The Thomas-Fermi model: momentum expectation values

Transcription

1 The ThomasFermi model: momentum expectation values I.K. Dmitrieva, G.I. Plindov To cite this version: I.K. Dmitrieva, G.I. Plindov. The ThomasFermi model: momentum expectation values. Journal de Physique, 1983, 44 (3), pp < /jphys: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1983 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Within Recently In J. Physique 44 (1983) MARS 1983, 333 Classification Physics Abstracts L The ThomasFermi model: momentum expectation values I. K. Dmitrieva and G. I. Plindov Heat and Mass Transfer Institute, BSSR Academy of Sciences, Minsk, USSR (Reçu le 12 juillet 1982, accepté le 30 novembre 1982) Résumé Les expressions analytiques de toutes les valeurs moyennes des impulsions pb> et de quelques puissances de la densité électronique 03C1m> pour les atomes dans un degré d ionisation arbitraire sont obtenues dans le cadre du modèle de ThomasFermi compte tenu des corrections d des à l échange et à la contribution des électrons fortement liés. On montre que le traitement correct de celleci permet d obtenir une estimation quantitative de pb > et 03C1m> lorsque 3 ~ b 5 et 1 ~ m 5/3. La dépendance des coefficients du développement de pb > et 03C1m> en Z1 est donnée explicitement en fonction du nombre d électrons. Abstract the ThomasFermi model including the exchange interaction and contributions of strongly bound electrons, analytical expressions are obtained for all momentum expectation values pb> and for some of the expectation values of powers of the electron density pm> for an atom with an arbitrary degree of ionization. It is shown that a correct treatment of strongly bound electrons gives a quantitative estimate of pb> and 03C1m> within 3 ~ b 5 and 1 ~ m 5/3. The Z1 expansion coefficients for pb> and 03C1m> are given as an explicit function of the electron number. 1. Introduction. [ 1 ], asymptotic estimates of the expectation values of electron positions ( r a > and of momentum pb) have been obtained for a neutral atom and for an atom without electronelectron interaction within the ThomasFermi model. In the previous work [2], study was made of ro ) for atoms with an arbitrary degree of ionization on the basis of the improved TF model. Here we shall study ( p > and related expectation values of powers of the electron density ( p " ). The quantum determination of ( pm ) equation in the momentum space. In (2) Io is the electron momentum density. The range of the validity of (2) is restricted by the behaviour of lo(p) at p + 0, p oo namely, lo(p + 0) const. [3] and lo(p + oo) = 8 Z. p(o) p 6 [4], and is given as 3 b 5. Alternative determination of pb ), relating pb > with the isotropic Compton profile, Jo(q), [4] : i is a rather tedious problem requiring the solution of the Nparticle Schrodinger equation in the coordinate space. Still more difficult is the search for ( pb > (p = I p I) which requires either the Fourier transformation of a spatial wave function or solution of the Schrodinger allows ( pb ) to be found from the experimental Compton profiles. Both these methods cannot give an analytical dependence of p > on the electron number N and nucleus charge Z. The present work is aimed at obtaining analytical estimates of expectation values ( pb ) and ( p " ) by using the ThomasFermi model with account for the exchange interaction and contributions of strongly bound electrons. Systematic trends in ( pb ) and ( p" ) will also be analysed. 2. Statistical model. the frame work of the ThomasFermi (TF) and ThomasFermiDirac (TFD) Article published online by EDP Sciences and available at

3 334 models, the state density in the phase space [5] is equal to : finite radius, ro, beyond which the electron density is equal to zero. Therefore, in the TF model for a neutral atom we have : where pf(r) is the Fermi momentum at a distance r from the nucleus and 0 the Heaviside thetafunction. Integrating (4) over momentum, it is easy to establish the relationship between the particle number density and the Fermi momentum : The value of ( pb > (b > 3), based on (4), is : This range includes 4 moments of the momentum distribution, namely, b 1, 0, 1, 2. The = zero moment is reduced to the normalization integral. The values ( p2 > and ( p > determine kinetic and exchange energy (in a local approximation) and ( p1 ) is proportional to Jo(0), (2). The virial theorem links ( p2 with the binding energy which has been earlier investigated in detail within the TF model [6]. Therefore p2 > values will be considered here only for the completeness of the analysis. For an ion, the range of the validity of (6) is extended to : Here and below, atomic units are used. With regard to (5), the value of pb > expressed in terms of pm) may be which allows an estimate of p 2 ). When the exchange is taken into account, the neutral atom is bounded and the range of validity of (6) coincides with (9) within this model. Let us use the TF equation Equation (7) gives a simple relationship between momentum and electron density expectation values; it is exact if the electrostatic and exchange interaction is taken into account. This relationship is broken by allowing for contributions of strongly bound electrons, or for the inhomogeneity and the oscillation of the electron density. Expression (6) is the basis to study pb > as a function of N and Z within the statistical model. The applicability of (6) is specified by the behaviour of the statistical density at r 0 and r oo. In the simple TF model for a neutral atom p(r + 0) r 3/2 and p(r oo) r 6. The ion in the TF model has a or the TF equation with regard for the exchange in the first order with respect to fl [2, 8] : where f3 = 2 (6 nz)2/3, the dimensionless radius x and the screening function 03C8(x) are related to r and p(r) by : Let us present (6) in the following form : or without and with account for the exchange interaction, respectively. In (12)(13), xo is the boundary ion radius in the TF model (for a neutral atom xo oo) and xex is the boundary radius when including the electron exchange interaction.

4 335 Within 3/2 b 3, ( pb > for a neutral atom may be given as a sum : where Here ( p6 >TF has a universal form [1, 7] : B( 1) = ; B(1) = ; B(2) = Special consideration must be made of the exchange contribution. Using the expansion of qlo(x) and 1 (x) at x > 1 [8] : 6 = ( 73 7)/2, it is easy to see that for 1/2 b 2, the main part of p6 )ex may be given as : Numerical integration in (15a) yields : The situation is more complex for the negative moments of the momentum distribution. Expression (14) may be used for estimating ( p1 ). However, since the integrals in the P >ex estimate substantially depend on xex, their analytical estimation is impossible. Because of xex Z 1/3 [8], expression (15) gives only a qualitative asymptotic dependence (Z > 1) : When estimating p I > for a neutral atom, (14) is not valid since the integrals in (12) and (15) diverge; when passing to simultaneous consideration of the integrals in (13) and taking into account that the integral on the RHS is mainly determined by x xex, we find a qualitative asymptotic dependence at Z> 1 : being filled. The SCFdata obtained from HF Compton profiles [ 15] exhibit an explicit periodic dependence of A p > = P /HF B p 1 /TF and C p 2 )HF on Z 1/3 (Fig. 1). The major maxima of the curves Ap1 > (Z 113 ) and p2 )HF(ZI/3) correspond to alkaliearth atoms. The positions of major minima correspond to noble gases. As is seen from figure 1, the oscillation amplitude for heavy atoms is independent on Z. Thus, the oscillation contribution of (pl > has a relative order Z 1/3, being a leading term. for (p2 ) Now we pass to the estimate of (ph) for ions. Expression (14) is valid for all b within 3 b 3. In order to obtain ( p > as an explicit function of N and Z, we use the expansions of the screening function, qlo(x), in the TF model and exchange correction, t/j 1 (x), into series in the parameter, which confirms the result given in [ 1]. The expectation values p 1 > ex and p > are determined by the external regions of atoms, and for real atoms must oscillate as the last electron shell is

5 The Values 336 The closed form of the functions cpk(x/xo) and xk(x/xo) enables one to obtain the exact values of Bk(b) and Bk X (b). Here Box(b) is given for arbitrary b(3b4) : Bl (b) is found in a closed form for integer b. This value being very bulky, we present only numerical values together with B2(b) and BlX(b) (Table I). Comparison of Bk(b) and Bke (b) leads to the conclusion that the exchange interaction increases ( pb > for b > 0 and decreases it for b 0. From table I, expressions (19) and (16) it is easy to see that (19) with three expansion terms well reproduces ( p2 )TF and p >TF, including the neutral atom (error does not exceed 0.6 % and 0.9 %, respectively). Based on (19), the values of p2 >TF and P >TF may be calculated with good accuracy only for small N/Z. The value of ( p6 >TF for a slightly ionized atom must be studied to improve (19) for ions with NIZ 1. At NIZ 1, #o(x) may be given as : Fig. 1. values of p >,, and Ap > = P >HF P 1 /TF obtained from HF Compton profiles [15] as a function of Z 1/3 : 1, p 2>HF; 2, A p >. The functions qji(y) and Xi(y) are presented in [2, 9]. Substitution of (17) and (18) into (12) and (13) and regard to J, and xo as a function of N and Z (cf. 2) give ( pb )TF and p6 > ex as an N/Z series expansion : where t/loo(x) is the function for a neutral atom and t/lol (x) is the correction function. Using the asymptotic expressions for t/loo(x) and t/lol(x) [2], #oo(x) = 144 x3(1 + O(xU)); t/lo 1 (x) Ax4+u(1 + O(xU)) and taking into account from (12), we obtain the asymptotic expressions : The value of Bo(b) equal to was obtained in [1]. Table I. of Bk(b) and B;X(b).

6 Values 337 Expressions (23a) and (23b) determine an approximate type of singularity at N/Z 1. We think it expedient to present p2 )TF at NIZ I as : Table II. of p /3 > for the isoelectronic series of Ne and Ar ; TF model with exchange (26a). where f 1 (N/Z) is a function having no singularities at N/Z 0 and N/Z 1. = = Equations (19), (20) and (23) give systematic trends of ( p > to be studied at a large electron number. It is easy to see that ( pb > obtained from (19) and (20) may be presented as the Z > expansion : the asymptotic expressions for the coefficients Dk(N, b) within 3 b 3 being of the form : For ( p2 ), Dk(N, b) as a function of N and k is well studied in [6]. It is shown that the TF model gives a reliable estimate of the Z expansion coefficients. In the present work, the values of the three first Z 1 expansion coefficients for p > for b =1= 2 are obtained for the first time. The coefficient of the higher power of N in (25) is exact (see, next section), thus the quality of Dk(N, b) increases with growing N. The deviation at moderate N is related to the fact that (25) does not HF data [11] are given in brackets. involve the contributions of strongly bound electrons, the inhomogeneity of the electron density and oscillations. The first of them may be very substantial and will be considered in section 4. The correction for the electron density inhomogeneity has the same relative order as the exchange contribution but with a smaller factor; this correction being neglected, a small error will be made in the estimate of ( pb ) at any b and N/Z. The oscillation contribution will be briefly discussed in the next section. Equation (25) gives an important property of the Z 1 expansion coefficients for ( p > : the ratios Dk + 1 (N, b)/ndk(n, b) quickly tend to a constant determined by the TF model and equal to Bk, l(b)l Bk(b). We studied pb) in detail. Expression (7) shows that all results obtained in this section are, to the same extent, related to expectation values p" ). For are used to estimate Z 1 example, (25) and (24) expansion coefficients for ( p > : The value of Go(N, m) coincides with the one obtained in [10]. Expressions (25) and (26) are the only estimates of the Z 1 coefficients for ( pb ) and pm ) expansions for manyelectron atoms at k > 0 (except ( p2 >). Therefore, we could not perform a direct comparison with other data. To illustrate the quality of (25) and (26), we made a systematic comparison of pl /3 > calculated by : with HartreeFock data [11] for isoelectronic series 10 N 54 and N Z 20 + N. The maximum error of (26a) does not exceed 8 % (isoelectronic series of Ar). The main error of (26a) is due to the absence of oscillation effects being essential for open shell isoelectronic series as in studying binding energy (or p2» [6]. A typical behaviour of ( pl/3 > for closed shell ions and open shell ones is demonstrated in table II. The data of table II show that (25) and (26) may be used to reliably estimate ( pb > and pm > for an atom with an arbitrary degree of ionization for 0 b 3 (0 m 1). 3. Noninteracting electron model. If an atom is considered to be with no electronelectron interaction, then ( p6 ) and pm > are found by summing over all occupied hydrogenlike orbitals :

7 338 Here 0,,,(r) are the orthonormalized radial wave functions and q,,, are the occupancy number for the orbitals with quantum numbers n and l. The values (.pb )H and ( pm >H determine exact quantum values of Do(N, b) and Go(N, b). The last quantity may be calculated only numerically. The analysis of Do(N, b) allows exact analytical expressions to be obtained for closed shells. The estimate Of pb )nl is given by the expression [3] : where Fnl( p) is the normalized radial function of the momentum distribution : Cm(x) are Gegenbauer s polynomials [12]. Replacement of (n2 p2 1 ) (n2 p2 + 1 ) 1 = u in (29) gives : and use of the symmetry property of Gegenbauer s polynomials C"(x) _ ( 1 )" Cm( x) results in a relationship between expectation values pb )nl for different b : Thus, for integer b the problem is reduced to calculation of only four moments, one of which, pl ), is trivial. To obtain pb )n, a calculation must be made of the integrals J(n, I, b) equal to : The use of the explicit expression for Gegenbauer s polynomials [12] yields : The exact expressions for ( p2 >n and p4 >n were derived in [3] : Using (30), from (32a) we have : Summing (32a) and (32b) over I for closed shells, ( pb >H may be presented for even b as : Numerical summation of (27) using (29), (31 a) and (31 b) for the four first electron shells shows that, for odd b = 1, 1, 3, DO(N, b) is also described by (33).

8 A 339 To find Do(N, b) as a function of N, summation is performed over n in (33) with regard to the relationship between the maximum main quantum number nm and N for closed electron shell atoms : When limiting to the terms of the relative order of N 2/3, we have Here C = is Euler s constant. It is easy to see that for 1 b 2 the coefficients with the leading power of N in (34c)434e) coincide with the values of Bo(b) found by the TF theory, equation (21). Comparing (34a, b) with the estimates of ( p4 > and ( p3 ) obtained by using the KompaneetsPavlovskii (KP) model [1], one may be convinced that the KP model gives a qualitatively correct description of ( pb ) within 3 b 5. This is due to cutting off the. electron density within KP model at small distances from nucleus and due to the dependence of the internal boundary radius on Z, XJ(Z) _ z 2/3. The asymptotic expressions (34af) perfectly describe Do(N, Z) for closed electron shells (33). For open shells account must be taken of the effect of oscillations, whose amplitude is of relative order of N 2/3 for positive and of N 1/3 for negative b. The oscillation effect is most substantial for expectation value p2 > (Fig. 2). due to the fact that the discrete quantum state electron distribution differs from the continuous one defined by The oscillation effects appear in ( p > (34af). The analytical estimate of these effects may be made using simple algebra as it was done for the energy [6]. 4. Strongly bound electrons. correct estimate of pb > for 3 b 5 and of ( p" ) for I K m K 5/3 may be made only if the quantum effects near the nucleus are allowed for. With these effects taken into account, the estimates of ( pb > for 1 b 3 and of p" ) for 1/3 m 1 may be essentially improved. A clear physical picture of strongly bound electrons has been recently elucidated by Schwinger [13]. Based on this method, the expectation value ( pb ) is given as a sum of two contributions : The first contribution is caused by strongly bound electrons (with binding energy B, 8 1 Z ) and is calculated by the summation over the states of pb >nl for noninteracting electrons (27) : Fig pb >H/DS(N, b) as a function of N 1/3 : 1, b = 1 ; 2,b=2.

9 x/xoo 340 Here n is the main quantum number of strongly bound electrons ; n (Z 2/2 E) is not obligatorily integer while [n ] is the integer part of n. The second contribution is calculated by the TF model; the strongly bound electron contribution incorrectly described by the TF model must be eliminated from (12) : Here xm is the region of localization of strongly bound = electrons, xm Z( E,u)1 2 n 2(Z p) 1. Let us show that a similar result may be obtained if the TF contribution of Nnoninteracting electrons is eliminated from (12) and replaced by a quantummechanical quantity. This approach as applied to pb > yields : is the TF screen Here in the first integral, 1 ing function of noninteracting electrons and is the dimensionless radius of the TF ion with rio electronelectron interaction. Integration in (35) and (36) and allowance for the relationship between xm and n demonstrate that both approaches coincide correctly to the terms of the relative order of N 2/3. Approximately the same method was used by Scott [14] to estimate the binding energy of a neutral atom. The approach based on (35) allows combination of the advantages of the quantummechanical model for noninteracting electrons and the TF model. The first model correctly takes into account a contribution of strongly bound electrons and partially another quantum contributions (oscillations, inhomogeneity of the electron density, etc.). The second model gives an exact asymptotic value of the contribution due to the electronelectron interaction. Equation (36) may be supplemented with the corrections for electron exchange interaction based on (13). Bearing this in mind, pb > (I K b 3) may be given as : From (15) it follows that for b > 4 the first integral on the RHS diverges, which shows that a strongly bound electron contribution to ( pb >ex (b > 4) must be taken into account. This approach is especially convenient when combined with the Z 1 perturbation theory. To determine the expectation value of the local operator, for example, ( pb ) it is sufficient to replace Do(N, b) in (25) by the exact quantum quantity conserving Dk(N, b) at k > 0 from (25), i.e. p4 >TF and p3 > TF are calculated to determine Dk(N, b) as a function of N for b = 3, 4 at k a 1. Partial integration of (36) gives : Substitution of (17) into (36a)(36b), with regard to the dependence of xo and )B on Z and N [2], yields Dk(N, 3) and Dk(N, 4) at k > 1 in the form of (25). The values of Bk(b) at b = 3, 4 and of Bkex 1 (3) at k 1, 2 = are listed in table I. Expressions (38), (24) and table I give asymptotically exact (at N > 1) values of the Z 1 expansion coefficients for all moments of the momentum distribution. For a neutral atom, the calculation of the integrals in ( 15a) and (36) and using the values Of pb >H from (34) result in : In (39d), the term Z incorporates the contributions both of the exchange interaction and of strongly bound electrons. Note that the amplitude of

10 Values Values The 341 Table III. of pb > for a neutral atom (39). Table IV. of p >.10 2 for the electronic series ofne and Ar ; TF model with account for exchange and strongly bound electron contributions (42). the oscillation, not taken into account in (39d), is also proportional to Z. Expression (39a) was for the first time obtained in [ 1], and the second term of this expression was found in [13] when the leading relativistic correction to the binding energy of a neutral atom was calculated. p3 > and ( p > as functions of Z are first obtained here. Comparison with HF data (Table III) shows that the error of (39) does not exceed 10 % at Z > 10 and falls with increasing Z. We have calculated pb )HF (Table III) on the basis of (3) using the isotropic HF Compton profiles [15]. We think that the accuracy of the HF data is about 0.5 % since for their calculation the interpolation procedures involving the exact asymptotic expression Substitution of (17) into (40) gives ( pm > for an isoelectronic series : Expression (41) may be used to obtain pm > for an atom with an arbitrary degree of ionization. To check the validity of (41), we have calculated p > for the isoelectronic series of Ne and Ar : were used; p(o) is the electron density at the nucleus [11]. Since data on ( pb > for ions are absent in literature, it is impossible to qualitatively estimate an error of (38). However, the fact that the limit of a highly ionized atom is described by this expression exactly and the limit of a neutral atom, quite accurately, (38) may be recommended to estimate ( pb > for b > 0 in an atom with an arbitrary degree of ionization. Now we discuss the expectation values p" ) with regard to the contributions of strongly bound electrons. Similar to (36), ( pm > may be presented as : Comparison shows that (42) well reproduces the HF data for isoelectronic series (Table IV). It is easily seen from table IV that the open shell isoelectronic series are described worse than the closed shell ones. This is because of the absence of oscillation contributions to the coefficients D 1 ( 18, 3) and D2(18, 3). The above consideration has shown that the inclusion of a strongly bound electron contribution allows not only investigation of systematic trends in the expectation values pb > and pm > but also reliable quantitative estimates for 0 b 5 and 0 m 5/3. For negative exponents 3 b 0 and 1 m 0, a strongly bound electron contribution is negligibly small while the oscillation contribution to pb > and pm > is very important and its inclusion requires special considerations. 5. Conclusions. main results obtained are : On the RHS of (40), the first integral diverges for m > 5/3, which limits the range of the validity of (40) to 1 K m 5/3. In quantummechanical consideration, ( pm > exists at all m within 1 m oo. The limit of the range of m in (40) is due to the incorrect TF contribution of strongly bound electrons not completely eliminated for m > 5/3. ( 1 ) the expectation values pb ) and pm > are found as functions of b, m and degree of ionization within the TF model with account for exchange interaction ; (2) ( p6 ) is obtained as a function of N within the noninteracting electron model; it is proved that for a great electron number the TF model gives the leading term in pb > which is identical to an exact quantum quantity within 2 b 2;

11 342 (3) it is shown that a correct treatment of strongly bound electrons gives a reliable estimate of pb >, within 3 b 5 and of ( p" ) within 1 m 5/3 and essentially improves them within 1 b 3 and within 1 /3 m 1; (4) the expectation values ( p > and ( p > for neutral atoms are found to obey the periodic law; (5) the three first Z 1 expansion coefficients for pb >, pm > are defined as functions of N. These results together with the data of [2] and [6] prove that the improved TF model provides a quantitative estimate of such atomic properties, whose values are basically determined by bulky and strongly bound electron contributions. Acknowledgment The authors would like to thank S. K. Pogrebnya for his programming assistance in calculations. References [1] DMITRIEVA, I. K. and PLINDOV, G. I., Z. Phys. 305 (1982) 103. [2] DMITRIEVA, I. K., PLINDOV, G. I. and POGREBNYA, S. K., J. Physique 43 (1982) [3] BETHE, H. A. and SALPETER, E. E., Quantum Mechanics of One and TwoElectron Atoms (SpringerVerlag, Berlin) [4] BENESCH, R. and SMITH, V. H., Wave Mechanics The First Fifty Years, ed. W. C. Price, S. Chissick and T. Ravensdale (Butterworths) [5] KIRZHNITS, D. A., Field Methods of the Theory of Many Particles (Gosatomizdat, Moscow) [6] PLINDOV, G. I. and DMITRIEVA, I. K., J. Physique 38 (1977) [7] PATHAK, R. K. and GADRE, S. K., J. Chem. Phys. 74 (1981) [8] DMITRIEVA, I. K. and PLINDOV, G. I., Izv. Akad. Nauk SSSR, Ser. Fiz. 41 (1977) [9] DMITRIEVA, I. K., PLINDOV, G. I. and CHEVGANOV, B. A., Opt. Spectrosk. 42 (1977) 7. [10] TAL, I. and BARTOLOTTI, L. J., J. Chem. Phys. 76 (1982) [11] BARTOLOTTI, L. J. (unpublished). [12] WHITTAKER, E. T. and WATSON, G. N., A Course of Modern Analysis, 4th edition (Cambridge) [13] SCHWINGER, J., Phys. Rev. A 22 (1981) [14] SCOTT, J. M. C., Philos. Mag. 43 (1952) 859. [15] BIGGS, F., MENDELSOHN, L. B. and MANN, J. B., Atom Data, Nuclear Data Tables 16 (1975) 201.