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![Symmetry of the orbitals used in this work for the H2O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {H}_{2}\hbox {O}$$\end{document} at equilibrium geometry. ±0.11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm 0.11$$\end{document} a.u. isosurfaces in all plots](publication/303531319/figure/fig2/AS:960035379228672@1605901679021/Symmetry-of-the-orbitals-used-in-this-work-for-the-H2Odocumentclass12ptminimal.gif)
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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 coe...
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... The formulation of compact wave functions, be it exact or approximate formulations, is a central part of both quantum physics and chemistry. The guiding principles behind these formulations are often vague though attempts at stricter measures of compactness from entropy measures are currently being developed [1][2][3][4][5]. While correlation between entropy and compactness is seen for the methods investigated with the entropy measures these methods are, unlike full configuration interaction (FCI), not invariant to orbital rotations, and the result may therefore be dependent on the choice of orbitals. ...
We show the transformation from a one-particle basis to a geminal basis, transformations between different geminal bases demonstrate the Lie algebra of a geminal basis. From the basis transformations, we express both the wave function and Hamiltonian in the geminal basis. The necessary and sufficient conditions of the exact wave function expanded in a geminal basis are shown to be a Brillouin theorem of geminals. The variational optimization of the geminals in the antisymmetrized geminal power (AGP), antisymmetrized product of geminals (APG) and the full geminal product (FGP) wave function ansätze are discussed. We show that using a geminal replacement operator to describe geminal rotations introduce both primary and secondary rotations. The secondary rotations rotate two geminals in the reference at the same time due to the composite boson nature of geminals. Due to the completeness of the FGP, where all possible geminal combinations are present, the FGP is exact. The number of parameters in the FGP scale exponentially with the number of particles, like the full configuration interaction (FCI). Truncation in the FGP expansion can give compact representations of the wave function since the reference function in the FGP can be either the AGP or APG wave function.
... It is mostly investigated nowadays for analysis of electron densities [4][5][6] and has numerous applications. It has been used in atomic avoided crossing [7], electron correlation [8], orbital free density functional theory [9], configuration interactions [10], entanglement in artificial atoms [11], etc. The negative value of Shannon entropy specifies intense localization. ...
We calculate Shannon entropy for Modified Hulthen potential (MHP). Dependence of Shannon entropy on various parameters of the potential is investigated. For solving Schrodinger equation, i.e., for calculation of the wavefunctions, Numerov method has been employed which is fast and efficient. Numerical simulation is performed for the purpose. Novel data for Shannon entropy is presented for a wide range of the parameters of MHP.
... Therefore, in this occasion, disequilibrium and information energy have same form. Like the previous measures, it is also utilized in orbital-free DFT [105], testing normality [136], electron correlation [110], Colin conjecture [100,101], configuration interaction [137] etc. By definition, E refers to the 2nd-order entropic moment [127]; for central 32 ...
In this chapter we shall discuss the recent progresses of information theoretic tools in the context of free and confined harmonic oscillator. Confined quantum systems have provided appreciable interest in areas of physics, chemistry, biology,etc., since its inception. A particle under extreme pressure environment unfolds many fascinating, notable physical and chemical changes. The desired effect is achieved by reducing the spatial boundary from infinity to a finite region. Similarly, in the last decade, information measures were investigated extensively in diverse quantum problems, in both free and constrained situations. The most prominent amongst these are: Fisher information, Shannon entropy, Renyi entropy , Tsallis entropy, Onicescu energy and several complexities. Arguably, these are the most effective measures of uncertainty, as they do not make any reference to some specific points of a respective Hilbert space. These have been invoked to explain several physico-chemical properties of a system under investigation. Kullback-Leibler divergence or relative entropy describes how a given probability distribution shifts from a reference distribution function. This characterizes a measure of discrimination between two states. In other words, it extracts the change of information in going from one state to another.
... where S is a subset of the Slater determinants within the given basis. The Slater determinants can be selected by seniority, such as the doubly occupied CI (DOCI) wavefunction [11][12][13][14][15][16][17][18][19][20][21][22], or by excitation-level relative to a reference Slater determinant, such as CI singles and doubles (CISD) wavefunction [23]. Alternatively, we can select all (or many) of the Slater determinants in a given set of orbitals. ...
We present a Flexible Ansatz for N-body Configuration Interaction (FANCI) that includes any multideterminant wavefunction. This ansatz is a generalization of the Configuration Interaction (CI) wavefunction, where the coefficients are replaced by a specified function of certain parameters. By making an appropriate choice for this function, we can reproduce popular wavefunction structures like CI, Coupled-Cluster, Tensor Network States, and geminal-product wavefunctions. The universality of this framework suggests a programming structure that allows for the easy construction and optimization of arbitrary wavefunctions. Here, we will discuss the structures of the FANCI framework and its implications for wavefunction properties, particularly accuracy, cost, and size-consistency. We demonstrate the flexibility of this framework by reconstructing popular wavefunction ans\"{a}tze and modifying them to construct novel wavefunction forms. FANCI provides a powerful framework for exploring, developing, and testing new wavefunction forms.
... For example, increasing Shannon entropy increases the spreading of the probability density, but the increase in Onicescu energy decreases the spreading of the probability density and vice-versa. The implementations of Shannon entropy are in atomic avoided crossing [30], illuminating Colin conjecture, electron correlation [31], orbital free density functional theory [32], aromaticity in many-electron systems [33], configuration interaction [34], entanglement in artificial atoms [35] etc. Onicescu energy has also been commonly used to estimate correlation energy and first ionization potential [21]. Over the past twenty years, information measures have been thoroughly studied with different quantum systems. ...
Various information theoretic measures, complexities and their inequalities are studied for hydrogen atom under modified Hulthen potential which is a combination of Hulthen potential and inverse square potential. In addition, spherical confinement is also considered. Few of these results are compared with the results are available in the literature. In particular the Shannon entropy for free and confined atom is detailed for many states. The effects of inverse square potential parameter (’a’ which is either positive or negative) on Shannon entropy and Fisher information is studied in confined and free region of atom. The Bialynicki-Birula-Mycielski (BBM) inequality is verified for the confined system.
... Importantly, being reciprocally connected to S, E usually upholds the inferences obtained from S. Particularly, in a given space, the increase of spreading in a density distribution is quantified by an increment in S and decay in E. Both SandE are successfully employed to quantify various density-distributions produced from relevant theoretical or experimental processes. S has a connection to Colin conjecture [53,54], atomic avoided crossing [55], electron correlation effect [54], configuration interaction [56], quantum entanglement in artificial atom [57,58], bond formation [59], elementary chemical reactions [60], orbital-free DFT [61], aromaticity [62] etc. Some of these like the avoided crossing occur under a confinement condition. ...
Confinement of atoms inside various cavities has been studied for nearly eight decades. However, the Compton profile for such systems has not yet been investigated. Here we construct the Compton profile (CP) for a H atom radially confined inside a hard spherical enclosure, as well as in free condition. Some exact analytical relations for the CP's of circular or nodeless states of free atom is presented. By means of a scaling idea, this has been further extended to the study of an H-like atom trapped inside an impenetrable cavity. The accuracy of these constructed CP has been confirmed by computing various momentum moments. Apart from that, several information theoretical measures, like Shannon entropy (S) and Onicescu energy (E) have been exploited to characterize these profiles. Exact closed form expressions are derived for S and E using the ground state CP in free H-like atoms. A detailed study reveals that, increase in confinement inhibits the rate of dissipation of kinetic energy. At a fixed ℓ, this rate diminishes with rise in n. However, at a certain n, this rate accelerates with progress in ℓ. A similar analysis on the respective free counterpart displays an exactly opposite trend as that in confined system. However, in both free and confined environments, CP generally gets broadened with rise in Z. Representative calculations are done numerically for low-lying states of the confined systems, taking two forms of position-space wave functions: (a) exact (b) highly accurate eigenfunctions through a generalized pseudospectral method. In essence, CPs are reported for confined H atom (and isoelectronic series) and investigated adopting an information-theoretic framework.
... This procedure aims to produce orbitals with better compressibility in representing the many-body wave function. The optimality of natural orbital has been questioned in several works, [20][21][22][23] which proposed various optimization procedures under different definitions of optimalities. One short coming of all these works, however, is that all these orbital rotations build on top of the many-body wave function with orbitals of the same size as that of the original molecular orbitals; thus it does not save much computational cost when we start with a large basis set. ...
Full configuration interaction (FCI) solvers are limited to small basis sets due to their expensive computational costs. An optimal orbital selection for FCI (OptOrbFCI) is proposed to boost the power of existing FCI solvers to pursue the basis set limit under computational budget. The method effectively finds an optimal rotation matrix to compress the orbitals of large basis sets to one with a manageable size, conducts FCI calculations only on rotated orbital sets, and produces a variational ground-state energy and its wave function. Coupled with coordinate descent full configuration interaction (CDFCI), we demonstrate the efficiency and accuracy of the method on carbon dimer and nitrogen dimer under basis sets up to cc-pV5Z. We also benchmark the binding curve of nitrogen dimer under cc-pVQZ basis set with 28 selected orbitals, which provide consistently lower ground-state energies than the FCI results under cc-pVDZ basis set. The dissociation energy in this case is found to be of higher accuracy.
... Therefore, in this occasion, disequilibrium and information energy have same form. Like the previous measures, it is also utilized in orbital-free DFT [105], testing normality [136], electron correlation [110], Colin conjecture [100, 101], configuration interaction [137] etc. By definition, E refers to the 2nd-order entropic moment [127]; for central potential it assumes the form (E t is the Onicescu energy product), ...
... Here a point to note that equation (136) and equation (137) are coupled equations. The solution scheme will be presented in the context of quantum dynamics. ...
... This is exactly the similar equation to that of equation (137). Here a point to note that equation (149) and (150) are coupled differential equations. ...
This book gathers state-of-the-art advances on harmonic oscillators including their types, functions, and applications. In Chapter 1, Neetik and Amlan have discussed the recent progresses of information theoretic tools in the context of free and confined harmonic oscillator. Confined quantum systems have provided appreciable interest in areas of physics, chemistry, biology, etc., since its inception. A particle under extreme pressure environment unfolds many fascinating, notable physical and chemical changes.
The desired effect is achieved by reducing the spatial boundary from infinity to a finite region. Similarly, in the last decade, information measures were investigated extensively in diverse quantum problems, in both free and constrained situations. The most prominent amongst these are: Fisher information, Shannon entropy, Renyi entropy, Tsallis entropy, Onicescu energy and several complexities. Arguably, these are the most effective measures of uncertainty, as they do not make any reference to some specific points of a respective Hilbert space. These have been invoked to explain several physic-chemical properties of a system under investigation. Kullback Leibler divergence or relative entropy describes how a given probability distribution shifts from a reference distribution function. This characterizes a measure of discrimination between two states. In other words, it extracts the change of information in going from one state to another.
In Chapter 2, Nabakumar, Subhasree, and Paulami have revisited classical-quantum correspondence in the context of linear Simple Harmonic Oscillator (SHO). According to Bohr’s correspondence principle, quantum mechanically calculated results match with the classically expected results when quantum number is very high. Classical quantum correspondence may also be visualized in the limit when the action integral is much greater than Planck’s constant. When de-Broglie wave length associated with a particle is much larger than system size, then quantum mechanical results also match with the classical results. In the context of dynamics, Ehrenfest equation of motion is used in quantum domain, which is analogous to classical Newton’s equation of motion. SHO is one of the most important systems for several reasons. It is one of the few exactly solvable problems. Any stable molecular potential can be approximated by SHO near the equilibrium point. This builds the foundation for the understanding of complex modes of vibration in large molecules, the motion of atoms in a solid lattice, the theory of heat capacity, vibration motion of nuclei in molecule etc. The authors have revisited the common solution techniques and important properties of both classical and quantum linear SHO. Then they focused on probability distribution, quantum mechanical tunneling, classical and quantum dynamics of position, momentum and their actuations, viral theorems, etc. and also analyzed how quantum mechanical results finally tend to classical results in the high quantum number limit.
In Chapter 3, Neeraj has discussed the nature of atomic motions, sometimes referred to as lattice vibrations. The lattice dynamics deals with the vibrations of the atoms inside the crystals. In order to write the dynamic equations of the motion of crystal atoms, we need to describe an inter-atomic interaction. Therefore, it is natural to start the study of the lattice dynamics with the case of small harmonic vibrations. The dynamics of one-dimensional and two-dimensional vibrations of monatomic and diatomic crystals can be understood by using the simple model forces based on harmonic approximation. This harmonic approximation is related to a simple ball-spring model. According to this model, each atom is coupled with the neighboring atoms by spring constants. The collective motion of atoms leads to a distinct traveling wave over the whole crystal, leading to the collective motion, so-called phonon. The simple ball-spring model enlightens us some of the significant common features of lattice dynamics that have been discussed throughout this chapter. Further, this chapter helps in understanding the quantization energy of a harmonic oscillation and the concept of phonon.
... Recently, sparsity of the wavefunction has been the focus of significant research, drawing upon compressed sensing as well as deterministic and stochastic sampling. [16][17][18][19][20][21][22][23][24][25][26] The present work provides a novel, non-perturbative approach to the issue through constraints on the 1-RDM. With calculations on atoms and diatomics, linear hydrogen chains and conjugated π systems, we examine the accuracy of the sparse wavefunction Ansätze arising from the GPCs. ...
... Sparsity of the wavefunction has recently been explored in both deterministic and stochastic algorithms because the use of a small subset of all possible determinants has the potential to accelerate accurate electronic structure calculations significantly. [16][17][18][19][20][21][22][23][24][25][26] The approach to sparsity presented here is unique in that we employ non-trivial pure N-representability conditions of the 1-RDM (GPCs), 4,34 which generalizes the Pauli exclusion principle. Because of the energy minimization, the ground-state 2-RDM of any electronic system lies on the boundary of its N-representable set. ...
Electron occupations that arise from pure quantum states are restricted by a stringent set of conditions that are said to generalize the Pauli exclusion principle. These generalized Pauli constraints (GPCs) define the boundary of the set of one-electron reduced density matrices (1-RDMs) that are derivable from at least one N-electron wavefunction. In this paper, we investigate the sparsity of the Slater-determinant representation of the wavefunction that is a necessary, albeit not sufficient, condition for its 1-RDM to lie on the boundary of the set of pure N-representable 1-RDMs or in other words saturate one of the GPCs. The sparse wavefunction, we show, is exact not only for 3 electrons in 6 orbitals but also for 3 electrons in 8 orbitals. For larger numbers of electrons and/or orbitals in the lowest spin state, the exact wavefunction does not generally saturate one of the GPCs, and hence, the sparse representation is typically an approximation. Because the sparsity of the wavefunction is a necessary but not sufficient condition for saturation of one of the GPCs, optimization of the sparse wavefunction Ansatz to minimize the ground-state energy does not necessarily produce a wavefunction whose 1-RDM exactly saturates one of the GPCs. While the sparse Ansatz can be employed with arbitrary orbitals or optimized orbitals, in this paper, we explore the Ansatz with the natural orbitals from full configuration interaction, which yields an upper bound to the ground-state energy that equals the exact energy for a given basis set if the full-configuration-interaction wavefunction saturates the Ansatz’s GPC. With calculations on the boron isoelectronic sequence, the dinitrogen cation N2+, hydrogen chains, and cyclic conjugated π systems, we examine the quality of the sparse wavefunction Ansatz from the amount of correlation energy recovered.
... In the case of hm equals to zero, the value of 0log2(0) is taken to 0 [27]. We calculated Shannon entropies of all subbands obtained from DWPT, and dubbed the results as discrete wavelet packet entropy (DWPE). ...
In order to develop an efficient computer-aided diagnosis system for detecting left-sided and right-sided sensorineural hearing loss, we used artificial intelligence in this study. First, 49 subjects were enrolled by magnetic resonance imaging scans. Second, the discrete wavelet packet entropy (DWPE) was utilized to extract global texture features from brain images. Third, single-hidden layer neural network (SLNN) was used as the classifier with training algorithm of adaptive learning-rate back propagation (ALBP). The 10 times of 5-fold cross validation demonstrated our proposed method yielded an overall accuracy of 95.31%, higher than standard back propagation method with accuracy of 87.14%. Besides, our method also outperforms the “FRFT + PCA (Yang, 2016)”, “WE + DT (Kale, 2013)”, and “WE + MRF (Vasta 2016)”. In closing, our method is efficient.
