Distribution of $r_{12} \cdot p_{12}$ in quantum systems

2 MB·English

by  Yves A. Bernard

Distribution of r ·p in quantum systems
12 12
Yves A. Bernard, Pierre-Fran¸cois Loos,∗ and Peter M. W. Gill†
Research School of Chemistry, Australian National University, Canberra, ACT 0200, Australia
(Dated: February 1, 2013)
We introduce the two-particle probability density X(x) of x = r ·p = (r −r )·(p −p ).
12 12 1 2 1 2
We show how to derive X(x), which we call the Posmom intracule, from the many-particle wave-
function. We contrast it with the Dot intracule [Y. A. Bernard, D. L. Crittenden, P. M. W. Gill,
Phys. Chem. Chem. Phys., 10, 3447 (2008)] which can be derived from the Wigner distribution
and show the relationships between the Posmom intracule and the one-particle Posmom density
[Y. A. Bernard, D. L. Crittenden, P. M. W. Gill, J. Phys. Chem. A, 114, 11984 (2010)]. To illus-
trate the usefulness of X(x), we construct and discuss it for a number of two-electron systems.
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PACSnumbers: 31.15.X-,31.15.ve,31.15.vj,81.07.Ta
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Keywords: Two-particledensitydistribution,Wignerdistribution,Intracule,Posmom
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∗ [email protected]
† Correspondingauthor;[email protected]
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I. INTRODUCTION
Intracules are two-particle density distributions obtained from the spinless second-order reduced density matrix [1]
(cid:18)r , r(cid:48)(cid:19) (cid:90)
ρ 1 1 = Ψ∗(r ,r ,r ,...,r )Ψ(r(cid:48),r(cid:48),r ,...,r )dr ···dr , (1)
2 r , r(cid:48) 1 2 3 N 1 2 3 N 3 N
2 2
where Ψ(r ,r ,r ,...,r ) is the N-particle position wave function. Intracules are usually normalized to the number
1 2 3 N
of particle pairs N(N −1)/2.
The seminal intracule is the Position intracule
(cid:90) (cid:18) (cid:19)
r , r
P(u)= ρ drdΩ , (2)
2 r+u , r+u u
which was introduced long ago by Coulson and Neilson [2] to study correlation effects in the helium atom. In (2),
u = r −r , u = |u| ≡ r and Ω is the angular part of u. P(u) gives the probability density for finding two
1 2 12 u
particles separated by a distance u and has been widely studied [3–21].
The corresponding Momentum intracule [22, 23] is
1 (cid:90) (cid:18) r , r+q(cid:19)
M(v)= ρ eiq·vdrdqdudΩ , (3)
(2π)3 2 r+u+q , r+u v
where v =p −p , v =|v|≡p and Ω is the angular part of v. M(v) gives the probability density for finding two
1 2 12 v
particles moving with a relative momentum v.
Starting with the Wigner distribution [24, 25], one can construct a family of intracules [15, 16], which provide
two-electron position and/or momentum information. Within this family, the patriarch is the Omega intracule
1 (cid:90) (cid:18) r , r+q(cid:19)
Ω(u,v,ω)= ρ eiq·vδ(ω−θ )drdqdΩ dΩ , (4)
(2π)3 2 r+u+q , r+u uv u v
where ω ≡ θ is the dynamical angle between the vector u and v. Ω(u,v,ω) can be interpreted as the joint quasi-
uv
probability density for u, v and ω. The quasi prefix emphasizes that Ω(u,v,ω) is not a rigorous probability density
and, indeed, it may take negative values [24]. Based on the observation of Rassolov [26] that both relative position
and relative momentum are important to describe the correlation between pairs of electrons, and because the Omega
intraculecontainsinformationonbothquantities,Ω(u,v,ω)hasbeenextensivelyusedinIntraculeFunctionalTheory
(IFT) [16, 17, 27–33].
Appropriate integrations [16] reduce the Omega intracule to lower-order intracules such as P(u), M(v) and the
Angle intracule [27, 28]
(cid:90) ∞(cid:90) ∞
Υ(ω)= Ω(u,v,ω)dudv, (5)
0 0
which provides information on the angle ω between u and v. A similar reduction yields the Dot intracule [17, 30]
(cid:90) ∞(cid:90) ∞ Ω(u,z/u,ω)
D(x)= dzdu. (6)
uzsinω
0 x
The variable x=u·v =uvcosω combines information on the relative position and momentum of the particles, and
it is easy to show that it gives the rate of change of u2, i.e.
1 d
x= u2. (7)
2dt
In this way, x sheds light on the motion of the electrons. For example, x = 0 implies that the electrons are moving
in such a way that their separation is constant. This could arise, for example, if they were in a circular orbit around
their centre of mass.
AlthoughD(x)isusuallyanon-negativefunctionandhasprovenusefulforunderstandingelectronicbehaviour[27]
andforestimatingelectroncorrelationenergiesinatomicandmolecularsystems[29,30],itsconnectiontotheOmega
suggests that it is not a rigorous probability density. However, in the following Section, we show how to derive the
exact probability distribution of x.
3
II. THE POSMOM INTRACULE
We define the Posmom intracule X(x) to be the exact probability density for the variable x = u·v. It is the
two-particle version of the Posmom density S(s) where s = r ·p [34–36] and, as we have argued that s describes
particle trajectories, we now propose that x likewise characterizes pair trajectories.
The quantum mechanical operator
(cid:18) (cid:19)
3
s¯=−i(cid:126) +r·∇ (8)
2 r
is known to be an unbounded self-adjoint operator [37, 38] and its two-particle equivalent is
(cid:18) (cid:19)
3
x¯=−2i(cid:126) +u·∇ , (9)
2 u
where ∇ is the gradient operator. Both s¯and x¯ correspond to quantum mechanical observables.
Following the same approach used in [34] to obtain S(s) from s¯, one can show that the Posmom intracule can be
expressed as the Fourier transform
1 (cid:90) ∞
X(x)= X(cid:98)(k)eikxdk (10)
2π
−∞
of the two-particle hyperbolic autocorrelation function
(cid:90) (cid:18) r , r+sinh(k(cid:126))u(cid:19)
X(cid:98)(k)= ρ2 r+ek(cid:126)u , r+cosh(k(cid:126))u drdu. (11)
This expression can be simplified after defining the intracule density matrix
ρ (cid:0)u , u(cid:48)(cid:1)=(cid:90) ρ (cid:18)U −u/2 , U −u(cid:48)/2(cid:19)dU, (12)
u 2 U +u/2 , U +u(cid:48)/2
where U =r +r is the extracule vector, and yields
1 2
(cid:90)
X(cid:98)(k)= ρu(cid:0)e+k(cid:126)u , e−k(cid:126)u(cid:1)du. (13)
In an entirely analogous way, the Dot intracule can be expressed as the Fourier transform
1 (cid:90) ∞
D(x)= D(cid:98)(k)eikxdk (14)
2π
−∞
of the f-Dot function [30]
(cid:90) (cid:18) r , r+k(cid:126)u(cid:19)
D(cid:98)(k)= ρ2 r+u+k(cid:126)u , r+u drdu, (15)
and the latter can be reduced to
(cid:90)
D(cid:98)(k)= ρu(cid:0)(1+k(cid:126))u , (1−k(cid:126))u(cid:1)du. (16)
Comparing (13) with (16) and the Taylor expansion of the exponential function
k2
e±k(cid:126) =1±k(cid:126)+ (cid:126)2+... (17)
2
reveals that the probability density D(x) derived from the Wigner distribution is a first-order approximation to the
exact density X(x). Thus, the quasi-intracule is correct to O((cid:126)) and becomes exact in the classical limit (cid:126) → 0.
Remarkably, one can construct the exact density from the approximate density using the mapping
D(cid:98)(tanh(k(cid:126)))
X(cid:98)(k)= . (18)
cosh3(k(cid:126))
4
TABLE I. One- and two-particle hyperbolic autocorrelation functions.
Density Intracule
(cid:90) (cid:16) (cid:17) (cid:90) (cid:16) (cid:17)
Dot S(cid:98)W(k)= ρ1 (1+k(cid:126)/2)r , (1−k(cid:126)/2)r dr D(cid:98)(k)= ρu (1+k(cid:126))u , (1−k(cid:126))u du
(cid:90) (cid:16) (cid:17) (cid:90) (cid:16) (cid:17)
Posmom S(cid:98)(k)= ρ1 e+k(cid:126)/2r , e−k(cid:126)/2r dr X(cid:98)(k)= ρu e+k(cid:126)u , e−k(cid:126)u du
S(cid:98)(k)= S(cid:98)W(2tanh(k(cid:126)/2)) X(cid:98)(k)= D(cid:98)(tanh(k(cid:126)))
Relation cosh3(k(cid:126)/2) cosh3(k(cid:126))
=S(cid:98)W(k)+O((cid:126)2) =D(cid:98)(k)+O((cid:126)2)
TableIgivestheone-andtwo-particlehyperbolicautocorrelationfunctions, thecorrespondingfirst-order(Wigner)
approximations, and the relations between them. In Table I, the one-particle density matrix is given by
(cid:32) (cid:33)
(cid:16) (cid:17) 2 (cid:90) r , r(cid:48)
ρ r , r(cid:48) = ρ 1 1 dr . (19)
1 1 1 N −1 2 r , r 2
2 2
Ifthewavefunctionisexpandedinone-electronfunctions, φ (r), thereducedtwo-particledensitymatrixbecomes
a
(cid:32) (cid:33)
r , r(cid:48) (cid:88)
ρ 1 1 = P φ (r )φ (r(cid:48))φ (r )φ (r(cid:48)), (20)
2 r , r(cid:48) abcd a 1 b 1 c 2 d 2
2 2 abcd
where P is a two-particle density matrix element. In this case, (11) is given by
abcd
(cid:88)
X(cid:98)(k)= Pabcd[abcd]X(cid:98), (21)
abcd
where we have introduced the two-particle hyperbolic autocorrelation integral [abcd] . For example, if the basis
X(cid:98)
functions are s-type Gaussians, we obtain (in atomic units)
(cid:90)
[ssss] = e−α|r−A|2e−β|r+sinh(k)u−B|2e−γ|r+exp(k)u−C|2e−δ|r+cosh(k)u−D|2drdu
X(cid:98)
π3 (cid:20)1(cid:16)|H|2 (cid:17)(cid:21)
= exp −F , (22)
J3/2 ξ J
where
ξ =α+β+γ+δ, (23a)
J =ξ[βsinh2(k)+γexp2(k)+δcosh2(k)]−[βsinh(k)+γexp(k)+δcosh(k)]2, (23b)
F =αβ|A−B|2+αγ|A−C|2+αδ|A−D|2+βγ|B−C|2+βδ|B−D|2+γδ|C−D|2, (23c)
and
H =sinh(k)G +exp(k)G +cosh(k)G , (24a)
B C D
G =αβ(B−A)+βγ(B−C)+βδ(B−D), (24b)
B
G =αγ(C−A)+βγ(C−B)+γδ(C−D), (24c)
C
G =αδ(D−A)+βδ(D−B)+γδ(D−C). (24d)
D
The scalars and vectors above are independent of the choice of origin and X(cid:98)(k) and X(x) are therefore likewise
independent. This contrasts with the annoying origin-dependence [35] of the one-electron posmom density S(s).
Integrals of higher angular momentum can be generated by differentiating [ssss] with respect to the Cartesian
X(cid:98)
coordinatesofthebasisfunctioncenters,asfirstsuggestedbyBoys[39],or,moreefficiently,usingrecurrencerelations
[31]. We have written a program to compute X(x) within an spd Gaussian basis set and implemented this in a
development version of the Q-Chem 3.2 quantum chemistry package [40].
5
Eqs (21) – (24) are easily modified to generate P(u), M(v), Υ(ω) and D(cid:98)(k). In particular, if the functions sinh(k),
exp(k) and cosh(k) are replaced by their first-order approximations (k, 1+k and 1, respectively) in the expressions
for J and H, one obtains the f-Dot integrals [ssss] [30].
D(cid:98)
In the special case of concentric s-type Gaussians, the intracule integrals become
4π5/2 (cid:18) µ (cid:19)
[ssss] = u2exp − u2 , (25a)
P ξ3/2 ξ
4π5/2 (cid:18) ν (cid:19)
[ssss] = v2exp − v2 , (25b)
M χ3/2 χ
π3(cid:0)λ−2ηcos2ω(cid:1)
[ssss] = sinω, (25c)
Υ 2ζ3/2(λ+ηcos2ω)5/2
π3
[ssss] = , (25d)
D(cid:98) K3/2
π3
[ssss] = , (25e)
X(cid:98) J3/2
where
µ=(α+β)(γ+δ), (26a)
ν =(α+γ)(β+δ), (26b)
ζ =(α+δ)(β+γ), (26c)
χ=4(αβγ+αβδ+αγδ+βγδ), (26d)
(cid:18) (cid:19)(cid:18) (cid:19)
1 1 αδ βγ
λ= + + , (26e)
α+δ β+γ α+δ β+γ
(cid:18) α β (cid:19)2
η = − , (26f)
α+δ β+γ
K =ξ(cid:2)βk2+γ(1+k)2+δ(cid:3)−[βk+γ(1+k)+δ]2. (26g)
Inthecalculationsdescribedbelow,wehavecomputedX(x)andD(x)numericallyusingEqs. (10)and(14). D(cid:98)(k),
X(cid:98)(k), D(x) and X(x) are all even functions and we will therefore focus only on x≥0 and k ≥0.
Physical interpretations of the variables u, v, ω and x are summarized in Fig. 1. The three limiting configurations
ω =0,π/2andπ (whichcorrespondtox=uv,0and−uv)aredepictedfortheweak(uandv large),medium(where
one of u and v is large and the other is small) and strong correlation (u and v small) regimes. A faithful description
of electron correlation requires information about the relative position u and momentum v, but also on the mutual
orientation ω of these two vectors, which gives insight into the nature of the electrons’ mutual orbit. The Dot and
Posmom intracules provide information about the distribution of values of x = uvcosω, and thus about the type of
correlation regime (weak, medium or strong). However, as noted above, being a first-order approximation of X(x),
the information gathered in D(x) is slightly biased. The effects of this approximation will be investigated below.
In Section III, the Posmom intracule is investigated alongside D(x), Υ(ω), P(u) and M(v) for the two electrons in
a parabolic quantum dot. In Section IV, we turn our attention to the electrons in a helium atom or helium-like ion.
We also compare the Posmom intracules for ground and excited states and study the effect of the dimensionality of
the space D. Atomic units are used throughout.
III. PARABOLIC QUANTUM DOTS
In our study of the Posmom intracule in parabolic quantum dots [41], we consider three different treatments of the
Coulomb interaction between the two electrons. First, the non-interacting case, in which it is simply ignored; second,
the Hartree-Fock (HF) case [42] in which it is approximated in a mean-field sense; third, the exact treatment which
is possible for certain values of the harmonic confinement force constant [43, 44].
6
!=0 !=0
x =u v (medium) x =u v (large)
!=!/2 !=!/2
large u
x =0 x =0
!=! !=!
x =-u v (medium) x =-u v (large)
medium correlation weak correlation
!=0 !=0
x =u v (small) x =u v (medium)
!=!/2 !=!/2
small u
x =0 x =0
!=! !=!
x =-u v (small) x =-u v (medium)
strong correlation medium correlation
small v large v
FIG. 1. Physical interpretation of the variables u, v, ω and x in the weak, medium and strong correlation regimes.
A. Hamiltonian and wave functions
The Hamiltonian is
1 1
H=− (∇2+∇2)+V(r )+V(r )+ , (27)
2 1 2 1 2 r
12
where
r2
V(r)= (28)
2κ2
is the external harmonic potential and 1/κ2 is the force constant.
The 1S ground state of the non-interacting system has the wave function
Ψ (r ,r )=ψ (r )ψ (r ), (29)
0 1 2 0 1 0 2
(cid:18) r2(cid:19)
ψ (r)=(πκ)−3/4exp − , (30)
0 2κ
and the energy
3
E = . (31)
0 κ
The more accurate HF wave function
Ψ (r ,r )=ψ (r )ψ (r ) (32)
HF 1 2 HF 1 HF 2
7
TABLE II. Energies of parabolic quantum dots for different treatments of the interelectronic interaction.
N κ=2 κ=10
G
E — 1.5 0.3
0
1 2.04 0.53
2 2.038439 0.52904
3 2.0384389 0.5290415
E
HF
4 2.0384388718 0.5290415256
5 2.038438871755 0.52904152556
O’Neill and Gilla 2.03843887 —
Ragotb 2.03843887176 —
E — 2. 0.5
exact
a Reference[14]: 7basisfunctions.
b Reference[46]: 11basisfunctions.
is not known in closed form, but can be efficiently treated numerically by expanding ψ (r) in a Gaussian basis
HF
(cid:88)NG
ψ (r)= c exp(−α r2). (33)
HF j j
j=1
TheHFenergycanbedirectlyminimizedwithrespecttothecoefficientsc andexponentsα usinganumericalsolver
j j
[45], thus avoiding the self-consistent field procedure usually needed for this kind of calculation [46, 47].
Theexactwavefunctionandenergycanbefoundinclosedform[44]forcertainvaluesofκ. Forexample, forκ=2
(cid:16) r (cid:17)
Ψ (r ,r )= 1+ 12 ψ (r )ψ (r ), (34a)
2 1 2 2 0 1 0 2
E =2, (34b)
2
and, for κ=10,
(cid:18) r r2 (cid:19)
Ψ (r ,r )= 1+ 12 + 12 ψ (r )ψ (r ), (35a)
10 1 2 2 20 0 1 0 2
E =1/2. (35b)
10
Table II shows the convergence of E with N for κ=2 and κ=10. The correlation energy
HF G
E =E −E (36)
c exact HF
for κ=2 (E =38.438871755 mE ) agrees with earlier work [14, 46]. For κ=10, we find E =29.04152556 mE .
c h c h
B. Position Intracule
The non-interacting Position intracule is
(cid:114) 2 (cid:18) u2(cid:19)
P (u)= u2exp − . (37)
0 πκ3 2κ
and the HF intracule P (u) is found from (25a) and (33). For κ=2 and κ=10, the exact intracules are given by
HF,κ
(1+u/2)2 (cid:18) u2(cid:19)
P (u)= √ u2exp − , (38)
2 8+5 π 4
(1+u/2+u2/20)2 (cid:18) u2(cid:19)
P (u)= √ u2exp − . (39)
10 5/2(240+61 5π) 20
Equation (38) has been reported previously [14].
8
C. Momentum Intracule
Using the same notation as above, the non-interacting Momentum intracule is
2 (cid:18) κv2(cid:19)
M (v)= κ3/2v2exp − , (40)
0 π 2
M (v) and M (v) are obtained from (25b) and (33), and the exact Momentum intracules are
HF,2 HF,10
M2(v)= 8+8v52√π (cid:34)(cid:114)π2 +e−v22 +(cid:18)i1v +iv(cid:19)erf(cid:18)√iv2(cid:19)e−v22(cid:35)2, (41)
M10(v)= 48√850√+56v12√π (cid:34)(cid:114)1π0 +(cid:0)4−5v2(cid:1)e−5v22 +(cid:18)i1v +5iv(cid:19)erf(cid:32)(cid:114)52iv(cid:33)e−52v2(cid:35)2, (42)
where erf(z) is the error function [48]. Equation (41) has been reported previously [14].
D. Angle Intracule
The Angle intracule of two non-interacting particles is entirely determined by the Jacobian factor and is [28]
1
Υ (ω)= sinω. (43)
0 2
Υ (ω) and Υ (ω) are obtained from (25c) and (33). Υ (ω) and Υ (ω) have been obtained by numerical
HF,2 HF,10 2 10
integration of (4) and (5). Eq. (4) can be reduced to a two-dimensional integral, and the resulting four-dimensional
numerical integration in Eq. (5) was performed carefully to ensure accuracy of O(10−3) for each value of ω.
E. Dot and Posmom intracules
D(cid:98)HF,2(k) and D(cid:98)HF,10(k) [Eqs. (33) and (25d)], as well as X(cid:98)HF,2(k) and X(cid:98)HF,10(k) [Eqs. (33) and (25e)] have been
obtained numerically. Table III gathers the non-interacting and exact (κ= 2 and 10) Dot and Posmom intracules in
Fourier and real space. The similarity between the Dot and Posmom expressions is striking.
F. Holes
The correlation hole was originally defined [2] as the difference between the exact and HF Position intracule
∆P(u)=P(u)−P (u), (44)
HF
but this can be extended to any intracule I
∆I =I−I . (45)
HF
One can also define the HF hole as the difference between the HF and non-interacting intracules
∆I =I −I . (46)
HF HF 0
Fig. 2showsalloftheintraculesforκ=2andFig.3showstheholescreatedastheCoulombinteractionisintroduced.
One can see from P(u) and M(v) in Figs. 2(a) or 2(b) that the electrons are found at larger separations and move
withlowerrelativemomentaintheHFapproximationthaninthenon-interactingcase. However, thenon-interacting
and HF intracules, Υ(ω), D(x) and X(x), are almost identical. The fact that Υ(ω), D(x) and X(x) are all invariant
under a uniform scaling leads us to conclude that the introduction of the Coulomb operator at the mean-field level
leads to an almost exact dilation of the system.
Fig. 3 reveals that the HF holes and correlation holes of P(u) and M(v) are surprisingly similar in size and shape.
It also shows that the introduction of correlation decreases the probabilities of ω ≈ π/2 and x ≈ 0, indicating that
9
(cid:21)
)
is (πxh4
Γ n
rierandrealspacesfortwoparabolicquantumdots.Notethatofthesecondkind[48]. Posmom 13/2cosh(2k) (cid:104)(cid:105)√√8cosh(k)2π3π1++√23/25/28+5πcosh(2k)cosh(2k)cosh(2k) (cid:104)(cid:105)√√√80cosh(k)160cosh(k)165π305π155π++++233/25/27/2cosh(2k)cosh(2k)5πcosh(2k)cosh(2k)cosh(2k) (cid:12)(cid:12)(cid:1)(cid:0)213+ixΓ√(cid:12)(cid:12)432π(cid:20)(cid:21)√()()πxπx(cid:12)(cid:12)(cid:12)(cid:12)(cid:0)(cid:1)(cid:0)(cid:1)xsinh+cosh223+ix25+ix44Γ+2Γ+π(cid:12)(cid:12)(cid:12)(cid:12)√)()πx44+5ππcosh2(cid:20)2()()πx(cid:12)(cid:12)(cid:12)(cid:12)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)x+5cosh+4xsi2223+ix5+ix10π4x+25Γ+40Γ+√(cid:12)(cid:12)(cid:12)(cid:12)()πx445cosh2
ntsinFoufunction 1√240+61 (8 √5√)40+615π
eatmeBessel √(2π2
trd (cid:111)
exactmodifie (cid:21) (cid:105)˜K(x)3
Posmomintraculesforthenon-interactingandthe˜n8]andK=xK,whereKrepresentsthenthnnn Dot 13/22(1+k)(cid:20)(cid:21)√√ππ681−++√23/25/28+5π()()()2221+k1+k1+k(cid:20)√√803206015π5π−++√233/27/2()()+615π()()22221+k1+k1+k1+k 1xK(x)1π √(cid:104)(cid:105)˜˜˜1−K(x)+22K(x)+2K(x)√123/28π+5π √√(cid:110)(cid:104)(cid:105)(cid:104)˜˜˜202−K(x)+K(x)+5K(x)+413/25/25π
4 0 √
d[ 4 1
tantion 2 1√π+6
Donc 40
u 2
f
EIII.mma n-int. =2 =10 n-int. =2 =10
BLga No κ κ No κ κ
TAhe ecaps reiruoF ecaps laeR
t
10
P(cid:72)u(cid:76) M(cid:72)v(cid:76)
0.4
0.8
0.3
0.6
0.2
0.4
0.1
0.2
u v
1 2 3 4 5 6 0.5 1.0 1.5 2.0 2.5 3.0
(a)Position (b)Momentum
(cid:85)(cid:72)Ω(cid:76)
0.5
0.4
0.3
0.2
0.1
Ω
0.5 1.0 1.5 2.0 2.5 3.0
(c)Angle
D(cid:72)x(cid:76) X(cid:72)x(cid:76)
0.30
0.25 0.15
0.20
0.10
0.15
0.10
0.05
0.05
x x
1 2 3 4 5 6 7 1 2 3 4 5 6 7
(d)Dot (e)Posmom
FIG. 2. Intracules for a parabolic quantum dot with κ=2: non-interacting(—), HF (- - -) and exact (···).
the correlated electrons spend less time circularly orbiting their centre of mass. This conclusion is supported by both
the non-rigorous Υ(ω) and D(x) intracules and the rigorous X(x) intracule.
However, the differences between D(x) and X(x) are significant. D(x) comes from the Wigner distribution and, as
Eqs (13) and (16) show, it is the O((cid:126)) approximation to X(x). Its Fourier transform D(cid:98)(k) decays as k−3 for large
k and this creates a discontinuity in the second derivative D(cid:48)(cid:48)(x) at x = 0 [30, 49]. In contrast, X(x) is smooth at
x=0. One of the consequences of this misbehaviour at x=0 is that, for the κ=2 quantum dot, the Dot intracule’s
prediction D(0)=0.219 overestimates the exact value X(0)=0.168 by 30%.