Bound-State Energies, Oscillator Strengths, and Multipole

CHINESE JOURNAL OF PHYSICS

VOL. 51, NO. 1

February 2013

Bound-State Energies, Oscillator Strengths, and Multipole Polarizabilities for the Hydrogen Atom with Exponential-Cosine Screened Coulomb Potentials H. F. Lai, Y. C. Lin, C. Y. Lin,∗ and Y. K. Ho† Institute of Atomic and Molecular Sciences, Academia Sinica, P. O. Box 23-166, Taipei 106, Taiwan (Received December 29, 2011) A finite basis set of B-spline functions is adopted to investigate atomic hydrogen with exponential-cosine screened Coulomb potentials. Bound-state energies, oscillator strengths, and multipole polarizabilities are calculated and compared with the available theoretical data. Our results, varying with the screening parameters of exponential-cosine screened Coulomb potentials, are presented. The influence of the screening effect on the energies, oscillators, and polarizabilities is discussed. DOI: 10.6122/CJP.51.73

PACS numbers: 03.65.Ge, 32.70.Cs, 31.15.ap

I. INTRODUCTION

The exponential-cosine screened Coulomb (ECSC) potential is well known for its wide applications in various branches of physics. In solid-state physics, for instance, the potential has been derived to describe the screening of impurities caused by conduction electrons in metals and semiconductors [1, 2]. Recently, it has been proposed by Shukla and Eliasson [3] that dense quantum plasmas can be depicted by the ECSC potential. Much effort has been devoted to the exploration of energy spectra and oscillator strengths for the ECSC potential [4–11] using the numerical and approximate analytic methods, such as the hypervirial Padé shceme, the shifted 1/N expansion method, the asymptotic iteration method, and the J-matrix approach. Within the framework of the complex-coordinate rotation method, the features of photoionization for the ECSC potential have been discussed in the literatures [12–14]. The influence of the ECSC potential on two-electron systems has been explored by Ghoshal and Ho [15–17]. The expression of the ECSC potential is given as 1 V (r) = − exp(−µr) cos(µr), r

(1)

where µ is the screening parameter. The purpose of the present work is to study the variation of bound-state energies, oscillator strengths, and multipole polarizabilities with the screening effect of ECSC potentials. Although bound-state energies have been reported previously in many articles, results for the oscillator strengths and multipole polarizabilities ∗ †

Electronic address: [email protected] Electronic address: [email protected]

http://PSROC.phys.ntu.edu.tw/cjp

73

c 2013 THE PHYSICAL SOCIETY ⃝ OF THE REPUBLIC OF CHINA

74

BOUND-STATE ENERGIES, OSCILLATOR STRENGTHS, . . .

VOL. 51

are scarce. To shed further light on these topics, calculations are performed on the basis of the configuration interaction method combined with the B-spline algorithm. The features of B-spline functions have been detailed by de Boor [18]. The B-spline basis functions are flexible regarding the adjustment of the distribution of basis functions, so that the small- and large-r behaviour of wave functions can be taken into account simultaneously. Compared to the Slater-type basis, the selection of non-linear parameters is not necessary for the usage of the B-spline-based finite basis set. In this paper, the theoretical approaches involved in the present work are presented in Sec. II. The results are reported in Sec. III with discussions. Some concluding remarks are given in Sec. IV. Atomic units are used throughout unless otherwise noted.

II. THEORETICAL METHOD

The non-relativistic radial Schrödinger equation for a hydrogen atom with the ECSC potential is expressed as ] [ l(l + 1) 1 d2 + + V (r) ϕ(r) = Eϕ(r), (2) − 2 dr2 2r2 where l is the angular momentum quantum number and V (r) is given in (1). The radial wave functions ϕ(r) are expanded in terms of the B-spline basis functions Bi,k (r), i.e., ϕ(r) =

N ∑

Ci Bi,k (r).

(3)

i=1

Along the r-axis with endpoints r = 0 and r = R, we select a knot sequence {ti } (i = 1,2,3,..,N+k), in which ti ≤ ti+1 (see Ref. [18]). The B-spline basis functions of order k are defined on the knot sequence by the following recurrence relations: { 1 for ti ≤ r < ti+1 , Bi,1 (r) = (4) 0 otherwise, and Bi,k (r) =

r − ti ti+k − r Bi,k−1 (r) + Bi+1,k−1 (r). ti+k−1 − ti ti+k − ti+1

(5)

To satisfy the boundary conditions of the radial wave functions, ϕ(0) = ϕ(R) = 0, the coefficients C1 and CN are required to vanish because of the properties of the B-splines, B1,k (0) = 1 and BN,k (R) = 1. Utilizing the expansion of radial wave functions in (3), a generalized eigenvalue equation, HC = EN C,

(6)

VOL. 51

H. F. LAI, Y. C. LIN, C. Y. LIN, AND Y. K. HO

is obtained from (2), where the Hamiltonian matrix elements are given by ∫ 1 R d2 H mn = − Bm,k (r) 2 Bn,k (r)dr 2 0 dr ∫ R l(l + 1) 1 + Bm,k (r) 2 Bn,k (r)dr 2 r 0 ∫ R + Bm,k (r)V (r)Bn,k (r)dr,

75

(7)

0

and the overlap matrix elements are defined as ∫ R N mn = Bm,k (r)Bn,k (r)dr,

(8)

0

which are non-zero due to the non-orthogonality of the B-spline basis. Within the framework of the configuration interaction, a finite basis approach using the B-spline basis functions is carried out for computing the energies, oscillator strengths, and multipole polarizabilities of hydrogen atoms with the ECSC potentials. The average oscillator strength of the dipole transition from an initial state |nl⟩ to a final state |n′ l′ ⟩ is defined as 2m ω ∑ fn′ l′ ,nl = |⟨nlm|D|n′ l′ m′ ⟩|2 (9) 3¯h 2l + 1 ′ m,m (∫ ∞ )2 2m l> = ω ϕnl (r)rϕn′ l′ (r)dr , (10) 3¯h 2l + 1 0 where ω = (Enl − En′ l′ )/¯h, D the electric dipole operator, and l> the larger angular momentum of l and l′ . The multipole polarizabilities are calculated using the well-known formula α=2

∑ |⟨m|T |n⟩|2 , Em − En

(11)

m̸=n

in which T , the transition operator, is taken to be z, z 2 , and z 3 for the dipole, quadrupole, and octupole polarizabilty, respectively.

III. RESULTS AND DISCUSSION

III-1. Bound-state energies The influence of the screening parameter µ on the bound-state energies of the ECSC potential is investigated using the finite B-spline basis set. In the present work, the total number and the order of the basis functions are taken to be N = 500 and k = 13, respectively. The maximum of the radial distance R adopted is 120a0 . The results as functions of

76


VOL. 51

FIG. 1: Energy eigenvalues of the 1s, 2s, and 2p states as a function of the screening parameter µ.

the screening parameter µ compared to the existing predictions are given in Tables I–IV for the principal quantum number n = 1–4, respectively. The good agreement of our results with other calculations demonstrates the feasibility and accuracy of the current computing scheme. In Figure 1(a) the energies of the ground state varying with the screening parameters are displayed. The energies increase with increasing screening parameter, and finally merge into a continuum for screening parameters larger than a critical value, i.e., a critical screening parameter. Figure 1(b) shows the energy variation of the 2s and 2p states with the parameter µ. The two states are degenerate at µ = 0, but gradually splitting with an increase of the parameter µ. The bound-state energies of n = 3 and 4 are shown in Figure 2. The degeneracy of the states is broken by the screening effect, leading to the split of the s-, p-, and d-states for each n. The state with a larger angular momentum l has a smaller critical screening parameter, due to the reduction of the attractive potential well and the enhancement of the centrifugal potential by the screening effect.

VOL. 51


77

TABLE I: Energy eigenvalues (in atomic units) as a function of the screening parameter µ (in a0 ) for the 1s state. µ

Lai [4]

Singh &

Dutt et al. [7] Ikhdair &

Bayrak &

Nasser et al. [11] present work

Varshni [5]

Sever [8]

Boztosun [9]

0.02

−0.48000780 −0.480008

−0.4800078 −0.48000780 −0.480007802609 −0.48000780

0.04

−0.46006089 −0.460061

−0.4600608 −0.46006088 −0.460060889023 −0.46006089

0.06 −0.440201 −0.44020051 −0.440201

−0.4402000 −0.44020051 −0.440200510290 −0.44020051

0.08 −0.420464 −0.42046391 −0.420464

−0.4204617 −0.42046390 −0.420463909843 −0.42046391

0.10 −0.400885 −0.40088477 −0.400884

−0.4008785 −0.40088476 −0.400884774639 −0.40088478

0.20 −0.306335

−0.30633449

0.30 −0.219416

−0.21941569

0.40 −0.142439

−0.14243918

0.50 −0.077680

−0.07768369

0.60 −0.028244

−0.02830496

TABLE II: Energy eigenvalues (in atomic units) as a function of the screening parameter µ (in a0 ) for the 2s and 2p states. µ

Lai [4]

Singh &

Ikhdair &

Bayrak &

Paul &

Varshni [5]

Sever [8]

Boztosun [9] Ho [10]

Nasser et al. [11] present work

2s 0.02 −0.105104

−0.1051033 −0.10510358 −0.10510359 −0.10510358769 −0.10510359

0.04 −0.085769

−0.0857621 −0.08576899 −0.08576899 −0.08576899889 −0.08576900

0.06 −0.067421

−0.0673900 −0.06742085 −0.06742173 −0.06742110514 −0.06742111

0.08 −0.050387

−0.0503576 −0.05038349 −0.05039222 −0.05038656192 −0.05038656

0.10 −0.034941

−0.0351880 −0.03491580 −0.03496764 −0.03494131194 −0.03494131 2p

0.02 −0.105075 −0.10507464 −0.105075 −0.1050744 −0.10507463 −0.105074638306 −0.10507464 0.04 −0.085591 −0.08555914 −0.085559 −0.0855520 −0.08555913 −0.085559137218 −0.08555914 0.06 −0.066778 −0.06677752 −0.066777 −0.0667611 −0.06677739 −0.066777520524 −0.06677752 0.08 −0.048997 −0.04899725 −0.048997 −0.0489939 −0.04899567 −0.048997247808 −0.04899725 0.10 −0.032469 −0.03246881 −0.032470 −0.0326733 −0.03245500 −0.032468805180 −0.03246881 0.12

−0.01747646 −0.017489

−0.017476455652 −0.01747646

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VOL. 51

FIG. 2: (Color online) Energy eigenvalues of the 3s, 3p, 3d, 4s, 4p, and 4d states as functions of the screening parameter µ.

III-2. Oscillator strengths

FIG. 3: (Color online) Oscillator strengths of the transitions from 1s to 2p, 3p, and 4p as functions of the screening parameter µ.

The results of the oscillator strengths for the transitions from 1s to 2p, 3p, and 4p states are listed in Table V and compared to the existing predictions for the 1s-2p transition. Our results are in very good agreement with the data by Singh and Varshni [5].

VOL. 51


79

TABLE III: Energy eigenvalues (in atomic units) as a function of the screening parameter µ (in a0 ) for the 3s, 3p, and 3d states. µ

Lai [4]

Ikhdair &

Bayrak &

Sever [8]

Boztosun [9]

Nasser et al. [11]

present work

3s 0.02

−0.036025

−0.0360213

−0.036025

−0.036025105113

−0.0360251

0.04

−0.018823

−0.0188586

−0.018821

−0.018823063334

−0.0188231

0.05

−0.011576

−0.0119523

−0.011559

−0.011575564207

−0.0115756

0.06

−0.005461

−0.0070778

−0.005349

−0.005462087488

−0.0054621

0.07

−0.000740

−0.000774004

−0.0007738

3p 0.02

−0.035968

−0.0359640

−0.035967

−0.035967603433

−0.0359676

0.04

−0.018453

−0.0184505

−0.018452

−0.018453352989

−0.0184534

0.05

−0.010929

−0.0111117

−0.010917

−0.010929329822

−0.0109293

0.06

−0.004471

−0.0054058

−0.004382

−0.004472575112

−0.0044726

3d 0.02

−0.035851

−0.0358490

−0.035850

−0.03585066231088

−0.0358507

0.04

−0.017682

−0.0176910

−0.017681

−0.01768206425572

−0.0176821

0.05

−0.009555

−0.0096940

−0.009549

−0.00955487932273

−0.0095549

0.06

−0.002309

−0.0029240

−0.002261

−0.00231094773317

−0.0023109

Figure 3 displays the oscillator strengths f1s2p , f1s3p , and f1s4p as functions of the screening parameter µ. As seen in (10), the oscillator strengths are associated with the transition energies and transition matrix elements between the initial and final states. Compared to the transition matrix elements, however, the variation of the transition energies with µ is relatively small. It has been known that the screening effect causes np wave functions to be shifted outward more than the 1s wave function. As a result, the oscillator strengths of the 1s-np transitions are decreased with increasing µ because the overlap of the 1s and np wave functions is diminished as the screening is increased. Table VI gives the oscillator strengths of the transitions from the 2s and 2p states to np (n = 2–4) and md (m = 3 and 4), respectively. The screening effects of the ECSC potentials on the oscillator strengths f2snp are illustrated in Figure 4. The oscillator strengths of the 2s-3p and 2s-4p transitions are decreased with an increase of the screening parameter µ. However, the 2s-2p transition has the opposite trend. For µ = 0, i.e., pure hydrogen, the energies of the 2s and 2p states are degenerate, and the maximum of the 2s wave function is located slightly farther away from the center than that of the 2p wave function. The degeneracy leads to a zero of the

80


VOL. 51

TABLE IV: Energy eigenvalues (in atomic units) as a function of the screening parameter µ (in a0 ) for the 4s, 4p, and 4d states. µ

Lai [4]

Bayrak &

Nasser et al. [11]

present work

Boztosun [9] 4s 0.01

−0.021438

−0.021437

−0.021437

0.02

−0.012572

−0.012572

−0.012572

0.03

−0.005270

−0.005259

−0.005270

0.04

−0.000119

−0.000136 4p

0.01

−0.021424

−0.021424

−0.021424

0.02

−0.012486

−0.012486

−0.012486

0.03

−0.005033

−0.005023

−0.005033 4d

0.01

−0.021398

−0.021398

−0.021398

0.02

−0.012310

−0.012310

−0.01231026647655

−0.012310

0.03

−0.004539

−0.004533

−0.00453927323299

−0.004539

FIG. 4: (Color online) Oscillator strengths of the transitions from 2s to 2p, 3p, and 4p as functions of the screening parameter µ.

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81

FIG. 5: (Color online) Oscillator strengths of the transitions from 2p to 3d and 4d as functions of the screening parameter µ.

oscillator functions oscillator in Figure

strength at µ = 0. With an increase of µ, the overlap of the 2s and 2p wave is enhanced due to the 2p wave function being pushed outward accordingly. The strengths f2p3d and f2p4d varying with the screening parameter µ are presented 5.

III-3. Multipole polarizabilities The dipole, quadrupole, and octupole polarizabilities calculated by (11) for the ground state of the ECSC potential are listed as functions of the screening parameters µ in Table VII. In Figure 6, the variations of the multipole polarizabilities with µ are displayed. The enhancement of the dipole, quadrupole, and octupole polarizabilities with increasing the parameter µ indicates that the stronger screening effect of ECSC potentials makes atoms more easily polarized. Our results are in good agreement with the available theoretical results [19, 20] for the unscreened case of µ = 0. For the screened cases of the ECSC potential, the results of the multipole polarizabilities are reported for the first time to our best knowledge. Although the bound-bound transitions are diminished with increasing the parameter µ, the polarizabilities are increased with increasing µ due to the contributions from the bound-free transitions, mainly the resonant transitions.

IV. CONCLUSIONS

We investigated atomic hydrogen with exponential-cosine screened Coulomb potentials using the B-spline basis functions. The bound-state energies, oscillator strengths,

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VOL. 51

TABLE V: Oscillator strengths f1snp as a function of the screening parameter µ (in a0 ) for the 1s-np transitions (n = 2–4).

µ

Singh & Varshni [5]

Dutt et al. [7]

f1s2p

f1s2p

0.01 0.02

0.415785

0.415951

0.03 0.04

0.413204

0.414353

0.05 0.06

0.406867

0.410322

0.07

present work f1s2p

f1s3p

f1s4p

0.416143

0.0789645

0.0287059

0.415785

0.0781209

0.0270645

0.414874

0.0760912

0.0231306

0.413203

0.0724707

0.410591

0.0667297

0.406867

0.0577837

0.401853

0.08

0.395350

0.402988

0.395350

0.09

0.387111

0.10

0.376809

0.391577

0.376809

0.11

0.363971

0.12

0.347845

0.375306

0.347845

TABLE VI: Oscillator strengths f2snp and f2pmd as a function of the screening parameter µ (in a0 ) for the 2s-np (n = 2–4) and 2p-md (m = 3 and 4) transitions, respectively. A(B) denotes A × 10B . µ

f2s2p

f2s3p

f2s4p

f2p3d

f2p4d

0.01

0.684943(−4)

0.434243

0.101963

0.695121

0.121060

0.02

0.521962(−3)

0.430395

0.973143(−1)

0.690946

0.116711

0.03

0.168257(−2)

0.421047

0.856312(−1)

0.680552

0.105036

0.04

0.382344(−2)

0.404122

0.661005

0.05

0.719122(−2)

0.376683

0.627345

0.06

0.120287(−1)

0.332418

0.566626

0.07

0.185991(−1)

0.08

0.272151(−1)

0.09

0.382806(−1)

0.10

0.523588(−1)

0.11

0.703027(−1)

0.12

0.935476(−1)

VOL. 51


83

FIG. 6: (Color online) Dipole, quadrupole, and octupole polarizabilities as functions of the screening parameter µ.

TABLE VII: Multipole polarizability (in a30 ) as a function of screening parameter µ (in a0 ) for the ground of ECSC potential. A(B) denotes A × 10B . µ

Dipole

Quadrupole

Octupole

0.00

4.50000

1.45000(1)

1.31250(2)

0.05

4.50839

1.50590(1)

1.32141(2)

0.10

4.56066

1.54231(1)

1.37603(2)

0.15

4.68918

1.63244(1)

1.51304(2)

0.20

4.92576

1.80343(1)

1.78395(2)

0.25

5.31470

2.10137(1)

2.29179(2)

0.30

5.93093

2.61792(1)

3.27256(2)

0.35

6.91550

3.55506(1)

5.33083(2)

0.40

8.56077

5.41114(1)

1.02437(3)

0.45

1.15488(1)

9.61247(1)

2.42796(3)

0.50

1.77459(1)

2.11634(2)

7.58995(3)

0.55

3.35925(1)

6.37611(2)

3.53411(4)

0.60

9.13104(1)

3.21403(3)

3.20149(5)

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VOL. 51

and multipole polarizabilities are calculated. Our results are in good agreement with the existing theoretical predictions. The results varying with the screening parameter of the exponential-cosine screened Coulomb potentials are presented. It should be noted that the oscillator strengths associated with the transitions 2s-np (n=2–4) and 2p-md (m=3 and 4) and multipole polarizabilities as functions of the screening parameters are reported for the first time. The shifting of bound-state energies toward the ionization thresholds is caused by the screening effect. The critical screening parameters of states with the same principal quantum number are diminished with increasing the angular momentum quantum number. The screening influence on the oscillator strengths is mainly through the transition energies and matrix elements. While the oscillator strengths for transitions between the states of different principal quantum numbers are decreased with increasing the screening parameter, the reverse trends are observed for transitions between the states of identical principal quantum numbers. The enhancement of the dipole, quadrupole, and octupole polarizabilities with increasing the screening parameter illustrates a tendency toward being polarized for atoms with larger screening parameters.

Acknowledgements This work has been supported by the National Science Council of Taiwan.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

N. Takimoto, J. Phys. Soc. Japan 14, 1142 (1959). E. C. McIrvine, J. Phys. Soc. Japan 15, 928 (1960). P. K. Shukla and B. Eliasson, Phys. Lett. A 372, 2897 (2008). C. S. Lai, Phys. Rev. A 26, 2245 (1982). D. Singh and Y. P. Varshni, Phys. Rev. A 28, 2606 (1983). A. Chatterjee, Phys. Rev. A 34, 2470 (1986). R. Dutt, U. Mukherji, and Y. P. Varshni, J. Phys. B: At. Mol. Phys. 19, 3411 (1986). S. Ikhdair and R. Sever, J. Math. Chem. 41, 329 (2007). O. Bayrak and I. Boztosun, Int. J. Quantum Chem. 107, 1040 (2007). S. Paul and Y. K. Ho, Comput. Phys. Commun. 182, 130 (2011). I. Nasser, M. S. Abdelmonem, and A. Abdel-Hady, Phys. Scr. 84, 045001 (2011). C. Y. Lin and Y. K. Ho, Eur. Phys. J. D 57, 21 (2010). C. Y. Lin and Y. K. Ho, Comput. Phys. Commun. 182, 125 (2011). C. Y. Lin and Y. K. Ho, Phys. Scr. T143, 014051 (2011). A. Ghoshal and Y. K. Ho, Phys. Rev. A 79, 062514 (2009). A. Ghoshal and Y. K. Ho, Phys. Rev. E 81, 016403 (2010). A. Ghoshal and Y. K. Ho, Comput. Phys. Commun. 182, 122 (2011). C. de Boor, A Practical Guide to Splines (Springer, New York, 2001). R. J. Bell, Proc. Phys. Soc. London 92, 842 (1967). K. T. Tang, J. M. Norbeck, and P. R. Certain, J. Chem. Phys. 64, 3063 (1976).