Hydrogen-Atom Ground State

Two ground-state hydrogen atoms, for example, interact via the and f> 5 electronic states of H2. The... 

In the atomic system of units, the energy of a ground-state hydrogen atom is... 

Another way of stating the exclusion principle is that no two electrons in an atom have the same four quantum numbers. This important idea means that each electron in an atom has its own unique set of four quantum numbers. For example, compare the quantum numbers that distinguish a ground state hydrogen atom from a helium atom. (Recall that a helium atom has two electrons. Note also that mg quantum number is given as +. It could just as easily have a value of —By convention, chemists usually use the positive value first.)... 

In the atomic system of units, the energy of a ground-state hydrogen atom is —j Eh and so we would expect the potential curve to tend asymptotically to -ljEh This is obviously not the case, and analysis of the results shows that the HF wavefunction describes the following process, for large R ... 

Production of ground state hydrogen atoms, and their transport to an interaction region. 

Exotic ground-state hydrogen atom Antiprotonic Helium is an exotic... 

Fig. 10. Radial probability density, 4 ry V, in ground state hydrogen atom...

Figure 7.21 indicates that the electron in a ground-state hydrogen atom must be in the Is orbital, so its electron configuration is 1 ... 

FIGURE 11.2 Atomic-orbital energies as a function of atomic number for neutral atoms, as calculated by Latter. [Figure redrawn by M. Kasha from R. Latter, Phys. Rev., 99,510 (1955). Used by permission.] Note the logarithmic scales. h is the ground-state hydrogen-atom energy, -13.6 eV. 

Consider what the H2 wave function would look like for large values of the inter-nuclear separation R. When the electron is neeu nucleus a, nucleus b is so far away that we essentially have a hydrogen atom with origin at a. Thus, when r is small, the ground-state electronic wave function should resemble the ground-state hydrogen-atom wave function of Eq. (6.104). We have Z = 1, and the Bohr radius Oq has the numerical value 1 in atomic units hence (6.104) becomes... 

The first step in the Heitler-London treatment of the H2 ground state is to approximate the molecule as two ground-state hydrogen atoms. The wave function for two such noninteracting atoms is... 

The quantity -1 hartree in these expressions is the energy of two ground-state hydrogen atoms. To obtain the U R) potential-energy curves we add the intemuclear repulsion 1/R to these expressions. 

Since the dimensional generalization introduces D upstairs in the Laplacians, and downstairs in the Coulomb terms, one might expect the two sets of terms to get out of balance as D is varied, with the Laplacians dominating at high D, and the Coulomb terms at low D. This is of course a rather simplistic perspective, but it does describe qualitatively what happens to the solutions as D is varied. A ground state hydrogen atom E = for example, falls apart as... 

This expression yields the known result 2 = 4.5 for the ground state hydrogen atom Z = 1). 

Fig. 9.14. The infinite chain td" ground-state hydrogen atoms and the influence of bonding and antibonding effects, (a) All interactions are bonding (b) introductim of a single node results in an energy increase (c) two nodes increase the eneigy even more (d) with the maximum number nodes, the eneigy is at its maximum.

---