Dipole polarizabilities for a hydrogen atom confined in a penetrable sphere

Physics Letters A 376 (2012) 1992–1996

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Physics Letters A www.elsevier.com/locate/pla

Dipole polarizabilities for a hydrogen atom confined in a penetrable sphere H.E. Montgomery Jr. a , K.D. Sen b,∗ a b

Chemistry Program, Centre College, 600 West Walnut Street, Danville, KY 40422-1394, USA School of Chemistry, University of Hyderabad, Hyderabad 500 046, India

a r t i c l e

i n f o

Article history: Received 4 April 2012 Received in revised form 29 April 2012 Accepted 30 April 2012 Available online 2 May 2012 Communicated by V.M. Agranovich

a b s t r a c t Benchmark numerical results on the ground and excited state eigenvalues and the ground state static and dynamic dipole polarizabilities are reported for a hydrogen atom confined at the center of a spherical box with penetrable walls. The dynamic polarizabilities are negative except when the frequency of incident radiation is below the 1s–2p transition frequency or in the frequencies immediately below a 1s–np transition. © 2012 Published by Elsevier B.V.

Keywords: Confined electronic systems Static polarizability Dynamic polarizability Dirichlet boundary condition Spherically confined hydrogen atom

1. Introduction Confined quantum mechanical systems are a useful model for simulating the effect of external conditions on an enclosed atom. Over seventy years ago Michels et al. [1] studied a hydrogen atom confined at the center of an impenetrable cavity and calculated the effects of pressure on kinetic energy and static dipole polarizability. This model has subsequently been applied to a wide range of physical problems [2–13]. Spatial confinement causes changes in the observable properties of the system such as the energy spectrum and polarizability. With the development of technology to construct atomic scale confinements, the study of confined systems has become increasingly relevant. To a first approximation, the behavior of a confined atom can be simulated as a particle at the center of a spherical box with an impenetrable wall. Under these conditions the wavefunction of the particle must satisfy Dirichlet boundary conditions. To give a more realistic model of confinement so as to account for interactions between particles such as van der Waals forces, the confinement requirement must be modified to consider a finite potential at the spherical boundary. In this Letter we report the new benchmark calculations of the electronic properties of a hydrogen atom confined in a spherical box with penetrable walls. Eigenvalues were determined by matching the logarithmic derivatives of the wavefunction at the spherical boundary. Development of Computer Algebra Systems (such as

*

Corresponding author. Tel.: +91 40 23134809; fax: +91 40 23012420. E-mail address: [email protected] (K.D. Sen).

0375-9601/$ – see front matter © 2012 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.physleta.2012.04.056

Maple 15, used in this work) has greatly facilitated this type of calculation, both by allowing increased precision and by providing stable algorithms for evaluation of the required special functions. Ley-Koo and Rubinstein [7] calculated static dipole polarizabilities using the Kirkwood lower bound [15]. Since our code facilitated calculation of excited state energies and wavefunctions, we used the perturbation sum over bound states to calculate static and dynamic polarizabilities with very high precision. This work offers the new possibility of studying the van der Waals dispersion forces within the spherically confined conditions with penetrable walls. Section 2 outlines the procedures for solving the Schrödinger equation inside and outside the confining surface and for matching logarithmic derivatives. Section 3 evaluates eigenvalues and static and dynamic polarizabilities for representative values of the confinement radius and potential. Section 4 summarizes the conclusions. 2. Computational procedures In atomic units, the Schrödinger equation for a hydrogen atom confined inside a sphere of radius rc by a constant potential V 0 is



 1 − ∇ 2 + V (r ) Ψ (r , θ, φ) = E Ψ (r , θ, φ), 2

(1)

where V (r ) is given by

1 V (r ) = − , 0  r  rc , r V (r ) = V 0 , r  rc .

(2)

H.E. Montgomery Jr., K.D. Sen / Physics Letters A 376 (2012) 1992–1996

Variables were separated using

Ψ (r , θ, φ) = R  (r )Y ,m (θ, φ),

Table 1 Energy levels of a hydrogen atom confined in a spherical box with V 0 = 10E h .

(3)

where Y ,m (θ, φ) is a normalized spherical harmonic with characteristic value ( + 1),  = 0, 1, 2, . . . , and R  (r ) is a radial function (i ) normalized with weight r 2 and composed of R  (r ), the wavefunc(o)

tion inside the sphere and R  (r ), the wavefunction outside the sphere. Following Ley-Koo and Rubinstein [7] with the change of variables

x=

r

κ

κ2 = ±

,

1 2E

(4)

,

Eq. (1) becomes



d2 dx2

+

2 d

( + 1)

−

x dx

x2

−

 (i ) (i ) ± 1 R  (x) = E R  (x),

2ν x

(5)

for the region inside rc . Changing variables to

k2 = 2( V 0 − E ),

y = kr ,

rc /a0

Energy/ E h

1 2 4 10

1s 1.1857771023 −0.2403853752 −0.4880275264 −0.4999995040

1 2 4 10

d2 dy 2

+

2 d y dy

( + 1)

−

y2

 (o)

(7)

for the region outside rc . Eq. (5) was solved by separating the behavior at r = 0 and ex(i ) panding R  (x) in the power series (i )

(i )

R  (x) = A κ  x

∞ 

()

c s xs ,

(8)

s =0

to obtain the coefficients ()

()

c 0 = 1,

c1 = − ()

()

c s +1 = −

κ +1

(s + 1)(s + 2 + 2)

for s > 1,

(9)

(i ) where A κ  is a normalization constant.

(o)

(o)

1 y +1

Energy/ E h

1 2 4 10

1s 1.7379420666 −0.1832974273 −0.4856172983 −0.4999993837

e − y 1 F 1 (−; −2; 2 y ),

(10)

(o)

 n =2

1 2 4 10

4.8720835540 1.0989353292 0.0415636466

2s 13.3103072397 2.9410453221 0.3807199929 −0.1135166630

3s 32.5079316725 8.3424886085 1.7603151992 0.0864595240

2p 6.6205187177 1.3852305543 0.1243084085 −0.1192247503

3p 22.1596065109 5.6222188496 1.1858364104 0.0456942673 3d 12.16599243155 2.9789051191 0.5812880236 −0.0087766899

1 2 4 10

f np

( E np − E 1s )2 − (hν )2

,

where f np , the oscillator strength for the 1s–np transition, is given by

2 3

rc ( E np − E 1s )

(11)

(i )

∞ 2

R np r R 1s r dr + 0

where B k is a normalization constant and 1 F 1 is the confluent hypergeometric function. The energies were determined by the requirement that the total wavefunction must be continuous with continuous first derivatives at r = rc . This is most easily accomplished by matching logarithmic derivatives at r = rc , while the wavefunctions are determined by the continuity requirements at r = rc and by the normalization condition. The eigenvalues of the Schrödinger equation can then be located using standard rootfinding techniques. The first node in the  = 0 radial function gives E 1s ( V 0 , rc ), the second node gives the E 2s ( V 0 , rc ), etc. Similarly, the first node in the  = 1 radial function gives E 2p ( V 0 , rc ), the next node gives E 3p ( V 0 , rc ), etc. For n = 1, 150 terms were used in Eq. (8) to maintain 15-digit accuracy in the calculated eigenvalues. As n increased, the number of terms was increased to maintain the accuracy. All calculations were performed using 150-digit arithmetic. At frequency ν , the 1s state dipole polarizability α (ν ) is given by perturbation theory as

α (ν ) =

3p

3d 9.1533087073 2.6045307928 0.5360785599 −0.0107591266

rc /a0

f np =

The solution outside the box was

R k ( y ) = B k

2p 5.0809281397 1.1864655933 0.1029683978 −0.1196441135

7.1014183250 1.6294316970 0.0805951539

,

()

2κ c s ± c s −1

3s

Table 2 Energies of a hydrogen atom confined in a spherical box with V 0 = 50E h .

(6)

− 1 R  ( y ) = 0,

2s 9.3560333028 2.5227144279 0.3365555108 −0.1143332300

1 2 4 10

transforms Eq. (1) into



1993

2 (o)

2

R np r R 1s r dr

.

(12)

rc

Since R 1s and R np are power series in r, all of the integrals needed to compute f np are analytic. The summation over n in Eq. (11) includes all excited p-states including the continuum wavefunctions. Having determined that accurate evaluation of the integrals over continuum states was beyond our current numerical capability, we terminated the sum after including all bound states. Additional justification for this choice of termination will be provided in Section 3.2. 3. Results and discussion 3.1. Eigenvalues and minimum confinement radii Illustrative results of the eigenvalue calculations for the 1s, 2s, 2p, 3s, 3p and 3d states are contained in Tables 1–3. Each table is associated with a specific value of the confining potential V 0 chosen so as to complement the energies in Tables 1–4 of Ley-Koo and Rubinstein [7]. For a specific confining potential the energies of all states increase monotonically as the confining radius, rc , decreases and the wavefunction becomes more localized. For a specific value of rc , the energies of all states increase as V 0 increases and the wavefunction becomes more localized. The changes in energy resulting from increased localization result from an increase

1994

H.E. Montgomery Jr., K.D. Sen / Physics Letters A 376 (2012) 1992–1996

α=

Table 3 Energies of a hydrogen atom confined in a spherical box with V 0 = 100E h . rc /a0

Energy/ E h

1 2 4 10

1s 1.9036518774 −0.1674228154 −0.4849669655 −0.4999993507

1 2 4 10

n =2

2s 14.2095199738 3.0488935737 0.3919313318 −0.1133131310

3s 35.0988185774 8.6204333790 1.7924792984 0.0878930039

2p 7.0482957740 1.4380719121 0.1297541732 −0.1191201505

3p 23.6898771516 5.8049088617 1.2074442743 0.0467042937 3d 12.9274954642 3.0760941880 0.5926888046 −0.0082906867

1 2 4 10

Table 4 Minimum confinement radius to support the existence of a bound 1s state. V 0/Eh

rc /a0

0 1 10 50 100

1s 0.722898245368 0.550374307424 0.277574663740 0.141367901300 0.103101795506

0 1 10 50 100

2s 3.808907792958 2.171653027066 0.919738810657 0.443676141186 0.319352116809

3s 9.360875848837 4.094793637297 1.594875285654 0.752527102088 0.538861572338

2p 3.296827053395 1.607195961716 0.633841444676 0.300069560830 0.215056986240

3p 8.856249864887 3.487866810834 1.302229054800 0.607434736282 0.433807731956 3d 7.197867612911 2.566740262951 0.939230799210 0.436011691311 0.311041073778

0 1 10 50 100

in momentum and thus can be understood as consequences of the Heisenberg uncertainty principle. As discussed in Goldman and Joslin’s analysis of confinement in an impenetrable sphere [10], the presence of the confining potential splits the angular momentum sublevel degeneracy. For a given rc and V 0 , the sublevel with the largest  has the lowest energy. For a given rc , the splitting increases with increasing V 0 , while for a given V 0 the splitting decreases with increasing rc . We also sought to determine the minimum confinement radius that would support the existence of a bound state. This is equivalent to finding the confinement radius where E = V 0 . The calculation is somewhat simplified by substituting

lim

E →V 0

+1 d  (o) ln R k = − , r

dr



(13)

for the logarithmic derivative of the wavefunction outside the sphere. The results of these calculations are shown in Table 4 for V 0 = 0, 1, 10, 50 and 100E h . An increase in V 0 results in a smaller minimum confinement radius, a feature also explained by the Heisenberg uncertainty principle. The minimum confinement radius increases significantly as n, the principal quantum number, increases with the greatest increase being observed in the s states. 3.2. Static dipole polarizabilities For the 1s ground state, the static (ν = 0) dipole polarizability is given by

f np

( E np − E 1s

)2

=



αn .

(14)

n =2

Having determined that including continuum wavefunctions in the sum was not practical, we first sought to determine if the sum over bound states gave a good approximation of the static polarizability. As a test case, we considered a hydrogen atom confined in an impenetrable sphere and evaluated Eq. (14) through the 20p contribution. The Thomas–Reiche–Kuhn oscillator strength sum rule ∞ 

f np = 1

(15)

n =2

was also evaluated through the 20p contribution. Since exact values of the static dipole polarizability for this system are available in the literature [14], we sought to determine if the proximity of the truncated oscillator strength sum to unity was a good indicator of the accuracy of the polarizability calculated using the same set of oscillator strengths. Table 5 shows a set of oscillator strength and polarizability sums calculated over a range of impenetrable confinement radii from 0 to 10a0 . The polarizability is also compared to Kirkwood lower bound [15]

 2

αLB = 4 z2 ,

(16)

the Buckingham lower bound [16]

αLB =

2 6r 2 3 − 8r r 2 r 3  + 3r 3 2 3

9r 2  − 8r 2

,

(17)

and the Dalgarno–Lewis upper bound [17]

αUB =

r 2  . 3 ( E 2p − E 1s ) 2

(18)

When the sum of oscillator strengths converged to within 1 part in 104 , the calculated polarizability typically agreed with the exact value to ∼ 10−7 a30 . The polarizabilities calculated using the truncated sum are in better agreement with the exact polarizabilities than those given by the Buckingham lower bound or the Dalgarno– Lewis upper bound. For large values of rc , the confined atom approaches free-atom behavior and the oscillator strength and polarizability sums approach the exact values more slowly. As a numerical experiment, we extended the sum for rc = 10a0 to include all terms through 40p. This gave an oscillator strength sum of 0.999993 and a polarizability of 4.49681418a30 , in eight decimal agreement with the polarizability calculated by Burrows and Cohen [14]. Thus encouraged by the polarizability calculations for systems confined by an impenetrable boundary, we extended our calculations to static polarizabilities of hydrogenic systems confined by finite potentials. For a given V 0 , all  = 1 terms up to nmax , the np term with the largest energy bound by V 0 , were included. The number of terms in the polarizability sum ranged from one for rc = 1a0 , V 0 = 10E h to 44 for rc = 10a0 , V 0 = 100E h . The results of those calculations are shown in Table 6. The Buckingham lower bound, derived 75 years ago, gives excellent estimates of the dipole polarizability, even at large rc where the Kirkwood lower bound and the Dalgarno–Lewis upper bound are less useful. For finite potentials, the polarizability is largely determined by rc , with the polarizability decreasing as the radius of confinement decreases. At a given value of rc the polarizability decreases as V 0 increases. This is consistent with the view that increasing the localization of the ground state results in a less polarizable electron distribution. Although relatively few bound states enter into the polarizability sum at small values of rc and V 0 , those few states are the

H.E. Montgomery Jr., K.D. Sen / Physics Letters A 376 (2012) 1992–1996

1995

Table 5 Oscillator strength sum, Kirkwood lower bound, αK , Buckingham lower bound, αB , static dipole polarizability calculated using a sum through 20p, upper bound, αDL compared to the exact polarizability [14] for a hydrogen atom confined in an impenetrable sphere of radius rc .

α , and the Dalgarno–Lewis

rc /a0

Σf

αK /a30

αB /a30

α /a30

αexact /a30

αDL /a30

1 2 4 10

0.999983 0.999988 0.999994 0.999813

0.0285 0.3401 2.2908 3.9986

0.0285 0.3402 2.3578 4.4968

0.0287920226 0.342558108 2.37798232 4.49681393

0.0287920226 0.342558111 2.37798233 4.49681418

0.0289 0.3429 2.4147 5.2759

Table 6 Static dipole polarizabilities for hydrogen confined by a finite spherical potential V 0 located at radius rc . Polarizabilities calculated using nmax terms are compared to the Kirkwood lower bound, αK , the Buckingham lower bound, αB , and the Dalgarno–Lewis upper bound, αDL . rc /a0

V 0/Eh

nmax

Σf

αK /a30

αB /a30

α /a30

αDL /a30

1

10 50 100

2 4 5

0.996623 0.999372 0.999325

0.06554 0.04108 0.03691

0.06557 0.04129 0.03713

0.06569 0.04144 0.03727

0.06573 0.04151 0.03734

2

10 50 100

4 7 10

0.997406 0.999791 0.999962

0.48349 0.39990 0.38171

0.48364 0.39990 0.38171

0.486766 0.402778 0.384462

0.48732 0.40317 0.38482

4

10 50 100

6 13 19

0.999034 0.999964 0.999994

2.52541 2.39803 2.36694

2.61693 2.47544 2.44121

2.63757714 2.49604994 2.46172232

2.68894 2.53893 2.50274

10

10 50 100

15 32 45

0.999199 0.999979 0.999996

3.99897 3.99876 3.99897

4.49763 4.49718 4.49706

4.497667338 4.497232416 4.497116024

5.25756 5.25164 5.25016

Table 7 Contributions of bound p-states to the oscillator strength sum and to the polarizability for rc = 1a0 , V 0 = 100E h . State

En /E h

fn

αn /a30

1s 2p 3p 4p 5p

1.90365188 7.04829577 23.68987715 48.67794808 80.98346234

0.98611074 0.03725762 0.00578877 0.00067251

0.03725762 0.00001423 0.00000265 0.00000011

0.99932499

0.03727460

Sum(2p–5p)

dominant contributions to the polarizability. This is illustrated in Table 7 which shows the contribution to Σ f and α for rc = 1a0 , V 0 = 100E h . 3.3. Dynamic dipole polarizabilities For the 1s ground state, the dynamic dipole polarizability is given by Eq. (11). Since the energies and oscillator strengths had been determined for the static polarizabilities, calculation of the dynamic polarizabilities was straightforward. α (ν ) has a pole wherever hν = E np − E 1s . The shape of the dynamic polarizability curve is shown in Fig. 1, where the polarizability for rc = 10a0 , V 0 = 10E h is shown for hν = 0–1.5E h . This range of ν encompasses the first four 1s − np excitations. The energy separation between successive np states increases with increasing n. This behavior of the confined atom differs from the free atom where adjacent excitations become progressively closer. α (ν ) < 0 except when hν < E 2p − E 1s or when ν is immediately below an excitation. This results from the dominance of the absorption terms in the complex expression for the first-order wavefunction as discussed in Karplus and Kolker’s [18] formulation of time dependent variational perturbation theory. Table 8 shows the dynamic dipole polarizabilities of the 1s ground state of the confined hydrogen atom for confinement radii rc = 1, 2, 4, 10a0 and confining potentials V 0 = 10, 50, 100E h and frequencies within the interval 0  hν  0.5E h . As V 0 increases, E np − E 1s increases and f np decreases, resulting in a decrease in the polarizability. The behavior for rc = 1a0 and hν = 0.5E h

Fig. 1. Dynamic polarizability for rc = 10a0 , V 0 = 10E h . The transition energies (indicated by the dotted vertical lines) are given in the legends. The transition energy for the unconfined atom is shown in parentheses.

is representative as the polarizability decreases from 0.06679 to 0.03763a30 as V 0 increases from 10 to 100a0 . 4. Conclusions We have presented data for static and dynamic polarizabilities of a hydrogen atom confined at the center of a penetrable spherical box. These calculations are improved models of confinement in that they can be extended to problems such as van der Waals dispersion forces. Our calculations used a perturbation theory expansion including all bound states but neglecting continuum contributions. These calculations should serve as benchmarks for further work that includes continuum wavefunctions either through explicit calculation or by a variational approach. We note that it is possible to control the polarizability for a confined system by a combination of confining radius, confining potential and the frequency of incident electromagnetic radiation. Since decreasing rc or increasing V 0 reduces the polarizability while the polarizability can be increased by increasing ν , it should

1996

H.E. Montgomery Jr., K.D. Sen / Physics Letters A 376 (2012) 1992–1996

Table 8 Dynamic dipole polarizabilities in a30 of the 1s ground state of the confined hydrogen atom for selected confinement radii rc and potentials V 0 within the frequency interval 0  hν  0.5E h . rc /a0 1

2

4

10

hν / E h

V 0 = 10E h

V 0 = 50E h

V 0 = 100E h

0.00

0.06569

0.04144

0.03727

0.10

0.06573

0.04146

0.03729

0.20

0.06586

0.04151

0.03733

0.30

0.06608

0.04160

0.03740

0.40

0.06639

0.04172

0.03750

0.50

0.06679

0.04188

0.03763

0.00

0.486766

0.402778

0.384462

0.10

0.489168

0.404422

0.385959

0.20

0.496519

0.409434

0.390521

0.30

0.509275

0.418070

0.398370

0.40

0.528276

0.430790

0.409903

0.50

0.554895

0.448329

0.425750

0.00

2.63757714

2.49604994

2.46172232

0.10

2.71463719

2.56448210

2.52815442

0.20

2.97568328

2.79448966

2.75102361

0.30

3.54486534

3.28649036

3.22561112

0.40

4.84649902

4.36488940

4.25572424

0.50

9.21569033

7.56876450

7.23645044

0.00

4.497667338

4.497232416

4.497116024

0.10

4.781065272

4.780481614

4.780326319

0.20

5.931819278

5.930243883

5.929831337

0.30

10.438856262

10.424055840

10.420282474

0.40

−29.350330009 1.729212560

−30.354319350 1.128780033

−30.609241132 0.995556476

0.50

glass–glass interfaces, can be cited as one of the recent examples wherein the magnetic properties are dramatically altered. This raises the possibility of new methods to investigate the spectral properties of confined atoms.

be possible to control the electronic properties of a confined atom by manipulating its confinement. This raises the possibility of new methods to investigate the spectral properties of confined atoms. The recent experimental results [19] on nanoglasses proposed to consist of a confined network of nanometer-sized glassy regions with interfaces of relatively lower density between them, called

Acknowledgements HEM would like to thank V.I. Pupyshev for insightful conversations. KDS is grateful to Department of Science and Technology, New Delhi for the J.C. Bose Fellowship and to Herbert Gleiter for fruitful discussions. The authors would also like to acknowledge the anonymous referee whose recommendations significantly improved the clarity of this work. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

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