Studies on some exponential-screened Coulomb potentials

# Studies on some exponential-screened Coulomb potentials

## Abstract

The generalized pseudospectral method is employed to study the bound-state spectra of some of the exponentially screened Coulomb potentials, viz., the exponential cosine screened Coulomb (ECSC) and general exponential screened Coulomb (GESC) potential, with special emphasis on higher states and stronger interaction. Eigenvalues accurate up to eleven significant figures are obtained through a non-uniform optimal spatial discretization of the radial Schrödinger equation. All the 55 eigenstates of ECSC potential with and 36 eigenstates of GESC potential with are considered for arbitrary values of the screening parameter, covering a wide range of interaction. Excited states as high as up to have been computed with high accuracy for the first time. Excellent agreement with the literature data has been observed in all cases. All the GESC eigenstates are calculated with much greater accuracy than the existing methods available in literature. Many states of this potential are reported here. In both cases, a detailed variation of energies with respect to the parameters in potential is monitored.

## I Introduction

Realistic potentials which describe quantum mechanical systems, are not usually exactly solvable in the Schrödinger picture, except for a few occasions such as Harmonic oscillator, Coulomb potential, etc. Therefore, finding exact analytical solution of Schrödinger equation for a given potential corresponding to a physical system of interest, constitutes one of the major challenges in quantum mechanics. This is a common problem, and often encountered in almost every branches, such as atomic, molecular, solid-state, nuclear, particle and plasma physics, etc. A large number of attractive promising approximate formalisms have been developed ever since the inception of theory, which can provide highly accurate or near-exact results in some cases. However, the same for a general potential for any allowed values of quantum numbers, for arbitrary values of potential parameters (if present in the system) still remains elusive, and thus always has been an active area of research.

Here we are concerned with the accurate bound states of two central singular potentials, namely (i) the generalized exponential cosine screened Coulomb potential, given by,

 v(r)=−Ar e−δ1r cos(gδ2r)=−Ar e−δr cos(gδr)   (when δ1=δ2=δ), (1)

and (ii) a much less frequently studied, general exponential screened Coulomb (GESC) potential of the form,

 v(r)=−ar [1+(1+br)e−2br]. (2)

In Eq. (1), represents the coupling strength constant while are two screening parameters. This potential reduces to the familiar Yukawa potential for , which has wide applications in nuclear, solid-state and plasma physics. For , this is termed as the exponential cosine screened Coulomb (ECSC) potential, and it is in this form that this potential has been studied maximum. One distinctive feature of this oscillating potential (in contrast to the Coulomb potential) is the finite number of bound states, i.e., such states exist only for certain values of the screening parameter below a threshold limit (the so-called critical ). In other words, the total number of different energy levels is finite for a given value of . Similarly, the two potential parameters in Eq. (2) signify coupling strength and screening parameters respectively. Throughout the whole article, is fixed at unity. These two potentials can be used to represent the effective interaction in many-electron atoms; also they have important applications in solid-state, nuclear and plasma physics as well as in field theory (1); (2); (3); (4); (5); (6); (7); (8); (9). Lately, the effect of screening on atomic photoionization in H and He has been studied by means of Yukawa and ECSC potential (10). Also, the ground and excited resonances in two-electron systems such as He, molecular H in ECSC as well as generalized screened potential have been investigated (11); (12); (13); (14).

None of these potentials admits exact analytical result. Therefore, over five decades, a considerably large number of attempts have been made to calculate their eigen spectra accurately. Here we mention a few of them. Perturbation and variational methods were used (15) to produce eigenvalues with reasonable accuracy, as well as the number of bound states for a given value of . The states were reported (16); (17) via the representation of ECSC potential by a Hulthén potential with an energy-dependent strength parameter, through the use of an Ecker-Weizel approximation. The states have been calculated by means of a hyper-virial Padé approximation (18), dynamical group approach (19), hyper-virial equation with Hellman-Feynman theorem (20), etc. Using a numerical method (21), all the 36 states below , as well as the critical screening parameters were obtained within an accuracy of eight to six significant figures. Analytical expressions for eigenvalues and eigenfunctions of ground and first excited states, up to 14 terms, were presented by a large- expansion method (22). Eigenvalues of to states have been reported by means of a shifted expansion technique (23). An iterative solution for eigenvalues belonging to arbitrary quantum numbers has been put forth by employing an asymptotic iteration method (24). A novel perturbation method (25) has also been proposed for this potential where the radial Schrödinger equation is decomposed into two parts, one of them being exactly solvable while the other part leading to closed analytical solution or an approximate treatment depending on the potential in question. Lately, a J-matrix approach (26) with a Gaussian quadrature scheme has offered high-quality results for bound and continuum states. A Ritz variation method with hydrogenic wave function as the trial function (27) has also produced promising results for such potential. Recently, an analytical scheme (28) inspired by the J-matrix method, has been quite successful for such potentials. The values have been estimated by numerical (15); (21); (23); (26), as well as analytical methods (29).

The GESC potential, on the other hand, has not received much attention. I am aware of only two studies. Energy eigenvalues of ground and first excited states were presented up to 14 terms using a large- expansion method (30). In another attempt (31), the perturbative method of (25) was used to obtain the states of this potential with decent accuracy.

In this work, we study the eigenspectra of both these potentials in Eqs. (1) and (2) using a generalized pseudospectral (GPS) method, which has been quite successful for a variety of systems such as the spiked harmonic oscillator, Hulthén and Yukawa potentials, power-law and logarithmic potentials, ground and excited states (low- and high-lying Rydberg states) of atoms as well as other singular systems (32); (33); (34); (35); (36); (38); (39). Potential parameters are scanned over a large domain. In few occasions, for some of the methodologies mentioned above for ECSC potential, it so happens that, eigenvalues and eigenfunctions are quite difficult to calculate for high-lying states and also at certain region of the potential parameter, especially near the . Here we pay special attention to both these issues for a better understanding of their spectra and also to judge the validity and efficacy of the method. To this end, accurate energies and wave functions are presented for all the 55 levels belonging to states of ECSC potential to extend the domain of applicability of the GPS procedure. Variation of the same with respect to are also monitored. For the GESC potential, only some low-lying states have been considered so far in the literature; no results are available for states. So here, we report all the states up to for a wide range of parameters in the potential for the first time, and some higher states as well. A detailed comparison with the available results in the literature has been made, wherever possible. The article is organized as follows. An outline of the theory and method of calculation is presented in Section II. A discussion of the results is given in Section III, while a few concluding remarks are made in Section IV.

## Ii The GPS method

The section gives the essential steps of GPS approach, as implemented here for the solution of single-particle Schrödinger equation for a non-relativistic Hamiltonian containing an exponentially screened potential term. The key advantage of the approach is that it offers a non-uniform, optimal spatial discretization. That means one can use a finer grid at small and coarser grid at large , maintaining high-accuracy at both these regions. This also implies only a small number of spatial points suffices to achieve convergence. Thus compared to standard finite difference/finite element methods, the GPS scheme is both accurate and efficient. Other details could be found in the references (32); (33); (34); (35); (36); (37); (38); (39). Unless otherwise mentioned, atomic unit is used throughout the article.

The radial Schrödinger equation can be written in the following working form,

 [−12 d2dr2+ℓ(ℓ+1)2r2+v(r)]Rn,ℓ(r)=En,ℓ Rn,ℓ(r) (3)

where is as given in Eq. (1) or (2), whereas , signify the usual radial and angular momentum quantum numbers respectively.

As a key step, a function , defined in the interval , is approximated by the N-th order polynomial, , through a cardinal function , as follows:

 f(x)≅fN(x)=N∑j=0f(xj) gj(x). (4)

This guarantees that the approximation is exact at the collocation points , i.e., , and requires that the cardinal function satisfies . Here we use the Legendre pseudospectral method, where , , and the are obtained from roots of first derivatives of the Legendre polynomial, with respect to , as . The are given by,

 gj(x)=−1N(N+1)PN(xj)  (1−x2) P′N(x)x−xj, (5)

Now, the semi-infinite domain is mapped onto a finite domain via the transformation . At this stage, one could introduce an algebraic nonlinear mapping of the form,

 r=r(x)=L 1+x1−x+α, (6)

with L and as two mapping parameters, to obtain a transformed differential equation as: . Now, one applies the Legendre pseudospectral method to this equation and finally a symmetrization procedure to yield the following symmetric eigenvalue problem,

 N−1∑j=1[−12Dij+ujδij]χj=ϵnlχi. (7)

This is readily solved by standard available routines such as that in NAG Fortran library, giving highly accurate eigenvalues and eigenfunctions. After some straightforward algebra, one finds that,

 χi=Rnl(ri) √(r′i)/PN(xi),   ui=l(l+1)/2r2i+v(ri), (8)

with , while denotes the symmetrized second derivative of cardinal function, given as follows,

 Dij = −2r′i(xi−xj)2r′j,    i≠j, (9) = −N(N+1)3r′2i(1−x2i),   i=j.

A series of calculation was performed for various potential parameters with respect to the grid mapping parameters to ascertain the accuracy and reliability of the current method. In this way, a “stable” grid was found, which appears to be sufficient for all the converged results presented in this article. Unless otherwise mentioned, all the reported results correspond to this consistent set of parameters, and . There are some instances, where this set is not adequate, and appropriate variations are allowed; these are mentioned appropriately in the text. Current results are reported only up to the precision that maintained stability. Eigenvalues are truncated rather than rounded-off, and hence may be considered as correct up to all the decimal places they are reported.

At this stage, a few remarks may be made regarding the GPS method. Typically in direct numerical methods, one truncates the semi-infinite domain into a finite domain to deal with the problems of singularity at , and infinite domain. In order for this, and need to be chosen sufficiently small and large respectively. This consequently results in a rather large number of grid points and also, in general, introduces some truncation error. To overcome this problem, one can map the semi-infinite domain exactly into the finite domain [1,1] using the mapping (Eq. 6) so that the Legendre pseudopotential technique can be applied. This introduces an additional undesirable feature; namely it leads to an unsymmetric or generalized eigenvalue problem, which in turn, is bypassed via the symmetrization procedure mentioned above. The method has been successfully applied to resonance states as well. For these and many other features of the method, the interested reader is referred to the references (40); (41); (42) and those therein.

## Iii Results and Discussion

At first, in Table I, we report some states of the ECSC potential for low as well as high excitations. For all the ECSC potential calculations throughout the article, parameter is set to unity. A wide range of screening parameters is considered–including low, intermediate and high values, signifying small, intermediate and large interaction respectively. Critical values of the screening parameter, taken from (26), are also mentioned in the table for for easy understanding. As evident, a large number of results are available in the literature for comparison, which we quote accordingly. One of the very first definitive calculations of this potential was reported in (15). All the states considered here with =1,4 (except in the high-screening region) were estimated by first-order perturbation treatment with (a) Coulomb potential as unperturbed potential (b) Hulthén potential as unperturbed potential, and (c) a one-parameter variational calculation with reasonably good accuracy. An Ecker-Weizel approach has been used for 1–4 states in the low-screening region through an approximation of the ECSC potential by Hulthén potential with modest accuracy (17). Both and states of were treated in intermediate coupling region within the hyper-virial Padé approximation (18), a dynamical group approach (19), and an asymptotic iteration method (24). The ground and first excited states of the ECSC potential for medium values of screening parameter have been obtained within the large- expansion method (22) as well. Energies correct up to six to eight significant figures were reported by means of a shifted expansion (23), for both as well as states. Lately, a new perturbative scheme (25) has been put forth for the states with decent success. However, it seems that, so far the most accurate eigenvalues are reported by a J-matrix method (26). In the neighborhood of low and moderate coupling, the present results are of very similar accuracy as those from (26) (in many occasions they coincide; otherwise they differ in the 11th or 12th place of decimal). In some states, for s near the threshold limit, their results were reported for somewhat lesser accuracy. Through the present method, we are able to obtain eigenvalues of consistently better accuracy near the strong-coupling region (see, for example, and for , 0.072 respectively). For higher-lying states, the reference literature values dramatically reduce in number so much so that for , our results could only be compared with the lone two-parameter variational calculation (15), where the present results are visibly improved. And for states with , no results could be found, and we report some sample results for and to emphasize the ease and ability of our method for higher states. Note that, for these higher states, however, the value needs to be suitably increased to achieve the desired convergence, e.g., for , an a.u., was used, whereas the other two parameters and needed no adjustments. This is reminiscent of a situation encountered earlier for Hulthén and Yukawa potentials (35) in the stronger coupling regime. Some numerical results are also available for the low-lying states (21); (27), which have been quoted as well.

Next in Table II, we report all the states belonging to , at selected values of . Here also the values are chosen so as to reflect both weak and strong couplings of the interaction, with respective values mentioned at the top of table. While the literature results are clearly quite scanty in comparison to case in Table I, wherever available, these are quoted appropriately. Once again, the existing best result is apparently the one from J-matrix calculation (26). As seen before, in all these cases again, our energy values are virtually identical to this method. And whenever these are not available, the present work produces quite superior eigenvalues compared to the other existing values. Note that, for all these eigenstates, we have chosen at least one value, which is very close to the value, as difficulties are encountered in these areas, with some of the methods. Near the critical values of , in general, one needs to extend to some larger values. For example, for a converged result for state at , an of 1100 a.u., was employed. This, again, is similar to a situation we came across for higher states in the previous paragraph, and also for some other central potentials (35).

Next in Fig. 1, variation of energy eigenvalues with respect to screening parameters are depicted for all the states belonging to (left) and (right) respectively, in the neighborhood of zero energy. Energy values increase monotonically with ; for each , they make a distinct family and for a particular value of the quantum number , the separation between states with different values of tends to increase with an increase in . Additionally, in Table III, calculated eigenvalues of all states are given at selected values of (0.005 and 0.003 for and 10 respectively). For sake of completeness, the available values for are mentioned in column 1 in parentheses. Only a two-parameter variational calculation (15) has been reported for the state. While these are reasonable first estimates in absence of any other result, our GPS results are significantly better than these. And for , there are no results to quote for direct comparison, and it is hoped that these would be helpful for the purpose of future referencing.

Now we turn to the GESC potential. Table IV reports energies for some low- and high-lying states of the same for some representative values, keeping fixed at 1. For lower states, several values are considered to understand the dependence on potential parameters. Reference results are much scarce in this case compared to the ECSC potential. The -expansion up to 14 terms (30) were obtained for ground and first excited state energies and wave functions. While these are reasonable initial estimates, clearly improved energies would be highly desirable. In another treatment, bound-state energies of first three states have been reported within a new perturbation technique (31). For smaller screening parameters, there is, in general, a decent agreement between our result and theirs. However, the discrepancy starts to grow quite fast as is increased. Finally, in Table V, eigenvalues of all the states having are reported for the first time, for two values of parameter. No results could be found for such states in the literature for comparison. This dependence of GESC eigenvalues on is pictorially shown in Fig. 2 for all the states belonging to (a), (b), (c) and (d) respectively. In all cases, energies gradually increase and then tend to assume a constant value. For smaller in (a), (b), it is seen that, all the states belonging to a particular form a characteristic family of the curve. Moreover, the states corresponding to a given do not mix with the states with a different . However, this scenario changes dramatically as we go for higher . Thus, as we move to (c), appreciable complex ordering and inter-state mixing is observed for ( mixing with ) and as we finally reach in (d), we encounter heavy mixing among the and states at around , making accurate determinate of these eigenvalues more and more difficult. Such complex level crossings have also been observed earlier for Hulthén and Yukawa potentials (35).

## Iv conclusion

Accurate bound states of ECSC and GESC potential have been presented by means of a GPS method. For both these cases, zero and non-zero angular momentum states are calculated easily with high accuracy. The methodology is simple, efficient and, as shown, produces eigenvalues and eigenfunctions of comparable accuracy to those of the best available methods found in the literature. All the 55 states lying with for the former, and 36 states with for the latter, are calculated up to eleven significant figures covering wide ranges of interaction. For the former, our results are superior to all the existing methods except that of the J-matrix formalism, while for the latter potential, our results surpass the accuracy of all existing methods. A detailed analysis of the variation of energies with respect to potential parameters show quite different trends for these two potentials. For higher , complex level crossing and inter-state mixing has been observed for the GESC potential. Special attention was paid for the high-lying states and regions of strong screening parameters. Many states are presented here for the first time. This offers a simple reliable method for the accurate calculation of these and other potentials in quantum mechanics.

###### Acknowledgements.
The two anonymous referees are thanked for their constructive comments.

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