Lower Excited States of Helium Atom
The ground state of the helium atom has the electronic configuration S. In the lowest excited states, an electron is promoted from the to a or orbital. Although the hydrogenic and orbitals are degenerate, the configuration of helium has a lower energy than the . This is attributed to the greater shielding of the nuclear charge experienced by the orbital. Each of these lowest excited configurations is further split into a singlet and a triplet state, depending on whether the electron spins are antiparallel or parallel. (They must be antiparallel in the 1sInlineMathItalic2 configuration, by virtue of the Pauli exclusion principle, so that the ground state is a singlet.) The energies of the four lowest excited states can be calculated to fairly reasonable accuracy by self-consistent field (SCF) theory. In this Demonstration, the orbitals are approximated by the simple forms: , and . According to the Hartree-Fock method, the energy of a two-electron state is given by . The one-electron integrals account for the kinetic energy and nuclear attraction. Atomic units are used, with . The Coulomb integrals represent the electrostatic repulsion between the two orbital charge distributions. The exchange integrals have no classical analog. They are a consequence of the quantum-mechanical exchange symmetry of the atomic wavefunction. The + and signs in the energy expression pertain to the singlet and triplet states, respectively. The SCF energy can be optimized by variation of the parameters , , , which you can do in the Demonstration. All energies are expressed in electron volts above the ground state. See how closely you can approximate the exact S, S, P and P energies, which are represented by black dashes. The S, P and P states are the lowest of their symmetry type, thus the calculated energies are upper bounds to the corresponding exact values, by virtue of the variational principle. This is not true for the S computation, however, since there exists a lower state of the same symmetry; it is just an approximation which can be higher or lower than the exact value.
Snapshot 1: optimized values of , , in a Hartree-Fock computation
Snapshot 2: setting the exchange integrals equal to zero collapses the singlet and triplet states of each configuration
Snapshot 3: optimized results using Hartree's original SCF method, without exchange
References: S. M. Blinder, Introduction to Quantum Mechanics, Amsterdam: Elsevier, 2004, pp. 119-121
H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, New York: Academic Press, 1957, pp. 157-160.