A critical theory of quantum
entanglement for the Hydrogen molecule
Tina A.C. Maiolo∗, Luigi Martina†, Giulio Soliani‡
Abstract
In this paper we investigate some entanglement properties for the Hy-
drogen molecule considered as a two interacting spin 1
(qubit) model.
The entanglement related to the H2 molecule is evaluated both using the
von Neumann entropy and the Concurrence and it is compared with the
corresponding quantities for the two interacting spin system. Many as-
pects of these functions are examinated employing in part analytical and,
essentially, numerical techniques. We have compared analogous results
obtained by Huang and Kais a few years ago. In this respect, some pos-
sible controversial situations are presented and discussed.
1 Introduction and the model
Entanglement is a physical observable measured by the von Neumann entropy
or, alternatively, by the Concurrence of the system under consideration.
The concept of entanglement gives a physical meaning to the electron cor-
relation energy in structures of interacting electrons. The electron correlation
is not directly observable, since it is defined as the difference between the ex-
act ground state energy of the many electrons Schrödinger equation and the
Hartree–Fock energy.
In this paper we discuss the Hamiltonian which describes the Hydrogen
molecule regarded as a two interacting spin 1
(qubit) model.
In [1] it was argued that the entanglement (a quantum observable) can be
used in analyzing the so–called correlation energy which is not directly observ-
able. From our point of view, the Hydrogen molecule is dealt with a bipartite
system governed by the Hamiltonian
HH2 = −
(1 + g)σ1 ⊗ σ1 −
(1− g)σ2 ⊗ σ2 − B(σ3 ⊗ σ3 + σ0 ⊗ σ3), (1)
∗Dipartimento di Fisica dell’Università del Salento and Sezione INFN di Lecce, 73100
Lecce, Italy; e–mail: tina.maiolo@le.infn.it
†Dipartimento di Fisica dell’Università del Salento and Sezione INFN di Lecce, 73100
Lecce, Italy; e–mail: luigi.martina@le.infn.it
‡Dipartimento di Fisica dell’Università del Salento and Sezione INFN di Lecce, 73100
Lecce, Italy; e–mail: giulio.soliani@le.infn.it
http://arxiv.org/abs/0704.0520v2
where σi stand for the Pauli matrices (σ0 = I). Actually, this model was con-
sidered in [1] in order to illustrate their method. However, here we will make
some interpretative changes. Indeed, from our point of view, the states of an
isolated atom are strongly reduced to a system with two energy levels related to
the intensity of the magnetic field B. Relatively to this scale, the exchange in-
teraction constant J is usually smaller than B, in order to represent the residual
interatomic interactions. From the point of view of quantum chemistry, one may
interpret the discrete spectrum as provided by the Hartree–Fock calculations,
while the interaction coupling J models the residual multielectronic effects, not
taken into account by the mean field approximation.
For simplicity we limit ourselves to the ferromagnetic phase with J > 0.
The parameter g, such that 0 ≤ g ≤ 1, describes the degree of anisotropy
corresponding for g = 0 to the completely isotropic XY spin model. Conversely,
g = 1 provides the anisotropic XY spin model, the so-called Ising model.
We notice that when the atoms are far apart, their interaction is quite weak.
This corresponds to a vanishing value of J . In this situation the state of the
system is completely factorized in the product state of the ground states of the
indipendent spins. The corresponding total energy, in unit of B, is just the sum
of the two fundamental levels, E0 = −2, which we may consider as the Hartree-
Fock approximated fundamental level in molecular structure calculations.
When J 6= 0, the fundamental energy eigenvalue is E= −
4 + g2λ2 in Re-
gion I defined by 0 < λ ≤ 2√
, otherwise E = −λ (λ means the coupling
constant) in Region II, which is the complement of I which respect to pos-
itive real axis. The corresponding (non normalized) eigenstates are |ΨI〉 =
g2λ2+4+2
, 0, 0, 1
and |ΨII〉 =
0, 1, 1, 0
, respectively. In both cases the
state is entangled.
Since we are dealing with pure states, the von Neumann entropy [2]
SvN = −Tr
ρ1log2ρ1
is chosen to be a measurement of the entanglement, where ρ1 is the 1-particle
reduced density matrix. However, for general mixed states other entanglement
estimators (for instance, the Concurrence [4]) have to be used. In the considered
case, one has
SvN,I = −
g2λ2 + 4 log
g2λ2 + 4
g2λ2 + 4 + 4
λ2 + 8
g2λ2 + 4 + 2
g2λ2 + 4
g2λ2 + 4
g2λ2 + 2
g2λ2 + 4 + 4
log(4)
SvN,II = 1. (4)
Scrutinizing Eq. (3) and Eq. (4) it emerges that the entropy is an increasing
function of the coupling constant λ in Region I, but the state is maximally
entangled in Region II independently from the anisotropy parameter g. One
sees that, as it arises graphycally, for g = 1 the entanglement is a monotonic
increasing function of the interaction coupling λ. Moreover for weak (< 1)
coupling values it is always less than the 30%. Of course, for large coupling
constants the entropy approaches 1, meaning that all levels are equiprobably
visited by the considered spin.
Limiting all further considerations to the case of weak interaction, we observe
that at the boundary point λb =
a discontinuity occurs, signaling a
crossing of the lowest eigenvalues and, in a more general context, a quantum
phase transition [5].
As it was pointed out in [6], for quantifying the entanglement we can resort
to the reduced density matrix. Furthermore, in [7], Wootters has shown that
for a pair of binary qubits one can use the concept of Concurrence C to measure
the entanglement.
The Concurrence reads
C(ρ) = max(0, ν1 − ν2 − ν3 − ν4), (5)
where the νi’s are the eigenvalues of the Hermitian matrix
where ρ̃ = (σy ⊗ σy)ρ∗(σy ⊗ σy), ρ∗ being the complex conjugate of ρ taken in
the standard basis [7].
Some interesting results on the simple model (1) of the Hydrogen molecule
can be achieved by realizing a comparative study of the von Neumann entropy
and the Concurrence.
To this aim, we compute the Concurrence CI and CII, i. e.
CI = gλ
g2λ2 + 4
, CII = 1. (6)
where I and II refer to Regions I and II, where 0 ≤ λ ≤ 2
1−g2 , and E = −λ,
respectively.
In Figure 1 a comparison between the Concurrence and the von Neumann
entropy for two spins system as a function of the coupling λ for g = 1 is pre-
sented.
Sec. 2 contains a comparison between the entanglement and the correlation
energy. In Sec. 3 the Configuration Interaction method is introduced to compare
entanglement and correlation energy. In Sec. 4 some differences between the
Configuration Interaction approach and the two spin Ising model are presented.
Finally, our main results are summarized in Sec. 5.
1 2 3 4 5
Conc.
Figure 1: Comparison between the Concurrence and the von Neumann entropy
for the two spins system as a function of the coupling constant λ for g = 1.
2 A comparison between the entanglement and
the correlation energy
Now we look for a comparison between the entanglement with the energy cor-
relation, which as we have already recalled, it is understood as the difference of
the fundamental energy level compared with respect to the corresponding value
at vanishing coupling constant λ.
For g = 1 and in unities of B it is given by
Ecorr = |E0| − 2 =
4 + λ2 − 2. (7)
We observe that the entanglement measure is always bounded, while Ecorr is
a divergent function of λ. So it does not make much sense to look for simple
relations valid on the entire λ-axes. Consequently, limiting ourselves to weak
couplings, for 0 ≤ λ ≤ 1, we minimize the mean squared deviation
∆S2α dλ, with ∆Sα = Ecorr − αSvN . (8)
Thus the minimizing parameter αmin will be given by
αmin =
EcorrSvN dλ
≈ −0.691217. (9)
A formula analogous to (9) can be obtained by using the Concurrence as
a measure of entanglement. In this case, by minimizing the mean squared
deviation we have
∆C2α′ dλ, with ∆Cα′ = Ecorr − α′ C. (10)
Now, in order to estimate the relative deviation of SvN with respect to Ecorr,
let us report |∆Sαmin |/SvN and |∆Sαmin/Ecorr| as functions of λ at the optimal
value αmin. The graphs of these functions are shown in Figure 2.
0.2 0.4 0.6 0.8 1
ÈDSmin�SvN È
0.2 0.4 0.6 0.8 1
ÈDSmin�EcorrÈ
Figure 2: The relative quadratic deviation between the von Neumann entropy
and the correlation energy with respect to the former and the latter, respectively,
at the optimal value αmin as a function of the coupling constant λ for g = 1.
In Figure 3, the relative quadratic deviation between the Concurrence and
the correlation energy with respect to the former and the latter, at the optimal
values α′min, is represented.
0.2 0.4 0.6 0.8 1
ÈDCminÈ�C
0.2 0.4 0.6 0.8 1
ÈDCmin�EcorrÈ
Figure 3: The relative quadratic deviation between the Concurrence and the
correlation energy with respect to the former and the latter, respectively, at the
optimal value α′min as a function of the coupling constant λ for g = 1.
Remark 1
From these graphs, one can argue that the agreement between the two func-
tions SvN and Ecorr is only qualitatively good, in fact, for very small λ, it is not
good at all. However, in an intermediate range of values, i. e., 0.6 ≤ λ ≤ 1 the
two functions are almost proportional within the 10%. Analogously, the same
is true between energy and Concurrence. Even, the agreement becomes worst
comparing the relative deviation of the Concurrence with respect to the corre-
lation energy, since the range in which the relative deviations become smaller
than 10% are narrower. Then, the question is whether the above results are
i) sufficient to justify the conjecture advanced in [1], i.e., entanglement can be
considered as an estimation of correlation energy; ii) if such a relation has a
more concrete physical meaning, in particular whether the minimizing parame-
ter αmin and the vanishing point of ∆Sαmin does possess any physical meaning
(or α′min and the vanishing point of ∆Cα′min). Notice that in the case of the
comparison for the Concurrence simpler analytical expressions appear. For in-
stance one finds ∆Cα′
0.383249 λ√
λ2 + 4 + 2
Remark 2
We note that in an interval of values around αmin, the deviation function
(8) possesses a minimum in the interval of interest 0 ≤ λ ≤ 1, otherwise the
minimum is achieved at larger value of λ, or the function is monotonically
increasing (see Figure 4).
0.2 0.4 0.6 0.8 1
-0.05
0.2 0.4 0.6 0.8 1
-0.05
d DSΑ � dΛ
Figure 4: The deviation ∆Sα and its derivative with respect to λ are computed
for values of −1.29(red) ≤ α ≤ −0.091(violet), for steps of 0.06. The curve
drawn thicker corresponds to αmin
This behavior suggests to consider the function ∆Sαmin as a sort of ”free en-
ergy” , where αmin mimics the ”temperature” specific of the system. If, for some
reason, we allow λ to change, then we expect that spontaneously the interaction
coupling adjusts itself to the minimum of ∆Sαmin . Similar considerations can
be made looking at the graphs drawn for the function ∆Cα′
and its derivative
with respect to λ (see Figure 5).
The function ∆Sαmin or, alternatively, the minimum of ∆Cα′
can be ob-
tained algebraically. Such a minimum is at the value of the coupling constant
λSvNmin ≈ 0.485 and λCmin ≈ 0.371, respectively.
The authors in [1] studied numerically the von Neumann entropy and the
correlation function for a Hydrogen molecule, using an old result by Herring and
Flicker [8], going back to an oldest idea by Heitler and London [9], which con-
sists in substituting the molecular binding with a position dependent exchange
coupling:
J(r) ≈ 1.641 r
2 e−2 r Ry, (11)
where r is given in Bohr radius, see Figure 6. The maximum value taken by this
function is at the point rmax = 1.25. Assuming B = 0.5 Ry, i.e. 12 of the funda-
mental level of the Hydrogen atom, the maximum value λ′max = J(rmax)/B ≈
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1
dDCΑ'� dΛ
Figure 5: The deviation ∆Cα′ and its derivative with respect to λ are computed
for values of −0.98(red) ≤ α′ ≤ 0.22(violet), for steps of 0.06. The curve drawn
thicker corresponds to α′min
0.5 1 1.5 2 2.5 3
JHrLHRyL
Figure 6: The effective interaction Hydrogen-Hydrogen atom
0.470628 < λSvNmin , i.e. the value of the effective interaction value is less than
the minimum for the deviation function ∆Sαmin . Then, the equilibrium bal-
ance between entanglement (as von Neumann entropy) and correlation energy
predicts a length of the molecule equal to rmax (see the first panel of Figure 7).
On the other hand, if we consider the energy gap 2B = 3/4 Ry, i.e. the energy
step to the first excited state, one obtains the new value λ′′max ≈ 0.628, which
goes beyond λmin, even if it is always less than 1. Now, the deviation function
∆Sαmin has two minima as seen in the second panel of Figure 7, one of which
is at r′′− ≈ 0.76 , the other one being at r′′+ ≈ 1.91.
These results should be compared with the experimental equilibrium length
of the Hydrogen molecule, which is rexp ≈ 2.0.
We point out that although the spin–model described by the Hamiltonian
(1) is characterized by features which are essentially rough, however we are
induced to answer positively to the quest for a physical meaning of the deviation
function ∆Sαmin . Indeed, the results elucidated in Figure 7 encourage, on one
part, improvement of the computation of r in order to make more accurate the
comparison with the experimental value rexp.
0.5 1 1.5 2 2.5 3
-0.015
-0.0125
-0.01
-0.0075
-0.005
-0.0025
DS minHr; B= .5 RyL
0.5 1 1.5 2 2.5 3
-0.015
-0.0125
-0.01
-0.0075
-0.005
-0.0025
DS minHr; B= .375 RyL
r''- r''+
Figure 7: The von Neumann entropy for the 2-spin model for B = .5 Ry (left
panel) and for B = .375 Ry (right panel) and the position depending interaction
given by (11).
The first question to answer is whether this draft works also for the Concur-
rence. A statement about it is not obvious, since the von Neumann entropy is
a nonlinear function of the Concurrence in the 2-qubits case.
However, from Figure 8 one can see that the minimized deviation of the
Concurrence takes one minimum for relatively large intensity of the magnetic
field ( say B ≥ 0.6 Ry), while for weak fields two minima appear, corresponding
to the situation depicted nearby.
0 0.2 0.4 0.6 0.8 1
r HBohrL
B HRyL
0.2 0.4 0.6 0.8 1
       
         B(Ry)
r(Bohr)
Figure 8: Two contour plots of the minimized deviation of the Concurrence as a
function of the magnetic field B (Ry) and of the internuclear distance r, as given
by (11). The range of values divided by the contour lines is [−0.038, 0, 04] for
the left panel and [−0.03705, −0, 03000] for the right one that approximatively
corresponding to the black area in the left panel.
In correspondence of the same values considered above, for B = 0.5 Ry
the function ∆Cα′
(r) has two minima at r = 0.79 and r = 1.88, while for
B = 0.375 Ry they are located at r = 0.60 and r = 2.25. So one sees that the
resulting equilibrium configurations are not much very close to the experimental
one. The equilibrium configuration more closest to the experimental one is the
minimum occurring at r = 1.88 (B = 1
Ry) for the function ∆Cα′
One sees that one of the resulting equilibrium configurations is only roughly
close to the experimental one.
In other words, to conclude monitoring numerically B the equilibrium config-
uration more closest to experimental one in the minimum occurring at r = 1.88
for B = 1
Ry and at r = 2.25 for B = 0.375 Ry for the function ∆Cα′
3 A quantum chemical framework to compare
entanglement and correlation energy
In this Section we represent the results produced in [1], where the electron entan-
glement in the Hydrogen molecule, calculated by the von Neumann entropy of
the reduced density matrix ρ1, is obtained starting by the excitation coefficients
of the wave function expanded by a configuration interaction method:
ρCISD1
= −Tr
ρCISD1 log2ρ
|c2i+11 |2 +
|c2i+1,2i+21,2 |2
|c2i+11 |2 +
|c2i+1,2i+21,2 |2
|c0|2 +
|c2i+22 |2
|c0|2 +
|c2i+22 |2
, (12)
where c1 is the coefficient for a single excitation, and c1,2 is the double
excitation (in Appendix A of [10] more details are shown).
In this framework, entanglement (S) and correlation energy (Ecorr), as func-
tions of nucleus – nucleus separation are those in Figure 9
0 1 2 3 4 5
R ( Å ) 
S ( ρ1
Figure 9: Comparison between the entanglement, calculated by the von Neu-
mann entropy of the reduced density matrix, and the electron correlation energy
in the Hydrogen molecule.
By the results given by this model, we want to discuss and to suggest some
answers to the questions i) and ii) presented in Remark 1. Even if, in order
to represent correlation energy and entanglement, we use two different scales,
in Figure 9 we can see that entanglement has a small value in the united atom
limit after it is growing for small distances till it arrives at a maximum value
then it decrease till it assumes zero value at the separated atom limit and it is
exactly the progress of the correlation curve.
In order to compare the entropy S with the electron correlation energy Ecorr,
we rescale S with the parameter αmin calculated with some procedure illustrated
in Eq. (8) and Eq. (9) replacing the integration variable λ with R; in this way
we extract
EcorrSvNdR
SvNdR
≈ 0.009. (13)
The corresponding ∆Sαmin = Ecorr−αS allows us to answer to the question
ii); in fact, as it is shown in Figure 10, the vanishing point of ∆Sαmin is,
according to the two –spin Ising model, nearby R ≈ 2 Å that corresponds to
the equilibrium configuration of the Hydrogen molecule.
0 1 2 3 4 5
−0.01
Figure 10: ∆Sαmin for theH2 molecule as a function of nucleus–nucleus distance.
4 Differences between the Configuration Inter-
action approach and the two–spin Ising model
The model proposed in Sec. 1 provides us with a measurement of entanglement:
indeed, Eq. (3) describes the von Neumann entropy as a function of coupling
constant λ, for small λ. By using Eq. (7), we can express λ in terms of corre-
lation energy and substituting it in Eq. (3) we can obtain the variation of SvN
in terms of Ecorr.
SvN = −
EcorrLog
Ecorr
2(Ecorr+2)
+ (Ecorr + 4)Log
Ecorr+4
2(Ecorr+2)
(Ecorr + 2)Log4
. (14)
In order to calculated the coefficient of proportionality among SvN and Ecorr
we make an expansion of SvN for Ecorr → 0 (or equivalently for λ → 0) at the
first order, obtaining a straight line characterized by an angular coefficient given
by mSvN (Ecorr) = (
)(1 + 1
). Since this behavior is uncorrect to represent
the logatithmic singularity of SvN in the origin, we make an expansion of Eq.
(14), preserving the logarithmic deviation, and we obtain an expression of the
SvN = AEcorr +BEcorrLog(Ecorr), (15)
where A = 1/2 and B = −1/(4Log2).
0.02 0.04 0.06 0.08 0.1
0.025
0.075
0.125
Ecorr   
Linear 
AE+BELogE
Figure 11: A comparison among the behavior of Eq. (14) and its linear approx-
imation and the logarithmic one, for the Ising model.
In order to compare the behavior of SvN in Eq. (14), we have organized
the numerical data, calculated with the method proposed in [1], by making a
correspondence between each value of Ecorr and its respective value of SvN ,
obtaining the plot in Figure 12
0 0.01 0.02 0.03 0.04 0.05
Figure 12: A correspondence of Ecorr and SvN calculated by the numerical
procedure suggested by [1]
Of particular significance is the fact that, in the range where S is monotoni-
cally increasing, the correlation energy has its maximum, consequently S seems
to be not a function. Moreover, it is important to note that Ecorr begins to
decrease for R > 1 Å, region where the states become mixed, i. e. ,Trρ 6= Trρ2;
as depicted in Figure 13.
0 1 2 3 4
R(Å )
Figure 13: The increasing of the degree of mixing in the two electron state: in
black we depict the trace of ρ, in red the trace of ρ2.
Probably, for this reason, the procedure adopted in [1] seems to be not cor-
rect: the density matrix, in fact, is calculated starting by the excitation coeffi-
cient of a wave function obtained developping with the Configuration Interaction
Single Double method a pure two electrons state.
However, even if we consider only the first branch of the plot in Figure 12,
i.e. , the numerical values of SvN corresponding with increasing values of Ecorr,
and we fit the values around Ecorr → 0 with a F = AEcorr + BEcorrLog(Ecorr)
we draw out numerical values of the coefficient different from the ones used in
Eq. (15). This result is shown in Figure 14.
0.01 0.02 0.03 0.04 0.05
0.35 S
Ecorr
A=17.1
B=3.3
Figure 14: A fit of SvN as a function of Ecorr, around the origin, with a function
of the form F = AEcorr +BEcorrLog(Ecorr) whose coefficients A and B assume
the numerical values in Figure.
In particular the arithmetic sign of the coefficient B in the two models are
opposite and this implies the opposite concavity of the curve.
This fact, clearly demonstrates a not satisfactory agreement between the
Ising model and the one proposed in [1].
5 Concluding remarks
We have explored the role of entanglement in the model of two qubits describing
the Hydrogen molecule (1), considered as a bipartite system. In our discussion
we have limited to the ferromagnetic case governed by the interaction coupling
parameter J > 0.
The concept of entanglement gives a physical meaning to the electron cor-
relation energy in structures of interacting electrons. The entanglement can be
measured by using the von Neumann entropy or, alternatively, the notion of
Concurrence [7]. To compute the entanglement it is convenient to consider two
Regions, say I and II, which provide two different reduced density matrices.
The entropy turns out to be an increasing function of the coupling constant λ
in Region I, but the state under consideration is maximally entangled in Region
II indipendently from the anisotropy parameter g.
An interesting result is that for large coupling constants the entropy ap-
proach 1, meaning that all levels are equiprobably visited by the considered
spin.
For weak interactions, at the boundary point λb =
the von Neumann
entropy admits a discontinuity, indicating a crossing of the lowest eigenvalues
and, in a more general constext, a quantum phase transition [5].
In Sec. 2 a comparison between the entanglement and the correlation energy
is performed.
To quantifying the entanglement we resort to the reduced density matrix.
The entanglement can also be measured by exploiting the concept of Concur-
rence.
The entanglement measure is always bounded, while the energy correlation,
Ecorr = |E0| − 2 =
4 + λ2 − 2, is a divergent function of λ. This fact tells us
that to look for simple relations valid on the whole λ−axes has no sense.
Thus, by limiting ourselves to weak couplings, we have minimized the mean
square deviation given by Eq. (8). This procedure leads to the value αmin ≈
−0.691217 for the minimizing parameter (see Eq. (9)).
Sec. 1 contains a comparison between the von Neumann entropy and the
Concurrence.
Such a comparison is illustrated in Figure 1, for two spin system as a function
of the coupling λ for g = 1.
Some important points are commented in Remark 1 and Remark 2 .
In Figure 4 the deviation ∆Sα and its derivatives with respect to λ are
computed and αmin is evaluated for α ranging in the interval −1.29 ≤ α ≤
−0.091.
In Figure 5 the minimized Concurrence deviation ∆C
for the four eigen-
states of the 2-spin model is shown.
We point out the existence of a perfect symmetry among the Concurrence
deviations for pairs of eigenstates of opposite eigenvalues.
Formula (11), due to Heitler–London [9], is reported, where the position
dependent exchange coupling J(r) is expressed in term of the length r of the
nucleus–nucleus separation in the Hydrogen molecule.
To conclude, the magnetic field B has been monitored such that the equi-
librium configuration more closest to the experimental one, r ≈ 2.00, is the
minimum occurring at r = 1.88 for B = 1
Ry and r = 2.25 for B = 0.375 Ry
for the function ∆Cα′
We observe also that in the intermediate range of values, i. e., for 0.6 ≤ λ ≤
1, the two functions SvN and the correlation energy are almost proportional
within the 10%.
However, when we organized the pairs of points (Ecorr, SvN ) calculated by
following the procedure described by [1], it is clear that the von Neumann en-
tropy cannot be considered a function of correlation energy. The principle cause
is that the function Ecorr presents a maximum in the region where SvN is mono-
tonically increasing.
The reversing behavior of correlation energy occurs in correspondence with
an increase of the mixing degree of the two electrons state. The function Ecorr
in terms of the nucleus – nucleus distance R, increases till the state is pure, on
the contrary, when Tr(ρ2) becomes discordant from Tr(ρ), the function Ecorr
decreases.
This fact suggests us that the numerical model based on the calculation of
SvN starting by the excitation coefficients ci, isn’t completley correct because
the density matrix is obtained as a product of two electron pure states. However,
even if we consider only a branch of the plot in Figure 12, the function obtained
by the two spin Ising model, i. e., Eq. (14), is unsuitable for fitting these
numerical data.
On the basis of our results, essentially grounded on numerical considerations,
in the near feature we would explore more complicated systems of molecules,
such as for example the ethylene or other hydrocarbons, and compare these
studies with the goals obtained for the Hydrogen molecule.
Acknowledgments
The authors acknowledge the Italian Ministry of Scientific Researches (MIUR)
for partial support of the present work under the project SINTESI 2004/06 and
the INFN for partial support under the project Iniziativa Specifica LE41.
References
[1] Z. Huang, S. Kais, Chem. Phys. Lett. 413, 1 (2005).
[2] M. A. Nielsen and I. L. Chuang Quantum Computation and Quantum In-
formation, Cambridge Univ. Press, Cambridge, 2000.
[3] D. M. Collin, Z. Naturforsch A 48, 68 (1993).
[4] P. Rungta and C. M. Caves Phys. Rev. A 67, 012307 (2003).
[5] S. Sachdev Quantum Phase Transition, Cambridge University Press, 2001.
[6] O. Osenda, Z. Huang and S. Kais Phys. Rev A 67, 062321 (2003).
[7] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
[8] C. Herring and M. Flicker, Phys. Rev. A 134, 362 (1964).
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[10] T. Maiolo, F. Della Sala, L. Martina, G. Soliani arXiv: quant–ph/ 0610238
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http://arxiv.org/abs/quant--ph/0610238
	Introduction and the model 
	A comparison between the entanglement and the correlation energy 
	A quantum chemical framework to compare entanglement and correlation energy 
	Differences between the Configuration Interaction approach and the two–spin Ising model 
	Concluding remarks

  In this paper we investigate some entanglement properties for the Hydrogen
molecule considered as a two interacting spin 1/2 (qubit) model. The
entanglement related to the $H_{2}$ molecule is evaluated both using the von
Neumann entropy and the Concurrence and it is compared with the corresponding
quantities for the two interacting spin system. Many aspects of these functions
are examinated employing in part analytical and, essentially, numerical
techniques. We have compared analogous results obtained by Huang and Kais a few
years ago. In this respect, some possible controversial situations are
presented and discussed.

Introduction and the model
Entanglement is a physical observable measured by the von Neumann entropy
or, alternatively, by the Concurrence of the system under consideration.
The concept of entanglement gives a physical meaning to the electron cor-
relation energy in structures of interacting electrons. The electron correlation
is not directly observable, since it is defined as the difference between the ex-
act ground state energy of the many electrons Schrödinger equation and the
Hartree–Fock energy.
In this paper we discuss the Hamiltonian which describes the Hydrogen
molecule regarded as a two interacting spin 1
(qubit) model.
In [1] it was argued that the entanglement (a quantum observable) can be
used in analyzing the so–called correlation energy which is not directly observ-
able. From our point of view, the Hydrogen molecule is dealt with a bipartite
system governed by the Hamiltonian
HH2 = −
(1 + g)σ1 ⊗ σ1 −
(1− g)σ2 ⊗ σ2 − B(σ3 ⊗ σ3 + σ0 ⊗ σ3), (1)
∗Dipartimento di Fisica dell’Università del Salento and Sezione INFN di Lecce, 73100
Lecce, Italy; e–mail: tina.maiolo@le.infn.it
†Dipartimento di Fisica dell’Università del Salento and Sezione INFN di Lecce, 73100
Lecce, Italy; e–mail: luigi.martina@le.infn.it
‡Dipartimento di Fisica dell’Università del Salento and Sezione INFN di Lecce, 73100
Lecce, Italy; e–mail: giulio.soliani@le.infn.it
http://arxiv.org/abs/0704.0520v2
where σi stand for the Pauli matrices (σ0 = I). Actually, this model was con-
sidered in [1] in order to illustrate their method. However, here we will make
some interpretative changes. Indeed, from our point of view, the states of an
isolated atom are strongly reduced to a system with two energy levels related to
the intensity of the magnetic field B. Relatively to this scale, the exchange in-
teraction constant J is usually smaller than B, in order to represent the residual
interatomic interactions. From the point of view of quantum chemistry, one may
interpret the discrete spectrum as provided by the Hartree–Fock calculations,
while the interaction coupling J models the residual multielectronic effects, not
taken into account by the mean field approximation.
For simplicity we limit ourselves to the ferromagnetic phase with J > 0.
The parameter g, such that 0 ≤ g ≤ 1, describes the degree of anisotropy
corresponding for g = 0 to the completely isotropic XY spin model. Conversely,
g = 1 provides the anisotropic XY spin model, the so-called Ising model.
We notice that when the atoms are far apart, their interaction is quite weak.
This corresponds to a vanishing value of J . In this situation the state of the
system is completely factorized in the product state of the ground states of the
indipendent spins. The corresponding total energy, in unit of B, is just the sum
of the two fundamental levels, E0 = −2, which we may consider as the Hartree-
Fock approximated fundamental level in molecular structure calculations.
When J 6= 0, the fundamental energy eigenvalue is E= −
4 + g2λ2 in Re-
gion I defined by 0 < λ ≤ 2√
, otherwise E = −λ (λ means the coupling
constant) in Region II, which is the complement of I which respect to pos-
itive real axis. The corresponding (non normalized) eigenstates are |ΨI〉 =
g2λ2+4+2
, 0, 0, 1
and |ΨII〉 =
0, 1, 1, 0
, respectively. In both cases the
state is entangled.
Since we are dealing with pure states, the von Neumann entropy [2]
SvN = −Tr
ρ1log2ρ1
is chosen to be a measurement of the entanglement, where ρ1 is the 1-particle
reduced density matrix. However, for general mixed states other entanglement
estimators (for instance, the Concurrence [4]) have to be used. In the considered
case, one has
SvN,I = −
g2λ2 + 4 log
g2λ2 + 4
g2λ2 + 4 + 4
λ2 + 8
g2λ2 + 4 + 2
g2λ2 + 4
g2λ2 + 4
g2λ2 + 2
g2λ2 + 4 + 4
log(4)
SvN,II = 1. (4)
Scrutinizing Eq. (3) and Eq. (4) it emerges that the entropy is an increasing
function of the coupling constant λ in Region I, but the state is maximally
entangled in Region II independently from the anisotropy parameter g. One
sees that, as it arises graphycally, for g = 1 the entanglement is a monotonic
increasing function of the interaction coupling λ. Moreover for weak (< 1)
coupling values it is always less than the 30%. Of course, for large coupling
constants the entropy approaches 1, meaning that all levels are equiprobably
visited by the considered spin.
Limiting all further considerations to the case of weak interaction, we observe
that at the boundary point λb =
a discontinuity occurs, signaling a
crossing of the lowest eigenvalues and, in a more general context, a quantum
phase transition [5].
As it was pointed out in [6], for quantifying the entanglement we can resort
to the reduced density matrix. Furthermore, in [7], Wootters has shown that
for a pair of binary qubits one can use the concept of Concurrence C to measure
the entanglement.
The Concurrence reads
C(ρ) = max(0, ν1 − ν2 − ν3 − ν4), (5)
where the νi’s are the eigenvalues of the Hermitian matrix
where ρ̃ = (σy ⊗ σy)ρ∗(σy ⊗ σy), ρ∗ being the complex conjugate of ρ taken in
the standard basis [7].
Some interesting results on the simple model (1) of the Hydrogen molecule
can be achieved by realizing a comparative study of the von Neumann entropy
and the Concurrence.
To this aim, we compute the Concurrence CI and CII, i. e.
CI = gλ
g2λ2 + 4
, CII = 1. (6)
where I and II refer to Regions I and II, where 0 ≤ λ ≤ 2
1−g2 , and E = −λ,
respectively.
In Figure 1 a comparison between the Concurrence and the von Neumann
entropy for two spins system as a function of the coupling λ for g = 1 is pre-
sented.
Sec. 2 contains a comparison between the entanglement and the correlation
energy. In Sec. 3 the Configuration Interaction method is introduced to compare
entanglement and correlation energy. In Sec. 4 some differences between the
Configuration Interaction approach and the two spin Ising model are presented.
Finally, our main results are summarized in Sec. 5.
1 2 3 4 5
Conc.
Figure 1: Comparison between the Concurrence and the von Neumann entropy
for the two spins system as a function of the coupling constant λ for g = 1.
2 A comparison between the entanglement and
the correlation energy
Now we look for a comparison between the entanglement with the energy cor-
relation, which as we have already recalled, it is understood as the difference of
the fundamental energy level compared with respect to the corresponding value
at vanishing coupling constant λ.
For g = 1 and in unities of B it is given by
Ecorr = |E0| − 2 =
4 + λ2 − 2. (7)
We observe that the entanglement measure is always bounded, while Ecorr is
a divergent function of λ. So it does not make much sense to look for simple
relations valid on the entire λ-axes. Consequently, limiting ourselves to weak
couplings, for 0 ≤ λ ≤ 1, we minimize the mean squared deviation
∆S2α dλ, with ∆Sα = Ecorr − αSvN . (8)
Thus the minimizing parameter αmin will be given by
αmin =
EcorrSvN dλ
≈ −0.691217. (9)
A formula analogous to (9) can be obtained by using the Concurrence as
a measure of entanglement. In this case, by minimizing the mean squared
deviation we have
∆C2α′ dλ, with ∆Cα′ = Ecorr − α′ C. (10)
Now, in order to estimate the relative deviation of SvN with respect to Ecorr,
let us report |∆Sαmin |/SvN and |∆Sαmin/Ecorr| as functions of λ at the optimal
value αmin. The graphs of these functions are shown in Figure 2.
0.2 0.4 0.6 0.8 1
ÈDSmin�SvN È
0.2 0.4 0.6 0.8 1
ÈDSmin�EcorrÈ
Figure 2: The relative quadratic deviation between the von Neumann entropy
and the correlation energy with respect to the former and the latter, respectively,
at the optimal value αmin as a function of the coupling constant λ for g = 1.
In Figure 3, the relative quadratic deviation between the Concurrence and
the correlation energy with respect to the former and the latter, at the optimal
values α′min, is represented.
0.2 0.4 0.6 0.8 1
ÈDCminÈ�C
0.2 0.4 0.6 0.8 1
ÈDCmin�EcorrÈ
Figure 3: The relative quadratic deviation between the Concurrence and the
correlation energy with respect to the former and the latter, respectively, at the
optimal value α′min as a function of the coupling constant λ for g = 1.
Remark 1
From these graphs, one can argue that the agreement between the two func-
tions SvN and Ecorr is only qualitatively good, in fact, for very small λ, it is not
good at all. However, in an intermediate range of values, i. e., 0.6 ≤ λ ≤ 1 the
two functions are almost proportional within the 10%. Analogously, the same
is true between energy and Concurrence. Even, the agreement becomes worst
comparing the relative deviation of the Concurrence with respect to the corre-
lation energy, since the range in which the relative deviations become smaller
than 10% are narrower. Then, the question is whether the above results are
i) sufficient to justify the conjecture advanced in [1], i.e., entanglement can be
considered as an estimation of correlation energy; ii) if such a relation has a
more concrete physical meaning, in particular whether the minimizing parame-
ter αmin and the vanishing point of ∆Sαmin does possess any physical meaning
(or α′min and the vanishing point of ∆Cα′min). Notice that in the case of the
comparison for the Concurrence simpler analytical expressions appear. For in-
stance one finds ∆Cα′
0.383249 λ√
λ2 + 4 + 2
Remark 2
We note that in an interval of values around αmin, the deviation function
(8) possesses a minimum in the interval of interest 0 ≤ λ ≤ 1, otherwise the
minimum is achieved at larger value of λ, or the function is monotonically
increasing (see Figure 4).
0.2 0.4 0.6 0.8 1
-0.05
0.2 0.4 0.6 0.8 1
-0.05
d DSΑ � dΛ
Figure 4: The deviation ∆Sα and its derivative with respect to λ are computed
for values of −1.29(red) ≤ α ≤ −0.091(violet), for steps of 0.06. The curve
drawn thicker corresponds to αmin
This behavior suggests to consider the function ∆Sαmin as a sort of ”free en-
ergy” , where αmin mimics the ”temperature” specific of the system. If, for some
reason, we allow λ to change, then we expect that spontaneously the interaction
coupling adjusts itself to the minimum of ∆Sαmin . Similar considerations can
be made looking at the graphs drawn for the function ∆Cα′
and its derivative
with respect to λ (see Figure 5).
The function ∆Sαmin or, alternatively, the minimum of ∆Cα′
can be ob-
tained algebraically. Such a minimum is at the value of the coupling constant
λSvNmin ≈ 0.485 and λCmin ≈ 0.371, respectively.
The authors in [1] studied numerically the von Neumann entropy and the
correlation function for a Hydrogen molecule, using an old result by Herring and
Flicker [8], going back to an oldest idea by Heitler and London [9], which con-
sists in substituting the molecular binding with a position dependent exchange
coupling:
J(r) ≈ 1.641 r
2 e−2 r Ry, (11)
where r is given in Bohr radius, see Figure 6. The maximum value taken by this
function is at the point rmax = 1.25. Assuming B = 0.5 Ry, i.e. 12 of the funda-
mental level of the Hydrogen atom, the maximum value λ′max = J(rmax)/B ≈
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1
dDCΑ'� dΛ
Figure 5: The deviation ∆Cα′ and its derivative with respect to λ are computed
for values of −0.98(red) ≤ α′ ≤ 0.22(violet), for steps of 0.06. The curve drawn
thicker corresponds to α′min
0.5 1 1.5 2 2.5 3
JHrLHRyL
Figure 6: The effective interaction Hydrogen-Hydrogen atom
0.470628 < λSvNmin , i.e. the value of the effective interaction value is less than
the minimum for the deviation function ∆Sαmin . Then, the equilibrium bal-
ance between entanglement (as von Neumann entropy) and correlation energy
predicts a length of the molecule equal to rmax (see the first panel of Figure 7).
On the other hand, if we consider the energy gap 2B = 3/4 Ry, i.e. the energy
step to the first excited state, one obtains the new value λ′′max ≈ 0.628, which
goes beyond λmin, even if it is always less than 1. Now, the deviation function
∆Sαmin has two minima as seen in the second panel of Figure 7, one of which
is at r′′− ≈ 0.76 , the other one being at r′′+ ≈ 1.91.
These results should be compared with the experimental equilibrium length
of the Hydrogen molecule, which is rexp ≈ 2.0.
We point out that although the spin–model described by the Hamiltonian
(1) is characterized by features which are essentially rough, however we are
induced to answer positively to the quest for a physical meaning of the deviation
function ∆Sαmin . Indeed, the results elucidated in Figure 7 encourage, on one
part, improvement of the computation of r in order to make more accurate the
comparison with the experimental value rexp.
0.5 1 1.5 2 2.5 3
-0.015
-0.0125
-0.01
-0.0075
-0.005
-0.0025
DS minHr; B= .5 RyL
0.5 1 1.5 2 2.5 3
-0.015
-0.0125
-0.01
-0.0075
-0.005
-0.0025
DS minHr; B= .375 RyL
r''- r''+
Figure 7: The von Neumann entropy for the 2-spin model for B = .5 Ry (left
panel) and for B = .375 Ry (right panel) and the position depending interaction
given by (11).
The first question to answer is whether this draft works also for the Concur-
rence. A statement about it is not obvious, since the von Neumann entropy is
a nonlinear function of the Concurrence in the 2-qubits case.
However, from Figure 8 one can see that the minimized deviation of the
Concurrence takes one minimum for relatively large intensity of the magnetic
field ( say B ≥ 0.6 Ry), while for weak fields two minima appear, corresponding
to the situation depicted nearby.
0 0.2 0.4 0.6 0.8 1
r HBohrL
B HRyL
0.2 0.4 0.6 0.8 1
       
         B(Ry)
r(Bohr)
Figure 8: Two contour plots of the minimized deviation of the Concurrence as a
function of the magnetic field B (Ry) and of the internuclear distance r, as given
by (11). The range of values divided by the contour lines is [−0.038, 0, 04] for
the left panel and [−0.03705, −0, 03000] for the right one that approximatively
corresponding to the black area in the left panel.
In correspondence of the same values considered above, for B = 0.5 Ry
the function ∆Cα′
(r) has two minima at r = 0.79 and r = 1.88, while for
B = 0.375 Ry they are located at r = 0.60 and r = 2.25. So one sees that the
resulting equilibrium configurations are not much very close to the experimental
one. The equilibrium configuration more closest to the experimental one is the
minimum occurring at r = 1.88 (B = 1
Ry) for the function ∆Cα′
One sees that one of the resulting equilibrium configurations is only roughly
close to the experimental one.
In other words, to conclude monitoring numerically B the equilibrium config-
uration more closest to experimental one in the minimum occurring at r = 1.88
for B = 1
Ry and at r = 2.25 for B = 0.375 Ry for the function ∆Cα′
3 A quantum chemical framework to compare
entanglement and correlation energy
In this Section we represent the results produced in [1], where the electron entan-
glement in the Hydrogen molecule, calculated by the von Neumann entropy of
the reduced density matrix ρ1, is obtained starting by the excitation coefficients
of the wave function expanded by a configuration interaction method:
ρCISD1
= −Tr
ρCISD1 log2ρ
|c2i+11 |2 +
|c2i+1,2i+21,2 |2
|c2i+11 |2 +
|c2i+1,2i+21,2 |2
|c0|2 +
|c2i+22 |2
|c0|2 +
|c2i+22 |2
, (12)
where c1 is the coefficient for a single excitation, and c1,2 is the double
excitation (in Appendix A of [10] more details are shown).
In this framework, entanglement (S) and correlation energy (Ecorr), as func-
tions of nucleus – nucleus separation are those in Figure 9
0 1 2 3 4 5
R ( Å ) 
S ( ρ1
Figure 9: Comparison between the entanglement, calculated by the von Neu-
mann entropy of the reduced density matrix, and the electron correlation energy
in the Hydrogen molecule.
By the results given by this model, we want to discuss and to suggest some
answers to the questions i) and ii) presented in Remark 1. Even if, in order
to represent correlation energy and entanglement, we use two different scales,
in Figure 9 we can see that entanglement has a small value in the united atom
limit after it is growing for small distances till it arrives at a maximum value
then it decrease till it assumes zero value at the separated atom limit and it is
exactly the progress of the correlation curve.
In order to compare the entropy S with the electron correlation energy Ecorr,
we rescale S with the parameter αmin calculated with some procedure illustrated
in Eq. (8) and Eq. (9) replacing the integration variable λ with R; in this way
we extract
EcorrSvNdR
SvNdR
≈ 0.009. (13)
The corresponding ∆Sαmin = Ecorr−αS allows us to answer to the question
ii); in fact, as it is shown in Figure 10, the vanishing point of ∆Sαmin is,
according to the two –spin Ising model, nearby R ≈ 2 Å that corresponds to
the equilibrium configuration of the Hydrogen molecule.
0 1 2 3 4 5
−0.01
Figure 10: ∆Sαmin for theH2 molecule as a function of nucleus–nucleus distance.
4 Differences between the Configuration Inter-
action approach and the two–spin Ising model
The model proposed in Sec. 1 provides us with a measurement of entanglement:
indeed, Eq. (3) describes the von Neumann entropy as a function of coupling
constant λ, for small λ. By using Eq. (7), we can express λ in terms of corre-
lation energy and substituting it in Eq. (3) we can obtain the variation of SvN
in terms of Ecorr.
SvN = −
EcorrLog
Ecorr
2(Ecorr+2)
+ (Ecorr + 4)Log
Ecorr+4
2(Ecorr+2)
(Ecorr + 2)Log4
. (14)
In order to calculated the coefficient of proportionality among SvN and Ecorr
we make an expansion of SvN for Ecorr → 0 (or equivalently for λ → 0) at the
first order, obtaining a straight line characterized by an angular coefficient given
by mSvN (Ecorr) = (
)(1 + 1
). Since this behavior is uncorrect to represent
the logatithmic singularity of SvN in the origin, we make an expansion of Eq.
(14), preserving the logarithmic deviation, and we obtain an expression of the
SvN = AEcorr +BEcorrLog(Ecorr), (15)
where A = 1/2 and B = −1/(4Log2).
0.02 0.04 0.06 0.08 0.1
0.025
0.075
0.125
Ecorr   
Linear 
AE+BELogE
Figure 11: A comparison among the behavior of Eq. (14) and its linear approx-
imation and the logarithmic one, for the Ising model.
In order to compare the behavior of SvN in Eq. (14), we have organized
the numerical data, calculated with the method proposed in [1], by making a
correspondence between each value of Ecorr and its respective value of SvN ,
obtaining the plot in Figure 12
0 0.01 0.02 0.03 0.04 0.05
Figure 12: A correspondence of Ecorr and SvN calculated by the numerical
procedure suggested by [1]
Of particular significance is the fact that, in the range where S is monotoni-
cally increasing, the correlation energy has its maximum, consequently S seems
to be not a function. Moreover, it is important to note that Ecorr begins to
decrease for R > 1 Å, region where the states become mixed, i. e. ,Trρ 6= Trρ2;
as depicted in Figure 13.
0 1 2 3 4
R(Å )
Figure 13: The increasing of the degree of mixing in the two electron state: in
black we depict the trace of ρ, in red the trace of ρ2.
Probably, for this reason, the procedure adopted in [1] seems to be not cor-
rect: the density matrix, in fact, is calculated starting by the excitation coeffi-
cient of a wave function obtained developping with the Configuration Interaction
Single Double method a pure two electrons state.
However, even if we consider only the first branch of the plot in Figure 12,
i.e. , the numerical values of SvN corresponding with increasing values of Ecorr,
and we fit the values around Ecorr → 0 with a F = AEcorr + BEcorrLog(Ecorr)
we draw out numerical values of the coefficient different from the ones used in
Eq. (15). This result is shown in Figure 14.
0.01 0.02 0.03 0.04 0.05
0.35 S
Ecorr
A=17.1
B=3.3
Figure 14: A fit of SvN as a function of Ecorr, around the origin, with a function
of the form F = AEcorr +BEcorrLog(Ecorr) whose coefficients A and B assume
the numerical values in Figure.
In particular the arithmetic sign of the coefficient B in the two models are
opposite and this implies the opposite concavity of the curve.
This fact, clearly demonstrates a not satisfactory agreement between the
Ising model and the one proposed in [1].
5 Concluding remarks
We have explored the role of entanglement in the model of two qubits describing
the Hydrogen molecule (1), considered as a bipartite system. In our discussion
we have limited to the ferromagnetic case governed by the interaction coupling
parameter J > 0.
The concept of entanglement gives a physical meaning to the electron cor-
relation energy in structures of interacting electrons. The entanglement can be
measured by using the von Neumann entropy or, alternatively, the notion of
Concurrence [7]. To compute the entanglement it is convenient to consider two
Regions, say I and II, which provide two different reduced density matrices.
The entropy turns out to be an increasing function of the coupling constant λ
in Region I, but the state under consideration is maximally entangled in Region
II indipendently from the anisotropy parameter g.
An interesting result is that for large coupling constants the entropy ap-
proach 1, meaning that all levels are equiprobably visited by the considered
spin.
For weak interactions, at the boundary point λb =
the von Neumann
entropy admits a discontinuity, indicating a crossing of the lowest eigenvalues
and, in a more general constext, a quantum phase transition [5].
In Sec. 2 a comparison between the entanglement and the correlation energy
is performed.
To quantifying the entanglement we resort to the reduced density matrix.
The entanglement can also be measured by exploiting the concept of Concur-
rence.
The entanglement measure is always bounded, while the energy correlation,
Ecorr = |E0| − 2 =
4 + λ2 − 2, is a divergent function of λ. This fact tells us
that to look for simple relations valid on the whole λ−axes has no sense.
Thus, by limiting ourselves to weak couplings, we have minimized the mean
square deviation given by Eq. (8). This procedure leads to the value αmin ≈
−0.691217 for the minimizing parameter (see Eq. (9)).
Sec. 1 contains a comparison between the von Neumann entropy and the
Concurrence.
Such a comparison is illustrated in Figure 1, for two spin system as a function
of the coupling λ for g = 1.
Some important points are commented in Remark 1 and Remark 2 .
In Figure 4 the deviation ∆Sα and its derivatives with respect to λ are
computed and αmin is evaluated for α ranging in the interval −1.29 ≤ α ≤
−0.091.
In Figure 5 the minimized Concurrence deviation ∆C
for the four eigen-
states of the 2-spin model is shown.
We point out the existence of a perfect symmetry among the Concurrence
deviations for pairs of eigenstates of opposite eigenvalues.
Formula (11), due to Heitler–London [9], is reported, where the position
dependent exchange coupling J(r) is expressed in term of the length r of the
nucleus–nucleus separation in the Hydrogen molecule.
To conclude, the magnetic field B has been monitored such that the equi-
librium configuration more closest to the experimental one, r ≈ 2.00, is the
minimum occurring at r = 1.88 for B = 1
Ry and r = 2.25 for B = 0.375 Ry
for the function ∆Cα′
We observe also that in the intermediate range of values, i. e., for 0.6 ≤ λ ≤
1, the two functions SvN and the correlation energy are almost proportional
within the 10%.
However, when we organized the pairs of points (Ecorr, SvN ) calculated by
following the procedure described by [1], it is clear that the von Neumann en-
tropy cannot be considered a function of correlation energy. The principle cause
is that the function Ecorr presents a maximum in the region where SvN is mono-
tonically increasing.
The reversing behavior of correlation energy occurs in correspondence with
an increase of the mixing degree of the two electrons state. The function Ecorr
in terms of the nucleus – nucleus distance R, increases till the state is pure, on
the contrary, when Tr(ρ2) becomes discordant from Tr(ρ), the function Ecorr
decreases.
This fact suggests us that the numerical model based on the calculation of
SvN starting by the excitation coefficients ci, isn’t completley correct because
the density matrix is obtained as a product of two electron pure states. However,
even if we consider only a branch of the plot in Figure 12, the function obtained
by the two spin Ising model, i. e., Eq. (14), is unsuitable for fitting these
numerical data.
On the basis of our results, essentially grounded on numerical considerations,
in the near feature we would explore more complicated systems of molecules,
such as for example the ethylene or other hydrocarbons, and compare these
studies with the goals obtained for the Hydrogen molecule.
Acknowledgments
The authors acknowledge the Italian Ministry of Scientific Researches (MIUR)
for partial support of the present work under the project SINTESI 2004/06 and
the INFN for partial support under the project Iniziativa Specifica LE41.
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http://arxiv.org/abs/quant--ph/0610238
	Introduction and the model 
	A comparison between the entanglement and the correlation energy 
	A quantum chemical framework to compare entanglement and correlation energy 
	Differences between the Configuration Interaction approach and the two–spin Ising model 
	Concluding remarks
