János Pipek

Correlation and Localization

Electron Correlation in Small Molecules: Grafting CI onto CC.- Extremal Electron Pairs - Application to Electron Correlation, Especially the R12 Method.- Many-Body Perturbation Theory with Localized Orbitals - Kapuys Approach.- An Introduction to the Theory of Geminals.- Extended Geminal Models.- Ab Initio Modern Valence Bond Theory.- Modern Correlation Theories for Extended, Periodic Systems.- Local Space Approximation Methods for Correlated Electronic Structure Calculations in Large Delocalized Systems that are Locally Perturbed.- Local Electron Densities and Functional Groups in Quantum Chemistry.- Electron Correlation and Reduced Density Matrices.- Localization via Density Functionals.

Elementary formula for entanglement entropies of fermionic systems

A generalized skew information is defined and a generalized uncertainty relation is established with the help of a trace inequality which was recently proven by Fujii. In addition, we prove the trace inequality conjectured by Luo and Zhang. Finally, we point out that Theorem 1 in S. Luo and Q. Zhang, IEEE Trans. Inf. Theory, vol. 50, pp. 1778-1782, no. 8, Aug. 2004 is incorrect in general, by giving a simple counter-example.An elementary formula for the von Neumann and Renyi entropies describing quantum correlations in two-fermionic systems having four single-particle states is presented. An interesting geometric structure of fermionic entanglement is revealed. A connection with the generalized Pauli principle is established.

Refinement trajectory and determination of eigenstates by a wavelet based adaptive method

The detail structure of the wave function is analyzed at various refinement levels using the methods of wavelet analysis. The eigenvalue problem of a model system is solved in granular Hilbert spaces, and the trajectory of the eigenstates is traced in terms of the resolution. An adaptive method is developed for identifying the fine structure localization regions, where further refinement of the wave function is necessary.

An economic prediction of refinement coefficients in wavelet-based adaptive methods for electron structure calculations.

The wave function of a many electron system contains inhomogeneously distributed spatial details, which allows to reduce the number of fine detail wavelets in multiresolution analysis approximations. Finding a method for decimating the unnecessary basis functions plays an essential role in avoiding an exponential increase of computational demand in wavelet‐based calculations. We describe an effective prediction algorithm for the next resolution level wavelet coefficients, based on the approximate wave function expanded up to a given level. The prediction results in a reasonable approximation of the wave function and allows to sort out the unnecessary wavelets with a great reliability.

Heat treatment parameters effecting the fractal dimensions of AuGe metallization on GaAs

Correlation was detected between the thermal treatment parameters of the AuGe–GaAs system and surface fractal structure. Structural entropic calculations were used to confirm the results obtained by fractal calculations.

Adaptive local refinement of the electron density, one-particle density matrices, and electron orbitals by hierarchical wavelet decomposition

The common experience that the distribution and interaction of electrons widely vary by scanning over various parts of a molecule is incorporated in the atomic-orbital expansion of wave functions. The application of Gaussian-type atomic orbitals suffers from the poor representation of nuclear cusps, as well as asymptotic regions, whereas Slater-type orbitals lead to unmanageable computational difficulties. In this contribution we show that using the toolkit of wavelet analysis it is possible to find an expansion of the electron density and density operators which is sufficiently precise, but at the same time avoids unnecessary complications at smooth and slightly detailed parts of the system. The basic idea of wavelet analysis is a coarse description of the system on a rough grid and a consecutive application of refinement steps by introducing new basis functions on a finer grid. This step could highly increase the number of required basis functions, however, in this work we apply an adaptive refinement only in those regions of the molecule, where the details of the electron structure require it. A molecule is split into three regions with different detail characteristics. The neighborhood of a nuclear cusp is extremely well represented by a moderately fine wavelet expansion; the domains of the chemical bonds are reproduced at an even coarser resolution level, whereas the asymptotic tails of the electron structure are surprisingly precise already at a grid distance of 0.5 a.u. The strict localization property of wavelet functions leads to an especially simple calculation of the electron integrals.

An economic prediction of the finer resolution level wavelet coefficients in electronic structure calculations

In wavelet based electronic structure calculations, introducing a new, finer resolution level is usually an expensive task, this is why often a two-level approximation is used with very fine starting resolution level. This process results in large matrices to calculate with and a large number of coefficients to be stored. In our previous work we have developed an adaptively refined solution scheme that determines the indices, where the refined basis functions are to be included, and later a method for predicting the next, finer resolution coefficients in a very economic way. In the present contribution, we would like to determine whether the method can be applied for predicting not only the first, but also the other, higher resolution level coefficients. Also the energy expectation values of the predicted wave functions are studied, as well as the scaling behaviour of the coefficients in the fine resolution limit.

Local expansion of N-representable one-particle density matrices yielding a prescribed electron density

Multiresolution (or wavelet) analysis offers a strictly local basis set for a systematic introduction of new details into Hilbert space operators. Using this tool we have previously developed an expansion method for density matrices. The set of density operators providing a given electron density plays an essential role in density functional theory, in the minimization of energy expectation values with the constraint that the electron density is fixed. In this contribution, using multiresolution analysis, we present an excellent quality density matrix expansion yielding a prescribed electron density, and compare it to other known methods. Due to the strictly local nature of the applied basis functions, our construction has the specific advantage that the resulting density matrix is correlated and N-representable in the infinite resolution limit. As a further consequence of this scheme we can conclude that the deviation of the exact kinetic energy functional from the Weizsacker term is not a necessary consequence of the particle statistics.

Localization maps by orbital partitioning of the electron density

SummaryWe define a localization measure for one-determinantal wave-functions based on the partitioning of the total electron density to orbital contributions. The set of occupied orbitals is the more localized the fewer terms are necessary to describe the total density. This measure varies from point to point in space which leads to characteristic localization maps for molecules.

The coupled cluster method and entanglement in three fermion systems

The Coupled Cluster (CC) and full CI expansions are studied for three fermions with six and seven modes. Surprisingly the CC expansion is tailor made to characterize the usual stochastic local operations and classical communication (SLOCC) entanglement classes. It means that the notion of a SLOCC transformation shows up quite naturally as a one relating the CC and CI expansions, and going from the CI expansion to the CC one is equivalent to obtaining a form for the state where the structure of the entanglement classes is transparent. In this picture, entanglement is characterized by the parameters of the cluster operators describing transitions from occupied states to singles, doubles, and triples of non-occupied ones. Using the CC parametrization of states in the seven-mode case, we give a simple formula for the unique SLOCC invariant J . Then we consider a perturbation problem featuring a state from the unique SLOCC class characterized by J ≠ 0 . For this state with entanglement generated by doubles, we...