The Van der Waals energy Sticking with the two Ar atoms, we can ask why these two atoms attract each other, when they are electrically neutral, and have closed shells. In other words, how does the R-6 energy arise? A simple 'classical' explanation is often given in terms of fluctuating dipoles. A dipole moment, p1 on atom(1), gives an electric field E21 at atom(2) which is proportional to p1/R3. This picture of the Van der Waals energy is that the dipole induced on atom(2) reacts back to give a dipole on atom(1), so that the energy (gain) is -p1.E12. But since there is no initial dipole, just a concerted dance of dipole moments and fields, then the energy gain is -p2/R6. In class we discussed the corresponding quantum description, as outlined below. The ground state of the rare gases is a filled shell with a 1S0 spectroscopic term. The is the 1s2 configuration in He, or the 3s23p6 configuration in Ar. The first dipole-allowed excited state has a 1P1 spectroscopic term, with the 1s2p configuration in He, or either the 3p54s or 3p53d configuration in Ar. Now consider two separated atoms and their interaction, with a product wavefunction y = y1.y2. Nuclei and electrons interact in both atoms, with a perturbation Hamiltonian (energy) This formalism is very general but of course complicated. So we expand the potential in a multipole (dipole, quadrupole, etc) series, such that the perturbation H1 is given by In the quantum case the picture of the dipole is a superposition of ground and excited states, and the Van der Waals energy consists of these "virtual" excited states, which are mixed into the ground state by dipole transitions, in second order perturbation theory. Thus the product ground state wave function yn = y10.y20, and the excited state wave function yk = y1k.y2k', where the excited states k and k' can be different. The second order energy is given by The resulting matrix elements have angular components based on the Ylm (does this sound familiar?). The radial integrals for the dipole-dipole terms vary as R-6. Dipole-quadrupole terms vary as R-8, and quadrupole-quadrupole terms as R-10. More details as applied to rare gas interactions can be found in the handout from chapter 2 of Rare Gas Solids, vol I.

Relation to Matrix methods: a rotational example On problem set #3, we considered the Hamiltonians of rotating molecules, and the solution of these problems by both Operator and Matrix methods. We recalled that the diagonal matrix elements relate to rotation about the axes x, y and z, and that the off-diagonal elements depend on the differences between such terms. As such, the off-diagonal elements are just those terms which appear in perturbation theory. This topic was discussed in the lectures, with examples from angular momentum quantum numbers 1 (3x3) and 2 (5x5) matrices. By rearranging rows and columns, these matrices can be rearranged into block-diagonal form. From this, one can show the correspondence both with perturbation theory, and with singlets, doublets and triplets, etc., i.e. with spectroscopy in general. This set of issues was the subject of a workshop in class time, and the rotating molecule problem also appears on problem set #5. The specific case of l = 1 and 2 was done in 2001 as a project by Jason Harris, which can be inspected on his website. His description includes a few-line MATLAB5.3 code, which can be downloaded (script1.m) and used to solve rotational problems. The code produces the analytic form of the matrix H, from which you can find the eigenvalues by typing the command eig(H): all very satisfying and simple, especially if you understand what you are doing.