Effective interactions and large-scale diagonalization for quantum dots

The widely used large-scale diagonalization method using harmonic oscillator basis functions (an instance of the Rayleigh-Ritz method, also called a spectral method, configuration-interaction method, or ``exact diagonalization'' method) is systematically analyzed using results for the convergence of Hermite function series. We apply this theory to a Hamiltonian for a one-dimensional model of a quantum dot. The method is shown to converge slowly, and the non-smooth character of the interaction potential is identified as the main problem with the chosen basis, while on the other hand its important advantages are pointed out. An effective interaction obtained by a similarity transformation is proposed for improving the convergence of the diagonalization scheme, and numerical experiments are performed to demonstrate the improvement. Generalizations to more particles and dimensions are discussed.Comment: 7 figures, submitted to Physical Review B Single reference error fixe

Casimir-Foucault interaction: Free energy and entropy at low temperature

It was recently found that thermodynamic anomalies which arise in the Casimir effect between metals described by the Drude model can be attributed to the interaction of fluctuating Foucault (or eddy) currents [Phys. Rev. Lett. 103, 130405 (2009)]. We show explicitly that the two leading terms of the low-temperature correction to the Casimir free energy of interaction between two plates, are identical to those pertaining to the Foucault current interaction alone, up to a correction which is very small for good metals. Moreover, a mode density along real frequencies is introduced, showing that the Casimir free energy, as given by the Lifshitz theory, separates in a natural manner in contributions from eddy currents and propagating cavity modes, respectively. The latter have long been known to be of little importance to the low-temperature Casimir anomalies. This convincingly demonstrates that eddy current modes are responsible for the large temperature correction to the Casimir effect between Drude metals, predicted by the Lifshitz theory, but not observed in experiments.Comment: 10 pages, 1 figur

Toward an analog neural substrate for production systems

Symbolic, rule-based systems seem essential for modeling high-level cognition. Subsymbolic dynamical systems, in con-trast, seem essential for modeling low-level perception and ac-tion, and can be mapped more easily onto the brain. Here we review existing work showing that critical features of sym-bolic production systems can be implemented in a subsym-bolic, dynamical systems substrate, and that optimal tuning of connections between that substrate’s analog circuit elements accounts for fundamental laws of behavior in psychology. We then show that emergent properties of these elements are re-flected in behavioral and electrophysiological data, lending support to a theory about the physical substructure of produc-tions. The theory states that: 1) productions are defined by connection strengths between circuit elements; 2) conflict res-olution among competing productions is equivalent to optimal hypothesis testing; 3) sequential process timing is parallel and distributed; 4) memory allocation and representational binding are controlled by competing relaxation oscillators

Analytical and Numerical Demonstration of How the Drude Dispersive Model Satisfies Nernst's Theorem for the Casimir Entropy

In view of the current discussion on the subject, an effort is made to show very accurately both analytically and numerically how the Drude dispersive model, assuming the relaxation is nonzero at zero temperature (which is the case when impurities are present), gives consistent results for the Casimir free energy at low temperatures. Specifically, we find that the free energy consists essentially of two terms, one leading term proportional to T^2, and a next term proportional to T^{5/2}. Both these terms give rise to zero Casimir entropy as T -> 0, thus in accordance with Nernst's theorem.Comment: 11 pages, 4 figures; minor changes in the discussion. Contribution to the QFEXT07 proceedings; matches version to be published in J. Phys.

Harmonic Infrared and Raman Spectra in Molecular Environments Using the Polarizable Embedding Model

We present a fully analytic approach to calculate infrared (IR) and Raman spectra of molecules embedded in complex molecular environments modeled using the fragment-based polarizable embedding (PE) model. We provide the theory for the calculation of analytic second-order geometric derivatives of molecular energies and first-order geometric derivatives of electric dipole moments and dipole–dipole polarizabilities within the PE model. The derivatives are implemented using a general open-ended response theory framework, thus allowing for an extension to higher-order derivatives. The embedding-potential parameters used to describe the environment in the PE model are derived through first-principles calculations, thus allowing a wide variety of systems to be modeled, including solvents, proteins, and other large and complex molecular environments. Here, we present proof-of-principle calculations of IR and Raman spectra of acetone in different solvents. This work is an important step toward calculating accurate vibrational spectra of molecules embedded in realistic environments