Bounds for the second and the third derivatives of the electron density at the nucleus

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by  Zhixin Qian

Bounds for the second and the third derivatives of the electron density at the nucleus
Zhixin Qian∗
Department of Physics, Peking University, Beijing 100871, China
(Dated: February 2, 2008)
Lowerboundforρ¯′′(0),thesecondderivativeofthesphericallyaveragedatomicelectronicdensity
at the nucleus, and upper bound for ρ¯′′′(0), the third derivative, are obtained respectively. It is
shown that,forthegroundstate, ρ¯′′(0)≥ 10Z2ρ(0)and ρ¯′′′(0)≤−14Z3ρ(0) whereZ isthecharge
3 3
of thenucleus,and ρ(0) is theelectron densityat thenucleus. Tighter boundsfor ρ¯′′(0) and ρ¯′′′(0)
are also reported which are valid for both the ground state and excited states. Explicit illustration
8 with theexample of one-electron atomic ions is given.
0
0 PACSnumbers:
2

n
I. INTRODUCTION ρ¯′′′(0) are also reported (see Eqs. (23) and (24) in Sec.
a
J III) which are valid for the excited states as well.
In Sec. II, we discuss the near nucleus behavior of
2 Rigorous knowledge of the electron density is rather
valuable in understanding the electronic structure of the wavefunction of the Schr¨odinger equation and the

] atoms,molecules,andsolids. Itisalsohelpful inguiding density. The derivation for the bounds is presented in
h
Sec. III. In Sec. IV explicit illustration for the case of
p theconstructionofthe approximationsfortheexchange-
one-electron atomic ions is given. Summarizing remarks
- correlation energy functional and the corresponding po-
m are made in Sec. V.
tentialintheapproachofdensityfunctionaltheory[1]to
o theproblemsofinhomogeneouselectronsystems. Oneof
t the well known exact results is the so-called Kato theo-
a II. NEAR NUCLEUS BEHAVIOR OF THE
. rem [2] which states that, near any nucleus,
s WAVEFUNCTION AND THE DENSITY
c
si ρ¯′(r)|r=0 =−2Zρ(0). (1) The Schr¨odinger equation for N-electrons in an exter-
y nal potential v(r) arising from their interactionwith nu-
h Here Z is the charge of the nucleus which is taken as
p the origin; f¯(r) means the spherical average of function clei is (in a.u.)
[ f(r); primes denote the derivatives with respect to r in HˆΨ=(Tˆ+Vˆ +Uˆ)Ψ=EΨ, (4)
this paper.
2
v Inrecentwork[3],by investigatingthebehaviorofthe where Tˆ = −12 i▽2i, Vˆ = iv(ri), Uˆ =
1 wavefunctions ofthe interacting Schr¨odingersystem and 1 1 , Ψ thPe wavefunction, anPd E the energy.
1 the corresponding noninteracting Kohn-Sham system in 2 i6=j |ri−rj|
FoPllowing Ref. [3], we write, for limiting small r, the
3 the vicinity of the nucleus, we established relations be-
many-body wavefunction as
4
tween the second and the third derivatives of the spheri-
9. callyaverageddensity atthe nucleus. They couldbe un- Ψ(r,X) =Ψ(0,X)+a(X)r+b(X)r2+c(X)r3+...
0 derstood as extentions of the cusp condition of Eq. (1) 1
07 to higher orders of derivatives. In this paper, we derive + [a1m(X)r+b1m(X)r2+...]Y1m(rˆ)
: rigorously a lower bound for the second derivative and mX=−1
v an upper bound for the third derivative, respectively, of 2
Xi the spherically averaged density at the nucleus for the + [b2m(X)r2+...]Y2m(rˆ)
r ground state. The bounds are given as follows: mX=−2
a +..., (5)
10
ρ¯′′(0)≥ 3 Z2ρ(0), (2) where rˆ= r/r, X denotes s,r2s2,...,rNsN, and Ylm(rˆ)
are the spherical harmonics. For r →0, the Schr¨odinger
and equation (4) can be rewritten as [3]
14 1
ρ¯′′′(0)≤− 3 Z3ρ(0). (3) [ − 1 ▽2−Z +r Y1m(rˆ)gm(X)]Ψ(r,X)
2 r
mX=−1
The results hold in general whether the system is an + HN−1(X)Ψ(r,X)=EΨ(r,X), (6)
atom, molecule or a solid. Tighter bounds for ρ¯′′(0) and Z−1
where
N
4π 1
g (X)= Y∗ (rˆ), (7)
∗Electronicaddress: [email protected] m 3 Xi=2 ri2 1m i
2
and N dX Re[Ψ∗(0,X)c(X)]
Z
N
1 1 1
HN−1(X)= [− ▽2+v(r )+ ] ≤ Z(−Z2+4E−4EN−1 )ρ(0), (17)
Z−1 2 i i r 18 Z−1,0
Xi=2 i
1 N 1 where EN−1 is the ground state energy of the Hamil-
+ . (8) Z−1,0
2i6=Xj6=1|ri−rj| tonian HZN−−11(X). Inequalities (16) and (17) are critical
equations in the following derivation. Comparing them
Substituting the expression of Eq. (5) into Eq. (6) and with Eq. (14) and Eq. (15), respectively, one has
equating the coefficients of the terms of r−1, r0Y1m(rˆ),
r0, and r1, respectively, one has ρ¯′′(0)≥ 10Z2ρ(0)+ 4(EN−1 −E)ρ(0)
3 3 Z−1,0
a(X)+ZΨ(0,X)=0, (9) 1
1
2b1m(X)+Za1m(X)=0, (10) +2NZ dX 4π|a1m(X)|2, (18)
3b(X)−Z2Ψ(0,X)=[HN−1(X)−E]Ψ(0,X),(11) mX=−1
Z−1
6c(X)+Zb(X)=Z[E−HN−1(X)]Ψ(0,X). (12) and
Z−1
We note that the behavior of the wavefunction in the ρ¯′′′(0)≤ −14Z3ρ(0)− 20Z(EN−1 −E)ρ(0)
vicinity of the nucleus has been extensively investigated 3 3 Z−1,0
[2, 3, 4]. In fact, Eqs. (9) and (10) have been obtained 1
1
previously. On the other side, Eqs. (11) and (12), which −6ZN dX |a1m(X)|2. (19)
Z 4π
shall play a key role in the derivation in Sec. III, have mX=−1
not been previously reported in this separate form.
The density is defined as Inpassing,wementionthat,atthenucleus,thekinetic
energy density, which is defined as
ρ(r)=N dX|Ψ(r,X)|2, (13)
Z 1
t(r)= N dX▽Ψ∗(r,X)·▽Ψ(r,X), (20)
where dX denotes s dx2...dxN. With the rela- 2 Z
tions inREqs. (9) and P(10)R, it can be shown that
has been shown in Ref. [3] as
ρ¯′′(0)= 2Z2ρ(0)+4N dXRe[Ψ∗(0,X)b(X)] 1
Z 1 3
+2N dX 1 1 |a1m(X)|2 (14) t¯(0)− 2Z2ρ(0)=NZ dXmX=−18π|a1m(X)|2. (21)
Z 4π
mX=−1
Equation (21) indicates that
and
1
t¯(0)≥ Z2ρ(0). (22)
ρ¯′′′(0)= 12N dXRe[Ψ∗(0,X)(c(X)−Zb(X))] 2
Z
1 The existence of the term on the right hand side of Eq.
1
−6ZN dX |a1m(X)|2, (15) (21) had not been recognized before [5].
Z 4π
mX=−1 From Eqs. (18) and (19), one obtains
(see also Ref. [3].) This completes the discussion of the
10 4
behavior of the wavefunction and the density near the ρ¯′′(0)≥ 3 Z2ρ(0)+ 3(EZN−−11,0−E)ρ(0), (23)
nucleus.
and
III. DERIVATION OF EQS. (2) AND (3) 14 20
ρ¯′′′(0)≤− Z3ρ(0)− Z(EN−1 −E)ρ(0). (24)
3 3 Z−1,0
We observe that Eqs. (11), (12) and Eq. (13) lead to
Up to this point, all the calculations are in fact not re-
stricted to the ground state. The bounds shown in Eqs.
N dX Re[Ψ∗(0,X)b(X)]
Z (23) and (24) are valid for both the ground state and ex-
1 cited states. For the groundstate we further obtain Eqs.
≥ 3(Z2−E+EZN−−11,0)ρ(0), (16) (2) and (3) from Eqs. (23) and (24).
3
IV. ILLUSTRATION FOR ONE-ELECTRON The inequalities of Eqs. (25) and (26) are satisfied. No-
ATOMIC IONS tice that in this case a1m 6=0 in Eqs. (18) and (19).
For one-electron atomic ions (including hydrogen
atom), Eqs. (23) and (24) become
V. SUMMARY
2
ρ¯′′(0)≥ (5Z2−2E)ρ(0), (25)
3 In summary, in this paper, we have established the
lowerboundforρ¯′′(0),thesecondderivativeofthespher-
and
ically averaged electron density at the nucleus, in Eq.
ρ¯′′′(0)≤−2Z(7Z2−10E)ρ(0). (26) (2), and the upper bound for ρ¯′′′(0), the third deriva-
3 tive, in Eq. (3). Tighter bounds for ρ¯′′(0) and ρ¯′′′(0),
valid for both the ground state and excited states, are
Thewavefunctionsareexactlyknownforboththeground
also reported in Eqs. (23) and (24). These results add
state and excited states: Ψ(r)=R (r)Y (rˆ), with [6]
nl lm some to our rigorousinformation of the electrondensity,
which might be valuable in the calculation of the elec-
1/2
(n−l−1)!
R (r)=− α3+2l e−αnr/2 tronic structure of atomic systems.
nl (cid:26) n 2n[(n+l)!]3(cid:27) Whethertheoppositeboundsexistforρ¯′′(0)andρ¯′′′(0)
rlL2l+1(α r), (27) remains an interesting question. Not unrelated to this
n+l n
question is the fact that both upper and lower bounds
whereα =2Z/n,andL2l+1(α r)aretheLaguerrepoly- have been extensively explored for ρ(0) (and, according
nomials.nIt is easy to seen+thlat fnor l 6=1, a1m =0 in Eqs. to Eq. (1), equally for ρ¯′(0)) [7].
(18) and (19), and Eqs. (23) and (24) in fact become Note added. During the preparation of this
equalities: manuscript, the author became aware of related work
[8]. The bounds shown in Eqs. (2) and (3) in this paper
2
ρ¯′′(0)= (5Z2−2E)ρ(0), (28) areindeedthe sameasthoseinEqs. (1.25)and(1.32)in
3 Ref. [8] (up to a different form of Hamiltonians). There
are,however,minor differences between inequalities (23)
and
and(24)ofthe presentpaperandthe firstinequalitiesof
ρ¯′′′(0)=−2Z(7Z2−10E)ρ(0). (29) Eqs. (1.25)and(1.32)inRef. [8]. Infact,EZN−−11,0−E can
3 berewrittenasEN−1 −EN−1−µ,whereµ=E−EN−1
Z−1,0 Z,0 Z,0
is the minus the ionization energy and denoted as −ǫ in
Wenextgiveanexplicitillustrationoftheresultswith
Eq. (1.20) in Ref. [8]. Since EN−1 −EN−1 ≥ 0, it is
the exactly known wavefunctions. Obviously for l ≥ 2 Z−1,0 Z,0
Eqs. (28) and(29)aretrivially true sinceallρ(0), ρ¯′′(0), easy to see that the bounds in Eqs. (23) and (24) are
and ρ¯′′′(0) are zero. For l = 0, one can obtain from Eq. tighter than those given by the first inequalities of Eqs.
(27) (1.25) and (1.32). The differences were also recognized
andanalyzedinRemark1.4inRef. [8],thoughequations
Z3 1 1 like inequalities (23) and (24) were not explicitly given
ρ¯(r)= 1 − 2Zr+ Z2(5+ )r2
πn3(cid:20) 3 n2 there.
The author is indebted to Profs. S. Fournais,
1 5
− Z3(7+ )r3+... . (30) M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and T.
9 n2 (cid:21)
Østergaard Sørensen for confirming that the bounds
shown in Eqs. (2) and (3) are the same as those in Eqs.
Equations (28) and (29) are hence confirmed with the
help of the fact that E =−Z2/2n2. For l =1, one has (1.25) and (1.32) in Ref. [8]. Enlightening comments on
this paper from these colleagues are also gratefully ac-
5 knowledged. This work was supported by the National
1 Z
ρ¯(r)= (n2−1)r2(1−Zr+...). (31) ScienceFoundationofChinaunderGrantNo. 10474001.
9π(cid:18)n(cid:19)
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