Measuring Multi-Configurational Character by Orbital Entanglement

One of the most critical tasks at the very beginning of a quantum chemical investigation is the choice of either a multi- or single-configurational method. Naturally, many proposals exist to define a suitable diagnostic of the multi-configurational character for various types of wave functions in order to assist this crucial decision. Here, we present a new orbital-entanglement based multi-configurational diagnostic termed $Z_{s(1)}$. The correspondence of orbital entanglement and static (or nondynamic) electron correlation permits the definition of such a diagnostic. We chose our diagnostic to meet important requirements such as well-defined limits for pure single-configurational and multi-configurational wave functions. The $Z_{s(1)}$ diagnostic can be evaluated from a partially converged, but qualitatively correct, and therefore inexpensive density matrix renormalization group wave function as in our recently presented automated active orbital selection protocol. Its robustness and the fact that it can be evaluated at low cost make this diagnostic a practical tool for routine applications.Comment: 8 pages, 2 figure, 3 table

Full Spin and Spatial Symmetry Adapted Technique for Correlated Electronic Hamiltonians: Application to an Icosahedral Cluster

One of the long standing problems in quantum chemistry had been the inability to exploit full spatial and spin symmetry of an electronic Hamiltonian belonging to a non-Abelian point group. Here we present a general technique which can utilize all the symmetries of an electronic (magnetic) Hamiltonian to obtain its full eigenvalue spectrum. This is a hybrid method based on Valence Bond basis and the basis of constant z-component of the total spin. This technique is applicable to systems with any point group symmetry and is easy to implement on a computer. We illustrate the power of the method by applying it to a model icosahedral half-filled electronic system. This model spans a huge Hilbert space (dimension 1,778,966) and in the largest non-Abelian point group. The $C_{60}$ molecule has this symmetry and hence our calculation throw light on the higher energy excited states of the bucky ball. This method can also be utilized to study finite temperature properties of strongly correlated systems within an exact diagonalization approach.Comment: 21 pages, 7 figures, abstract rewritten, a few changes in text, to appear in International Journal of Quantum Chemistr

An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group

The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combinining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained from a polar decomposition of SU(2) and analysed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach.Comment: Topical review (40 pages). Dedicated to the memory of Yurii Fedorovich Smirno

The density-matrix renormalization group

The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly acquired the status of method of choice for numerical studies of one-dimensional quantum systems. Its applications to the calculation of static, dynamic and thermodynamic quantities in such systems are reviewed. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and non-equilibrium statistical physics, and time-dependent phenomena is discussed. This review also considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by DMRG.Comment: accepted by Rev. Mod. Phys. in July 2004; scheduled to appear in the January 2005 issu

Representation theory and Wigner-Racah algebra of the SU(2) group in a noncanonical basis

The Lie algebra su(2) of the classical group SU(2) is built from two commuting quon algebras for which the deformation parameter is a common root of unity. This construction leads to (i) a not very well-known polar decomposition of the ladder generators of the SU(2) group, in terms of a unitary operator and a Hermitean operator, and (ii) a nonstandard quantization scheme, alternative to the usual quantization scheme correponding to the diagonalization of the Casimir of su(2) and of the z-generator. The representation theory of the SU(2) group can be developed in this nonstandard scheme. The key ideas for developing the Wigner-Racah algebra of the SU(2) group in the nonstandard scheme are given. In particular, some properties of the coupling and recoupling coefficients as well as the Wigner-Eckart theorem in the nonstandard scheme are examined in great detail.Comment: To be presented at ICSSUR'05 (9th International Conference on Squeezed States and Uncertainty Relations, France, 2-6 May 2005). Dedicated to Professor Josef Paldus on the occasion of his 70th birthday. To be published in Collection of Czechoslovak Chemical Communication

Performance of Shannon-entropy compacted N-electron wave functions for configuration interaction methods

The coefficients of full configuration interaction wave functions (FCI) for N-electron systems expanded in N-electron Slater determinants depend on the orthonormal one-particle basis chosen although the total energy remains invariant. Some bases result in more compact wave functions, i.e. result in fewer determinants with significant expansion coefficients. In this work, the Shannon entropy, as a measure of information content, is evaluated for such wave functions to examine whether there is a relationship between the FCI Shannon entropy of a given basis and the performance of that basis in truncated CI approaches. The results obtained for a set of randomly picked bases are compared to those obtained using the traditional canonical molecular orbitals, natural orbitals, seniority minimising orbitals and a basis that derives from direct minimisation of the Shannon entropy. FCI calculations for selected atomic and molecular systems clearly reflect the influence of the chosen basis. However, it is found that there is no direct relationship between the entropy computed for each basis and truncated CI energies.Fil: Alcoba, Diego Ricardo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Torre, Alicia. Universidad de Buenos Aires; Argentina. Universidad de Buenos Aires; ArgentinaFil: Lain, Luis. Universidad del País Vasco; España. Universidad del País Vasco; EspañaFil: Massaccesi, Gustavo Ernesto. Universidad del País Vasco; España. Universidad del País Vasco; EspañaFil: Oña, Ofelia Beatriz. Universidad de Buenos Aires; Argentina. Universidad de Buenos Aires; ArgentinaFil: Ayers, P. W.. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Van Raemdonck, M.. Mcmaster University; Canadá. Mcmaster University; CanadáFil: Bultinck, P.. University of Ghent; Bélgica. University of Ghent; BélgicaFil: Van Neck, D.. University of Ghent; Bélgica. University of Ghent; Bélgic