William J. Meath

Relativistic Intermolecular Forces, Moderately Long Range

The generalized Breit—Pauli Hamiltonian is used to give a systematic treatment of magnetic and other relativistic intermolecular energies through O(α2) (where α is the fine‐structure constant) for intermolecular separations, R, sufficiently large that the charge distributions of the two molecules do not overlap, but sufficiently small that R<λ/0=(αΔe)−1, where Δe is the excitation energy of the first allowed transition of one of the molecules.The theory is discussed in general and many types of interaction energies are obtained which depend on the spin and orbital angular‐momentum states of the molecules. The interaction of two nondegenerate atoms (L=0, S=0) is considered specifically. Of particular interest is an interaction‐energy term which varies as α2/R4.

Long‐Range (Retarded) Intermolecular Forces

The Casimir and Polder retarded dipole—dipole energy of interaction between two ground‐state (non‐degenerate) atoms is expressed in terms of sine and cosine integrals. This result should be accurate for all interatomic separations R. In the range of moderately large separations (discussed in the preceding paper), where the charge distributions do not overlap and where R is small compared to λ/0=(αΔe)−1, the Casimir and Polder results can be expanded in the form Eint=R−6C6+α2R−4W4+α3R−3A+α4R−2B+···. This expansion is only accurate for R/λ/0<0.6. Here α is the fine‐structure constant. The R−6C6 term is the usual London dispersion energy. The α2R−4W4 term was obtained in the preceding paper by taking the expectation value of the Breit—Pauli Hamiltonian using the wavefunction for the two‐atom system corrected for the classical electrostatic dipole—dipole interactions. Thus, at least in the dipole—dipole approximation, the Breit—Pauli Hamiltonian gives the energy of interaction accurate through O(α2). For larg...

RETARDED INTERACTION ENERGIES BETWEEN LIKE ATOMS IN DIFFERENT ENERGY STATES.

The long‐range interaction energy between two like atoms in different energy states is discussed in general, through O (α2), using the Breit‐Pauli approximation. Selection rules are derived for possible interactions where retardation or relativistic effects may be important. The S‐P resonant interaction is considered as a specific example. The interaction energies for the resulting Σ and Π molecular states are calculated in both the Breit‐Pauli approximation and from the exact dipole‐dipole results of McLone and Power, Stephen and others. The R−1 expansion of the exact result and the validity of the Breit‐Pauli approximation are discussed in detail. The interaction energies are also compared with the linewidths for the various molecular states. It is shown that the Breit‐Pauli approximation can give a good representation (especially for the Σ states) of the interaction for quite large values of R < λ, where λ is the characteristic wavelength for the S‐P transition.

One- and two-center expansions of the Breit- Pauli Hamiltonian.

The orbit‐orbit, spin‐spin, and spin‐orbit Hamiltonians of the Breit‐Pauli approximation are expressed in terms of irreducible tensors. One‐ and two‐center expansions are given in a form in which the coordinate variables of the interacting particles are separated. In the one‐center expansions of the orbit‐orbit and spin‐orbit Hamiltonians the use of the gradient formula reduces some of the infinite sums to finite ones. Two‐center expansions are discussed in detail for the case of nonoverlapping charge distributions. The angular parts of the matrix elements of these Hamiltonians are evaluated for product wavefunctions.

Relativistic Interaction Energies between Atoms in Degenerate States

The interaction of two like atoms in degenerate quantum states of the same energy and the interaction of two unlike atoms in arbitrary states is considered in the Breit—Pauli approximation. For these interactions the calculation of the relativistic long‐range interaction energy, through O(α2), is discussed with specific allowance for degeneracy in the interacting atoms. Possible interactions where relativistic effects may be important are discussed. As a specific example the interaction of two spin‐degenerate atoms (L=0, S≠0) is calculated through O(α2/R6) (where R is the interatomic separation). The nonrelativistic energy is given by the usual London dispersion energy which varies as 1/R6 while relativistic effects introduce an interaction energy which varies as α2/R3.

Variational Solutions to the Brillouin—Wigner Perturbation Differential Equations

Variational techniques for the Brillouin—Wigner (BW) perturbation theory, analogous to the Hylleraas and Sinanoglu principles in Rayleigh—Schrodinger (RS) theory, are derived. A practical method of applying this approach to BW theory, which does not require the knowledge of the exact BW wavefunctions, is discussed. Using this method one obtains an upper bound to the exact total energy for systems in the lowest energy state of a given symmetry. Finally, a convenient matrix method of applying the variational principles is suggested and degenerate BW theory is discussed briefly.

Direct calculation of contributions to the second-order energy of helium.

The first‐order perturbed wavefunction of helium is written as a sum of terms due to various electron excitations. The portion of the first‐order wavefunction due to one‐electron excitations has been determined by several investigators. The portion due to two‐electron excitations with one of the electrons excited to the 2s orbital is determined exactly by solving a differential equation. The contributions of these functions to the second‐order energy are computed and found to be in exact agreement with the energies determined by direct summation of the spectral expansion.