arXiv subjects

Dudley R. Herschbach

Publications and source records attributed to Dudley R. Herschbach.

6 recordsLinked to original sources

arXiv2022

Geometrical picture of the electron-electron correlation at the large-D limit

In electronic structure calculations, the correlation energy is defined as the difference between the mean field and the exact solution of the non relativistic Schrödinger equation. Such an error in the different calculations is not directly observable as there is no simple quantum mechanical operator, apart from correlation functions, that correspond to such quantity. Here, we use the dimensional scaling approach, in which the electrons are localized at the large-dimensional scaled space, to describe a geometric picture of the electronic correlation. Both, the mean field, and the exact solutions at the large-D limit have distinct geometries. Thus, the difference might be used to describe the correlation effect. Moreover, correlations can be also described and quantified by the entanglement between the electrons, which is a strong correlation without a classical analog. Entanglement is directly observable and it is one of the most striking properties of quantum mechanics and bounded by the area law for local gapped Hamiltonians of interacting many-body systems. This study opens up the possibility of presenting a geometrical picture of the electron-electron correlations and might give a bound on the correlation energy. The results at the large-D limit and at D=3 indicate the feasibility of using the geometrical picture to get a bound on the electron-electron correlations.

quant-ph↗

arXiv2020

Dimensional interpolation for metallic hydrogen

We employ a simple and mostly accurate dimensional interpolation formula using dimensional limits $D=1$ and $D=\infty$ to obtain $D=3$ ground-state energy of metallic hydrogen. We also present results describing the phase transitions for different symmetries of three-dimensional structure lattices. The interpolation formula not only predicts fairly accurate energies but also predicts a correct functional form of the energy as a function of the lattice parameters. That allows us to calculate different physical quantities such as the bulk modulus, Debye temperature, and critical transition temperature, from the gradient and the curvature of the energy curve as a function of the lattice parameters. These theoretical calculations suggest that metallic hydrogen is a likely candidate for high temperature superconductivity. The dimensional interpolation formula is robust and might be useful to obtain the energies of complex many-body systems.

cond-mat.str-el↗

arXiv2020

Unorthodox dimensional interpolations for He, Li, Be atoms and hydrogen molecule

We present a simple interpolation formula using dimensional limits $D=1$ and $D=\infty$ to obtain the $D=3$ ground-state energies of atoms and molecules. For atoms, these limits are linked by first-order perturbation terms of electron-electron interactions. This unorthodox approach is illustrated by ground-states for two, three, and four electron atoms, with modest effort to obtain fairly accurate results. Also, we treat the ground-state of H$_2$ over a wide range of the internuclear distance R, and compares well with the standard exact results from the Full Configuration Interaction method. Similar dimensional interpolations may be useful for complex many-body systems.

quant-ph↗

arXiv2005

Bohr's 1913 molecular model revisited

It is generally believed that the old quantum theory, as presented by Niels Bohr in 1913, fails when applied to few electron systems, such as the H2 molecule. Here we find new solutions within the Bohr theory that describe the potential energy curve for the lowest singlet and triplet states of H2 about as well as the early wave mechanical treatment of Heitler and London. We also develop a new interpolation scheme which substantially improves the agreement with the exact ground state potential curve of H2 and provides a good description of more complicated molecules such as LiH, Li2, BeH and He2.

physics.chem-ph↗

arXiv2005

A simple and surprisingly accurate approach to the chemical bond obtained from dimensional scaling

We present a new dimensional scaling transformation of the Schrodinger equation for the two electron bond. This yields, for the first time, a good description of the two electron bond via D-scaling. There also emerges, in the large-D limit, an intuitively appealing semiclassical picture, akin to a molecular model proposed by Niels Bohr in 1913. In this limit, the electrons are confined to specific orbits in the scaled space, yet the uncertainty principle is maintained because the scaling leaves invariant the position-momentum commutator. A first-order perturbation correction, proportional to 1/D, substantially improves the agreement with the exact ground state potential energy curve. The present treatment is very simple mathematically, yet provides a strikingly accurate description of the potential energy curves for the lowest singlet, triplet and excited states of H_2. We find the modified D-scaling method also gives good results for other molecules. It can be combined advantageously with Hartree-Fock and other conventional methods.

physics.chem-ph↗

arXiv1994

Dimensional Perturbation Theory on the Connection Machine

A recently developed linear algebraic method for the computation of perturbation expansion coefficients to large order is applied to the problem of a hydrogenic atom in a magnetic field. We take as the zeroth order approximation the $D \rightarrow \infty$ limit, where $D$ is the number of spatial dimensions. In this pseudoclassical limit, the wavefunction is localized at the minimum of an effective potential surface. A perturbation expansion, corresponding to harmonic oscillations about this minimum and higher order anharmonic correction terms, is then developed in inverse powers of $(D-1)$ about this limit, to 30th order. To demonstrate the implicit parallelism of this method, which is crucial if it is to be successfully applied to problems with many degrees of freedom, we describe and analyze a particular implementation on massively parallel Connection Machine systems (CM-2 and CM-5). After presenting performance results, we conclude with a discussion of the prospects for extending this method to larger systems.

chem-ph↗