We go beyond the approximate series-expansions used in the dispersion theory
of finite size atoms. We demonstrate that a correct, and non-perturbative,
theory dramatically alters the dispersion selfenergies of atoms. The
non-perturbed theory gives as much as 100% corrections compared to the
traditional series expanded theory for the smaller noble gas atoms.Comment: 3 pages, no figures, 1 tabl

Boström, Mathias

Brevik, I.

Persson, C.

Sernelius, Bo E.

Thiyam, Priyadarshini

English

arXiv

We go beyond the approximate series expansions used in the dispersion theory of finite-size atoms. We demonstrate that a correct, and nonperturbative, theory dramatically alters the dispersion self-energies of atoms. The nonperturbed theory gives as much as 100% corrections compared to the traditional series-expanded theory for the smaller noble gas atoms

Persson, Clas

Brevik, Iver Håkon

Sernelius, Bo. E.

NORA - Norwegian Open Research Archives

Nonperturbative theory for the dispersion self-energy of atoms

We go beyond the approximate series expansions used in the dispersion theory of finite-size atoms. We demonstrate that a correct, and nonperturbative, theory dramatically alters the dispersion self-energies of atoms. The nonperturbed theory gives as much as 100% corrections compared to the traditional series-expanded theory for the smaller noble gas atoms.Funding Agencies|Research Council of Norway [221469]; Swedish Reseach Council [C0485101]; European Commission</p

Sernelius, Bo

Bostrom, Mathias

Publikationer från Linköpings universitet

We go beyond the approximate series expansions used in the dispersion theory of finite-size atoms. We demonstrate that a correct, and nonperturbative, theory dramatically alters the dispersion self-energies of atoms. The nonperturbed theory gives as much as 100% corrections compared to the traditional series-expanded theory for the smaller noble gas atoms.</p

Thiyam, Priyadarshini,

Persson, C.,

Brevik, I.,

Sernelius, Bo,

Bostrom, Mathias,

Swepub

Nonperturbative theory for the dispersion self-energy of atoms [Elektronisk resurs]

Persson, Clas,

We go beyond the approximate series expansions used in the dispersion theory of finite-size atoms. We demonstrate that a correct, and nonperturbative, theory dramatically alters the dispersion self-energies of atoms. The nonperturbed theory gives as much as 100% corrections compared to the traditional series-expanded theory for the smaller noble gas atoms.publishedVersion© 2014 American Physical Societ

NTNU Open (Norwegian University of Science and Technology)

PHYSICAL REVIEW A 90, 054502 (2014)Nonperturbative theory for the dispersion self-energy of atomsPriyadarshini Thiyam,1,* C. Persson,1,2,3 I. Brevik,4 Bo E. Sernelius,5,† and Mathias Boström3,‡1Department of Materials Science and Engineering, Royal Institute of Technology, SE-100 44 Stockholm, Sweden2Department of Physics, University of Oslo, P. O. Box 1048 Blindern, NO-0316 Oslo, Norway3Centre for Material Science and Nanotechnology, University of Oslo, P. O. Box 1048 Blindern, NO-0316 Oslo, Norway4Department of Energy and Process Engineering, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway5Division of Theory and Modeling, Department of Physics, Chemistry and Biology, Linköping University, SE-581 83 Linköping, Sweden(Received 7 October 2014; published 14 November 2014)We go beyond the approximate series expansions used in the dispersion theory of finite-size atoms. Wedemonstrate that a correct, and nonperturbative, theory dramatically alters the dispersion self-energies of atoms.The nonperturbed theory gives as much as 100% corrections compared to the traditional series-expanded theoryfor the smaller noble gas atoms.DOI: 10.1103/PhysRevA.90.054502 PACS number(s): 31.30.jh, 34.20.Cf, 42.50.Lc, 03.70.+kUntil recently most work, using either the quantum electro-dynamic (QED) or the semiclassical formalism of dispersionenergies, have relied upon a point dipolar description. Previouswork that indeed did incorporate the finite atomic size wasbased on series expansions, and suggested that the vander Waals interaction contributed to a short-range attractivebinding energy [1,2]. The aim of a recent work [3] was todemonstrate how keeping the full nonperturbative theory andtaking the finite atomic size into account strongly alter thenonretarded van der Waals force and resonance interactionenergy at contact distances. Including finite atomic size effectsin a nonperturbative theory opens up for the possibility ofhaving van der Waals repulsion when two atoms come veryclose. Series expansion is a valid approach if the coupling isvery weak.In the present Brief Report we present a derivation ofthe nonperturbative retarded self-energy of atoms and ions invacuum. Within our theory, taken without series expansions,we find substantial corrections to the self-energy of an atom.A useful analytical asymptote is derived and we presentsome illuminating numerical results considering the finite-sizeeffects for different atoms and ions.In the contributions by Mahanty and Ninham [1,2] theydemonstrated that the secular equation that gives the perturbedeigenmodes of the electromagnetic field due to the presenceof a polarizable particle with finite size is[Ī + 4πG̃(r̄ ,r̄ ′,ξ )] = 0, (1)whereG̃(r̄ ,r̄ ′,ξ ) = [ξ 2/c2Ī + ∇r∇r ]∫G̃(r̄ − r̄ ′,ξ )×α(r̄ ′′ − r̄ ′,ξ )d3r ′′, (2)and the free space Green function has the form [2]G̃(r̄ − r̄ ′,ξ ) = Ĩ(2π )3∫d3kexp[ik̄(r̄ − r̄ ′)](ξ 2/c2 − k2) , (3)where Ĩ is the unit tensor.*thiyam@kth.se†bos@ifm.liu.se‡mathias.bostrom@smn.uio.noThe dispersion self-energy of a finite-size isotropicmolecule with a Gaussian spread was derived by Mahanty andNinham [1,2] in a series-expanded theory. Here, we present anonperturbative theory that can be used for atoms and ions invacuum, and we show that there are often large corrections tothe results from the approximate series-expanded theory. Thedispersion self-energy is [1,2]Es = ∫ ∞0dξ2πln[Ĩ + 4πG̃(r̄ ,r̄,iξ )]. (4)As was pointed out by Mahanty and Ninham the finitespread of the polarization make the Green function, G̃(r̄ ,r̄,iξ ),convergent (here r̄ is the position of the polarization). Thepolarization cloud of real atoms has a finite spread and weconsider as an interesting case a spatial distribution of theatom following an isotropic Gaussian function. This gives aGreen’s tensor where the diagonal elements are equal (i.e.,j = x, y, and z components are equal) [2,4]. The choice forthe polarizability tensor isα̃(r̄ ,ξ ) = Ĩ (π−3/2a−3)e−r2/a2α(ξ ), (5)where a is the Gaussian radius. The fully retarded expressioncan be shown to beGRetj (iξ ) =α(iξ )3(2π )3I (iξ ), (6)whereI (iξ ) = π3/2(a/2)3(1 + ξ2a2c2)− 4π2 ξ3c3e(ξa/2c)2[1 − erf(ξa2c)]. (7)In the nonretarded limit this is reduced toGNRj (iξ ) = [α(iξ )]/[3π3/2a3]. (8)The traditional way to treat these integrands is to make aseries expansion of the logarithm in Eq. (4) and keep only thelowest-order term, i.e., ln(1 + x) ≈ x. However, the energyfrom the eigenmodes for an isotropic atom in vacuum is givenby the sum of the equal j = x, y, and z contributions from the1050-2947/2014/90(5)/054502(2) 054502-1 ©2014 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW A 90, 054502 (2014)TABLE I. Finite-size dispersion self-energy, Es, for noble gasatoms and two ions. The subscript expanded indicates that the resultis from using series expansion of the logarithm in the integrand. Allenergies are in eV. The input data were taken from Refs. [5–8].Element ENR,fulls ENR,expandedsHe 71.2 131.5Ne 104.9 220.3Ar 37.6 62.1Kr 29.9 47.5Na+ 20.2 22.4Cl− 5.2 6.0secular equation. We findEs =∑j=x,y,z∫ ∞0dξ2πln[1 + 4πGj (iξ )]. (9)Mahanty demonstrated that when the retarded Green’sfunction is substituted into the series-expanded expression,the main contribution in the integration must come from thecharacteristic absorption frequencies of the atomic system. Forthe corresponding values of aξ/2c ≈ a/2λc, where λc is thewavelength of a characteristic absorption line, the retardationeffects are negligible. In the retarded limit one must use acutoff as proposed by Mahanty in similar calculations for theLamb shift [1]. However, we focus on the nonretarded limitwhere there is no need for a cutoff and we obtain the followingexpression:ENRs = 3∫ ∞0dξ2πln[1 + 4π3π3/2a3α(iξ )]. (10)To illustrate the point we use a simple one oscillator model forthe atomic polarizability, α(iξ ) = α(0)/(1 + ξ 2/ω20). We findthat a nonperturbative theory gives the following expressionfor the dispersion self-energy of isotropic polarizable particles(with a Gaussian polarization spread) in vacuum:ENRs =3ω02[−1 +√1 + 4α(0)/(3a3√π )]. (11)We give in Table I the results found when using our nonpertur-bative theory and as comparison also the results from using theapproximate series-expanded theory. As input we have usedatomic radii and static polarizabilities given by Hohm andThakkar [5] and characteristic frequencies given by Mahanand Subbaswamy [6]. For the ions we use the input data fromParsons and Ninham [7,8].The effects of a nonperturbative theory ought in principleto be detectable experimentally, although it is a challenge tomeasure self-energies directly. What is practically possibleis to verify our results indirectly in experiment. Solvationenergies of atoms and ions in a dielectric medium (i.e., changesof the self-energy in a vacuum compared to in a medium)can be measured [9–11]. Latimer et al. [12] were able to fitexperimental solvation energies (or rather the related heatsof solvation) to the Born equation by increasing the effectiveradius of the ions. Self-energy changes have also been shown toinfluence permeabilities across dielectric membranes [13,14].The difference between perturbative and nonperturbativetheories will be much reduced in a dielectric medium due tothe factor 1/ε (where ε is the frequency-dependent dielectricfunction of a medium). Hence the results obtained for instancefor the permeability of atoms across a membrane will bechanged mainly due to changes in the self-energy in vacuum.A factor of 2 difference for these self-energies compared tothose obtained from a series-expanded theory ought thereforeto be measurable with existing experimental equipment forsolvation free energies and permeabilities. Large correctionshave been overlooked in the past when performing seriesexpansions of the logarithmic terms before including the finiteatomic size [4]. The self-energy is also known to give acontribution to the Lamb shift [1]. When series expandingEq. (10), we rederive the dispersion self-energy found byMahanty [1]. However, the validity of a series expansionassumes that α(0)/a3 is much smaller than unity which isnot always the case.M.B. and C.P. acknowledge support from the ResearchCouncil of Norway (Project No. 221469). C.P. acknowledgessupport from the Swedish Reseach Council (Contract No.C0485101). P.T. acknowledges support from the EuropeanCommission.[1] J. Mahanty, Nuovo Cimento 22B, 110 (1974).[2] J. Mahanty and B. W. Ninham, J. Chem. Phys. 59, 6157 (1973).[3] M. Boström, P. Thiyam, C. Persson, D. F. Parsons, S. Y.Buhmann, I. Brevik, and B. E. Sernelius (unpublished).[4] J. Mahanty and B. W. Ninham, Dispersion Forces (Academic,London, 1976).[5] U. Hohm and A. J. Thakkar, J. Phys. Chem. A 116, 697 (2012).[6] G. D. Mahan and K. R. Subbaswamy, Local Density Theoryof Polarizability (Plenum, New York and London, 1990),Table 4.11.[7] D. F. Parsons and B. W. Ninham, J. Phys. Chem. A 113, 1141(2009).[8] D. F. Parsons and B. W. Ninham, Langmuir 26, 1816 (2010).[9] A. A. Rashin and B. Honig, J. Phys. Chem. 89, 5588(1985).[10] M. Boström and B. W. Ninham, J. Phys. Chem. B 108, 12593(2004).[11] T. Duignan, D. F. Parsons, and B. W. Ninham, J. Phys. Chem. B118, 3122 (2014).[12] W. M. Latimer, K. S. Pitzer, and C. M. Slansky, J. Chem. Phys.7, 108 (1939).[13] A. Parsegian, Nature (London) 221, 844 (1969).[14] M. Boström and B. W. Ninham, Biophys. Chem. 114, 95(2005).054502-2

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Priyadarshini Thiyam

C. Persson

I. Brevik

Bo E. Sernelius

Mathias Boström

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Physical Review A

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