochrome c3. Two high energy electrons (2 e" ) plus two low energy protons (2 H+) convert the deprotonated one electron species (c3) into the protonated three-electron species (H2c3n). The cycle is completed by donating two low energy electrons (2e ) and two high energy protons (2H+ ). The decrease in electronic energy is coupled to the increase in acidity of the ionisable redox-Bohr groups...

Ms is obtained by algebraic summation of the values for individual electrons. One electron with 5 = 5 obviously has 5 = 5 with Ms —or — Ms for the multi-electron system is analogous to for the one-electron species. Two electrons lead to >S = 0 and —4 giving... 

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. 

What is the key distinction between sublevel energies in one-electron species, such as the H atom, and those in many-electron species, such as the C atom What factors lead to this distinction Would you expect the pattern of sublevel energies in Be " to be more like that in H or that in C Explain. 

How does the waveiength of a fast-pitched baseball compare to the wavelength of an electron traveling at 1/10 the speed of light What is the significance of this comparison See Example 7.3. The Bohr model only works for one electron species. Why do we discuss it in this text (what s good about it) ... 

Then, to provide for the analysis of one-electron species AB as well as create one and two-variable tables on the various worksheets, with. 

The importance of the hydrogen molecule ion for the theory of diatomic molecules is similar to the importance of the hydrogen atom for our understanding of atoms both H and H2 are one-electron systems for which the Schrbdinger equation can be solved exactly. The exact solution of the one-electron species is then used as a starting point for the discussion of polyelectron species for which exact solutions of the Schrodinger are unavailable. 

Shortly after the development of VBT, an alternative model, known as MOT, was introduced by the American physicist Robert Mulliken (and others) around 1932. MOT is a delocalized bonding model, where the nuclei in the molecule are held in fixed positions at their equilibrium geometries and the Schrodinger equation is solved for the entire molecule to yield a set of MOs. In practice, it is possible to solve the Schrodinger equation exactly only for one-electron species, such as H2. Whenever more than one electron is involved, the wave equation can only yield approximate solutions because of the e/ectron correlation problem that results from Heisenberg s principle of indeterminacy. If one cannot know precisely the position and momentum of an electron, it is impossible to calculate the force field that this one electron exerts on every other electron in the molecule. As a result of this mathematical limitation, an approximation method must be used to calculate the energies of the MOs. 

The wave function for an atom simultaneously depends on (describes) all the electrons in the atom. The Schrodinger equation is much more complicated for atoms with more than one electron than for a one-electron species such as a hydrogen atom, and an explicit solution to this equation is not possible even for helium, let alone for more complicated atoms. We must therefore rely on approximations to solutions of the many-electron Schrodinger equation. One of the most common and useful of these is the orbital approximation. In this approximation, the electron cloud of an atom is assumed to be the superposition of charge clouds, or orbitals, arising from the individual electrons these orbitals resemble the atomic orbitals of hydrogen (for which exact solutions are known), which we described in some detail in the previous section. Each electron is described by the same allowed combinations of quantum numbers (w, m(, and /,)... 

The Bohr model only works for one electron species. Why do we discuss it in this text (what s good about it) ... 

Bohr s model predicts only the spectrum of the H atom and other one-electron species. Despite this, Bohr was correct that an atom s energy is quantized. 

To formalize the concept of spin conservation we can borrow from Shuler who stated In order for reactants and products to correlate, it will be necessary that the intermediate conplex formed during the reaction have at least one electronic species in its term manifold (the collection of micro-states falling under one term s)mbol) which arises from the combination both of the reactants and of the products . This can be expressed via the reaction sequence shown in equation (8.1.11). 

The Energies of Principal Shells and Subshells in One-Electron Species... 

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