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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�. �jpa-00209602�
333
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 puissances 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 dévelop-
pement de pb > et 03C1m> en Z-1 est donnée explicitement en fonction du nombre d’électrons.
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.
J. Physique 44 (1983)333-342 MARS 1983,
Classification
Physics Abstracts
31.10 - 31.20L
1. Introduction. - Recently [ 1 ], asymptotic esti-
mates 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 electron- electron 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 )
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)
which requires either the Fourier transformation of
a spatial wave function or solution of the Schrodinger
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
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 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.
2. Statistical model. - In the frame work of the Thomas-Fermi (TF) and Thomas-Fermi-Dirac (TFD)
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004403033300
models, the state density in the phase space [5] is equal to :
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 esta- blish the relationship between the particle number density and the Fermi momentum :
The value of ( pb > (b > - 3), based on (4), is :
Here and below, atomic units are used.
With regard to (5), the value of pb > may be
expressed in terms of pm)
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 elec- trons, or for the inhomogeneity and the oscillation of the electron density.
Expression (6) is the basis to study pb > as a func-
tion of N and Z within the statistical model. The appli- cability of (6) is specified by the behaviour of the sta- tistical 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
finite radius, ro, beyond which the electron density is equal to zero. Therefore, in the TF model for a neutral
atom we have :
This range includes 4 moments of the momentum
distribution, namely, b = - 1, 0, 1, 2. The zero mo-
ment is reduced to the normalization integral. The
values ( p2 > and ( p > determine kinetic and exchange
energy (in a local approximation) and ( p-1 ) is pro-
portional to Jo(0), (2). The virial theorem links ( p2
with the binding energy which has been earlier inves-
tigated 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 :
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
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.
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) = 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] :
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 ( p-1 ). However, since the integrals in the P-’ >ex estimate substan-
tially 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 :
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
being filled. The SCF-data obtained from HF Comp-
ton 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 indepen-
dent on Z. Thus, the oscillation contribution of
(p-l > has a relative order Z - 1/3 , being a leading
term for (p-2 ). .
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 ,
Fig. 1. - The values of p-’>,, and Ap-’> = P-’ >HF - P-’ 1 /TF obtained from HF Compton pro-
files [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 :
The value of Bo(b) equal to
was obtained in [1].
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 conclu- sion 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 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 :
where t/loo(x) is the function for a neutral atom and
t/lOl (x) is the correction function. Using the asympto- 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
from (12), we obtain the asymptotic expressions :
Table I. - Values of Bk(b) and B;X(b).
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 :
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 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
Table II. - Values of p’ /3 > for the isoelectronic series of Ne and Ar ; TF model with exchange (26a).
HF data [11] are given in brackets.
involve the contributions of strongly bound electrons, the inhomogeneity of the electron density and oscilla-
tions. 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 dis-
cussed in the next section.
Equation (25) gives an important property of the Z -1 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 example, (25) and (24) are used to estimate Z -1 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 many-electron 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 Hartree-Fock 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 arbi-
trary degree of ionization for 0 b 3 (0 m 1).
3. Non-interacting electron model. - If an atom is considered to be with no electron-electron interaction,
then ( p6 ) and pm > are found by summing over all occupied hydrogen-like orbitals :
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 analy-
tical 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 relation-
ship 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).
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 = 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 Kompa-
neets-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 depen-
dence of the internal boundary radius on Z, XJ(Z) _ z - 2/3 .
Fig. 2. 2013 pb >H/DS(N, b) as a function of N 1/3 : 1, b = 1 ; 2,b=-2.
The asymptotic expressions (34a-f) perfectly des-
cribe 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 distri- bution 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. - A 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 non-interacting electrons (27) :
Here n’ is the main quantum number of strongly
bound electrons ; n’ - (Z 2/2 E) is not obligato-
rily 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 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 screen- ing function of non-interacting electrons and is the dimensionless radius of the TF ion with rio electron-electron interaction. Inte-
gration in (35) and (36) and allowance for the relation-
ship between xm and n’ demonstrate that both ap-
proaches 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 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 :
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 asymptoti- cally exact (at N > 1) values of the Z -1 expansion
coefficients for all moments of the momentum distri- bution.
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 contri-
butions both of the exchange interaction and of
strongly bound electrons. Note that the amplitude of
