Homolytic molecular dissociation in natural orbital functional theory

ISSN 1463-9076

Physical Chemistry Chemical Physics

www.rsc.org/pccp Volume 13 | Number 45 | 7 December 2011 | Pages 20023–20482

COVER ARTICLEMatxain et al.Homolytic molecular dissociation in natural orbital functional theory

HOT ARTICLEDe Kepper et al.Sustained self-organizing pH patterns in hydrogen peroxide driven aqueous redox systems

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This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 20129–20135 20129

Cite this: Phys. Chem. Chem. Phys., 2011, 13, 20129–20135

Homolytic molecular dissociation in natural orbital functional theory

J. M. Matxain,*aM. Piris,

abF. Ruiperez,

aX. Lopez

aand J. M. Ugalde

a

Received 25th May 2011, Accepted 2nd August 2011

DOI: 10.1039/c1cp21696a

The dissociation of diatomic molecules of the 14-electron isoelectronic series N2, O2+2 , CO, CN�

and NO+ is examined using the Piris natural orbital functional. It is found that the method

describes correctly the dissociation limit yielding an integer number of electrons on the

dissociated atoms, in contrast to the fractional charges obtained when using the variational

two-particle reduced density matrix method under the D, Q and G positivity necessary

N-representability conditions. The chemistry of the considered systems is discussed in terms

of their dipole moments, natural orbital occupations and bond orders as well as atomic

Mulliken populations at the dissociation limit. The values obtained agree well with

accurate multiconfigurational wave function based CASSCF results and the available

experimental data.

1 Introduction

Since the molecular Hamiltonian operator contains only

one- and two-electron operators, the energy of a molecule

can in principle be determined exactly from the one- and two-

particle reduced density matrices (1- and 2-RDMs). These RDMs

carry all the relevant information, ergo theN-particle dependence

can in principle be avoided.1 In this vein, it can be shown that

using the reduced Hamiltonian,2 the molecular energy is

expressed as a linear functional of the 2-RDM. Consequently,

variational minimizations of the energy functional can be

carried out in terms of the 2-RDM.3 This, however, is a

difficult task that has recently become computationally viable

through the use of both the contracted Schrodinger equation4

and the optimization techniques known as semidefinite

programming.5

There remains, however, the long-standing problem posed by

the fact that not every 2-RDM is derivable from an N-particle

wave function: the N-representability problem.6 Hence, when

applying the variational procedure we might go outside the

allowed domain and, consequently, the resulting minimum

energies might not be bounded from below by the exact energy,

violating, therefore, the variational principle. This yields a

number of unphysical features. Thus, it has been shown recently

that the variational 2-RDM approach, under the D, Q, and G

positivity necessary conditions,7 leads to incorrect dissociation

limits with a fractional number of electrons on the dissociated

atoms8 for a number of diatomic molecules. The performance

of the 2-RDM variational method for the 14-electron diatomic

isoelectronic series, N2, O2+2 , CN�, NO+ and CO, has been

recently studied in detail in ref. 9, and it has been demonstrated

that additional subsystem constraints need to be imposed to

cure this dissociation problem.10

This problem can alternatively be treated by a functional

theory based on the 1-RDM. One can employ the exact 2-RDM

energy functional but with an approximate two-electron density

matrix built from the 1-RDM using a reconstruction functional.

Relative to the 2-RDM, the 1-RDM is a much simpler object

and the ensemble N-representability conditions that have to

be imposed on variations of the 1-RDM are well-known.6

However, it is worth emphasizing that this does not overcome

the N-representability problem of the energy functional,11

which is a problem related to the N-representability of the

2-RDM, implicitly presented via the reconstruction functional.

A way to solve the N-representability problem by defining

a functional, in exact terms, was recently proposed.12 The

existence13 and properties14 of the energy functional of the

1-RDM are well-established and its development has been

greatly facilitated by imposing multiple constraints.15 A major

advantage of a 1-RDM formulation is that the kinetic energy is

explicitly defined. The unknown functional in a 1-RDM based

theory only needs to incorporate electron correlation effects.

The first explicit approximate relation between the 2-RDM and

the 1-RDM, containing one free parameter, was proposed by

Muller in 1984.16 In this vein, the spectral expansion of the

1-RDM, which renders the so-called Natural Orbital Functional

(NOF) theory, offers a convenient formulation to develop

practical functionals for the elucidation of the electronic struc-

ture. NOF theory has been reviewed in a recent revision paper

where further details may be found.17 Additional developments

in the field have been reported recently.18–26

Our chosen route to the above-mentioned reconstruction27

is based on the cumulant expansion28 of the 2-RDM. We shall

a Faculty of Chemistry, University of the Basque Country UPV/EHU,and Donostia International Physics Center (DIPC), P.K. 1072,20080 Donostia, Euskadi, Spain. E-mail: [email protected]

b IKERBASQUE, Basque Foundation for Science, 48011 Bilbao,Euskadi, Spain

PCCP Dynamic Article Links

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use hereafter the Piris reconstruction functional,29 PNOF, in

which the two-particle cumulant is explicitly constructed in

terms of two matrices, D(n) and P(n), n being the set of the

occupation numbers. Hence, the PNOF comes from the general

properties of the reduced density matrices, several properties

like antisymmetry, compatibility with the occupation numbers,

existence of the electron–electron potential only for occupations

that sum up to the number of electrons, etc., are satisfied by

construction. The D(n) and P(n) matrices have been selected in

order to satisfy N-representability conditions and sum rules by

the 2-RDM, or equivalently, by the functional. The uniform

coordinate scaling constraint is satisfied as our NOF does not

mix operators with different coordinate scaling. Moreover, the

precise constraints that the two-particle cumulant matrix must

fulfill in order to conserve the expectation values of the total spin

and its projection have been recently formulated.21 The resulting

corresponding constraints for the Piris reconstruction functional

yield an efficient implementation as was demonstrated by our

calculations.21 On the other hand, the size-consistency is, in

general, not satisfied.

The D(n) and P(n) matrices can be written in a form that

yields the exact ground-state two-electron NOF.29 Moreover, a

convenient approximation for the mean value of the Coulomb

interactions allows us to formulate a practical implementation

of the functional for N-electron systems (N 4 2),29 referred

to as PNOF1 in the literature. The PNOF1 approximate

functional was found to be accurate enough to predict mole-

cular properties,30 but unable to account for dispersion effects.

Imposing a priori the D and Q positivity conditions of the

2-RDM, the dispersion binding phenomena can be treated,

as it has been illustrated by our calculations of the helium

dimer.19 This functional, denoted as PNOF2, has also been

successful for the dissociation curve of the ground-state radical

helium dimer.20 However, PNOF2 is not as accurate for larger

systems.

More accurate results for larger systems can be obtained

with the PNOF3 formulation, which showed an outstanding

performance for atoms and molecules.22 Moreover, PNOF3

can describe the correct topology of even challenging potential

energy surfaces.24 Unfortunately, closer analysis of the disso-

ciation curves for various diatomics,23 as well as the description

of diradicals and diradicaloids,25 revealed that PNOF3 over-

estimates the amount of electron correlation, when orbital near-

degeneracy effects become important. We demonstrated that this

ill behavior is related to the violation of the N-representability

conditions, in particular, to the violation of the G positivity

conditions. Consequently, we developed a more restricted func-

tional, PNOF4,23 that meets rigorously the known three

necessary N-representability conditions (D, Q and G) of the

2-RDM. It has been shown that PNOF4 correctly describes

molecules with electrons occupying orbitals that become

increasingly degenerated.23,25 However, the formulation of

the PNOF4 functional contains a positive variable SF

which must be bounded from above by one (see eqn (7)–(11)

of ref. 23). Explicit calculations have revealed that these

conditions do not meet when four or more orbitals become

degenerate. Additionally, we have also found that PNOF4

leads frequently to incorrect diatomic dissociation limits with

a fractional number of electrons on the dissociated atoms.

In order to remedy these problems we have introduced the

PNOF5 functional26 based on a new ansatz for the D and Pmatrices to deal with multiconfigurational states. A preliminary

performance assessment has shown that PNOF5 predicts the

barrier for the ethylene torsion in an outstanding agreement

with highly accurate multiconfigurational perturbation theory

(CASPT2) calculations and describes correctly the dissociation

limit of a small diatomics, yielding an integer number of elec-

trons on the dissociated atoms. The size-consistency of the

PNOF5 has also been addressed numerically,26 and our findings

suggest that PNOF5 is near size-consistent for singlet states

of the spin-compensated systems studied in ref. 26. However,

its performance for diatomics that dissociate to open shell

atoms with multiple unpaired electrons is a challenging task

that remains to be assessed.

In the present paper we will address this issue by investigating

the reliability of PNOF5 towards the dissociation of the

14-electron diatomic isoelectronic series N2, O2+2 , CN�, NO+

and CO. Their calculated dissociation potential energy curves,

dipole moments, bond orders, natural orbital occupations and

Mulliken populations at the dissociation limit will be examined

and compared to multiconfigurational CASSCF (n,m) calcula-

tions, with various (n,m) active spaces and experimental results.

The article is organized as follows. In Section 2, we present

the basic concepts relevant to the formulation of PNOF5.

A detailed description of the formulation can be found in

ref. 26. Our results are presented and discussed in Section 3,

and conclusions are drawn in Section 4.

2 Theory

Assuming a real set of natural orbitals, the PNOF5 energy for

a singlet state of an N-electron system can be cast as:26

EPNOF5 ¼XN

p¼1½npð2Hpp þ JppÞ �

ffiffiffiffiffiffiffiffiffin~pnpp

Kp~p�

þXN

p;q¼1

00nqnpð2Jpq � KpqÞ

ð1Þ

where p denotes the spatial natural orbital and np its occupation.

Hpp is the matrix element of the kinetic energy and nuclear

attraction terms, whereas Jpq = hpq|pqi and Kpq = hpq|qpi arethe usual Coulomb and exchange integrals, respectively. The

p-state defines the coupled natural orbital to the orbital p,

namely, p = N � p + 1, N being the number of particles in

the system. It is worth noting that we look for the pairs of

coupled orbitals (p, p) which yield the minimum energy for the

functional of eqn (1). The actual p and p orbitals paired are not

constrained to remain fixed along the orbital optimization

process. Consequently, the pairing scheme of the orbitals is

allowed to vary along the optimization process until the most

favorable orbital interactions are found. In accordance with this

assumption, all occupancies vanish for p4N. The double prime

in eqn (1) indicates that both the q = p term and the coupled

one-particle state terms p = p are omitted from the last

summation. Additionally, the bounds on the off-diagonal

elements of D that stem from imposing the N-representability

D and Q positivity necessary conditions of the 2-RDM, as well

as the sum rule that D must fulfill, imply that the occupation of

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the p level must coincide with that of the hole of its coupled state

p, namely,

np = hp, np + np = 1 (2)

where hp denotes the hole 1 � np in the spatial orbital p.

The solution in NOF theory is established by optimizing

the energy functional with respect to the occupation numbers

and to the natural orbitals separately. The orbital optimiza-

tion constitutes the bottleneck of this algorithm since direct

minimization of the orbitals has been proven to be a costly

method.31 In the present study, the recent successful imple-

mentation of an iterative diagonalization procedure32 has

been employed. Relevant for the current investigation is that

the number of particles is always conserved (N = 2P

pnp) due

to the relation of eqn (2) for the occupation numbers of the

coupled one-particle states. Eqn (2) and the N-representability

bounds (0 r np r 1) of the 1-RDM are easily enforced

by setting np = cos2 gp and np = sin2 gp. Note that PNOF5

allows constraint-free minimization with respect to the auxiliary

variables {gp}, which yield substantial savings of the computa-

tional time.

All calculations were carried out with the PNOFID code.33

We have used the correlation-consistent valence double-z basisset (cc-pVDZ) developed by Dunning.34 For comparison, we

have also performed complete active space self-consistent field

calculations, CASSCF(n,m), where n electrons are distributed

in m orbitals, using MOLCAS 7.4 suite of programs.35

3 Results

In this section, the results obtained for the N2-isoelectronic

series, N2, O2+2 , CO, CN� and NO+, using PNOF5 and

CASSCF are presented and discussed. All these molecules have

the same electronic configuration at the minima, s2gp

4up

0gs

0u, but

have two different dissociation limits. Namely, the N2-like

dissociation, leading to two fragments with spin quartets and

7 electrons on each atom, and the CO-like dissociation, leading

to two fragments with spin triplets, and 6 and 8 electrons on

each atom, respectively. It is worth emphasizing that previous

calculations using functionals of both the 1-RDM and 2-RDM

do not predict correctly neither of these dissociation limits,

unless subsystem-dependent constraints are imposed.10

The dissociation curves for all of the molecules considered

here, obtained from PNOF5 and CASSCF (6,6), CASSCF

(10,8) and CASSCF (14,14) calculations, are shown in Fig. 1.

Fig. 2 displays the natural orbitals for N2, both at the equilibrium

distance and at the dissociation limit, along with the pseudo-

one-particle energies corresponding to the diagonal Lagrange

Multipliers. The natural orbitals for the remaining molecules

look similar to these and hence will not be shown for the sake

of brevity. Fig. 3 shows the occupation numbers as a function

of the internuclear distance, for N2 and NO+. Finally, Table 1

collects the calculated equilibrium distances, dissociation

energies, bond orders, dipole moments, andMulliken populations

at the dissociation limit for N2, NO+, CN� and CO. Additionally,

the energy barrier and the geometry of the dissociation transition

state found for O2+2 are also reported in Table 2.

Before proceeding further, a brief remark regarding the steps

found in the CASSCF(6,6) curve for CN� and the CASSCF(14,14)

curve for NO+ is worth done. These unphysical steps may be

due to the following two facts. On the one hand, the (6,6)

active space for CN�might be a bit too small to account for all

the subtle effects of the non-dynamical electron correlation for

this system, because in this case the optimal active space

should be made by distributing 10 valence electrons in the

8 valence molecular orbitals generated from the combinations

of the 2s and the 2p atomic orbitals of each of the two atoms,

which renders a (10,8) active space. In the case of NO+ we

observe that the (14,14) space gives a poorer description than

the smaller (10,8) active space. This illustrates the fact that

larger active spaces may lead to unbalanced calculations36

and not necessarily to better results. For these latter cases

incorporation of dynamical correlation effects via perturbation

theory (CASPT2) normally improves the quality of the results.

Inspection of the dissociation curves shown in Fig. 1 reveals

that the PNOF5 predicts a smooth correct behavior towards

the dissociation limit for all the molecules considered. Addi-

tionally, we observed that the PNOF5 energies are bounded

from below by the CASSCF energies, which reflects the fact

that the variational procedure does not go outside the allowed

N-representable domain and, consequently, the energies do

not violate the ‘‘exact’’ boundaries. The similarity between

PNOF5 and CASSCF potential energy curves shown here adds

to the mounting evidence that PNOF5 recovers essentially the

non-dynamical correlation energy.26

At the equilibrium region, PNOF5 performs well predicting

accurately bond distances (Re) and dipole moments (me). Thesevalues, shown in Tables 1 and 2, compare, in general, better

than their corresponding CASSCF values to experimental

marks. The case of the dipole moments of NO+ and CN� is

remarkable. PNOF5 predicts dipole moments of the same

order of magnitude as found experimentally, while CASSCF

predicts much larger values. The CASSCF dipole moments

depend very much on the active space chosen and the basis set

used, and therefore the CASSCF values reported here must be

taken with caution. Bond orders (BO), estimated as the half of

the difference between the sums of the occupation numbers

of bonding and antibonding orbitals, compare well with the

CASSCF results. Note that, for all these molecules, both

the HF and the Kohn–Sham bond order is 3. However, when

natural orbital occupation numbers are considered, the calcu-

lated bond orders of these molecules differ from 3, as a

consequence of including specific non-dynamical correlation

effects through both CASSCF and PNOF5.

Let us now focus on the dissociation limit of these systems.

Recall that 2-RDM based direct variational minimization

methods do not yield dissociated fragments with integer

numbers of electrons. As mentioned above, subsystem con-

straints must be put in place in order to achieve physically

meaningful dissociated atoms having an integer number of

electrons.9,10 On the other hand, methods based on the varia-

tional minimization of the spectral representation of the

1-RDM, like PNOF4, do also yield unphysical dissociated

atoms with non-integer numbers of electrons, albeit the latter

provides very accurate dissociation energies.23 Thus, for the

dissociation of N2, the calculated charges on the two nitrogen

atoms are +0.003 and �0.003, respectively. For the CO mole-

cule, the charges at the dissociated atoms are +0.004 and �0.004

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for C and O respectively. Furthermore, the occupation numbers

of the p-type valence natural orbitals at the dissociation limit are

1.10 and 0.9, far away from 1.0 which should be for the correct

dissociation limit. However, PNOF5 produces a more satisfactory

description of the dissociation process. The PNOF5 natural

orbitals of N2 at the dissociation limit are depicted in Fig. 2,

while their occupation numbers along the dissociation curve are

drawn in Fig. 3. PNOF5 yields six energy-degenerate symmetry-

adapted delocalized valence natural orbitals each of them with

occupation number 1.0, rendering seven electrons on each of the

dissociated nitrogen atoms. We conclude, therefore, that the

dissociation of N2 is correctly described by PNOF5, leading to

the physically meaningful result: the ground singlet spin state

of the N2 molecule dissociates to two ground quartet spin state

Fig. 1 Total energies of a N2 molecule calculated with different CASSCF active spaces and PNOF5 natural orbital functional.

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Fig. 2 Calculated PNOF5 sg, pu, pg and su N2 natural orbitals at the equilibrium bond distance (R= 1.099 A) on the left and at the dissociation

limit (R= 10 A) on the right. The graphical representation of the pseudo-one-particle energies corresponding to the diagonal Lagrange multipliers

for these orbitals is given as well.

Fig. 3 Occupations of the sg, pu, pg and su orbitals as a function of R calculated with PNOF5 and CASSCF(10,8).

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N atoms as it does our reference CASSCF method, shown for

comparison in Fig. 3. The similarity between the two sets of

curves along the whole dissociation process lends support to the

correctness of our PNOF5 calculations. Conversely, the closed

shell singlet state of the CO molecule is found to dissociate to

a triplet carbon atom and a triplet oxygen atom. CN� and

O2+2 (not shown) dissociate like N2. NO+ constitutes a special

case. We have found both dissociation limits for NO+, namely,

the dissociation to quartet spin N and quartet spin O+, and to

triplet spin N+ and triplet spin O atoms, as shown in Fig. 3. The

latter solution is found to be more stable by 1.16 eV, opposite to

the experimental data, based on ionization energy data.

Finally, we discuss the remarkable transition state of

O2+2 towards dissociation. This species is known to be kinetically

stable but thermodynamically unstable. Previous calculations37

at the QCISD(T)/6-311+G(3df) level predicted an equili-

brium bond length of 1.050 A and a dissociation energy of

�93.45 kcal mol�1. In the same work, QCISD(T)/6-31G*

calculations predicted a bond length of 1.079 A and a kinetic

barrier of 65.13 kcal mol�1 located at 1.52 A. Higher level

MRCI calculations, using the same basis set, yielded a barrier of

63.3 kcal mol�1 located at 1.598 A. In this work (see Table 2),

our CASSCF(10,8) calculations shortened this distance to

1.588 A, with a barrier of 85.5 kcal mol�1, and CASSCF(14,14)

worsens the estimation of the barrier raising it to 91.9 kcal mol�1.

Remarkably, our PNOF5 calculated barrier is a better esti-

mate than those obtained on the CASSCF calculations,

namely, 76.5 kcal mol�1. It lies closer to the reference MRCI

barrier of 63.3 kcal mol�1. Therefore, PNOF5 results improve

upon those obtained with CASSCF (10,8).

4 Conclusions

The PNOF5 natural orbital functional has been applied to the

dissociation of selected diatomic molecules: N2, O2+2 , CN�,

NO+ and CO. It is found that the method always describes the

dissociation limit with an integer number of electrons on the

well-separated atoms and, therefore, leading to physically

meaningful solutions. In addition to the correct behavior at

the dissociation limit, PNOF5 also represents well the equili-

brium region yielding accurate equilibrium bond distances and

dipole moments, as well as dissociation energies.

Acknowledgements

Financial support comes from Eusko Jaurlaritza and the

Spanish Office for Scientific Research. The SGI/IZO-SGIker

UPV/EHU is gratefully acknowledged for generous allocation

of computational resources. JMMwould like to thank Spanish

Ministry of Science and Innovation for funding through a

Ramon y Cajal fellow position (RYC 2008-03216).

References

1 K. Husimi, Proc. Phys.-Math. Soc. Jpn., 1940, 22, 264; P. O.Lowdin, Phys. Rev., 1955, 97, 1474.

2 C. Valdemoro, Phys. Rev. A, 1985, 31, 2123.3 D. A. Mazziotti, in In Reduced-Density-Matrix Mechanics: WithApplication to Many-Electron Atoms and Molecules, ed. D. A.Mazziotti, Wiley, New York, 2007, vol. 134 of Advances inChemical Physics, ch. 3, pp. 21–59.

4 C. Valdemoro, in In Reduced-Density-Matrix Mechanics: WithApplication to Many-Electron Atoms and Molecules, ed. D. A.Mazziotti, Wiley, New York, 2007, vol. 134 of Advances inChemical Physics, ch. 7, pp. 121–164.

Table 1 Equilibrium distances, Re (A), dissociation energies, De (kcal mol�1), bond orders (BO), dipole moments, me (Debye), at the equilibriumdistance and Mulliken populations, qi (i = N,C), at the dissociation limit, calculated with PNOF5, CASSCF(14,14), CASSCF(10,8) andCASSCF(6,6), for N2, CN

�, NO+ and CO molecules

N2 CN�

Re De BO me qN Re De BO me qN

PNOF5 1.099 229.9 2.87 0.000 7 1.180 247.6 2.89 0.900 7CASSCF(14,14) 1.115 210.4 2.85 0.000 7 1.196 235.4 2.86 2.360 7CASSCF(10,8) 1.117 205.0 2.85 0.000 7 1.200 220.0 2.86 2.241 7CASSCF(6,6) 1.115 196.6 2.86 0.000 7 1.198 281.4 2.87 2.253 8Exper. 1.098 225.1 — 0.000 7 1.177 — — — 7

NO+ CO

Re De BO me qN Re De BO me qC

PNOF5 1.059 228.2 2.87 0.337 6 1.130 221.0 2.92 0.209 6CASSCF(14,14) 1.076 261.7 2.83 2.260 6 1.145 247.0 2.86 �0.059 6CASSCF(10,8) 1.077 229.0 2.84 2.368 7 1.143 249.9 2.88 �0.259 6CASSCF(6,6) 1.075 219.8 2.85 2.382 7 1.141 255.1 2.89 �0.239 6Exper. 1.066 — — — 7 1.128 256.2 — 0.112 6

Experimental values from ref. 38–40.

Table 2 Equilibrium distances, Re (A), bond orders (BO), distance atthe maximum, Rmax (A), barrier energy DE, dissociation energies,De (kcal mol�1), bond orders (BO), dipole moments, me (Debye), at theequilibrium distance and Mulliken populations, qO, at the dissociationlimit, calculated with PNOF5, CASSCF(14,14), CASSCF(10,8) andCASSCF(6,6), for the O2+

2 molecule

O2+2

Re BO Rmax DE De me qO

PNOF5 1.038 2.78 1.66 76.5 �71.5 0.000 7CASSCF(14,14) 1.052 2.79 1.59 91.9 �92.0 0.000 7CASSCF(10,8) 1.054 2.79 1.59 85.5 �94.8 0.000 7CASSCF(6,6) 1.051 2.79 1.59 83.9 �90.9 0.000 7Exper. — — — — — 0.000 7

Experimental values from ref. 38–40.

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5 D. A. Mazziotti, in In Reduced-Density-Matrix Mechanics: WithApplication to Many-Electron Atoms and Molecules, ed. D. A.Mazziotti, Wiley, New York, 2007, vol. 134 of Advances inChemical Physics, ch. 4, pp. 61–92.

6 A. J. Coleman, Rev. Mod. Phys., 1963, 35, 668.7 C. Garrod and J. K. Percus, J. Math. Phys. (Cambridge, Mass.),1964, 5, 1756; R. M. Erdahl, Int. J. Quantum Chem., 1978, 13, 697;J. K. Percus, Int. J. Quantum Chem., 1978, 13, 79.

8 H. Van Aggelen, P. Bultinck, B. Verstichel, D. Van Neck andP. W. Ayers, Phys. Chem. Chem. Phys., 2009, 11, 5558.

9 H. Van Aggelen, B. Verstichel, P. Bultinck, D. Van Neck, P. W.Ayers and D. L. Cooper, J. Chem. Phys., 2010, 132, 114112.

10 B. Verstichel, H. Van Aggelen, D. Van Neck, P. W. Ayers andP. Bultinck, J. Chem. Phys., 2010, 132, 114113.

11 J. M. Herbert and J. E. Harriman, J. Chem. Phys., 2003,118, 10835; P. W. Ayers and S. Liu, Phys. Rev. A, 2007, 75, 022514.

12 P. W. Ayers, S. Golden and M. Levy, J. Chem. Phys., 2006,124, 054101.

13 T. L. Gilbert, Phys. Rev. B: Solid State, 1975, 12, 2111; M. Levy,Proc. Natl. Acad. Sci. U. S. A., 1979, 76, 6062; S. M. Valone,J. Chem. Phys., 1980, 73, 1344.

14 M. Levy, in In Density Matrices and Density Functionals,ed. R. Erdahl and V. H. Smith, Reidel, Dordrecht, 1987,pp. 479–498.

15 J. Cioslowski, J. Chem. Phys., 2005, 123, 164106.16 A. M. K. Muller, Phys. Lett. A, 1984, 105, 446.17 M. Piris, In Reduced-Density-Matrix Mechanics: With Applica-

tion to Many-Electron Atoms and Molecules, in Advances inChemical Physics, ed. D. A. Mazziotti, Wiley, New York, 2007,vol. 134, pp. 387–428.

18 K. Pernal, O. Gritsenko and E. J. Baerends, Phys. Rev. A, 2007,75, 012506; D. R. Rohr, K. Pernal, O. V. Gritsenko andE. J. Baerends, J. Chem. Phys., 2008, 129, 164105; K. J. H.Giesbertz, K. Pernal, O. V. Gritsenko and E. J. Baerends,J. Chem. Phys., 2009, 130, 114104; K. J. H. Giesbertz,O. V. Gritsenko and E. J. Baerends, Phys. Rev. Lett., 2010,105, 013002; N. N. Lathiotakis, S. Sharma, J. K. Dewhurst,F. G. Eich, M. A. L. Marques and E. K. U. Gross, Phys. Rev.A, 2009, 79, 040501; N. N. Lathiotakis, S. Sharma, N. Helbig,J. K. Dewhurst, M. A. L. Marques, F. Eich, T. Baldsiefen,A. Zacarias and E. K. U. Gross, Z. Phys. Chem., 2010, 224, 467;H. Appel and E. K. U. Gross, Europhys. Lett., 2010, 92, 23001;R. Requist and O. Pankratov, Phys. Rev. B: Condens. Matter,2008, 77, 235121; R. Requist and O. Pankratov, Phys. Rev. A,2010, 81, 042519.

19 M. Piris, X. Lopez and J. M. Ugalde, J. Chem. Phys., 2007,126, 214103; M. Piris, X. Lopez and J. M. Ugalde, Int. J. Quantum

Chem., 2008, 108, 1660; M. Piris, X. Lopez and J. M. Ugalde,J. Chem. Phys., 2008, 128, 134102.

20 M. Piris, J. M. Matxain and J. M. Ugalde, J. Chem. Phys., 2008,129, 014108.

21 M. Piris, J. M. Matxain, X. Lopez and J. M. Ugalde, J. Chem.Phys., 2009, 131, 021102.

22 M. Piris, J. M. Matxain, X. Lopez and J. M. Ugalde, J. Chem.Phys., 2010, 132, 031103; J. M. Matxain, M. Piris, X. Lopez andJ. M. Ugalde, Chem. Phys. Lett., 2010, 499, 164.

23 M. Piris, J. M. Matxain, X. Lopez and J. M. Ugalde, J. Chem.Phys., 2010, 133, 111101.

24 X. Lopez, M. Piris, J. M. Matxain and J. M. Ugalde, Phys. Chem.Chem. Phys., 2010, 12, 12931.

25 X. Lopez, F. Ruiperez, M. Piris, J. M. Matxain and J. M. Ugalde,ChemPhysChem, 2011, 12, 1061.

26 M. Piris, X. Lopez, F. Ruiperez, J. M. Matxain and J. M. Ugalde,J. Chem. Phys., 2011, 134, 164102.

27 M. Piris and P. Otto, Int. J. Quantum Chem., 2003, 94, 317.28 D. A. Mazziotti, Chem. Phys. Lett., 1998, 289, 419; W. Kutzelnigg

and D. Mukherjee, J. Chem. Phys., 1999, 110, 2800.29 M. Piris, Int. J. Quantum Chem., 2006, 106, 1093.30 P. Leiva and M. Piris, J. Chem. Phys., 2005, 123, 214102; P. Leiva

and M. Piris, J. Theor. Comput. Chem., 2005, 4, 1165; P. Leiva andM. Piris, J. Mol. Struct. (THEOCHEM), 2006, 770, 45; P. Leivaand M. Piris, Int. J. Quantum Chem., 2007, 107, 1.

31 A. J. Cohen and E. J. Baerends, Chem. Phys. Lett., 2002, 364, 409;J. M. Herbert and J. E. Harriman, Chem. Phys. Lett., 2003,382, 142.

32 M. Piris and J. M. Ugalde, J. Comput. Chem., 2009, 30, 2078.33 M. Piris, PNOFID, downloadable at http://www.ehu.es/mario.

piris/#Software.34 T. H. Dunning, J. Chem. Phys., 1989, 90, 1007.35 F. Aquilante, L. De Vico, N. Ferre, G. Ghigo, P. A. Malmqvist,

P. Neogrady, T. B. Pedersen, M. Pitonak, M. Reiher, B. O. Roos,L. Serrano-Andres, M. Urban, V. Veryazov and R. Lindh,J. Comput. Chem., 2010, 31, 224.

36 K. Anderson and A. J. Sadlej, Phys. Rev. A, 1992, 46, 2356.37 R. H. Nobes, D. Moncrieff, M. W. Wong, L. Radom, P. M. W.

Gill and J. A. Pople, Chem. Phys. Lett., 1991, 182, 216.38 K. P. Hubert and G. Herzberg, Molecular Spectra and Molecular

Structure, Van Nostrand-Reinhold, New York, ADDRESS, 1979,vol. 14.

39 NIST Computational Chemistry Comparison and BenchmarkDatabase, NIST Standard Reference Database Number 101,Release 15a, edited by R. D. Johnson III, April 2010, availableat http://cccbdb.nist.gov/.

40 M. W. Chase, Jr., J. Phys. Chem. Ref. Data Monogr., 1998, 9, 1.

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