Susceptibility and polarizability of atoms and ions S. H. Patil Citation: J. Chem. Phys. 83, 5764 (1985); doi: 10.1063/1.449654 View online: http://dx.doi.org/10.1063/1.449654 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v83/i11 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 01 Mar 2012 to 59.162.23.79. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions Susceptibility and polarizability of atoms and ions s. H. Patil Department of Physics, Indian Institute of Technology, Bombay-400 076, India (Received 29 August 1984; accepted 25 June 1985) With the help of some relations for the oscillator strengths and semiclassical relations, we have calculated diamagnetic susceptibilities and dipole polarizabilities for a large number of atoms and ions. We have also obtained simple expressions for susceptibility and polarizability in terms of ionization energy. I. INTRODUCTION The response of an atom to an external, weak static electric field is described mainly by its electric dipole polarizability. The average polarizability of an atom is given by (in atomic units) ~~ L 1(01 ~+)I' a p 3 k Ek -Eo (Ll) The calculation of this quantity encounters formidable difficulties because (a) the states required for the evaluation of the matrix elements are many-electron states, (b) the expression requires a summation over an infinite number of intermediate states. As a result, even with very elaborate computations, large uncertainties still remain in the theoretical estimations of atomic polarizabilities. Another property of interest, especially in the case of inert gases and their isoelectronic sequences, is the diamagnetic susceptibility. It is given by (1.2) It is intimately related to dipole polarizability and can play an essential role in the determination of polarizability. A. A very brief review The state of experimental and theoretical progress in the determination of atomic polarizabilities has been comprehensively reviewed by Miller and Bederson. 1 They also list some recommmended values for the polarizabilities along with estimated errors. Here we mention a few of the more recent theoretical efforts. The coupled Hartree-Fock approximation first proposed by Dalgarn0 2 has been used by Markiewicz, McEachran, and Stauffer3 to obtain accurate values for the polarizabilities of some atoms and ions with two electrons in the outermost shell. Using the self-consistent field method, Voegel, Hinze, and Tobin4 have calculated the polarizabilities of the atoms in the He-Ne series. Density dependent potentials have been used by Mahan 5 to calculate multipole polarizabilities of several atoms and ions with closed shells. For the alkali metals, accurate values have been obtained by using configuration interaction approach, 6 pseudopotential approach, 7 and effective quantum number approach. 8.9 More directly of interest to us are the calculations based on statistical approaches. It was shown by Bruch and Lehnen lO that the Thomas-Fermi-Dirac model gives polariza5764 J. Chem. Phys. 83 (11), 1 December 1985 bilities for inert gases, which are many times larger than the observed values although they do show improvement for larger atomic numbers. On the other hand, the calculated values for alkaline earths are smaller than the experimental values. Similar calculations have also been carried out by Shevelko and Vinogradov.ll Diamagnetic susceptibilities have been calculated 12 by using Hartree-Fock wave functions, for inert gases, to within about 10% accuracy. Recently, efforts have also been made 13 •14 to calculate susceptibilities from extensions of the Thomas-Fermi model. B. An outline of our approach The polarizability of an atom depends quite significantlyon the large r behavior of atomic wave functions. Therefore, statistical approaches do not readily yield accurate values for atomic polarizabilities. It has, however, been shown 14 that iflarge rbehavior is incorporated in the Thomas-Fermi model, one gets reliable values for diamagnetic susceptibilities of inert gases, values which are comparable in accuracy to those from Hartree-Fock calculations. With this encouragement, we have analyzed atomic polarizabilities and susceptibilities using a semiclassical approach. The analysis is presented in three parts. (a) We consider the well-known function Stu) defined as 15-22 Stu) = L (Ek - EotlkO' k (1.3) IkO = (Ek - Eo) 1 (01 ~ r Ik ) i 2 1 , in terms of which one has ap =~S( (1.4) - 2). It is known that Stu) is relatively easier to evaluate for u = - 1,0, 1, 2, and has a pole at u = 5/2. The values ofS (0) and S(2) are particularly simple to estimate, while S( - 1) and S (1) are related to the diamagnetic susceptibility and the energy of the system, and the two-particle probability function. (b) We obtain an expression for the two-particle probability function within a semiclassical approximation, which allows us to estimate S ( - 1) and S (1). The value of S ( - 1) depends on the diamagnetic susceptibility which is calculated within the framework of a modified Thomas-Fermi model which incorporates the correct asymptotic behavior. 14 (c)WiththehelpofthevaluesofS(u)foru = - 1,0,1,2, 0021-9606/85/235764-08$02.10 @ 1985 American Institute of Physics Downloaded 01 Mar 2012 to 59.162.23.79. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions S. H. Patil: Susceptibility and polarizability we develop a suitable expression for S (u), which is then used for extrapolation to u = - 2 and hence obtain dipole polarizabilityap- 5765 (2.7) In obtaining Eq. (2.6), we have made use ofVinti's relation 24 (2.8) c. Results The main results are summarized here. (a) Of general interest are the expressions we have obtained for S ( - 1) and S (1) based on a semiclhssical approximation. (b) We have calculated polarizabilities of a large number of atoms and ions. The agreement betweeen the calculated values and the experimental values is good over the range Ne-Ba. Especially striking is the agreement between the calculated values for the ten-electron isoelectronic sequence and the empirical values of Edlen. 23 (c) We have also calculated diamagnetic susceptibilities for these atoms and ions. Since the susceptibilities play an important role in the analysis, we have obtained a semiempirical expression for them. The agreement between the calculated values, those from the semiempirical expression, and the available experimental values is quite good. (d) Combining our results with the well-known relation a p = 16 X2/N, we have obtained two approximate relations for susceptibility and polarizability, x = N /4E;(lm + 1), a p =N/E~(lm + W, (1.6) II. EXPRESSIONS FOR S(u) It is known l5- 22 that while S( - 2) is quite difficult to evaluate,S(u) is relatively easier to analyze for u = - 1,0,1, 2. We first simplify S (u) for these values. A. S(u) for u = - 1, 0, 1, 2 The well-known 15-22 expressions for S (u) are (ol( ~ r;Ylo), = (2.1) S(O)=~, S(2) (2.2) (ol( ~ p;Ylo), (2.3) = 21TZ (01 ~ 8(r;) 10), (2.4) S(l) = where N is the number of electrons in the atom or ion and Z is the nuclear charge. In terms of the electron density PI(r) and the two-electron density P12(r,r'), the expressions for S( - 1), S(l), and S(2) reduce to S( - 1) = J PI(r)r d 3r (2.9) with Vtot being the total potential energy and Veo being the energy due to electron-electron interaction. Now the virial theorem implies that I Vtot I = 21E, I where E, is the total energy. Also I Vee I is expected to be relatively small. Assuming an electron density of D exp ( - 2r Z 1/3) as suggested by the Thomas-Fermi model, IVeo I is estimated to be about 20% of IVtot I for neutral atoms, and smaller for positive ions. One may therefore write S(1):::::2.4IE, 1 +Z + J pdr,r')r r' d 3rd 3r', (2.5) 0 f pdr,r') r~r' d 3rd 3r'. (2.10) Proceeding directly with Eq. (2.3), Dehmer et a/. 22 have obtained a somewhat different expression, Sill = IE,I + L (Olp;· PjIO). (2.11) ;#j (1.5) where N is the number of electrons, E; is the ionization energy and /m is the highest / value of the occupied states. The predictions ofEqs. (1.5) and (1.6) are in quite good agreement with the observed values for those neutral atoms which do not have a low-lying excited state. S( -1) The first term on the right-hand side of Eq. (2.6) represents the negative of the energy due to interaction of the electrons with the nucleus. Designating it by IVen lone has The advantage of using Eq. (2.6) is that S (1) can be expressed in terms of the density functions in the r space. One can estimate S (2) by noting that most of the contribution to PI(O) comes from the s-wave electrons in the n = 1,2 states. This leads to (2.12) S(2):::::4Z [(Z - O.4f +!(Z - 2.4)3], where we have used23 a screening charge of 0.4 for the n = 1 electrons and 2.4 for the n = 2 electrons. B. S(u) near u = 5/2 It is known 22 that S (u) diverges at u = 5/2. This divergence must come from the summation over the large values of Ek in Eq. (1.3). The matrix elements involving such states emphasize the behavior at small r. One therefore expects to get a good estimation for S (u) near the singular point by considering the innermost electrons. For Ek very large, one may ignore the interaction and use free-particle wave functions. Then for n = 1 states, we have the large-Ek matrix element (0Ixlk):::::(Z3/1T)1/2 J x exp(ik o r-rZ)d 3 r. (2.13) Hence the singular part of Stu) is given approximately by Sd(u):::::20481TZ 5 d 3k (k 2 + Z 2)U + (21T)3 2 J I k2 (k2+Z2)6' (2.14) where the subscript d indicates that we are considering the divergent part. It is observed that this integral diverges at u = 5/2. For later reference, we note that Sd(2)/Sd(1) = 3Z 2. (2.1S) J. Chern. Phys., Vol. 83, No. 11, 1 December 1985 Downloaded 01 Mar 2012 to 59.162.23.79. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 5766 S. H. Patil: Susceptibility and polarizability III. SEMICLASSICAL APPROACH In Sec. II we discussed the expressions for Stu) for - 1,0, 1, and 2. In particular, it was shown that S ( - 1) and S (1) can be expressed in terms of one- and two-electron density functions. Here. we discuss these functions within a semiclassical approach and obtain closed expressions for S( - 1) andS(!). u= where 8 12 is the angle between r and r'. This term may be expected to be significant mainly for r near r'. Using the identity (3.9) sin WI sin W2 = ![COS(WI - W2) - cos(w} + w2)] for the sin functions, and neglecting the second term which oscillates rapidly one gets 1 F 12(r,r'):::::-)' A. One- and two-electron densities 81T We consider wave functions of the form l/I(rl,r2' ... ) = (N!)-1/2£ij ••• ~;(rl)tPj(r2) .•• , = 2 2: ItPj(rW, (3.2) j where the factor of 2 is for the spin states of the electrons. Similarly, the two-electron density function is given by pdr,r') = PI(rlol(r') - 2 [Fdr,r')F, (3.3) 2: ~r(r)tPi(r') rr'p,.(R) P,(cos 8u) (3.10) where we have used the approximations f (3.4) p,,(r")dr" :::::p,.(R)(r - r'), [p,. (r)p" (r')] 1/2:::::p,,(R), (3.11) with R = (r + r')l2. The summation over n may now be carried out using Eq. (3.6) to yield , F I2 (r,r) = k/(R) where Fdr,r') = tf X cos [p,,{R l(r - r')], (3.1) where ~j(r) are orthonormal functions and N is the number of electrons. The one-electron density function is then given by PI(r) IC,. 12(21 + 1) = 1 ~ )' (21 + l)p/(cos 8121 sin [k/(R )(r- r')] 471rr' ." [2~ - (I + 1I2)2/R 2pn. r- r " (3.12) This expression and Eq. (3.7) can be used in Eq. (3.3) to obtain the two-electron density function. I and the factor of 2 takes into account the spin states of the electrons. B. WKB wave functions and densities The single-particle wave functions may be evaluated within the WKB approximation. They are given by25 ~n./.m(r)= C n r[Pn (r)] PrJ (r) = [2(E" 1/2sin[fpn(r')dr'+.E::.]Yi, " 4 (3.5) C. Expressions for S( - 1) and S(1) The one- and two-electron density functions are substituted in Eq. (2.5) to simplify the expression for S ( - 1). Using r' r' = rr' cos 8 12, and carrying out the angular integrations, we get S(-1)=~f2:(21+ l)k/(r)rdr 1T , +~) _ (I + ~/2)2] 1/2, 4 -- r where En is the energy and tP is the effective potential and rj are the turning points. The quantization condition and the normalization condition lead t025 Using the wave functions in Eq. (3.5), and Eq. (3.6), one obtains25 for the one-electron density PI(r) = 2~r ++ (21 1)[ ~ - (I +~/2)2r/2. (3.7) One may also carry out the summation over I by integrating over I but this is not necessary for our purpose. For obtaining the two-electron density function, we start with F r' 1" dr, ) = 41T f.t -2R dtJ 0 ." tJ 2 - ¥1 2 2 (3.14) ), wheretJ = r - r' and we have changed the integration variables from rand r' to R and 1:1. Now since the main contribution comes from the large R region (as in the case of diamagnetic susceptibility). one can take limits of 1:1 integration to ± 00 and neglect the last term 1:1 2 , to get S( -1):::::~f2:(2/+ l)k,(r)rdr 1T , -! + f 2Ik,(R)R 2 dR. k/(R):::::+(2/+ 1)k/(R)/+(21+ 1). p,.(r")dr" + 1T/4] One then obtains xSin[i~ p,.(rH)dr" + 1T/4]. (3.15) Note that the first term is equal to 6x [see Eqs. (2.5)]. In the second term we approximate k/(R ) by its weighted average 2 ICn I (2/+ 1) P 8 rr' [prJ (r)pn (r')] 1/2 /(cos 12) xSin[L ioo dR f2R X)' I sin[k,(R)tJ ]sin[k/_dR)tJ] x(R (3.6) (3.13) (3.8) S( - 1):::::6X [ 1 +Y+ (I + 112)], J. Chern. Phys., Vol. 83, No. 11, 1 December 1985 Downloaded 01 Mar 2012 to 59.162.23.79. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions (3.16) (3.17) 5767 S. H. Patil: Susceptibility and polarizability where the summation is over the occupied I values. The second term inside the bracket is equal to Im/(lm + 1) where 1m is the highest value of I for the occupied states, so that S( - 1)::::;6X/(/m + 1). (3.18) The analysis for S (1) is similar. Starting from Eq. (2.10) and going through the same steps as in the derivation of Eq. (3.15), we get S(I)=2.4IEt l -2Z 1T i oo 0 dR 2Ik /(R). -L R / (3.19) If we substitute Eq. (3.7) in the first term of Eq. (2.6), one observes that it is similar to the second term in Eq. (3.19) except for the factor of 21 + 1 in place of 2/. Since the first term in Eq. (2.6) was shown to lead to 2.4 lEt I, we have S(I) = 2.4[IEt 1- )' _2/_ IE (I)I], ~ (3.20) 2/+ 1 where E (I) is the contribution to energy from the angular momentum I states, and the summation is over the occupied I levels. The contribution of the second term in Eq. (3.20) is relatively small, being about 10%-25% of the first term. with a and b given in Eq. (3.24), and (3.26) 'T/(O)=N. What one does is to start with a suitable value of A and the asymptotic behavior in Eq.(3.25), and trace the solution to smaller values of r by using Eq. (3.24). We then choose that value of A for which the condition in Eq. (3.26) is satisfied. Empirically, we find that A for neutral atoms is given quite accurately, within about 10%, by the expression A = 21.4N 1/2E :.35IE)12 for Z = N. (3.27) The susceptibility of an atom is given in terms of electron density PI(r) by the relation X=i J rpI(r)d 3 r . (3.28) It is evaluated by using the density PI(r) = 3~ (2'T//r )3/2 + :1T rOe- br (3.29) which is obtained by solving Eq. (3.24). D. Diamagnetic susceptibility Since S ( - 1) in Eq. (3.17) is given in terms ofX' we need to develop a reliable method for calculating X. Experimentally, we have accurate values of X for only the inert gases. The diamagnetic susceptibility depends quite sensitively on the large-r behavior of atomic wave functions, so that a reliable approach to determine X should include the correct asymptotic behavior. It is known that the asymptotic behavior of electron density in atoms has the form 14,26 p(r)_rOe- br for r> 1, a = (Z - N + 1)12/E; 11/2 - 2, (3.21) b = 212E; 11/2 , where E; is the ionization energy. We had recently proposed 14 a modification of the Thomas-Fermi model which includes this asymptotic behavior, and gives quite satisfactory values for susceptibility. The proposed equation for the potential was l4 !!:.- (rt,6) = (...!..) (2rt,6)3/2 + Aya + Ie - br, (3.22) dr 31T r l/2 where the right-hand side is 41T r PI(r) and A is a constant. In this equation, the last term dominates the equation for large r. However for ions, one has rt,6--+Z -N for r--+ 00, (3.23) so that the first term in Eq. (3.22) dominates at large r and the density would be incorrectly represented. We instead propose an equation (...!..) d 2'T/ = (2'T/)3/2 dr 31T r l/2 'T/(r) = rt,6 - (Z - N), + Ar ° + Ie - br, (3.24) which is acceptable for neutral atoms as well as ions. The boundary conditions are (3.25) IV. DIPOLE POLARIZABILITIES We are now in a position to estimateS (u) for u = - 1,0, 1, and 2, from Eqs. (3.18), (2.2), (3.20), and (2.12). This provides us with sufficient information to obtain a reliable parametric expression for stu). A. Parametric expression for S(u) For a normal atom, the summation over the intermediate states involves roughly three characteristic energies. These are the ionization energy of the atom, average energy of the electrons in the atom, and the energy of the innermost electrons. This would correspond to Ek - Eo being equal to the ionization energy E;, often used in the single term approximations, average energy Eo and the energy of the innermost electrons which may be represented by Z2. We therefore consider a representation where B, C, D are constants for a given atom. The last term represents the singular part given in Eq. (2.14). Note that it is consistent with the ratio of the singular parts for u = 2 and 1, given in Eq. (2.15). Our representation is similar in spirit to the one used by Langhoff 27 and Cummings.21 In their scheme, the energies also are taken as parameters and are determined from various physical requirements, e.g., 1S ( - 2) should be equal to the observed polarizability. It is to be noted that our expression for S (u) is dominated by the first term for negative u, and by the remaining two terms for positive u. Therefore, since E; < Eo one expects a rapid changeinS(u) between u = - 1 and O. Th.e change is expected to be particularly sharp for "loose" atoms which have a small ionization energy. This is consistent with the behavior described by Dehmer et al. 22 The constants B, C, and D are to be determined from the J. Chern. Phys., Vol. 83, No. 11, 1 December 1985 Downloaded 01 Mar 2012 to 59.162.23.79. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 5768 S. H. PatH: Susceptibility and polarizability conditions discussed earlier and summarized here, S( - 1) = 6X/(lm S(O)=~, + 1), (5.3) (4.2) (4.3) S(I)=2.4[IEt l - + IE(l)ll/(I+ 112)]. S(2) = 4Z [(Z - 0.4)3 + ilZ - 2.4)3], (4.4) (r); = (4.5) where for simplicity, we have written the approximate relations as equalities. Actually we need only three of these equations to determine the constants B, C, and D. Equation (4.4) will be used only as a consistency check for the validity of the parametric form used for S(u). B. Polarizabillty a p It is found that the last term in Eq. (4.1) contributes a negligible amount for u = 0, - 1, - 2. For example, in the case of Xe, S(0)::::81 whereas the contribution of the last term in Eq. (4.1) is less than 1. Therefore, we may determine Band C from the relations + C::::6X/(/m + 1) BE; + CEa ::::~N. where the summation is over all the electrons. The (r); are evaluated with suitably screened Coulomb wave functions to give X X (4.7) X Determining Band C allows us to obtain S ( - 2) and hence electric polarizability a p as X 4(Ea + EJX/(lm + 1) - N (4.8) E;Ea The parametrization we have described is not valid for atoms in which there are excited states whose energies are close to the ground state energies, e.g., alkali metals. ap = V. ISOELECTRONIC SEQUENCES Within an isoelectronic sequence, the electronic structure is not expected to change drastically when we go from one member to another. Therefore one expects simple relations to exist between the properties of different members of a given sequence. 4 (10) _ 1 - (Z _ 0.4)2 (18) - - (54) - 44 (Z _ 4.41)2 ' + (Z _24910.0)2 ' (18) + 210 816 X (Z _ 10)2 + (Z _ 23.6)2 - X (36) - + - X (10) (36) 840 23.6f + (Z _ + l (5.1) Since the number of contributing electrons is the same for the different members of the isoelectronic sequence, the ratio of diamagnetic susceptibilities of two different members is given approximately by Xq (b q -X q' - (b q• + 4)(bq + 3) + 4)(bq• + 3) ' for N = 10, 18,36, and 54, respectively. C. Polarizabillties Polarizabilities can be calculated by using the above susceptibilities and Eq. (4.8). The ionization energy required for the use of Eq. (4.8) can be obtained either from experiments or from a parametrization E; = 2: 2 (Z - O'f (5.6) with n being the total quantum number for the outermost shell and (5.7) Let the wave function of an electron in the outermost shell be given by the square root of the density in Eq. (3.21). Then the contribution to susceptibility Xe by this electron is given by + 3~~b + 4) (5.5) 2025 (Z _ 37.2)2 ' A. Ratio of susceptibilities Xe::::i [(b (5.4) The screening charge for the outermost shell is critical in determining X and may be determined by requiring that the value given by Eq. (5.3) should agree with the observed value of X for the neutral member of the sequence. For the inert gases we take 0' = 0.4 for the n = 1 shell, and find 0' = 4.41 for n = 2, 0' = 10.0 for n = 3, 0' = 23.6 for n = 4, and 37.2 for n = 5. The susceptibility for the different sequences is then given by (4.6) B 1) - 113]}. n {~_1. [/(1 + 2 (Z -of 2 2 n (a- .)2 q aq (5.2) , where q and q' characterize the different values of Z - N. where ao and a 1 are determined by requiring that Eq. (5.6) gives the observed values of E; for, say, Z - N = 0 and 1. Similarly the average value of Ea required for the use ofEq. (4.8) can be estimated by 1 L (Z - O';f (5.8) Ea - N 2' ; 2n; where the summation is over all the electrons, with a suitable choice of values for the screening charge 0';. The precise values of 0'; in this case are not very important since a p does not depend sensitively on Ea. For Z _ 00, and N = constant, we can neglect 0' and 0'/ in Eqs.(5.6) and (5.8), use Eq. (5.3) to determine X and hence determine polarizability a p from Eq. (4.8) in terms Z, N, n, and I values. For example we get for Z - 00 (5.9) B. Susceptibilities One can also obtain a simple expression for susceptibility by using the standard relation withB = 770, 12 528,819712/13,320 778.4 for N = 10,18, 36, 54, respectively. J. Chern. Phys., Vol. 83, No. 11. 1 December 1985 Downloaded 01 Mar 2012 to 59.162.23.79. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 5. H. Patti: Susceptibility and polarizability TABLE I. Calculated values of diamagnetic susceptibilities and electric dipole polarizabilities for some neutral atoms, along with experimental values given in brackets. The letters a, b, c represent a = Ref. 28, b = Ref. 1, c=Ref.29. Z Ei E. A X ap 10 0.793 12.7 44.0 14 0.300 20.1 4.1 1.61 (1.42)" 5.06 15 0.404 22.3 11.9 4.2 16 0.381 24.6 10.1 4.5 17 0.478 26.6 21.0 3.9 18 0.579 28.8 35.5 31 0.221 61.9 1.52 32 0.290 64.6 5.4 8.9 33 0.361 67.3 12.5 7.6 34 0.359 70.0 12.5 7.9 35 0.435 72.8 24.2 7.1 36 0.515 75.5 40.0 42 0.261 92.8 52 53 0.331 0.384 126.8 130.0 11.4 19.9 10.9 10.1 54 0.446 133.3 32.4 9.14 (9.24)" 3.32 (2.67)b 31.9 (36.3)b 19.5 (24.5)b 22.3 (19.6)b 15.3 (14.7)b 11.3 (11.1)b 58.9 (54.9)b 39.4 (41.0)b 26.9 (29.1)b 28.1 (25.5)b 20.8 (20.6)b 15.6 (16.8)b 55.6 (61)" 42.8 34.1 (36)C 26.5 (27.3)b 3.88 3.51 (4.12)" 10.1 6.34 (6.06)" 11.2 VI. NUMERICAL RESULTS In this section, we present the numerical results which follow from our analysis. The polarizability calculations are based on Eq. (4.8). The susceptibilities required in using these equations and which are of independent interest are obtained from Eq. (3.28) by solving Eq. (3.24). A. X and a p for neutral atoms The values of diamagnetic susceptibility X calculated from Eq. (3.28) and polarizability a p calculated from Eq. (4.8) are given in Table I, for some neutral atoms, along with the values of the relevant parameters. The values of E; are taken from experiments,30 while those of Ea are taken either from experiments or are estimated from Eq. (5.8). The agreement between calculated values and experimental values is quite good over the entire range of Z values. Even in the case ofZ = 10, if we take the experimental valueofx = 1.42, one gets a p = 2.80 which is close to the observed polarizability. B. Isoelectronic sequences The diamagnetic susceptibilities and polarizabilities of isoelectronic sequences are of particular interest and our cal- 5769 culated values are listed in Table II, along with the experimental or empirically derived values. The agreement between our calculated values for the dipole polarizabilities and the empirically derived values of Refs. 23 and 32, for the N = 10 isoelectronic sequence, is excellent. There do not appear to be accurate empirical values of the polarizabilities for the other sequences. We urge that accurate experiments be done to determine the polarizabilities of the ions of the N = 18, 36, 54 isoelectronic sequences. With the susceptibilities tabulated in Table II, one can verify the accuracy of the approximate relation for the ratio of the susceptibilities given in Eq. (5.2). This relation gives, for example, xlo1xl1 = 1.56, (6.1) X17IXI8 = 1.17, xlo1x42 = 68.1, for the N = 10 series, where the SUbscript is the value of Z. The corresponding ratios from the calculated values in Table II are 1.41,1.16, and 46.9, respectively. For the higher N, Eq. (5.2) is not that accurate. For example, we have from Eq. (5.2), = 1.15 X3~X37 = 1.26 X2~X25 for N for N = 18, = 36, (6.2) X551X56 = 1.30 for N = 54, whereas the corresponding values calculated from Table II are 1.12, 1.15, and 1.14, respectively. The decrease in the accuracy of Eq. (5.2) for larger N values may be expected since the electronic structure becomes more complicated for higher values of N. The value of X is not only of independent interest, but is of critical importance in our evaluation of a p [see Eq. (4.8)]. Therefore, we would like to verify the accuracy of our calculated values by using the expressions in Eq. (5.5). Some of the extrapolated values from Eq. (5.5) are X12 = 0.77 (0.84), = 0.117 X42 = 0.032 X24 (0.128), (6.3) (0.034), for the N = 10 sequence, with the calculated values from Table II given in brackets. The agreement between the two sets of values is quite satisfactory. Even the small difference between the two is ascribed to the fact that the first expression in Eq.(5.5) is normalized to givexlO = 1.42, whereas our calculated value is X 10 = 1.61. Similar agreement is observed for the other isoelectronic sequences as well. For example, Eq. (5.5) gives X26 = 1.07 (1.09) for N = 18, X38 = 4.58 (4.83) for N = 36, (6.4) X57 = 6.87 (6.14) for N = 54, where the calculated values from Table II are given in brackets. The generally satisfactory agreement between the calculated values and the values extrapolated from Eq. (5.5) gives us confidence in the essential correctness of our approach. J. Chem. Phys., Vol. 83, No. 11, 1 December 1985 Downloaded 01 Mar 2012 to 59.162.23.79. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 5770 S. H. Patil: Susceptibility and polarizability TABLE II. Calculated values of diamagnetic susceptibilities and electric dipole polarizabilities for some members of isoelectronic series, along with experimental values given in brackets. The letters b, c, d, f, h represent b = Ref. 1, c = Ref. 29, d = Ref. 31, f= Ref. 32, h = Ref. 23. TABLE II. (continued). ap N Z Ei E. A 10 10 0.793 12.7 44.0 11 1.74 15.9 179 1.61 (1.42) 1.14 12 2.95 19.5 530 0.84 13 4.41 23.4 1.31 X 1Q3 0.66 14 6.13 27.7 2.85X 103 0.53 15 8.10 32.3 5.63X 103 0.44 16 10.33 37.3 1.02 X 10" 0.36 17 12.81 42.6 1.76 X 10" 0.31 18 15.48 48.3 2.89 X 10" 0.268 19 18.52 54.3 4.53 X 10" 0.230 X 20 21.76 60.9 6.9X 10" 0.202 21 25.24 68.0 1.01 X lOS 0.179 22 28.99 75.5 1.44 X 105 0.159 23 32.98 8304 2.04X WS 0.142 24 37.25 91.7 2.81 X lOS 0.128 25 41.76 10004 3.79X lOS 0.1l6 26 46.54 109.5 5.05XWS 0.105 27 51.58 119.0 6.58X 105 0.096 3.32 (2.67)b 1.09 (1.002)f 0.48 (0.47)h 0.259 (0.253)h 0.152 (0.15)h 0.0977 (0.095)h 0.0630 (0.063)h 0.0446 (0.0438)h 0.0323 (0.0313)h 0.0234 (0.0230)h 0.0177 (0.0 I 74)h 0.0136 (0.01 34)h 0.0106 (O.OI04)h 0.0084 (0.0083)h 0.0067 (0.0067)b 0.0055 (0.0054)h 0.0045 (O.OO44)h 0.0037 (0.0037t 18 36 54 36 108.2 222.5 4.61xl(f 0.0484 42 157.1 309.5 1.23 X 107 0.0343 35.5 3.51 18 0.579 28.8 19 1.17 32.6 120 2.90 20 1.88 36.6 342 2.42 21 22 23 24 25 2.72 3.67 4.75 5.93 7.21 41.0 45.6 50.4 55.5 60.8 840 1.88X loJ 3.90X 1Q3 7.55X loJ 1.38 X 10" 2.06 1.78 1.56 1.36 1.21 26 8.59 66.4 2.40X 10" 1.09 27 10.10 72.2 4.13X 10" 0.99 28 36 11.70 0.515 78.2 75.5 37 1.011 80.5 124 5.50 38 1.61 85.7 340 4.83 39 2.30 91.2 850 4.30 40 3.05 97.0 1830 3.83 54 0.446 133.3 55 0.923 138.8 114 7.83 56 1.50 144.7 343 6.85 57 2.14 150.7 853 6.14 6.7X 10" 40.0 32.4 0.90 6.34 9.14 0.00091 (0.00093)h 0.00045 (O.OOO46)b 11.3 (ll.l)b 4.66 (5.47)d 2.45 (3.l2)d 1.45 0.94 0.64 0.45 0.334 (0.29 ± 0.15)d 0.255 (0. 37)d 0.20 (0.26 ± O.04)d 0.157 15.6 (16.8)h 6.9 (9.5-12.2)< 3.81 (6.6)C 2.38 (3.71)C 1.61 (2.50)· 26.5 (27.3)b 11.0 (16.3-21.2)C 5.9 (1IA)C VII. SUMMARY AND DISCUSSION Here we summarize the main results of our analysis and discuss some of their implications. 3.71 accurate for the ions in the N = 36 and 54 isoelectronic sequences and we strongly urge that accurate experiments with modem techniques be carried out for these ions. B. Susceptibility and polarizablllty as functions of EI A.Summary The analysis is based on the expressions for S ( 1), S (0), S(l), and S(2) given in Eqs. (4.2)-(4.5). We have used these expressions to obtain polarizability a p as a = 4(E; p + Ea)l'/(lm + l)-N E.E I a (7.1) ' whereE; is the ionization energy, Ea is the average energy, N is the number of electrons, and 1m is the highest I value of occupied states. The susceptibilities are calculated by using an extension of the Thomas-Fermi equation given in Eq. (3.24) which incorporates the correct asymptotic behavior. Of particular interest are the susceptibilities and polarizabilities of isoelectronic sequences for which one has the simple relations given in Eqs. (5.2), (5.5), and (5.9). We find that the experimental polarizabilities available are not very There is an approximate relation 1 between dipole polarizability and diamagnetic susceptibility, a p = 16l'2 1N. (7.2) This relation is known to be fairly good for inert gases. We find this relation to be reasonably accurate for the atoms given in Table I as well. For example, in the case of As (Z = 33), using our value of X = 7.6, Eq. (7.2) gives a p = 28.0, whereas the experimental valuel is 29.1. However, Eq. (7.2) is generally unsatisfactory for atoms and ions with resonant states as also for highly ionized atoms. As an example note that for the ionized Ca (Z = 20, N = 10) with ten electrons, Eq. (7.2) on using our value ofl' = 0.202 gives a p = 0.065, whereas the experimental value 23 is 0.0174. We can combine Eq. (7.2) with our relation in Eq. (7.1) to obtain a p and X independently for atoms with no resonant J. Chem. Phys., Vol. 83, No. 11, 1 December 1985 Downloaded 01 Mar 2012 to 59.162.23.79. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions S. H. Patil: Susceptibility and polarizability states. On equating Eqs. (7.1) and (7.2) one gets X= N(Ea +Ej ) { 4EaEj(/m + [ 1+ 12 SEaEj(/m + 1) (Ea +Ej ) W]1I2} 5771 higher multipole polarizabilities, van der Waals constants, etc. Some of these will be considered separately. I thank Srinivas Krishnagopal and Shobhana Narasimhan for their help in numerical computations. (7.3) and a = p {I [1- + E j)2 + 4E~E~(/m + W N(Ea W]1I2}2, 4EaEj(/m + (Ea + E j)2 (7.4) where E j is the ionization energy, Ea is the average energy, and 1m is the highest I value of occupied states. They give quite satisfactory values for susceptibility and polarizability of neutral atoms with no resonant transitions. For example, they predict X = 3.63, 5.49, 9.S1 for A, Kr, and Xe respectively, whereas the experimental values are 4.12,6.06,9.24, respectively. Similarly, they predict a p = 35.3, 11.7, 39.2, 13.4, 65.4, 2S.5 for Z = 14, IS, 32, 36,42, 54, respectively, whereas the corresponding experimental values given in Table I are 36.3, 11.1,41.0, 16.S, 61, 27.3, respectively. In most situations, especially the heavier atoms, one finds that E j is much smaller than Ea' In these cases Eqs. (7.3) and (7.4) simplify to N 4Ej(/m X~, + 1) N a -----p- E~(/m + W' (7.5) (7.6) whose predictions are in reasonable agreement with the experimental values. For example, in the case of Xe, they predict X = 10.1 and a p = 30.2, whereas the experimental values are 9.24 and 27.3, respectively. It may be noted that a linear, parametric relation between In X and In E j has been found to be quite useful 33 in relating the energies of isoelectronic sequences, consistent with Eq. (7.5). C. Other applications The techniques of extrapolation and semiclassical ap· proach, and the resulting simplifications, can be used in the analysis of other atomic properties as well. For example, they can be used for analyzing dynamic dipole polarizability, IT. M. Miller and B. Bedcrson, Adv. Atom. Mol. Phys. 13, 1 (1977). 2A. Dalgarno, Proc. R. Soc. London Ser. A 251, 282 (1959). 3E. Markiewicz, R. P. McEachran, and A. D. Stauffer, J. Phys. B 14, 949 (1981). +r. Yoegel, J. Hinze, and F. Tobin, J. Chern. Phys. 70, 1107 (1979). sG. D. Mahan, Phys. Rev. A 22, 1780 (1980); J. Chern. Phys. 76, 493 (1982). 6p. S. Christiansen and K. S. Pitzer, Chern. Phys. Lett. 85, 434 (1982). 7p. Fuentealba, J. Phys. B 15, L555 (1982). A. Adelman and A. Szabo, J. Chern. Phys. 58, 687 (1973). 9K. T. Tang, J. M. Norbeck, and P. R. Certain, J. Chern. Phys. 64, 3063 (1976). IOL. W. Bruch and A. P. Lehnen, J. Chern. Phys. 64, 2065 (1976). "Y.P. Shevelko and A. Y. Yinogradov, Phys. Scr. 19,275 (1979). 12S. Fraga, J. Karwowski, and K. Saxena, Handbook ofAtomic Data (Elsevier, Amsterdam, 1976); J. P. Desclaux, Atom. Data Nucl. Data Tables 12,311 (1973). 13D-G. Englert and J. Schwinger, Phys. Rev. A 26,2322 (1982). 14S. H. Patil, J. Chern. Phys. SO, 5073 (1984). ISH. A. Bethe and E. E. Salpeter, Quantum Mechanics of One and Two Electron Atoms (Springer, Berlin, 1957), Sees. 61a and 62. 16A. Dalgarno, Rev. Mod. Phys. 35, 522 (1963). 17J. O. Hirschfelder, W. Byers Brown, and S. T. Epstein, Adv. Quantum Chern. 1,256 (1964). 18R. Jackiw, Phys. Rev. 157, 1220 (1967). I~. Fanoand J. W. Cooper, Rev. Mod. Phys. 40, 441 (1968). lOp. W. Langhoff, J. Chern. Phys. 57, 2604 (1972). 21F. E. Cummings, J. Chern. Phys. 63, 4960 (1975). 22J. L. Dehmer, M. Inokuti, and R. P. Saxon, Phys. Rev. A 12, 102 (1975). 23B. Edlen, Phys. Scr. 17, 565 (1978). 24J. P. Yinti, Phys. Rev. 58, 882 (1940). 25D. A. Kirzhnitz, Yu. E. Lozovik, and G. Y. Shpatakovskaya, Usp. Fiz. Nauk. 117, 3 (1975) [Sov. Phys. Usp. 18, 649 (1976)]. 26E. N. Lassettre, J. Chern. Phys. 43, 4475 (1965); J. Katriel and E.R. Davidson, Proc. Nat!. Acad. Sci. U.S.A. 77, 4403 (1980); R. Ahlrichs, M. Hoffmann-Ostenhoff, T. Holfmann-Ostenhoff, and J. D. Morgan, Phys. Rev. A 23,2106 (1981). 27p. W. Langhoff, Chern. Phys. Lett. 12,217 (1971). 28Handbook of Chemistry and Physics (Chernical Rubber, Cleveland, Ohio, 1979/80). ~. Fajans and G. Jaos, Z. Phys. 23, 1 (1924); J. R. Tessman, A. H. Kahn, and W. Shockley, Phys. Rev. 92, 890 (1953); Landolt-Bornstein, Zahlenwarte und Funktionen, B. I., Teil I (Springer, Berlin, 1950). 30C. E. Moore, Atomic Energy Levels, Nat!. Bur. Stand. (U.S. GPO, Washington, D.C., 1948). 31p. Yogel, Nuc1. Instrum. Methods 110, 241 (1973). 32R. R. Freeman and D. Kleppner, Phys. Rev. A 14, 1614 (1976). 33R. Pucci and N. H. March, J. Chern. Phys. 76, 6091 (1982). as. J. Chern. Phys., Vol. 83, No. 11,1 December 1985 Downloaded 01 Mar 2012 to 59.162.23.79. 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