Studies on the bound-state spectrum of hyperbolic potential

Bound states of hyperbolic potential is investigated by means of a generalized pseudospectral method. Significantly improved eigenvalues, eigenfunctions are obtained efficiently for arbitrary $n, \ell$ quantum states by solving the relevant non-relativistic Schr\"odinger 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.Comment: 13 pages, 3 figures, 2 table

Accurate calculation of the bound states of Hellmann potential

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 Schr\"odinger 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

Confinement in 3D polynomial oscillators through a generalized pseudospectral method

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 Schr\"odinger equation in an \emph{optimum, non-uniform} radial grid. Eigenvalues, eigenfunctions, position expectation values, radial densities in \emph{low and high-lying} states are presented in case of \emph{small, intermediate and large} confinement radius. The \emph{degeneracy breaking} in confined situation as well as correlation in its \emph{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.Comment: 17 pages, 5 figures, 6 table

A density functional method for general excited states in atoms

This chapter presents the development of a density functional theory (DFT)-based method for accurate, reliable treatment of various resonances in atoms. Many of these are known to be notorious for their strong correlation, proximity to more than one thresholds, degeneracy with more than one minima. Therefore these pose unusual challenges to both theoreticians and experimentalists. Our method uses a work-function-based exchange potential in conjunction with the popular gradient-corrected Lee-Yang-Parr correlation functional. The resulting Kohn-Sham equation, in the non-relativistic framework, is numerically solved accurately and efficiently by means of a generalized pseudospectral method through a non-uniform, optimal spatial discretization. This has been applied to a variety of excited states, such as low and high states; single, double, triple as well as multiple excitations; valence and core excitations; autoionizing states; satellites; hollow and doubly-hollow states; very high-lying Rydberg resonances; etc., of atoms and ions, with remarkable success. A thorough and systematic comparison with literature data reveals that, for all these systems, the exchange-only results are practically of Hartree-Fock quality; while with inclusion of correlation, this offers excellent agreement with available experimental data as well as those obtained from other sophisticated theoretical methods. Properties such as individual state energies, excitation energies, radial densities as well as various expectation values are studied. This helps us in predicting many states for the first time.Comment: 46 pages, 1 figure, 6 table

Spherical confinement of Coulombic systems inside an impenetrable box: H atom and the Hulth\'en potential

The generalized pseudospectral method is employed to study spherical confinement in two simple Coulombic systems: (i) well celebrated and heavily studied H atom (ii) relatively less explored Hulth\'en potential. In both instances, arbitrary cavity size, as well as low and higher states are considered. Apart from bound state eigenvalues, eigenfunctions, expectation values, quite accurate estimates of the critical cage radius for H atom for all the 55 states corresponding to $n \leq 10$, are also examined. Some of the latter are better than previously reported values. Degeneracy and energy ordering under the isotropic confinement situation are discussed as well. The method produces consistently high-quality results for both potentials for small as well as large cavity size. For the H atom, present results are comparable to best theoretical values, while for the latter, this work gives considerably better estimates than all existing work so far.Comment: 23 pages, 10 tables, 2 figure

Density functional studies on the hollow resonances in Li-isoelectronic sequence (Z=4--10

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 $2l2l'$n$l"$ (3$\leq$n$\leq$6) hollow resonance series; {\em viz.,} 2s$^2$ns $^2$S$^e$, 2s$^2$np $^2$P$^o$, 2s$^2$nd $^2$D$^e$, 2s2pns $^4$P$^o$, 2s2pnp $^4$D$^e$, 2p$^2$ns $^4$P$^e$, 2p$^2$np $^4$D$^o$ and 2p$^2$ns $^2$D$^e$, of all the 7 positive ions. Next, as an illustration, higher resonance positions of the 2s$^2$ns $^2$S$^e$ series are calculated for all the ions with a maximum of n=25. The excitation energies calculated from this single-determinantal approach are in excellent agreement with the available literature data (for the n=2 intrashell states the deviation is within 0.125% and excepting only one case, the same for the resonance series is well below 0.50%). With an increase in Z, the deviations tend to decrease. Radial densities are also presented for some of the selected states. The only result available in the literature for the lower resonances (corresponding to a maximum of n=17) have been reported very recently. The n$>$16 ($>17$ for F$^{6+}$) resonances are examined here for the first time. This gives a promising viable and general DFT scheme for the accurate calculation of these and other hollow resonances in many-electron atoms

A new density functional method for electronic structure calculation of atoms and molecules

This chapter concerns with the recent development of a new DFT methodology for accurate, reliable prediction of many-electron systems. Background, need for such a scheme, major difficulties encountered, as well as their potential remedies are discussed at some length. Within the realm of non relativistic Hohenberg-Kohn-Sham (HKS) DFT and making use of the familiar LCAO-MO principle, relevant KS eigenvalue problem is solved numerically. Unlike the commonly used atom-centered grid (ACG), here we employ a 3D cartesian coordinate grid (CCG) to build atom-centered localized basis set, electron density, as well as all the two-body potentials directly on grid. The Hartree potential is computed through a Fourier convolution technique via a decomposition in terms of short- and long-range interactions. Feasibility and viability of our proposed scheme is demonstrated for a series of chemical systems; first with homogeneous, local-density-approximated XC functionals followed by non-local, gradient- and Laplacian-dependent functionals. A detailed, systematic analysis on obtained results relevant to quantum chemistry, are made, \emph{for the first time}, using CCG, which clearly illustrates the significance of this alternative method in the present context. Quantities such as component energies, total energies, ionization energies, potential energy curve, atomization energies, etc., are addressed for pseudopotential calculations, along with a thorough comparison with literature data, wherever possible. Finally, some words on the future and prospect of this method are mentioned. In summary, we have presented a new CCG-based \emph{variational} DFT method for accurate, dependable calculation of atoms and molecules.Comment: 32 pages, 1 figure, 6 ptable

The generalized pseudospectral approach to the bound states of Hulthen and Yukawa potentials

The generalized pseudospectral method is employed to calculate the bound states of Hulth\'en 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 Schr\"odinger equation. Results accurate up to thirteen to fourteen significant figures are reported for all the 55 eigenstates of both these potentials with $n\leq$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 Hulth\'en potential are reported here for the first time. Excepting the $1s$ and $2s$ states of Yukawa potential, the present method surpasses in accuracy all other existing results in the stronger coupling region for all other states of both these systems. This offers a simple and efficient scheme for the accurate calculation of these and other screened Coulomb potentials

Studies on the hollow states of atomic lithium by a density functional approach

Density functional calculations are performed for twelve $2l2l'$n$l"$ (n$\geq$2) triply excited hollow resonance series of Li, {\em viz.,} 2s$^2$ns $^2$S$^e$, 2s$^2$np $^2$P$^o$, 2s$^2$nd $^2$D$^e$, 2p$^2$ns $^2$D$^e$,$^4$P$^e$, 2s2pns $^4$P$^o$, 2s2pnp $^4$D$^e$, 2p$^2$np $^2$F$^o$,$^4$D$^o$, 2p$^2$nd $^2$G$^e$, $^4$F$^e$ and 2s2pnd $^4$F$^o$, 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%, whereas the excitation energies deviate by 0.02--0.58% compared to the available experimental and other theoretical results. Additionally companion calculations are also presented for the 37 $3l3l'$n$l"$ (n$\geq$3) doubly hollow states (seven are n=3 intrashell type) of Li with both K and L shells empty (up to n=6) in the photon energy range 175.63--180.51 eV, with varying symmetries and multiplicities. Our calculation shows good agreement with the recent literature data for the only two such doubly hollow states reported so far, {\em viz.,} 3s$^2$3p $^2$P$^o$ and 3s3p$^2$ $^4$P$^e$. The resonance series are found to be inextricably entangled to each other, leading to complicated behavior in their positions. Many new states are reported here for the first time. This provides a simple, efficient and general scheme for the accurate calculation of these and other multiply excited Rydberg series of many-electron atomic systems within density functional theory

Ro-vibrational studies of diatomic molecules in a shifted Deng-Fan oscillator potential

Bound-state spectra of shifted Deng-Fan oscillator potential are studied by means of a generalized pseudospectral method. Very accurate results are obtained for \emph{both low as well as high} states by a non-uniform optimal discretization of the radial Schr\"odinger equation. Excellent agreement with literature data is observed in \emph{both $s$-wave and rotational} states. Detailed variation of energies with respect to potential parameters is discussed. Application is made to the ro-vibrational levels of four representative diatomic molecules (H$_2$, LiH, HCl, CO). Nine states having $\{n,\ell\} =0,1,2$ are calculated with good accuracy along with 15 other higher states for each of these molecules. Variation of energies with respect to state indices $n$, $\ell$ show \emph{behavior} similar to that in the Morse potential. Many new states are reported here for the first time. In short, a simple, accurate and efficient method is presented for this and other similar potentials in molecular physics.Comment: 16 pages, 2 figures, 5 table