Ab initio calculations of the pi electron hamiltonian: singlet-triplet splittings

Some results of approximate ab initio calculations of the “correlation” contribution to the true parameters of the pi electron hamiltonian are presented for the ethylene molecule. In particular, by using sum-of-the-pairs type generalized perturbation theory, it is shown that there is a large core “correlation” contribution to singlet-triplet splittings within pi electron theories that results from the difference in the degree of ionicity of the isoconfigurational states.     

Open-shell generalized perturbation theory

Open shell generalized perturbation theory (GPT) is a systematic scheme for the improvement of a multi-configurational zeroth order wavefunction that can be chosen so the lowest order corresponds to a sum-of-the-pairs- type theory of electron correlation (with or without pair-pair couplings) in the spirit of the theories of Kelly, Sinano?lu, and Nesbet for the single configurational case. Applications axe presented to the generalized perturbative solution for individual states and for the simultaneous solution for many nondegenerate states. The (2n+ 1)-rule of GPT and the interchange theorems of double Girl' are illustrated. A brief discussion is included concerning the use of GPT in ab initio evaluations of the true parameters that are customarily, only modeled and then fit to experiment in semi-empirical theories of molecular electronic structure. The conceptual value of expressing the results of ab initio calculations in terms of effective hamiltonians is stressed and the relationship between open shell GPT and other theories of electron correlation is also considered.     

Ab initio calculation of the effective valence shell hamiltonian of carbon: Simultaneous treatment of neutral and ion states

The n = 2 effective valence shell hamiltonian, $\text{H}$^v, of carbon is evaluated through second order using ³P Hartree—Fock orbitals (5s4p) with added d functions to provide results within a few percent of the spd convergence limits. The calculated $\text{H}$^v is employed to evaluate the n = 2 valence states of C, C^?, C⁺, C²⁺ and C³⁺ with an average deviation of the 21 excitation energies, ionization potentials and electron affinity from experimental values of 0.32 eV. Three-electron parts of $\text{H}$^v contribute substantially to a number of these excitation energies.     

Theory of correlation measurements of resonance light scattering

The boson nature of radiation is shown to give rise to a purely quantum mechanical exchange contribution to the intensity-intensity correlation function for resonant light scattering by an atomic or molecular system. The exchange contribution can be decomposed into three components, one involving the intensity correlation for a pair of coherently scattered photons (“resonant Raman” processes), another for a pair of incoherently scattered ones (“resonance fluorescence”), and the last involving the exchange correlation one of each. The intensity correlation measurements of Kimble et al., on optically pumped atomic beams of sodium atoms are interpreted with the theory, producing values of the decay rate of the excited sodium atoms and of the coherence time of the exciting radiation in good agreement with expectations.     

Chern-Simons theory with finite gauge group

We construct in detail a 2+1 dimensional gauge field theory with finite gauge group. In this case the path integral reduces to a finite sum, so there are no analytic problems with the quantization. The theory was originally introduced by Dijkgraaf and Witten without details. The point of working it out carefully is to focus on the algebraic structure, and particularly the construction of quantum Hilbert spaces on closed surfaces by cutting and pasting. This includes the Verlinde formula. The careful development may serve as a model for dealing with similar issues in more complicated cases.The first author is supported by NSF grant DMS-8805684, an Alfred P. Sloan Research Fellowship, a Presidential Young Investigators award, and by the O'Donnell Foundation. The second author is supported by NSF grant DMS-9207973     

Growth Rates and Explosions in Sandpiles

We study the abelian sandpile growth model, where n particles are added at the origin on a stable background configuration in ? ^d . Any site with at least 2d particles then topples by sending one particle to each neighbor. We find that with constant background height h≤2d?2, the diameter of the set of sites that topple has order n ^1/d . This was previously known only for h<d. Our proof uses a strong form of the least action principle for sandpiles, and a novel method of background modification. We can extend this diameter bound to certain backgrounds in which an arbitrarily high fraction of sites have height 2d?1. On the other hand, we show that if the background height 2d?2 is augmented by 1 at an arbitrarily small fraction of sites chosen independently at random, then adding finitely many particles creates an explosion (a sandpile that never stabilizes).     

Geometric nature of the environment-induced Berry phase and geometric dephasing

We investigate the geometric phase or Berry phase acquired by a spin half which is both subject to a slowly varying magnetic field and weakly coupled to a dissipative environment (either quantum or classical). We study how this phase is modified by the environment and find that the modification is of a geometric nature. While the original Berry phase (for an isolated system) is the flux of a monopole field through the loop traversed by the magnetic field, the environment-induced modification of the phase is the flux of a quadrupolelike field. We find that the environment-induced phase is complex, and its imaginary part is a geometric contribution to dephasing. Its sign depends on the direction of the loop. Unlike the Berry phase, this geometric dephasing is gauge invariant for open paths of the magnetic field.     

Beyond Flory-Huggins theory: new classes of blend miscibility associated with monomer structural asymmetry

Flory-Huggins (FH) theory is restricted to polymer mixtures whose monomers are structurally identical, a situation limited to isotopic blends and computer simulations. We investigate the influence of monomer structure on blend miscibility and scattering properties using the lattice cluster theory generalization of the FH model. Monomer structural asymmetry is shown to profoundly affect blend miscibility (T(c),phi(c)), chain swelling (T(theta)), and the scale (xi) and intensity [S(0)] of composition fluctuations. Four distinct blend miscibility classes are identified and experimental evidence for these classes is discussed.