Generalized Hopf algebra fundamental formula for non-orthogonal group functions

2005 ◽  
pp. 18-26
Author(s):  
Patrick Cassam-Chenaï
Keyword(s):  
Hopf Algebra ◽  
Orthogonal Group ◽  
Group Functions ◽  
Fundamental Formula

The density matrix in may-electron quantum mechanics III. Generalized product functions for beryllium and four-electron ions

1963 ◽  
Vol 273 (1352) ◽  
pp. 103-116 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Density Matrix ◽  
Orthogonal Group ◽  
Electron Quantum ◽  
Group Functions ◽  
Strong Orthogonality

The theory of generalized product functions is extended to non-orthogonal group functions in the case of two groups. Calculations on beryllium and on some four-electron ions are based on such group functions, and are compared with similar calculations using orthogonal group functions. I t is shown that, in the case of the systems examined, the strong orthogonality restriction is not severe.

scholarly journals Ab Initio Non-Covalent Crystal Field Theory for Lanthanide Complexes: A Multiconfigurational Non-Orthogonal Group Functions Approach

2020 ◽  
Author(s):  
Alessandro Soncini ◽  
MATTEO PICCARDO
Keyword(s):  
Magnetic Properties ◽  
Perturbation Theory ◽  
Ab Initio ◽  
Crystal Field ◽  
Orthogonal Group ◽  
Lanthanide Complexes ◽  
Field Energy ◽  
First Order ◽  
Group Functions ◽  
Metal Ligand

We present a non-orthogonal fragment ab initio methodology for the calculation of crystal field energy levels and magnetic properties in lanthanide complexes, implementing a systematic description of non-covalent contributions to metal-ligand bonding. The approach has two steps. In the first step, appropriate ab initio wavefunctions for the various ionic fragments (lanthanide ion and coordinating ligands) are separately optimized, accounting for the electrostatic influence of the surrounding environment, within various approximations. In the second and final step, the scalar relativistic (DKH2) electrostatic Hamiltonian of the whole molecule is represented on the basis of the optimized metal-ligand multiconfigurational non-orthogonal group functions (MC-NOGF), and reduced to an effective (2J+1)-dimensional non-orthogonal Configuration Interaction (CI) problem via L{\"o}wdin-partitioning. Within the proposed formalism, the projected Hamiltonian can be implemented to any desired order of perturbation theory in the fragment-localised excitations out of the degenerate space, and its eigenvalues and eigenfunctions are systematic approximations to the crystal field energies and wavefunctions. We present a preliminary implementation of the proposed MC-NOGF method to first-order degenerate perturbation theory within our own ab initio code CERES, and compare its performance both with the simpler non-covalent orthogonal ab initio approach Fragment Ab Initio Model Potential (FAIMP) approximation, and with the full CAHF/CASCI-SO method, accounting for metal-ligand covalency in a mean-field manner. We find that energies and magnetic properties for 44 complexes obtained via an iteratively optimized version of our MC-NOGF first-order non-covalent method, compare remarkably well to the full CAHF/CASCI-SO method including metal-ligand covalency, and are superior to the best purely electrostatic results achieved via an iteratively optimized version of the FAIMP approach.<br>

scholarly journals Ab Initio Non-Covalent Crystal Field Theory for Lanthanide Complexes: A Multiconfigurational Non-Orthogonal Group Functions Approach

2020 ◽  
Author(s):  
Alessandro Soncini ◽  
MATTEO PICCARDO
Keyword(s):  
Magnetic Properties ◽  
Perturbation Theory ◽  
Ab Initio ◽  
Crystal Field ◽  
Orthogonal Group ◽  
Lanthanide Complexes ◽  
Field Energy ◽  
First Order ◽  
Group Functions ◽  
Metal Ligand

We present a non-orthogonal fragment ab initio methodology for the calculation of crystal field energy levels and magnetic properties in lanthanide complexes, implementing a systematic description of non-covalent contributions to metal-ligand bonding. The approach has two steps. In the first step, appropriate ab initio wavefunctions for the various ionic fragments (lanthanide ion and coordinating ligands) are separately optimized, accounting for the electrostatic influence of the surrounding environment, within various approximations. In the second and final step, the scalar relativistic (DKH2) electrostatic Hamiltonian of the whole molecule is represented on the basis of the optimized metal-ligand multiconfigurational non-orthogonal group functions (MC-NOGF), and reduced to an effective (2J+1)-dimensional non-orthogonal Configuration Interaction (CI) problem via L{\"o}wdin-partitioning. Within the proposed formalism, the projected Hamiltonian can be implemented to any desired order of perturbation theory in the fragment-localised excitations out of the degenerate space, and its eigenvalues and eigenfunctions are systematic approximations to the crystal field energies and wavefunctions. We present a preliminary implementation of the proposed MC-NOGF method to first-order degenerate perturbation theory within our own ab initio code CERES, and compare its performance both with the simpler non-covalent orthogonal ab initio approach Fragment Ab Initio Model Potential (FAIMP) approximation, and with the full CAHF/CASCI-SO method, accounting for metal-ligand covalency in a mean-field manner. We find that energies and magnetic properties for 44 complexes obtained via an iteratively optimized version of our MC-NOGF first-order non-covalent method, compare remarkably well to the full CAHF/CASCI-SO method including metal-ligand covalency, and are superior to the best purely electrostatic results achieved via an iteratively optimized version of the FAIMP approach.<br>

scholarly journals Hopf algebra methods in graph theory

1995 ◽  
Vol 101 (1) ◽  
pp. 77-90 ◽  
Author(s):  
William R. Schmitt
Keyword(s):  
Graph Theory ◽  
Hopf Algebra

scholarly journals CLASSICAL MARKOV PROCESSES FROM QUANTUM LÉVY PROCESSES

1999 ◽  
Vol 02 (01) ◽  
pp. 105-129 ◽  
Author(s):  
UWE FRANZ
Keyword(s):  
Markov Process ◽  
Hopf Algebra ◽  
Markov Processes ◽  
Lévy Processes ◽  
Sufficient Conditions ◽  
Markov Property ◽  
Vacuum State ◽  
Levy Processes ◽  
Quantum Process ◽  
Quantum Markov Processes

We show how classical Markov processes can be obtained from quantum Lévy processes. It is shown that quantum Lévy processes are quantum Markov processes, and sufficient conditions for restrictions to subalgebras to remain quantum Markov processes are given. A classical Markov process (which has the same time-ordered moments as the quantum process in the vacuum state) exists whenever we can restrict to a commutative subalgebra without losing the quantum Markov property.8 Several examples, including the Azéma martingale, with explicit calculations are presented. In particular, the action of the generator of the classical Markov processes on polynomials or their moments are calculated using Hopf algebra duality.

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