Convergence of the multipole expansions of the polarization and dispersion interactions for atoms under confinement

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1 Convergence of the multipole expansions of the polarization and dispersion interactions for atoms under confinement Yong-Hui Zhang 1,2, Li-Yan Tang 2, Xian-Zhou Zhang 1, Jun Jiang 3 and J. Mitroy 3 1 Department of Physics, Henan Normal University, XinXiang , P. R. China 2 State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan , P. R. China and 3 School of Engineering, Charles Darwin University, Darwin NT 0909, Australia (Dated: March 31, 2012) The multipole expansion of the polarization interaction between a charged particle and an electrically neutral object has long been known to be asymptotic in nature, i.e. the multiple expansion diverges at any finite distance from the atom. However, the multipole expansion of the polarization potential of a confined hydrogen atom is shown to be absolutely convergent at a distance outside the confinement radius, R 0, of the atom. The multipole expansion of the dispersion potential between two confined hydrogen atoms is also shown to be absolutely convergent provided the two atoms satisfy R > 2R 0, where R is the inter-nuclear separation. These results were established analytically using oscillator strength sum rules and verified numerically using a B-spline description of the hydrogen ground state and its excitation spectrum. PACS numbers: ac, ap, Cf I. NATURE OF THE MULTIPOLE EXPANSIONS The long-range interaction between a charged particle and an atom is usually described by a polarization potential. This interaction is constructed by making a multipole expansion of the two-particle coulomb interaction leading to the following long-range expression for the second-order adiabatic polarization potential, V pol (R) l=1 V (l) pol (R) = α l. (1) 2R2+2l In this expression, the α l are the static multipole polarizabilities. The most important term arising from l = 1, the static dipole polarizability is sometimes called the polarizability. Equation (1) has been long known to be an asymptotic expansion. The polarization potential given by Eq. (1) eventually diverges as l increases at any finite value of R [1, 2]. Formal issues of a similar nature also lead to a problem in the multipole expansion of the atom-atom dispersion interaction. Dalgarno and Lewis [2] originally demonstrated that the multipole expansion of the H-H second-order dispersion interaction, V disp (R) n=3 l=1 C 2n, (2) R2n diverged in the Unsöld mean energy approximation [3], as n at any finite R. Sometime later it was shown that the exact multipole series for the H-H interaction was also asymptotic

2 2 [4, 5]. Koide had devised a momentum space treatment that resulted in each dispersion parameter being multiplied by a radial cutoff factor [6]. Although it could be shown that the resulting dispersion interaction was convergent at any finite R as n, the Koide method has only ever been applied to a few terms of the H-H dispersion interaction. Recently, B-spline basis sets have become very popular for solving atomic structure and collisions problems [7 11]. Such methods often impose the boundary condition that the wave function is zero at some finite radial distance, i.e. effectively solving the Schrödinger equation in a sphere of finite volume. One aspect of bounding the wave function is that the reaction of the wave function to an electric field also becomes bounded in space. This results in a multipole expansion of the adiabatic polarization interaction that can be shown to be absolutely convergent provided that the distance from the atom is larger than the confining radius of the atom. The imposition of a V = boundary will result in a multipole expansion that is mathematically well behaved at the cost of tolerating a small error in the magnitude of the polarization potential. A similar result holds for the dispersion interaction provided the two atoms are sufficiently far apart for there to be no overlap between the charge distributions of the two atoms. In this manuscript the Unsöld approximation is applied to estimate the multipole polarizabilities of a confined H atom. The resulting polarization expansion is then shown to be absolutely convergent when the radius is larger than the boundary of the confinement potential. This is supplemented by finite radius B-spline calculations of the hydrogen polarizabilities for a box size of R 0 = 12 a 0. The results of these explicit calculations confirm the Unsöld analysis. A similar analysis was also performed for the dispersion interaction. The resulting dispersion expansion is shown to be absolutely convergent when the distance between two nuclear centers as twice large as the radius of the confining potentials. This result was also confirmed by B-spline calculations of C 2n for two hydrogen atoms in their ground state. The present paper is restricted to the second-order expansions for the polarization and dispersion interactions. There has also been considerable discussion about the convergence of 1/R expansions of various parts of the interaction energy for molecular systems [12]. The idea of confining the electric charge distribution has proved itself useful in related applications. For example, intermolecular potentials can be computed directly from electronic charge densities by using the multipole expansion of the Coulomb interaction [13]. The use of the multipole expansion of these molecular electrostatic potentials can lead to an a series expansion that diverges as the multipole order increases [12, 14, 15]. However it has been shown that restricting the electron densities in molecules to have a finite range will result in a multipole expansion of the molecular electrostatic potential that is absolutely convergent in the region outside the range of the electrostatic charge distributions [14, 16 18]. II. POLARIZATION INTERACTION FOR A CONFINED HYDROGEN ATOM The static multipole polarizabilities of any spherically symmetric state are defined α l = i f (l) 0i (ε 0i ) 2, (3) where ε 0i is the excitation energy from state 0 to state i, the sum implicitly includes the continuum, and f (l) 0i is the oscillator strength of multipole l connecting the state 0 to the

3 3 excited state i. The f (l) 0i are defined [19, 20] f (l) 0i = 2 ψ 0 r l C l (ˆr) ψ i 2 ε 0i (2l + 1). (4) In this expression C l is the spherical tensor of rank l. Consider the oscillator strength sum rule S (l) (0) = i f (l) 0i. (5) This satisfies the identity S (l) (0) = Nl r 2l 2, (6) [21] where N is the number of electrons and r 2l 2 is a radial expectation value of the ground state wave function. This expression reduces to the well known Thomas-Reiche-Kuhn sum rule S (1) (0) = N for l = 1. For the hydrogen atom ground state, the general multipole sum rule is S (l) (0) = l(2l)!. (7) 22l 1 This exhibits a factorial growth as l increases. The exact value of the multipole polarizabilities for the hydrogen atom ground state can be written [22] α l = (2l + 2)!(l + 2) l(l + 1)2 2l+1. (8) For a confined atom, the radial expectation value in Eq. (6) will be bounded by the radius of the B-spline box, R 0, i.e. r 2l 2 R0 2l 2. (9) Note we have verified the correctness of Eq. (6) for calculations in a finite box to 20 significant figures. Applying Eq. (9) to Eq. (6) leads to the following rigorous inequality for the confined hydrogen atom ground state S (l) (0) lr0 2l 2. (10) The factorial growth present in Eq. (7) is eliminated once the wave function is confined. Confining a hydrogen atom also has an influence on the energies of the states. So the energies used to calculate the oscillator strengths and the energies used in the energy denominators of Eq. (3) are affected. As l increases in size, the centrifugal potential will tend to dominate the coulomb potential for r R 0. For l > 2ZR 0 the total potential is repulsive everywhere and increases monotonically for decreasing R < R 0. When this occurs, the minimum excitation energy will be larger than the potential at R = R 0, i.e ε = l(l + 1)/(2R 2 0) 1/R For sufficiently large l one can write ε l(l + 1)/(2R 2 0). Making the Unsöld mean energy approximation [3], and setting the mean excitation energy to the minimum value, leads to the following large l approximate expression for the hydrogen polarizabilities α l 4R2l+2 0 l(l + 1) 2. (11)

4 4 The ratio of two successive terms in Eq. (1), T = V (l+1) pol (R) V (l) pol (R) = l(l + 1)R2 0 (l + 2) 2 R 2. (12) This ratio is less than one for R 0 /R less than 1. Therefore the polarization series is absolutely convergent as long as R is greater than R 0. This absolute convergence has been substantiated by calculations of the multipole polarizabilities of the hydrogen ground state. Calculations were performed using a B-spline basis with the outer boundary set to R 0 = 12 a 0 and results are reported in Table I. The choice of R 0 = 12 a 0 contains most of the probability distribution of the unconfined H-atom ground state. Only % of the probability distribution of the unconfined H(1s) state lies outside this radius. This choice of R 0 leads to only a minor decrease in the ground state dipole polarizability. The polarizability for R 0 = 12 was a 3 0, a decrease of 0.004%. This is best put in perspective by noting that the change in polarizability due to the finite proton mass of 1 H is 0.11%. The static polarizabilities of hydrogen have been previously calculated using B-spline basis sets [23] but the intent of the present work is different from earlier investigations. Analytic expressions for the dipole polarizability of a confined hydrogen atom exist [24, 25]. The radial wave function is written as B-splines Ψ(r) = i c i B i (r), (13) where Ψ(r) is normalized such that 0 Ψ(r) 2 r 2 dr = 1. (14) The boundary conditions are such that the first B-spline is finite at the origin and zero at R = R V (R=15) pol V pol (a.u.) V 12 (R=15) pol L FIG. 1: (color online). The magnitude of the contribution to V pol (R = 15) from each successive multipole, L, of Eq. (1). Results for R 0 = 12 a 0 and the unconfined atom are depicted.

5 5 TABLE I: The multipole polarizabilities (in a.u.) of the hydrogen atom ground state. Both exact and B-spline polarizabilities calculated in a sphere of radius R 0 = 12 a 0 are given. The polarization potential at R = 15 a 0 is computed with Eq. (1) including terms of successively larger l for both sets of polarizabilities. The column E lowest gives the lowest B-spline energies for each value of l. l E lowest α l α 12 l : B-spline 105 Vpol (R = 15) 105 Vpol 12 (R = 15) The polarizabilities listed in Table I show the increasing difference between the B-spline calculation of α l from the exact calculation at the higher values of l. Polarizabilities are given to fifteen digits since this roughly approximates the precision achievable with a doubleprecision calculation. All the quoted digits are accurate. The polarizabilities are written as α R 0 l where R 0 specifies the outer radius of the B-spline box. The B-spline α R 0 l increases less rapidly than αl, and at l = 20 is times smaller than the exact polarizability. The convergence of the B-spline polarization potential at R = 15 a 0 is also shown in Fig. 1. The contributions of each individual multipole in Eq. (1), i.e. Vpol 12 (R), are plotted as a function of l. The divergence in the polarization series is apparent as the increments start to increase for l 15. However, the increments steadily decrease as l increases for the multipole expansion for the B-spline Vpol 12 (R = 15) providing a computational demonstration of the absolute convergence. The imposition of the boundary at R 0 = 12 a 0 has resulted in a polarization potential that is 0.005% smaller in magnitude at R = 15 a 0 than the expansion using the exact polarizabilities. Most of this difference arises from the dipole and quadrupole terms in the multipole expansion. The polarization potential calculated using αl 12 of Table I would give a polarization potential that is absolutely convergent and is accurate to better than 0.005% for R > 15 a 0. The demonstrated convergence of the B-spline expansion has some relevance to the earlier analysis by Brooks [26]. He stated that the origin of the polarization series divergence was the use of the expansion V = 1 r 1 r 2 = k=0 r< k r> k+1 C k (ˆr 1 ) C k (ˆr 2 ) (15) beyond its region of validity. This analysis recommended that all radial matrix elements involved in the evaluation of the polarizability have their radial integrations limited to a

6 6 finite value, e.g. X, which would be a lower bound on the region for which one writes the polarization potential using Eq. (1). The interaction, Eq. (15) is effectively reduced to V = k=0 r k X k+1ck (ˆr) C k ( ˆX) (r X), (16) where X is a lower bound to the radial coordinate used to evaluate V pol. The resulting multipole expansion was then shown to be convergent. Brookes concluded that the divergence in the polarization expansion was caused by using Eq. (15) in regions of space where it is not correct. This statement is correct, but cannot be used to imply that the evaluation of the second order polarization interaction using r 1 r 2 1 without making the multipole expansion will lead to a convergent expansion in powers of R 1. Indeed, it has been shown by an analysis in the Unsöld approximation that the 1/R n series expansion of the hydrogen atom polarization potential (without making a multipole expansion) is asymptotic [1, 2]. However, using a wave function that is limited in radial extent, or using an interaction that is limited in radial extent [26] will lead to a convergent expansion of the polarization interaction. III. THE DISPERSION COEFFICIENTS FOR THE HYDROGEN DIMER The second-order dispersion coefficients between two atoms in spherically symmetric states are defined C (2) 2n = l i =1 l j =1 (2n 2)! δ n 1,li +l j 4(2l i )!(2l j )! ij f (l i) A,0i f(l j) B,0j ε A,0i ε B,0j (ε A,0i + ε B,0j ), (17) where l i + l j + 1 = n and ε A,0i is the excitation energy from state 0 to state i for atom A. The sum implicitly includes the continuum, and f (l i) A,0i is the oscillator strength of multipole l i connecting the state 0 to the excited state i for atom A. Considerations of molecular symmetry do not have a direct effect on Eq. (17) when both atoms are in spherically symmetric states. The dispersion coefficients for a pair of ground state hydrogen atoms are the same regardless of whether they are in a gerade and ungerade molecular state. The following derivation treats the interaction between two hydrogen atoms. Setting all the excitation energies in the energy denominator to the minimum possible, ε 0,2p = 3/8 a.u., allows Eq. (17) to be simplified. The expression for the dispersion interaction reduces to the following expression, C 2n < n 1 l i =1 l j =1 δ n 1,li +l j (2n 2)! (2l i )!(2l j )! ij f (l i) 0i f (l j) 0j, (18) which is a rigorous upper bound. The sums over i,j can be replaced using Eq. (10) C 2n < l i =1 l j =1 δ n 1,li +l j (2l i + 2l j )! (2l i )!(2l j )! l ir 2l i 2 0 l j R 2l j 2 0. (19)

7 7 The product l i l j is less than n 2, terms involving R 0 can be collected, giving C 2n < l i =1 l j =1 The sum of the factorial products is easily evaluated l i =1 l j =1 The final inequality for C 2n is δ n 1,li +l j (2l i + 2l j )! (2l i )!(2l j )! n2 R 2n 6 0. (20) δ n 1,li +l j (2l i + 2l j )! (2l i )!(2l j )! = 22n 3 2 < 2 2n 3. (21) C 2n < 8n2 (2R 27R0 6 0 ) 2n. (22) Examining the ratio, T n = V 2n /V 6 where V 2n = C 2n /R 2n and V 6 = C 6 /R 6 gives T n = 8n2 27C 6 (2R 0 ) 2n 6 R 2n 6. (23) The lim n T n 0 exponentially when R > 2R 0. The dispersion series is absolutely convergent as long as there is no overlap between the charge distributions of the two atoms. Confining the wave function would act to increase the energy denominators in the expressions for the C 2n coefficients. This would also lead to improving the convergence of Eq. (2). This absolute convergence has been substantiated by calculations of the dispersion coefficients between two hydrogen atoms in their ground states. Calculations were performed using a B-spline basis with the outer boundary set to R 0 = 12 a 0 and results are reported in Table II. This R 0 value is sufficiently large to make only minor changes in the lowest order C 6 and C 8 coefficients V disp (a.u.) Vdisp (R=26) V 12 (R=26) disp n FIG. 2: (color online). The magnitude of the contribution to V disp (R = 26) from each successive term of Eq. (2). Results for R 0 = 12 a 0 and the unconfined atom are depicted. The dispersion coefficients listed in Table II show the increasing difference between the B- spline calculation of C 2n from the exact calculation at the higher values of n. The dispersion

8 8 TABLE II: The second-order dispersion coefficients (in a.u.) of the hydrogen atom ground state. Both exact and B-spline C 2n calculated in a sphere of radius R 0 = 12 a 0 are given. The dispersion potential at R = 26 a 0 is computed with Eq. (2) including terms of successively larger n for both sets of C 2n. 2n C2n C2n 12: 108 Vdisp (R = 26) 108 Vdisp 12 (R = 26) coefficients in the table are written as C R 0 2n where R 0 specifies the outer radius. The C2n values in Table II were computed by diagonalising the Hamiltonian with a B-spline basis with a much larger boundary radius. We also diagonalized the Hamiltonian in a large basis of Laguerre type orbitals [27] as a check. These coefficients are accurate to all digits for 2n 40 [28]. There is some degradation in precision for 2n > 40 and the value of C64 is accurate to about 6 digits. This lowering of precision has no influence on the conclusions that are deduced. The B-spline C2n 12 increases less rapidly than C2n, and at 2n = 62 is times smaller than C2n. The convergence of the B-spline dispersion potential at R = 26 a 0 is also shown in Fig. 2. The contributions of each individual multipole in Eq. (2), i.e. Vdisp 12 (R), are plotted as a function of 2n. The divergence in the dispersion series for Vdisp (R = 26) is just starting to occur for 2n > 50. However, the successive V 2n increments steadily decrease as n increases for the multipole expansion for the B-spline Vdisp 12. This provided a computational demonstration of the absolute convergence of the B-spline expansion. The imposition of the boundary at R 0 = 12 a 0 has resulted in a dispersion potential that is 0.004% smaller in magnitude at R = 26 a 0 than the expansion using the C2n coefficients. Most of this difference arises from the C 6 and C 8 terms.

9 9 IV. CONCLUSION It has been demonstrated that the multipole expansion of a confined hydrogen atom polarization and dispersion potentials leads to an inverse power series that is absolutely convergent provided the relevant distance parameter is sufficiently large. While the result is rigorous for hydrogen, one can reasonably assert that the result can be applied to any atom or pair of atoms. To a certain extent the asymptotic nature of the polarization and dispersion potentials is not surprising. The concepts of polarization and dispersion potentials implicitly assume interactions between particles that are spatially separated. So it should also not be surprising that spatially restricting the response of the atom to the perturbing field eliminates the asymptotic divergence. As noted [12], relatively little is known of the convergence properties of the multipole expansions of the induction and dispersion interactions. The present result adds considerable insight into the reason for their asymptotic natures and thus adds to the formal theory of these long-range interactions. It is interesting to note that Jansen [29] had suggested that the V pol multipole expansion would converge at sufficiently large separations provided the Slater or Gaussian type orbital basis set used in the evaluation of V pol was of finite dimension. This result has implications that go beyond the second-order perturbation expansion of these interactions. For example, the Stark effect is known to be divergent with respect to order of perturbation theory [30]. It is interesting to speculate whether bounding the wave function will also remove this divergence. The confinement of the atomic wave function to a finite region of space does of course constitute the imposition of an artificial constraint upon the wave function. However, such constraints are routinely applied in calculations of atomic structure based on B-spline basis sets. These descriptions of atomic structure will automatically have no formal issues with respect to the convergence of the multipole expansion of polarization and dispersion interactions. Acknowledgments This work was supported by NNSF of China under Grant Nos. 2012CBB21305, , , and by the National Basic Research Program of China under Grant No. 2010CB J.M. would like to thank the Wuhan Institute of Physics and Mathematics for its hospitality during his visits. The work of J.M was supported in part by the Australian Research Council Discovery Project DP [1] G. M. Roe, Phys. Rev. 88, 659 (1952). [2] A. Dalgarno and J. T. Lewis, Proc. Phys. Soc. London, A 69, 57 (1956). [3] A. Unsöld, Zeitschrift fur Physik 43, 563 (1927). [4] R. H. Young, Int. J. Quantum Chem. 9, 47 (1975). [5] R. H. Ahlrichs, Theor. Chim. Acta 41, 7 (1976). [6] A. Koide, J. Phys. B 9, 3173 (1976). [7] W. R. Johnson, S. A. Blundell, and J. Sapirstein, Phys. Rev. A 37, 307 (1988). [8] J. Sapirstein and W. R. Johnson, J. Phys. B 29, 5213 (1996).

10 [9] S. Ting-Yun, Q. Hao-Xue, and L. Bai-wen, J. Phys. B 33, L349 (2000). [10] S. Kang, J. Li, and T.-Y. Shi, J. Phys. B 39, 3491 (2006). [11] H. Bachau, E. Cormier, P. Decleva, J. E. Hansen, and F. Martín, Rep. Prog. Phys. 64, 1815 (2001). [12] B. Jeziorski, R. Moszynski, and K. Szalewicz, Chem. Rev. 94, 1887 (1994). [13] E. Scrocco and J. Tomasi, Top. Curr. Chem. 42, 95 (1973). [14] P. L. A. Popelier and M. Rafat, Chem. Phys. Lett. 376, 148 (2003). [15] C. Hattig and B. A. Hess, J. Phys. Chem. 100, 6243 (1996). [16] D. S. Kosov and P. L. A. Popelier, J. Chem. Phys. 113, 3969 (2000). [17] D. S. Kosov and P. L. A. Popelier, J. Phys. Chem. A 104, 7339 (2000). [18] M. Rafat and P. L. A. Popelier, J. Chem. Phys. 123, (2005). [19] Z. C. Yan, J. F. Babb, A. Dalgarno, and G. W. F. Drake, Phys. Rev. A 54, 2824 (1996). [20] J. Mitroy and M. W. J. Bromley, Phys. Rev. A 68, (2003). [21] D. Rosenthal, R. P. McEachran, and M. Cohen, Proc. R. Soc. London. Ser. A 337, 365 (1974). [22] A. Dalgarno and J. T. Lewis, Proc. R. Soc. London, Ser. A 233, 70 (1955). [23] M. I. Bhatti, K. D. Coleman, and W. F. Perger, Phys. Rev. A 68, (2003). [24] A. Sommerfeld and H. Welker, Ann. Phys. 32, 56 (1938). [25] B. L. Burrows and M. Cohen, Phys. Rev. A 72, (2005). [26] F. C. Brooks, Phys. Rev. 86, 92 (1952). [27] J. Mitroy and M. W. J. Bromley, Phys. Rev. A 71, (2005). [28] M. A. Cebim, M. Masili, and J. J. de Groote, Few-Body Systems 46, 75 (2009). [29] G. Jansen, Theor. Chem. Acc. 104, 499 (2000). [30] T. Kato, Perturbation Theory for Linear Operators, 2nd. Ed. (Springer, New York, 1976). 10