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The Thomas-Fermi model: momentum expectation values I.K. Dmitrieva, G.I. Plindov To cite this version: I.K. Dmitrieva, G.I. Plindov. The Thomas-Fermi model: momentum expectation values. Journal de Physique, 1983, 44 (3), pp.333-342. <10.1051/jphys:01983004403033300>. <jpa00209602> HAL Id: jpa-00209602 https://hal.archives-ouvertes.fr/jpa-00209602 Submitted on 1 Jan 1983 HAL is a multi-disciplinary 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. J. Physique 44 (1983)333-342 MARS 1983, 333 Classification Physics Abstracts 31.10 - 31.20L The Thomas-Fermi 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. 2014 Les expressions analytiques de toutes les valeurs moyennes des impulsions pb> et de quelques de la densité électronique 03C1m> pour les atomes dans un degré d’ionisation arbitraire sont obtenues dans le cadre du modèle de Thomas-Fermi compte tenu des corrections d des à l’échange et à la contribution des électrons fortement liés. On montre que le traitement correct de celle-ci 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 Z-1 est donnée explicitement en fonction du nombre d’électrons. puissances Abstract Within the Thomas-Fermi 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 Z-1 expansion coefficients for pb> and 03C1m> are given as an explicit function of the electron number. - 1. Introduction. Recently [ 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 Thomas-Fermi 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 ) - 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 equation is a rather tedious problem requiring the solution of the N-particle Schrodinger equation in the coordinate space. Still more difficult is the search for ( pb > (p =I p I) ( 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 Thomas-Fermi model with account for the exchange interaction and contributions of strongly bound electrons. Systematic trends in ( pb ) and ( p"‘ ) will also be analysed. requires either the Fourier transformation of spatial wave function or solution of the Schrodinger 2. Statistical model. In the frame work of the Thomas-Fermi (TF) and Thomas-Fermi-Dirac (TFD) allows which a - Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004403033300 334 models, the equal to : state density in the phase space [5] is equal The value of ( pb > (b > - 3), based on (4), is : beyond to zero. atom we where pF(r) is the Fermi momentum at a distance r from the nucleus and 0 the Heaviside theta-function. Integrating (4) over momentum, it is easy to establish the relationship between the particle number density and the Fermi momentum : which the electron density is in Therefore, the TF model for a neutral have : finite radius, ro, This range includes 4 moments of the momentum 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 ( p-1 ) 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 distribution, namely, b = - to : Here and below, atomic units are used. With regard to (5), the value of pb > may be expressed in terms of pm) 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 or the TF equation with regard for the exchange in simple relationship between the first order with respect to fl [2, 8] : 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. where the dimensionless radius x Expression (6) is the basis to study pb > as a func- and thef3 = 2 (6 nZ)-2/3, function screening 03C8(x) are related to r and tion of N and Z within the statistical model. The applicability of (6) is specified by the behaviour of the sta- p(r) by : 0 and r tistical density at r oo. In the simple TF Equation (7) gives a - - model for p(r - oo) Let us - neutral atom p(r -+ 0) - r- 3/2 and r- 6. The ion in the TF model has a a 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. 335 Within - b 3/2 3, ( pb > for a neutral atom may be given as a sum : where Here ( p6 >TF has a universal form [1, 7] : 9.175 8 ; B(1) 0.693 75 ; B(2) 1.537 5 . Special consideration must be made of the exchange contribution. Using the expansion of qlo(x) and 1 (x) at x > 1 [8] : B(- 1) 6 = = = ( 73 - 7)/2, it is easy to Numerical = see that for - 1/2 b The situation is for the negative Expression estimating ( p-1 ). However, more complex may be used for 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 (Z > 1) : a p6 )eX may be given as : integration in (15a) yields : moments of the momentum distribution. (14) 2, the main part of qualitative asymptotic dependence being filled. The SCF-data 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 Ap-1 > (Z 113 ) and p-2 )HF(ZI/3) correspond to alkali-earth 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 has a relative order Z - 1/3 , being a leading (p-l > for (p-2 ). term 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 : 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 . pass to the estimate of (ph) for ions. b 3. is valid for all b within - 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, Now we Expression (14) t/J1 (x), into series in the parameter , 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, and it is (19) (16) expressions 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 p-2 >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. - The values of p-’>,, and Ap-’> = obtained from HF Compton proa function of Z 1/3 : 1, p -2>HF; 2, A p-’ >. P-’>HF - P-’1 /TF files [15] as 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 : is the function for a neutral atom and is function. Using the asymptothe correction t/lOl (x) tic expressions for t/loo(x) and t/lOl(X) [2], #oo(x) 144 x-3(1 + O(x-U)); t/lo 1 (x) Ax4+u(1 + O(x-U)) and taking into account where = from The value of was Bo(b) equal obtained in Table I. - to [1]. Values of Bk(b) and B;X(b). t/loo(x) (12), we obtain the asymptotic expressions : 337 Expressions (23a) and (23b) determine an approximate type of singularity at N/Z - 1. We think it expedient to present p-2 )TF at NIZ - I as : series f 1 (N/Z) is a function having no singularities 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 : HF data Table II. - Values of Ne and Ar ; of p’ /3 > for TF model with the isoelectronic exchange (26a). where at = = [11] are given in brackets. --> the asymptotic expressions for the coefficients Dk(N, within - 3 b 3 being of the form : b) 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 -11 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 The value of Go(N, 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 example, (25) and (24) are used to estimate Z -1 expansion coefficients for ( p’ > : coincides with the m) Expressions (25) for many-electron 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 -11 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 and (26) are atoms at k > 0 the one obtained in [10]. only estimates of the Z -1 coefficients for ( pb ) and pm ) expansions (except ( p2 >). Therefore, other data. To illustrate the quality of we could not perform a direct comparison with (25) and (26), we made a systematic comparison of pl /3 > calculated by : 54 and N Z 20 + N. The maximum with Hartree-Fock data [11] for isoelectronic series 10 N 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 arbi3 (0 m 1). trary degree of ionization for 0 b 3. then Non-interacting electron model. - If an atom is considered to be with no electron-electron interaction, are found by summing over all occupied hydrogen-like orbitals : ( p6 ) and pm > 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 trivial. To to calculation of only four moments, one of which, pl ), is obtain pb )n’, a calculation must be made of the integrals J(n, I, b) equal to : The use The exact of the explicit expression for Gegenbauer’s polynomials [12] yields : 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), odd b 1, 1, 3, DO(N, b) is also described = - (31 a) and (31 b) for by (33). the four first electron shells shows that, for 339 To find Do(N, b) as a function of N, summation is performed over n in (33) with regard between the maximum main quantum number nm and N for closed electron shell atoms : When limiting Here C = to the terms of the relative order of N - 2/3, we to the relationship have 0.557 216 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 Kompaneets-Pavlovskii (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 (34a-f) 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 p-2 > (Fig. 2). The oscillation effects appear in ( p’ > due to the fact that the discrete quantum state electron distribution differs from the continuous one defined by (34a-f). The analytical estimate of these effects may be made using simple algebra as it was done for the energy [6]. 4. Strongly bound electrons. of pb > for 3 b - A correct estimate 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 b 3 and of account, the estimates of ( pb > for 1 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 non-interacting electrons (27) : Fig. 2. 2013 pb >H/DS(N, b) as a function of N 1/3 : 1, b = 1 ; 2,b=-2. 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) : 2 n 2(Z p)- 1. Here xm is the region of localization of strongly bound electrons, xm Z( - E,u)-1 Let us show that a similar result may be obtained if the TF contribution of N-non-interacting electrons is eliminated from (12) and replaced by a quantum-mechanical quantity. This approach as applied to pb > = yields : Here in the first integral, 1 x/xoo is the TF screenfunction of ing non-interacting electrons and - is the dimensionless radius of the From (15) it follows that for b > 4 the first integral 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. on as a = ion with rio electron-electron interaction. Intein and (36) and allowance for the relationgration (35) between and n’ demonstrate that both apship xm coincide correctly to the terms of the relative proaches 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 quantum-mechanical model for non-interacting 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 electron-electron 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 : TF are calculated to determine of N for b function 3, 4 at k a 1. Dk(N, b) Partial integration of (36) gives : p4 >TF and p3 >’TF 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 341 Table III. - Values of pb > for a neutral atom (39). Table IV. - Values of p >.10- 2 for the electronic ; TF model with account for exchange bound electron contributions (42). series ofNe and Ar and 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 were used; p(O) is the electron density strongly Substitution of (17) into (40) electronic series : Expression (41) gives ( pm > may be used to for an iso- obtain pm > for 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 : an 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 a is bound electron contribution 0, strongly small while the oscillation contribution to negligibly pb > and pm > is very important and its inclusion requires special considerations. 5. Conclusions. 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 quantum-mechanical 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. - The main results obtained are : ( 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 non-interacting 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; 342 is shown that a correct treatment of strongly bound electrons gives a reliable estimate of pb > 5 and of ( p"’ ) within 1 within 3 b m 5/3 b 3 and and essentially improves them within 1 within 1 /3 m 1; (3) it , (4) the expectation values ( p - ’ > and ( p - ’ > for neutral atoms are found to obey the periodic law; 1 (5) the three first Z -1 expansion coefficients for pb >, pm > are defined as functions of N. These results with the data of [2] and [6] provides a quantitative estimate of such atomic properties, whose values are basically determined by bulky and strongly prove that the together improved TF model 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) 1339. [3] BETHE, H. A. and SALPETER, E. E., Quantum Mechanics of One- and Two-Electron Atoms (Springer-Verlag, Berlin) 1957. [4] BENESCH, R. and SMITH, V. H., Wave Mechanics The First Fifty Years, ed. W. C. Price, S. Chissick and T. Ravensdale (Butterworths) 1973. [5] KIRZHNITS, D. A., Field Methods of the Theory of Many Particles (Gosatomizdat, Moscow) 1963. [6] PLINDOV, G. I. and DMITRIEVA, I. K., J. Physique 38 (1977) 1061. [7] PATHAK, R. K. and GADRE, S. K., J. Chem. Phys. 74 (1981) 5926. [8] DMITRIEVA, I. K. and PLINDOV, G. I., Izv. Akad. Nauk SSSR, Ser. Fiz. 41 (1977) 2639. [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) 2558. [11] BARTOLOTTI, [12] WHITTAKER, L. J. (unpublished). E. T. and WATSON, G. N., A Course of Analysis, 4th edition (Cambridge) [13] SCHWINGER, J., Phys. Rev. A 22 (1981) 1827. [14] SCOTT, J. M. C., Philos. Mag. 43 (1952) 859. [15] BIGGS, F., MENDELSOHN, L. B. and MANN, J. B., Data, Nuclear Data Tables 16 (1975) 201. Modern 1927. Atom