Gaussian Wave Functions and the Many-Body Problem

1972 ◽  
Vol 50 (4) ◽  
pp. 305-311 ◽  
Author(s):  
R. L. Hall
Keyword(s):  
Bound States ◽  
Wave Functions ◽  
Permutation Symmetry ◽  
Many Body ◽  
Single Product ◽  
Gaussian Functions ◽  
Energy Expectation ◽  
The Many ◽  
Body Energy ◽  
Exact Energy

A system of identical nonrelativistic particles is considered. It is shown that the wave functions for relative motion, which have the correct permutation symmetry, must satisfy two functional equations. In the case of bosons these equations are solved for those bound states where the wave function is also in a single-product form. The only solutions are Gaussian functions. Consequently these are the only functions which can reduce the N-body energy expectation to an integral over a single variable. Furthermore, we show that our reduced two-body Hamiltonian which in general gives energy lower bounds yields the exact energy of the entire system only for the Hooke's law interaction. Neither possibility is allowed by fermions.

The many-body expansion for aqueous systems revisited: III. Hofmeister ion–water interactions

2021 ◽  
Author(s):  
Kristina M. Herman ◽  
Joseph P. Heindel ◽  
Sotiris S. Xantheas
Keyword(s):  
Water Molecules ◽  
Hofmeister Series ◽  
Aqueous Systems ◽  
Many Body ◽  
Ionic Clusters ◽  
Anions And Cations ◽  
Chaotropic Anions ◽  
The Many ◽  
Body Energy

We report a Many Body Energy (MBE) analysis of aqueous ionic clusters containing kosmotropic and chaotropic anions and cations at the two opposite ends of the Hofmeister series to quantify how these ions alter the interaction between the water molecules in their immediate surroundings.

The potential model treatment of the scattering of electrons by atoms and the existence of negative ions

1961 ◽  
Vol 16 (5) ◽  
pp. 492-500
Author(s):  
F. B. Malik ◽  
E. Trefftz
Keyword(s):  
Bound States ◽  
Negative Ions ◽  
Potential Scattering ◽  
Point Of View ◽  
Neutral Atoms ◽  
Many Body ◽  
Scattered Electron ◽  
Scattered Particle ◽  
Potential Term ◽  
The Many

The low energy scattering of electrons by different neutral atoms has been treated by assuming that the atomic wave functions remain unchanged even at the presence of the scattered particle and by neglecting the exchange between the scattered electron and the bound electrons. The potential term in the differential equation of the scattered particle is exactly the atomic potential of the neutral atom and is approximated by analytical expressions, yielding the potential scattering equation. The variational treatments of Hulthén, Kohn and a related one suggested by Malik, are applied to solve this equation for a Hartree atom with l=0. The scattering by He, C and N is treated explicitly and the results of He indicate that in this way one may get some good result without going into the great complexity of the many body problem. It is further pointed out that the study of scattering by neutral atoms near zero energy under this model may serve as a possible mean to investigate the existence of different negative ions and their number of bound states. It seems from this point of view that He-, C- and N- for this model may exist and have one bound s-state.

scholarly journals Tunable discrete scale invariance in transition-metal pentatelluride flakes

2020 ◽  
Vol 5 (1) ◽  
Author(s):  
Yanzhao Liu ◽  
Huichao Wang ◽  
Haipeng Zhu ◽  
Yanan Li ◽  
Jun Ge ◽  
...  
Keyword(s):  
Transition Metal ◽  
Scale Invariance ◽  
Bound States ◽  
Scale Factor ◽  
Coulomb Interactions ◽  
Many Body ◽  
Body Effect ◽  
Discrete Scale Invariance ◽  
The Many ◽  
Discrete Scale

AbstractLog-periodic quantum oscillations discovered in transition-metal pentatelluride give a clear demonstration of discrete scale invariance (DSI) in solid-state materials. The peculiar phenomenon is convincingly interpreted as the presence of two-body quasi-bound states in a Coulomb potential. However, the modifications of the Coulomb interactions in many-body systems having a Dirac-like spectrum are not fully understood. Here, we report the observation of tunable log-periodic oscillations and DSI in ZrTe5 and HfTe5 flakes. By reducing the flakes thickness, the characteristic scale factor is tuned to a much smaller value due to the reduction of the vacuum polarization effect. The decreasing of the scale factor demonstrates the many-body effect on the DSI, which has rarely been discussed hitherto. Furthermore, the cut-offs of oscillations are quantitatively explained by considering the Thomas-Fermi screening effect. Our work clarifies the many-body effect on DSI and paves a way to tune the DSI in quantum materials.

Study of Unbound HePs Using Exact Diagonalization Technique

2012 ◽  
Vol 733 ◽  
pp. 38-42
Author(s):  
Asier Zubiaga ◽  
Filip Tuomisto ◽  
Martti Puska
Keyword(s):  
Variational Method ◽  
Interaction Potential ◽  
Exact Diagonalization ◽  
Wave Functions ◽  
Many Body ◽  
Strong Electron ◽  
Stochastic Variational Method ◽  
Explicitly Correlated ◽  
Pauli Repulsion ◽  
The Many

The many-body wavefunction of the unbound He-Ps system has been studied by the exact diagonalization of a explicitly correlated gaussians basis optimized by a stochastic variational method. The nucleus-positron distance has been varied by constraining the parameters of the nucleus-positron correlated gaussian term. The constraining technique allows to describe He and Ps interacting at different distances. The calculated wavefunction can be approximated as composed by weakly perturbed He and Ps atoms. The electron forming the Ps tends to be farther from the nucleus than the positron due to the strong electron-electron Pauli repulsion with the electrons of He. The described technique gives accurate energy and wave functions for Ps interacting with atoms that can be used to calculate the interaction potential of Ps with molecular matter.

scholarly journals A Simple Molecular Orbital Picture of RIXS Distilled from Many-Body Damped Response Theory

2020 ◽  
Author(s):  
Kaushik Nanda ◽  
Anna I. Krylov
Keyword(s):  
Molecular Orbital ◽  
Cross Sections ◽  
Transition Density ◽  
Wave Functions ◽  
Many Body ◽  
Response Theory ◽  
Electronic Density ◽  
Near Resonance ◽  
Alternative Approach ◽  
The Many

<div>Ab initio calculations of resonant inelastic X-ray scattering (RIXS) rely on the damped response theory, which prevents the divergence of response solutions in the resonant regime. Within the damped response theory formalism, RIXS moments are expressed as sum over all electronic states of the system (SOS expressions). By invoking resonance arguments, these expressions can be reduced to a few terms, an approximation commonly exploited for interpretation of the computed cross sections. We present an alternative approach: a rigorous formalism for deriving a simple molecular orbital picture of the RIXS process from the many-body calculations using damped</div><div>response theory. In practical implementations, the SOS expressions of RIXS moments are recast in terms of matrix elements between the zero-order wave functions and first-order frequency-dependent response wave functions of the initial and final states, such that the RIXS moments can be evaluated using complex response</div><div>one-particle transition density matrices (1PTDMs). Visualization of these 1PTDMs connects the RIXS process with the changes in electronic density. We demonstrate that the real and imaginary components of the response 1PTDMs can be interpreted as contributions of the undamped off-resonance and damped near-resonance SOS terms, respectively. By analyzing these 1PTDMs in terms of natural transition orbitals, we derive a rigorous, black-box mapping of the RIXS process into a molecular orbital picture. We illustrate the utility of the new tool by analyzing RIXS transitions in the OH radical, benzene, para-nitroaniline, and</div><div> 4-amino-4'-nitrostilbene. These examples highlight the significance of both near-resonance and off-resonance channels. </div>

scholarly journals EXACT SOLUTIONS OF THE MODIFIED KRATZER POTENTIAL PLUS RING-SHAPED POTENTIAL IN THE D-DIMENSIONAL SCHRÖDINGER EQUATION BY THE NIKIFOROV–UVAROV METHOD

2008 ◽  
Vol 19 (02) ◽  
pp. 221-235 ◽  
Author(s):  
SAMEER M. IKHDAIR ◽  
RAMAZAN SEVER
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Laguerre Polynomials ◽  
Three Dimensional ◽  
Wave Functions ◽  
Modified Kratzer Potential ◽  
Energy Bound ◽  
Uvarov Method ◽  
Exact Energy

We present analytically the exact energy bound-states solutions of the Schrödinger equation in D dimensions for a recently proposed modified Kratzer plus ring-shaped potential by means of the Nikiforov–Uvarov method. We obtain an explicit solution of the wave functions in terms of hyper-geometric functions (Laguerre polynomials). The results obtained in this work are more general and true for any dimension which can be reduced to the well-known three-dimensional forms given by other works.

scholarly journals The Valence-Bond (VB) Model and Its Intimate Relationship to the Symmetric or Permutation Group

Molecules ◽  
2021 ◽  
Vol 26 (15) ◽  
pp. 4524
Author(s):  
Marco Antonio Chaer Nascimento
Keyword(s):  
Permutation Group ◽  
Point Group ◽  
Valence Bond ◽  
Fundamental Difference ◽  
Wave Functions ◽  
Group Symmetry ◽  
Chemical Bonds ◽  
Permutation Symmetry ◽  
Independent Entity ◽  
The Many

VB and molecular orbital (MO) models are normally distinguished by the fact the first looks at molecules as a collection of atoms held together by chemical bonds while the latter adopts the view that each molecule should be regarded as an independent entity built up of electrons and nuclei and characterized by its molecular structure. Nevertheless, there is a much more fundamental difference between these two models which is only revealed when the symmetries of the many-electron Hamiltonian are fully taken into account: while the VB and MO wave functions exhibit the point-group symmetry, whenever present in the many-electron Hamiltonian, only VB wave functions exhibit the permutation symmetry, which is always present in the many-electron Hamiltonian. Practically all the conflicts among the practitioners of the two models can be traced down to the lack of permutation symmetry in the MO wave functions. Moreover, when examined from the permutation group perspective, it becomes clear that the concepts introduced by Pauling to deal with molecules can be equally applied to the study of the atomic structure. In other words, as strange as it may sound, VB can be extended to the study of atoms and, therefore, is a much more general model than MO.

Estimates on derivatives of Coulombic wave functions and their electron densities

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Søren Fournais ◽  
Thomas Østergaard Sørensen
Keyword(s):  
Ground State ◽  
A Priori ◽  
Wave Functions ◽  
Many Body ◽  
A Priori Bounds ◽  
Electron Densities ◽  
The Many ◽  
Body Potential ◽  
Derivatives Of

Abstract We prove a priori bounds for all derivatives of non-relativistic Coulombic eigenfunctions ψ, involving negative powers of the distance to the singularities of the many-body potential. We use these to derive bounds for all derivatives of the corresponding one-electron densities ρ, involving negative powers of the distance from the nuclei. The results are both natural and optimal, as seen from the ground state of Hydrogen.

scholarly journals Relativistic solution in D-dimensions to a spin-zero particle for equal scalar and vector ring-shaped Kratzer potential

Open Physics ◽  
2008 ◽  
Vol 6 (1) ◽  
Author(s):  
Sameer Ikhdair ◽  
Ramazan Sever
Keyword(s):  
Bound States ◽  
Gordon Equation ◽  
Wave Functions ◽  
Three Dimensions ◽  
Vector Potentials ◽  
Klein Gordon ◽  
Energy Bound ◽  
Uvarov Method ◽  
Equal Scalar ◽  
Exact Energy

AbstractThe Klein-Gordon equation in D-dimensions for a recently proposed ring-shaped Kratzer potential is solved analytically by means of the conventional Nikiforov-Uvarov method. The exact energy bound states and the corresponding wave functions of the Klein-Gordon are obtained in the presence of the non-central equal scalar and vector potentials. The results obtained in this work are more general and can be reduced to the standard forms in three dimensions given by other works.

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