The application of inner projection technique to many electron systems as compared with the coupled cluster expansion. A study of the Pariser-Parr-Pople model of the benzene molecule

1988 ◽  
Vol 53 (9) ◽  
pp. 1910-1918 ◽  
Author(s):  
Jiří Čížek ◽  
Francine Vinette
Keyword(s):  
Lower Bound ◽  
Ground State ◽  
State Energy ◽  
Benzene Molecule ◽  
Coupled Cluster ◽  
Electron Systems ◽  
Projection Technique ◽  
Electron Theory ◽  
First Time ◽  
Intermediate Hamiltonians

The technique of inner projection is used for the first time in a chemical context, namely for the study of the ground state energy of the Pariser-Parr-Pople model of the benzene molecule. The lower bound for the energy is calculated for three types of parametrization. These results are compared with the “exact” energies which are obtained from full configuration interaction. It is shown that the inner projection technique provides very good lower bounds for the energy. In addition, we compare these inner projection results with those obtained by other approximative techniques, namely with the results of the Coupled Pair Many Electron Theory. A discussion of the application of the method of intermediate hamiltonians and the inner projection technique is also included.

Energy bounds for the spiked harmonic oscillator

1995 ◽  
Vol 73 (7-8) ◽  
pp. 493-496 ◽  
Author(s):  
Richard L. Hall ◽  
Nasser Saad
Keyword(s):  
Lower Bound ◽  
Harmonic Oscillator ◽  
Ground State Energy ◽  
Upper Bound ◽  
Ground State ◽  
Trial Function ◽  
State Energy ◽  
Parameter Range ◽  
Complementary Energy ◽  
Energy Bounds

A three-parameter variational trial function is used to determine an upper bound to the ground-state energy of the spiked harmonic-oscillator Hamiltonian [Formula: see text]. The entire parameter range λ > 0 and α ≥ 1 is treated in a single elementary formulation. The method of potential envelopes is also employed to derive a complementary energy lower bound formula valid for all the discrete eigenvalues.

A Cluster Expansion Formalism for Direct Calculation of Ionisation Potential and Excitation Energy of Many Electron Systems Using H-F Ground State as Vacuum

1978 ◽  
Vol 33 (12) ◽  
pp. 1549-1551
Author(s):  
D. Mukherjee ◽  
A. Mukhopadhyay ◽  
R. K. Moitra
Keyword(s):  
Excitation Energy ◽  
Cluster Expansion ◽  
Ground State ◽  
Shell Theory ◽  
State Energy ◽  
Direct Calculation ◽  
Electron System ◽  
Open Shell ◽  
Electron Systems ◽  
Ionisation Potential

Abstract In this note, the authors’ recently developed non-perturbative open-shell theory is adapted for direct calculation o f ionisation potential and excitation energy of m any-electron systems. The H -F ground state is used as the “vacuum ” or “ core” in order to achieve a transparent separation o f the ground state energy. An application to a simple 4 π-electron system is discussed as an illustration o f the workability of the theory.

scholarly journals Remarks on the boundary set of spectral equipartitions

2014 ◽  
Vol 372 (2007) ◽  
pp. 20120492 ◽  
Author(s):  
P. Bérard ◽  
B. Helffer
Keyword(s):  
Riemannian Manifold ◽  
Lower Bound ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Two Dimensional ◽  
Nodal Sets ◽  
Nodal Domains ◽  
Open Set ◽  
Ground State Energies

Given a bounded open set in (or in a Riemannian manifold), and a partition of Ω by k open sets ω j , we consider the quantity , where λ ( ω j ) is the ground state energy of the Dirichlet realization of the Laplacian in ω j . We denote by ℒ k ( Ω ) the infimum of over all k -partitions. A minimal k -partition is a partition that realizes the infimum. Although the analysis of minimal k -partitions is rather standard when k =2 (we find the nodal domains of a second eigenfunction), the analysis for higher values of k becomes non-trivial and quite interesting. Minimal partitions are in particular spectral equipartitions, i.e. the ground state energies λ ( ω j ) are all equal. The purpose of this paper is to revisit various properties of nodal sets, and to explore if they are also true for minimal partitions, or more generally for spectral equipartitions. We prove a lower bound for the length of the boundary set of a partition in the two-dimensional situation. We consider estimates involving the cardinality of the partition.

scholarly journals Helium-like ions in d-dimensions: analyticity and generalized ground state Majorana solutions

2021 ◽  
Author(s):  
Adrian Mauricio Escobar ◽  
Horacio Olivares-Pilón ◽  
Norberto Aquino ◽  
Salvador Antonio Cruz-Jimenez
Keyword(s):  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Dimensional Space ◽  
Algebraic Function ◽  
Approximate Solutions ◽  
Algebraic Expression ◽  
Three Body ◽  
First Time ◽  
Analytical Expressions

Abstract Non-relativistic Helium-like ions (−e, −e, Ze) with static nucleus in a d−dimensional space (d > 1) are considered. Assuming r−1Coulomb interactions, a 2-parametric correlated Hylleraas-type trial function is used to calculate the ground state energy of the system in the domain Z ≤ 10. For odd d = 3, 5, the variational energy is given by a rational algebraic function of the variational parameters whilst for even d = 2, 4 it is shown for the first time that it corresponds to a more complicated non-algebraic expression. This twofold analyticity will hold for any d. It allows us to construct reasonably accurate approximate solutions for the ground state energy E0(Z, d) in the form of compact analytical expressions. We call them generalized Majorana solutions. They reproduce the first leading terms in the celebrated 1Z expansion, and serve as generating functions for certain correlation-dependent properties. The (first) critical charge Zc vs d and the Shannon entropy S(d)r vs Z are also calculated within the present variational approach. In the light of these results, for the physically important case d = 3 a more general 3-parametric correlated Hylleraas-type trial is used to compute the finite mass effects in the Majorana solution for a three-body Coulomb system with arbitrary charges and masses. It admits a straightforward generalization to any d as well. Concrete results for the systems e− e− e+, H+2 and H− are indicated explicitly. Our variational analytical results are in excellent agreement with the exact numerical values reported in the literature.

Lower Bound to the Ground-State Energy and Mass Polarization in Helium-Like Atoms

1956 ◽  
Vol 103 (1) ◽  
pp. 112-115 ◽  
Author(s):  
Lawrence Wilets ◽  
Ivan J. Cherry
Keyword(s):  
Lower Bound ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy

scholarly journals The Coupled Cluster Method Applied to the Spin-Half XXZ Model on the Honeycomb Lattice

1998 ◽  
Vol 12 (23) ◽  
pp. 2371-2383 ◽  
Author(s):  
R. F. Bishop ◽  
J. Rosenfeld
Keyword(s):  
Ground State ◽  
State Energy ◽  
Finite Lattice ◽  
Approximation Schemes ◽  
Cluster Method ◽  
Coupled Cluster ◽  
Series Expansions ◽  
Honeycomb Lattice ◽  
Coupled Cluster Method ◽  
Xxz Model

The coupled cluster method (CCM) is applied to the spin-half XXZ model on the honeycomb lattice. Hierarchical CCM approximation schemes are applied in order to obtain the ground-state properties. The possible critical behavior of this model is also examined. Reasonably good results for the ground-state energy and the staggered magnetization are obtained, in comparison to the results from Monte Carlo simulations, spin-wave techniques, series expansions, and finite lattice diagonalizations, that have already been performed on this model.

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