In this sequel to our work on triply excited hollow resonances in three-electron atomic systems, ... more In this sequel to our work on triply excited hollow resonances in three-electron atomic systems, a density functional theory (DFT)-based formalism is employed to investigate similar resonances in Li-isoelectronic series (Z=4--10). A combination of the work-function-based local nonvariational exchange potential and the popular gradient plus Laplacian included Lee-Yang-Parr correlation energy functional is used. First, all the 8 n=2 intrashell states of B^2+, N^4+ and F^6+ are presented, which are relatively less studied in the literature compared to the remaining 4 members. Then calculations are performed for the 8 2l2l'nl" (3≤n≤6) hollow resonance series; viz., 2s^2ns ^2S^e, 2s^2np ^2P^o, 2s^2nd ^2D^e, 2s2pns ^4P^o, 2s2pnp ^4D^e, 2p^2ns ^4P^e, 2p^2np ^4D^o and 2p^2ns ^2D^e, of all the 7 positive ions. Next, as an illustration, higher resonance positions of the 2s^2ns ^2S^e series are calculated for all the ions with a maximum of n=25. The excitation energies calculated from...
Density functional calculations are performed for twelve 2l2l'nl" (n≥2) triply excited h... more Density functional calculations are performed for twelve 2l2l'nl" (n≥2) triply excited hollow resonance series of Li, viz., 2s^2ns ^2S^e, 2s^2np ^2P^o, 2s^2nd ^2D^e, 2p^2ns ^2D^e,^4P^e, 2s2pns ^4P^o, 2s2pnp ^4D^e, 2p^2np ^2F^o,^4D^o, 2p^2nd ^2G^e, ^4F^e and 2s2pnd ^4F^o, covering a total of about 270 low-, moderately high- and high-lying states, with n as high as up to 25. The work-function-based exchange potential and the nonlinear gradient plus Laplacian included Lee-Yang-Parr correlation energy functional is used. The relevant Kohn-Sham-type equation is solved numerically using the generalized pseudospectral method offering nonuniform, optimal spatial discretization to obtain the orbitals and densities. Except for the one state, the discrepancy in the calculated state energies remains well within 0.98 available experimental and other theoretical results. Additionally companion calculations are also presented for the 37 3l3l'nl" (n≥3) doubly hollow states (seven ...
Bound states of the power-law and logarithmic potentials are calculated using a generalized pseud... more Bound states of the power-law and logarithmic potentials are calculated using a generalized pseudospectral method. The solution of the single-particle Schrödinger equation in a nonuniform and optimal spatial discretization offers accurate eigenvalues, densities and expectation values. The calculations are carried out for states with arbitrary n and ℓ quantum numbers. Comparisons are made with the available literature data and excellent agreement is observed. In all the cases, the present method yields considerably improved results over the other existing calculations. Some new states are reported.
The generalized pseudospectral method is employed to calculate the bound states of Hulthén and Yu... more The generalized pseudospectral method is employed to calculate the bound states of Hulthén and Yukawa potentials in quantum mechanics, with special emphases on higher excited states and stronger couplings. Accurate energy eigenvalues, expectation values and radial probability densities are obtained through a nonuniform and optimal spatial discretization of the radial Schrödinger equation. Results accurate up to thirteen to fourteen significant figures are reported for all the 55 eigenstates of both these potentials with n≤10 for arbitrary values of the screening parameters covering a wide range of interaction. Furthermore, excited states as high as up to n=17 have been computed with good accuracy for both these potentials. Excellent agreement with the available literature data has been observed in all cases. The n>6 states of Yukawa potential has been considerably improved over all other existing results currently available, while the same for Hulthén potential are reported here ...
Information based uncertainty measures like Rényi entropy (R), Shannon entropy (S) and Onicescu e... more Information based uncertainty measures like Rényi entropy (R), Shannon entropy (S) and Onicescu energy (E) (in both position and momentum space) are employed to understand the influence of radial confinement in isotropic harmonic oscillator. The transformation of Hamiltonian in to a dimensionless form gives an idea of the composite effect of oscillation frequency (ω) and confinement radius (r_c). For a given quantum state, accurate results are provided by applying respective exact analytical wave function in r space. The p-space wave functions are produced from Fourier transforms of radial functions. Pilot calculations are done taking order of entropic moments (α, β) as (3/5, 3) in r and p spaces. A detailed, systematic analysis is performed for confined harmonic oscillator (CHO) with respect to state indices n_r,l, and r_c. It has been found that, CHO acts as a bridge between particle in a spherical box (PISB) and free isotropic harmonic oscillator (IHO). At smaller r_c, E_ increas...
Fisher information (I) is investigated for confined hydrogen atom (CHA)-like systems in conjugate... more Fisher information (I) is investigated for confined hydrogen atom (CHA)-like systems in conjugate r and p spaces. A comparative study between CHA and free H atom (with respect to I) is pursued. In many aspects, inferences in CHA are significantly different from free counterpart; that includes its dependence on n, l, m. The role of atomic number and atomic radius is discussed. Further, a detailed systematic result of I with respect to variation of confinement radius r_c is presented, with particular emphasis on non-zero-(l,m) states. Several new interesting observations are recorded. Most of these results are of benchmark quality and presented for the first time.
A simple methodology is suggested for the efficient calculation of certain central potentials hav... more A simple methodology is suggested for the efficient calculation of certain central potentials having singularities. The generalized pseudospectral method used in this work facilitates nonuniform and optimal spatial discretization. Applications have been made to calculate the energies, densities and expectation values for two singular potentials of physical interest, viz., (i) the harmonic potential plus inverse quartic and sextic perturbation and (ii) the Coulomb potential with a linear and quadratic term for a broad range of parameters. The first 10 states belonging to a maximum of ℓ=8 and 5 for (i) and (ii) have been computed with good accuracy and compared with the most accurate available literature data. The calculated results are in excellent agreement, especially in the light of the difficulties encountered in these potentials. Some new states are reported here for the first time. This offers a general and efficient scheme for calculating these and other similar potentials of ...
We present bound state spectra of the 3D rational potential, V(r)=r^2 + λ r^2/(1+gr^2), g>0, b... more We present bound state spectra of the 3D rational potential, V(r)=r^2 + λ r^2/(1+gr^2), g>0, by means of the generalized pseudospectral method. All the thirty states corresponding to n=0--9 are considered for the first time for a broad range of coupling parameters. These results surpass the accuracy of all other existing calculations published so far except the finite-difference method, which yields similar accuracy as ours. Variation of energies and radial distribution functions is followed with respect to the interaction parameters. Special emphasis has been laid on higher excitations and negative values of the interaction, where relatively less work has been reported. The energy sequence is found to be different for positive and negative interaction; numerically following a mirror-image relationship usually, if not always. Additionally, twenty energy splittings arising from certain levels belonging to n=0--9 are systematically studied as functions of the potential parameters. ...
Starting from a time-dependent Schr\"odinger equation, stationary states of 3D central poten... more Starting from a time-dependent Schr\"odinger equation, stationary states of 3D central potentials are obtained. An imaginary-time evolution technique coupled with the minimization of energy expectation value, subject to the orthogonality constraint leads to ground and excited states. The desired diffusion equation is solved by means of a finite-difference approach to produce accurate wave functions, energies, probability densities and other expectation values. Applications in case of 3D isotropic harmonic oscillator, Morse as well the spiked harmonic oscillator are made. Comparison with literature data reveals that this is able to produce high-quality and competitive results. The method could be useful for this and other similar potentials of interest in quantum mechanics. Future and outlook of the method is briefly discussed.
Bound states of the Hellmann potential, which is a superposition of the attractive Coulomb (-A/r)... more Bound states of the Hellmann potential, which is a superposition of the attractive Coulomb (-A/r) and the Yukawa (Be^-Cr/r) potential, are calculated by using a generalized pseudospectral method. Energy eigenvalues accurate up to thirteen to fourteen significant figures, and densities are obtained through a nonuniform, optimal spatial discretization of the radial Schrödinger equation. Both ground and excited states are reported for arbitrary values of the potential parameters covering a wide range of interaction. Calculations have been made for higher states as well as for stronger couplings. Some new states are reported here for the first time, which could be useful for future works. The present results are significantly improved in accuracy over all other existing literature values and offers a simple, accurate and efficient scheme for these and other singular potentials in quantum mechanics.
Spherical confinement in 3D harmonic, quartic and other higher oscillators of even order is studi... more Spherical confinement in 3D harmonic, quartic and other higher oscillators of even order is studied. The generalized pseudospectral method is employed for accurate solution of relevant Schrödinger equation in an optimum, non-uniform radial grid. Eigenvalues, eigenfunctions, position expectation values, radial densities in low and high-lying states are presented in case of small, intermediate and large confinement radius. The degeneracy breaking in confined situation as well as correlation in its energy ordering with respect to the respective unconfined counterpart is discussed. For all instances, current results agree excellently with best available literature results. Many new states are reported here for first time. In essence, a simple, efficient method is provided for accurate solution of 3D polynomial potentials enclosed within spherical impenetrable walls.
Bound states of hyperbolic potential is investigated by means of a generalized pseudospectral met... more Bound states of hyperbolic potential is investigated by means of a generalized pseudospectral method. Significantly improved eigenvalues, eigenfunctions are obtained efficiently for arbitrary n, ℓ quantum states by solving the relevant non-relativistic Schrödinger equation allowing a non-uniform, optimal spatial discretization. Eigenvalues accurate up to tenth decimal place are reported for a large range of potential parameters; thus covering a wide range of interaction. Excellent agreement with available literature results is observed in all occasions. Special attention is paid for higher states. Some new states are given. Energy variations with respect to parameters in the potential are studied in considerable detail for the first time.
Exploratory variational pseudopotential density functional calculations are performed for the ele... more Exploratory variational pseudopotential density functional calculations are performed for the electronic properties of many-electron systems in the 3D cartesian coordinate grid (CCG). The atom-centered localized gaussian basis set, electronic density and the two-body potentials are set up in the 3D cubic box. The classical Hartree potential is calculated accurately and efficiently through a Fourier convolution technique. As a first step, simple local density functionals of homogeneous electron gas are used for the exchange-correlation potential, while Hay-Wadt-type effective core potentials are employed to eliminate the core electrons. No auxiliary basis set is invoked. Preliminary illustrative calculations on total energies, individual energy components, eigenvalues, potential energy curves, ionization energies, atomization energies of a set of 12 molecules show excellent agreement with the corresponding reference values of atom-centered grid as well as the grid-free calculation. R...
This is a follow-up of our recently proposed work on pseudopotential calculation (Ref. [21]) of a... more This is a follow-up of our recently proposed work on pseudopotential calculation (Ref. [21]) of atoms and molecules within DFT framework, using cartesian coordinate grid. Detailed results are presented to demonstrate the usefulness, applicability of the same for a larger set of species (5 atoms; 53 molecules) and exchange-correlation functionals (local, nonlocal). A thorough comparison on total, component, ionization, atomization energies, eigenvalues, potential energy curves with available literature data shows excellent agreement. Additionally, HOMO energies for a series of molecules show significant improvements by using the Leeuwen-Baerends exchange potential, compared to other functionals considered. Comparison with experiments has been made, wherever possible.
Several well-known statistical measures similar to LMC and Fisher-Shannon complexity have been co... more Several well-known statistical measures similar to LMC and Fisher-Shannon complexity have been computed for confined hydrogen atom in both position (r) and momentum (p) spaces. Further, a more generalized form of these quantities with Rényi entropy (R) is explored here. The role of scaling parameter in the exponential part is also pursued. R is evaluated taking order of entropic moments α, β as (2/3,3) in r and p spaces. Detailed systematic results of these measures with respect to variation of confinement radius r_c is presented for low-lying states such as, 1s-3d, 4f and 5g. For nodal states, such as 2s, 3s and 3p, as r_c progresses there appears a maximum followed by a minimum in r space, having certain values of the scaling parameter. However, the corresponding p-space results lack such distinct patterns. This study reveals many other interesting features.
... 3 Faculty of Engineering, Chiba University, Japan. Email: Abraham F. Jalbout (ajalbout@u. ari... more ... 3 Faculty of Engineering, Chiba University, Japan. Email: Abraham F. Jalbout (ajalbout@u. arizona.edu) Abul Haider Shipar (shipar7@yahoo.com). ... Correspondence: Abul Haider Shipar, Faculty of Engineering, Chiba University, Japan. Publication History. ...
In this sequel to our work on triply excited hollow resonances in three-electron atomic systems, ... more In this sequel to our work on triply excited hollow resonances in three-electron atomic systems, a density functional theory (DFT)-based formalism is employed to investigate similar resonances in Li-isoelectronic series (Z=4--10). A combination of the work-function-based local nonvariational exchange potential and the popular gradient plus Laplacian included Lee-Yang-Parr correlation energy functional is used. First, all the 8 n=2 intrashell states of B^2+, N^4+ and F^6+ are presented, which are relatively less studied in the literature compared to the remaining 4 members. Then calculations are performed for the 8 2l2l'nl" (3≤n≤6) hollow resonance series; viz., 2s^2ns ^2S^e, 2s^2np ^2P^o, 2s^2nd ^2D^e, 2s2pns ^4P^o, 2s2pnp ^4D^e, 2p^2ns ^4P^e, 2p^2np ^4D^o and 2p^2ns ^2D^e, of all the 7 positive ions. Next, as an illustration, higher resonance positions of the 2s^2ns ^2S^e series are calculated for all the ions with a maximum of n=25. The excitation energies calculated from...
Density functional calculations are performed for twelve 2l2l'nl" (n≥2) triply excited h... more Density functional calculations are performed for twelve 2l2l'nl" (n≥2) triply excited hollow resonance series of Li, viz., 2s^2ns ^2S^e, 2s^2np ^2P^o, 2s^2nd ^2D^e, 2p^2ns ^2D^e,^4P^e, 2s2pns ^4P^o, 2s2pnp ^4D^e, 2p^2np ^2F^o,^4D^o, 2p^2nd ^2G^e, ^4F^e and 2s2pnd ^4F^o, covering a total of about 270 low-, moderately high- and high-lying states, with n as high as up to 25. The work-function-based exchange potential and the nonlinear gradient plus Laplacian included Lee-Yang-Parr correlation energy functional is used. The relevant Kohn-Sham-type equation is solved numerically using the generalized pseudospectral method offering nonuniform, optimal spatial discretization to obtain the orbitals and densities. Except for the one state, the discrepancy in the calculated state energies remains well within 0.98 available experimental and other theoretical results. Additionally companion calculations are also presented for the 37 3l3l'nl" (n≥3) doubly hollow states (seven ...
Bound states of the power-law and logarithmic potentials are calculated using a generalized pseud... more Bound states of the power-law and logarithmic potentials are calculated using a generalized pseudospectral method. The solution of the single-particle Schrödinger equation in a nonuniform and optimal spatial discretization offers accurate eigenvalues, densities and expectation values. The calculations are carried out for states with arbitrary n and ℓ quantum numbers. Comparisons are made with the available literature data and excellent agreement is observed. In all the cases, the present method yields considerably improved results over the other existing calculations. Some new states are reported.
The generalized pseudospectral method is employed to calculate the bound states of Hulthén and Yu... more The generalized pseudospectral method is employed to calculate the bound states of Hulthén and Yukawa potentials in quantum mechanics, with special emphases on higher excited states and stronger couplings. Accurate energy eigenvalues, expectation values and radial probability densities are obtained through a nonuniform and optimal spatial discretization of the radial Schrödinger equation. Results accurate up to thirteen to fourteen significant figures are reported for all the 55 eigenstates of both these potentials with n≤10 for arbitrary values of the screening parameters covering a wide range of interaction. Furthermore, excited states as high as up to n=17 have been computed with good accuracy for both these potentials. Excellent agreement with the available literature data has been observed in all cases. The n>6 states of Yukawa potential has been considerably improved over all other existing results currently available, while the same for Hulthén potential are reported here ...
Information based uncertainty measures like Rényi entropy (R), Shannon entropy (S) and Onicescu e... more Information based uncertainty measures like Rényi entropy (R), Shannon entropy (S) and Onicescu energy (E) (in both position and momentum space) are employed to understand the influence of radial confinement in isotropic harmonic oscillator. The transformation of Hamiltonian in to a dimensionless form gives an idea of the composite effect of oscillation frequency (ω) and confinement radius (r_c). For a given quantum state, accurate results are provided by applying respective exact analytical wave function in r space. The p-space wave functions are produced from Fourier transforms of radial functions. Pilot calculations are done taking order of entropic moments (α, β) as (3/5, 3) in r and p spaces. A detailed, systematic analysis is performed for confined harmonic oscillator (CHO) with respect to state indices n_r,l, and r_c. It has been found that, CHO acts as a bridge between particle in a spherical box (PISB) and free isotropic harmonic oscillator (IHO). At smaller r_c, E_ increas...
Fisher information (I) is investigated for confined hydrogen atom (CHA)-like systems in conjugate... more Fisher information (I) is investigated for confined hydrogen atom (CHA)-like systems in conjugate r and p spaces. A comparative study between CHA and free H atom (with respect to I) is pursued. In many aspects, inferences in CHA are significantly different from free counterpart; that includes its dependence on n, l, m. The role of atomic number and atomic radius is discussed. Further, a detailed systematic result of I with respect to variation of confinement radius r_c is presented, with particular emphasis on non-zero-(l,m) states. Several new interesting observations are recorded. Most of these results are of benchmark quality and presented for the first time.
A simple methodology is suggested for the efficient calculation of certain central potentials hav... more A simple methodology is suggested for the efficient calculation of certain central potentials having singularities. The generalized pseudospectral method used in this work facilitates nonuniform and optimal spatial discretization. Applications have been made to calculate the energies, densities and expectation values for two singular potentials of physical interest, viz., (i) the harmonic potential plus inverse quartic and sextic perturbation and (ii) the Coulomb potential with a linear and quadratic term for a broad range of parameters. The first 10 states belonging to a maximum of ℓ=8 and 5 for (i) and (ii) have been computed with good accuracy and compared with the most accurate available literature data. The calculated results are in excellent agreement, especially in the light of the difficulties encountered in these potentials. Some new states are reported here for the first time. This offers a general and efficient scheme for calculating these and other similar potentials of ...
We present bound state spectra of the 3D rational potential, V(r)=r^2 + λ r^2/(1+gr^2), g>0, b... more We present bound state spectra of the 3D rational potential, V(r)=r^2 + λ r^2/(1+gr^2), g>0, by means of the generalized pseudospectral method. All the thirty states corresponding to n=0--9 are considered for the first time for a broad range of coupling parameters. These results surpass the accuracy of all other existing calculations published so far except the finite-difference method, which yields similar accuracy as ours. Variation of energies and radial distribution functions is followed with respect to the interaction parameters. Special emphasis has been laid on higher excitations and negative values of the interaction, where relatively less work has been reported. The energy sequence is found to be different for positive and negative interaction; numerically following a mirror-image relationship usually, if not always. Additionally, twenty energy splittings arising from certain levels belonging to n=0--9 are systematically studied as functions of the potential parameters. ...
Starting from a time-dependent Schr\"odinger equation, stationary states of 3D central poten... more Starting from a time-dependent Schr\"odinger equation, stationary states of 3D central potentials are obtained. An imaginary-time evolution technique coupled with the minimization of energy expectation value, subject to the orthogonality constraint leads to ground and excited states. The desired diffusion equation is solved by means of a finite-difference approach to produce accurate wave functions, energies, probability densities and other expectation values. Applications in case of 3D isotropic harmonic oscillator, Morse as well the spiked harmonic oscillator are made. Comparison with literature data reveals that this is able to produce high-quality and competitive results. The method could be useful for this and other similar potentials of interest in quantum mechanics. Future and outlook of the method is briefly discussed.
Bound states of the Hellmann potential, which is a superposition of the attractive Coulomb (-A/r)... more Bound states of the Hellmann potential, which is a superposition of the attractive Coulomb (-A/r) and the Yukawa (Be^-Cr/r) potential, are calculated by using a generalized pseudospectral method. Energy eigenvalues accurate up to thirteen to fourteen significant figures, and densities are obtained through a nonuniform, optimal spatial discretization of the radial Schrödinger equation. Both ground and excited states are reported for arbitrary values of the potential parameters covering a wide range of interaction. Calculations have been made for higher states as well as for stronger couplings. Some new states are reported here for the first time, which could be useful for future works. The present results are significantly improved in accuracy over all other existing literature values and offers a simple, accurate and efficient scheme for these and other singular potentials in quantum mechanics.
Spherical confinement in 3D harmonic, quartic and other higher oscillators of even order is studi... more Spherical confinement in 3D harmonic, quartic and other higher oscillators of even order is studied. The generalized pseudospectral method is employed for accurate solution of relevant Schrödinger equation in an optimum, non-uniform radial grid. Eigenvalues, eigenfunctions, position expectation values, radial densities in low and high-lying states are presented in case of small, intermediate and large confinement radius. The degeneracy breaking in confined situation as well as correlation in its energy ordering with respect to the respective unconfined counterpart is discussed. For all instances, current results agree excellently with best available literature results. Many new states are reported here for first time. In essence, a simple, efficient method is provided for accurate solution of 3D polynomial potentials enclosed within spherical impenetrable walls.
Bound states of hyperbolic potential is investigated by means of a generalized pseudospectral met... more Bound states of hyperbolic potential is investigated by means of a generalized pseudospectral method. Significantly improved eigenvalues, eigenfunctions are obtained efficiently for arbitrary n, ℓ quantum states by solving the relevant non-relativistic Schrödinger equation allowing a non-uniform, optimal spatial discretization. Eigenvalues accurate up to tenth decimal place are reported for a large range of potential parameters; thus covering a wide range of interaction. Excellent agreement with available literature results is observed in all occasions. Special attention is paid for higher states. Some new states are given. Energy variations with respect to parameters in the potential are studied in considerable detail for the first time.
Exploratory variational pseudopotential density functional calculations are performed for the ele... more Exploratory variational pseudopotential density functional calculations are performed for the electronic properties of many-electron systems in the 3D cartesian coordinate grid (CCG). The atom-centered localized gaussian basis set, electronic density and the two-body potentials are set up in the 3D cubic box. The classical Hartree potential is calculated accurately and efficiently through a Fourier convolution technique. As a first step, simple local density functionals of homogeneous electron gas are used for the exchange-correlation potential, while Hay-Wadt-type effective core potentials are employed to eliminate the core electrons. No auxiliary basis set is invoked. Preliminary illustrative calculations on total energies, individual energy components, eigenvalues, potential energy curves, ionization energies, atomization energies of a set of 12 molecules show excellent agreement with the corresponding reference values of atom-centered grid as well as the grid-free calculation. R...
This is a follow-up of our recently proposed work on pseudopotential calculation (Ref. [21]) of a... more This is a follow-up of our recently proposed work on pseudopotential calculation (Ref. [21]) of atoms and molecules within DFT framework, using cartesian coordinate grid. Detailed results are presented to demonstrate the usefulness, applicability of the same for a larger set of species (5 atoms; 53 molecules) and exchange-correlation functionals (local, nonlocal). A thorough comparison on total, component, ionization, atomization energies, eigenvalues, potential energy curves with available literature data shows excellent agreement. Additionally, HOMO energies for a series of molecules show significant improvements by using the Leeuwen-Baerends exchange potential, compared to other functionals considered. Comparison with experiments has been made, wherever possible.
Several well-known statistical measures similar to LMC and Fisher-Shannon complexity have been co... more Several well-known statistical measures similar to LMC and Fisher-Shannon complexity have been computed for confined hydrogen atom in both position (r) and momentum (p) spaces. Further, a more generalized form of these quantities with Rényi entropy (R) is explored here. The role of scaling parameter in the exponential part is also pursued. R is evaluated taking order of entropic moments α, β as (2/3,3) in r and p spaces. Detailed systematic results of these measures with respect to variation of confinement radius r_c is presented for low-lying states such as, 1s-3d, 4f and 5g. For nodal states, such as 2s, 3s and 3p, as r_c progresses there appears a maximum followed by a minimum in r space, having certain values of the scaling parameter. However, the corresponding p-space results lack such distinct patterns. This study reveals many other interesting features.
... 3 Faculty of Engineering, Chiba University, Japan. Email: Abraham F. Jalbout (ajalbout@u. ari... more ... 3 Faculty of Engineering, Chiba University, Japan. Email: Abraham F. Jalbout (ajalbout@u. arizona.edu) Abul Haider Shipar (shipar7@yahoo.com). ... Correspondence: Abul Haider Shipar, Faculty of Engineering, Chiba University, Japan. Publication History. ...
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