Low-Lying Isomers of Free-Space Halogen Clusters with Tetrahedral

Sep 22, 2015 - and Octahedral Symmetry in Relation to Stable Molecules Such as ... on some 20 stable tetrahedral (t) and octahedral (o) molecules, inc...
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Low-Lying Isomers of Free-Space Halogen Clusters with Tetrahedral and Octahedral Symmetry in Relation to Stable Molecules Such as SF6 M. Piris*,†,‡,⊥ and N. H. March†,§,∥ †

Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal, 4, 20018 Donostia, Euskadi, Spain Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU), Manuel Lardiazabal Pasealekua, 3, 20018 Donostia, Euskadi, Spain § Department of Physics, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium ∥ Department of Physics, Oxford University, Parks Road, OX1 3PU Oxford, England ⊥ IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48011 Bilbao, Euskadi, Spain ‡

ABSTRACT: Motivated by previous work involving one of us (N.H.M.) on some 20 stable tetrahedral (t) and octahedral (o) molecules, including XF4 (X = C, Si, Ge), the natural orbital functional PNOF6 is here used to study the free-space halogen cluster t-F4. We consider an extended functional PNOF6(Nc) by coupling Nc orbitals (Nc > 1) to each orbital below the Fermi level, which improves the description of the electron pairs. Similar studies are presented for t-Cl4. The successful calculation on the stable molecule BrF5 (Theor. Chem. Acc. 2013, 132, 1298) has prompted a study of the clusters o-F6 and o-Cl6 too. The size relation with calculated known stable molecules and the experimental data are finally considered. In the case of the o-SF6, the geometry optimization with fixed octahedral symmetry has also been performed at the PNOF6(3) level of theory, leading to an equilibrium distance of 2.95 au in perfect agreement with the experiment. Our results confirm the multireferential character of Z4 and Z6 compounds (Z = H, F, Cl), in contrast with the single-reference character of the XZ4 (X = C, Si, Ge) and YZ6 (Y = S, Se, U) compounds; therefore, despite the clear patterns within a group, it is not possible to extrapolate the results to the case when the atomic number, X or Y, becomes zero.

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INTRODUCTION In a previous study,1 one of us (N.H.M.) used the forerunner of modern density functional theory (DFT), namely, the Thomas−Fermi (TF) semiclassical method, to treat a variety of tetrahedral and octahedral molecules; however, to solve the nonlinear TF differential equation essentially exactly by numerical procedure, in GeH4 say the four protons were smeared uniformly over the surface of a sphere centered on the Ge point nucleus. This simplistic model revealed a remarkable scaling property of the equilibrium bond length between the center and a vertex of the tetrahedron.1 Taking the XH4 series as starting point, we display in Figure 1 the equilibrium X−H distances from experiment2 for X ranging from C to Pb. The plot is motivated by the TF length scaling factor, b, given by1 b=

0.8853 z1/3

Unfortunately, this simple model overestimates the role of z in determining Re, so the agreement with the nonrelativistic TF prediction1 increases with X becoming heavy, up to Pb in Figure 1. Other studies3 have explored the relation between Re and the total number of electrons N = z + n in these molecules. Transcending Thomas−Fermi Statistical Theory. Subsequently, Krishtal, Van Alsenoy, and March4 worked out the cases of some 20 tetrahedral molecules, without involving the so-called one-center expansion previously referred to for GeH4. The method used by these authors was basically Hartree−Fock (HF), but with a second-order Møller−Plesset (MP2) correction applied to determine the equilibrium distance between the center and a vertex of the tetrahedron. In Figure 2, we have taken these values for CF4, SiF4, and GeF4 and plotted them versus the charge (z) on the central nucleus (14 for Si say). Although there is fairly substantial variation with z, we can make a rough extrapolation to z = 0, to suggest a value of Re (now from the center of gravity of the tetrahedron to a vertex) as ∼2.1 au.

(1)

We plotted essentially Xe = Re/b versus n/z, where Re is the equilibrium X−H bond length, z is the charge on the central nucleus, and n is the total charge carried by the outer nuclei (4 for XH4). The actual plot in Figure 1 is Xe versus 4/z. It is worth noting that we use atomic units throughout this work. © 2015 American Chemical Society

Received: March 23, 2015 Revised: September 21, 2015 Published: September 22, 2015 10190

DOI: 10.1021/acs.jpca.5b02788 J. Phys. Chem. A 2015, 119, 10190−10194

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The Journal of Physical Chemistry A

level (F = N/2), with one orbital above it, so Ωg ≡ (g, g̃) is the subspace containing the orbital g (g≤ F) and its coupled orbital g̃ (g̃>F) . In the case of independent pairs, it was demonstrated9 that the description of each pair can be further improved by inclusion of more orbitals in each subspace, which gave rise to PNOF5e. Following this recipe, in this work we will consider a more general functional PNOF6(Nc) by coupling Nc orbitals to each orbital g. Accordingly, each subspace Ωg contains now an orbital g (g ≤ F), and its Nc coupled orbitals with q > F. If Nc = 1, PNOF6(1) corresponds obviously to the simplest formulation PNOF6.7 The PNOF6(Nc) energy for a singlet state of an N-electron molecule can be cast as

Figure 1. Experimental2 equilibrium radius Re for CH4 through to PbH4, in au, versus inverse atomic number of atom at tetrahedron center of gravity. The continuous line plotted is a guide to the eye.

F

E=

F

∑ Eg +

∑ ∑ ∑

g=1

f ≠ g p ∈Ω f q ∈Ωg

int Epq

(3)

The first term of the energy (eq 3) draws the system as independent F electron pairs described by the following NOF of two-electron systems Eg =

∑

np(2/pp + 1pp) +

p ∈Ωg

∑

int Epq

p , q ∈Ωg , p ≠ q

(4)

where /pp is the matrix element of the kinetic energy and nuclear attraction terms, whereas 1pp = is the Coulomb interaction between two electrons with opposite spins at the spatial orbital p. It is worth noting that the last terms of eqs 3 and 4 contain the interactions between the electrons in different pairs and inside a pair, respectively. The interaction energy Eint pq is given by

Figure 2. MP2 equilibrium distance4 between the center and a vertex of the tetrahedron, in au, in the XF4 series for X ranging from C to Ge. The continuous line plotted is a guide to the eye.

int Epq = (nqnp − Δqp)(21pq − 2pq) + Πqp3 pq

This has prompted us to use the natural orbital functional (NOF) theory of Piris,5,6 specifically the version PNOF6,7 to make a correlated calculation for the free-space halogen cluster F4 constrained, however, to tetrahedral geometry. Of course, this tetrahedron does not represent the lowest-lying isomer of four-atom fluorine cluster and is only of theoretical interest.

where 1pq = ⟨pq|pq⟩ and 2pq = ⟨pq|qp⟩ are the usual direct and exchange integrals, respectively. 3 pq = ⟨pp|qq⟩ is the exchange and time-inversion integral,10 which differs only in phases of the natural orbitals with respect to the exchange integrals, so 3 pq = 2pq for real orbitals. The PNOF6 ansatz for

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NOF THEORY A series of functionals has been proposed 5 using a reconstruction of the two-particle reduced-density matrix (2RDM) in terms of the one-particle RDM (1-RDM) by ensuring necessary N-representability positivity conditions on the 2RDM.8 In this work, we employ the PNOF6,7 which has proved a better treatment of both dynamic and nondynamic correlations than its predecessors. PNOF6 is an orbital-pairing approach, which is reflected in the sum rule for the occupation numbers, namely

∑ p ∈Ωg

np = 1,

(5)

the off-diagonal elements of Δ and Π matrices is Δqp

Πqp

orbitals

e−2Shqhp

−e−S(hqhp)1/2

q ≤ F, p ≤ F

γqγp Sγ e−2Snqnp

g = 1, F

γ −Π qp

e−S(nqnp)1/2

q ≤ F, p > F q > F, p ≤ F q > F, p > F

(6)

(2)

where hp denotes the hole 1 − np in the spatial orbital p. The magnitudes γ and Πγ are given by

where p denotes a spatial natural orbital and np is its occupation number. This involves coupling each orbital g, below the Fermi 10191

DOI: 10.1021/acs.jpca.5b02788 J. Phys. Chem. A 2015, 119, 10190−10194

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The Journal of Physical Chemistry A γp = nphp + αp2 − αpSα

1.5 au, which is much smaller than 1.77 and 2.14 au, obtained with PNOF6(3) and full CI,16 respectively. This result is not surprising considering the differences among the ground states that describe the free-space cluster (H4) of clusters with an X atom in the center (XH4, X = C, Si, Ge). In H4, the occupation numbers (>10−3) obtained with PNOF6(3) are 1.876, 1.152, 0.847, and 0.123, whereas full CI afforded values of 1.688, 1.173, 0.759, and 0.377. These partial occupancies indicate the multireferential character of the ground state in the free-space cluster, which contrasts with the well-known single-reference character in doped XH4 clusters. Furthermore, the potential energy curve of tetrahedron H4 is very flat when coming from large distances up to the equilibrium, yielding a binding energy of only 5.4 kcal/mol at the full CI level. This helps us to understand the significant difference in bond lengths (0.37 au) obtained from the complete CI and PNOF6 (3) calculations, although a qualitative correct description is reflected in the occupation numbers of the four strongest occupied spatial orbitals at the equilibrium obtained by PNOF6(3). It is worth noting that the global potential energy surface (PES) for the adiabatic ground state of the H4 system has been extensively studied in the past.17,18 These studies indicate that the global minimum is reached for a rectangular arrangement (D2h) of the four atoms. Besides, other planar (including C2v and D3h) and pyramidal (including C3v and Td) structures afford stationary points. Recently,19 we have computed the PES of the planar D4h/D2h H4 model. By comparison with full-CI calculations, we demonstrated that PNOF6 provides a qualitatively correct description of the D2h−D4h symmetry transition in H4. PNOF6 was shown to provide the correct smooth PES features and total and local spin properties along with the correct electron delocalization.19 We shall restrict ourselves below to recording simply the consequences of PNOF6 calculations, but no deviations from the corresponding symmetries, tetrahedral or octahedral, have been permitted. On the basis of the PES studies of the H4 molecule17,18 and the fact that the obtained structures have higher energies than those corresponding to infinite-separated halogen dimers, we expect that the most stable structures are planar (D2h). Hence, our symmetry-constrained structures do not represent the lowest-lying isomers and could only be stationary points. In Table 2, the results obtained with PNOF6(3) for lowlying isomers of free-space halogen clusters with tetrahedral (t)

⎧ e −S h , p ≤ F p ⎪ αp = ⎨ S − ⎪ e np , p > F ⎩ γ Π qp

1/2 1/2 ⎛ γqγp ⎞ ⎛ γqγp ⎞ ⎟⎟ ⎜⎜hqnp + ⎟⎟ = ⎜⎜nqhp + Sγ ⎠ ⎝ Sγ ⎠ ⎝

F + FNc

S=

∑ q=F+1

F + FNc

∑

hq , Sα =

q=F+1

F + FNc

αq , Sγ =

∑

γq (7)

q=F+1

According to eq 6, the auxiliary Δ and Π matrices do not differentiate between orbitals that belong to the same subspace (q, p ∈ Ωg) and orbitals belonging to two different subspaces (q ∈ Ωg1, p ∈ Ωg2, g1 ≠ g2). Therefore, the interactions between orbitals are independent of the set to which they belong, so the intrapair and interpair correlations are equally balanced in PNOF6. The interpair electron correlation in PNOF6 removes the symmetry-breaking artifacts that are present in independent-pair approaches such as PNOF5 when treating delocalized systems.7 The better description of correlation effects makes PNOF6 a good candidate to treat nondynamic correlated systems. The solution in NOF theory11 is established by optimizing the energy functional with respect to the occupation numbers and to the natural orbitals separately, for which the iterative diagonalization procedure proposed by Piris and Ugalde12 has been employed.

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RESULTS In this section, the optimal equilibrium distances obtained for fluorides and chlorides are presented and discussed. All calculations have been carried out using our computational code DoNOF with the contracted Gaussian basis set 631G*.13,14 The matrix element of the kinetic energy and nuclear attraction terms as well as the electron repulsion integrals are inputs to DoNOF. In the current implementation, we have used the GAMESS program15 for this task. Before proceeding with the results obtained for the halogen clusters, let us consider hydrogen instead of halogens in the simpler compounds XH4 with X = 0, C, Si, Ge, constrained to tetrahedral symmetry (Td). In Table 1, the equilibrium bond

Table 2. Equilibrium Bond Length (Re), in au, Calculated at the PNOF6(3) Level of Theory with the 6-31G* Basis Set

Table 1. Equilibrium Bond Length (Re), in au, Calculated at the PNOF6(3) and MP24 Levels of Theory with the 631G** Basis Set MP2 t-H4 CH4 SiH4 GeH4 a

2.05 2.79

PNOF6(3) 1.77 2.05 2.80 2.90

EXP2 2.14 2.05 2.80 2.87

a

Equilibrium bond length obtained at the full CI level of theory.16

cluster

Re

t-F4 o-F6 t-Cl4 o-Cl6

2.95 3.39 4.02 4.72

and octahedral (o) symmetry can be found. We next show in Figures 3 and 4 how the t- and o-halogen clusters relate to the sizes of the known stable molecules, respectively. All values shown in these Figures were previously obtained at the MP2 level of theory,4 except for heavy-uranium clusters o-UF6 and oUCl6,20 and the equilibrium bond length of the SF6 molecule. For the latter species, we have also performed a geometry optimization constrained to the octahedral symmetry Oh group.

length (Re) calculated with PNOF6(3) for these hydrogen clusters can be found. The optimized distances are very close to those previously obtained at the MP2 level of theory4 and the available experimental data.2 A rough extrapolation to z = 0 suggests a value of Re from the center of gravity to a vertex of the tetrahedron H4 lesser than 10192

DOI: 10.1021/acs.jpca.5b02788 J. Phys. Chem. A 2015, 119, 10190−10194

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reference method, which takes into account only the dynamic correlation. The found multiconfigurational nature of the wave function prevents the use of post-HF methods and poses important challenges to DFT. Even keeping the symmetry, the free-space halogen clusters dissociate into four or six halogen atoms with an unpaired electron, which implies that the nondynamic correlation is greatly important along the whole potential energy curve. DFT-based studies are inadequate to describe the potential surface, and spin contamination can be present, which determines that a single-reference representation is somewhat simplistic for the description of these molecules.

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CONCLUSIONS The main achievement of the present study involves relating low-lying isomers of halogen clusters, calculated with natural orbital functional theory from both static and dynamic electron correlation to properties of stable tetrahedral and octahedral molecules, from both experiment and theory. The natural orbital functional PNOF6(3) has been applied to determine the equilibrium geometries of free-space halogen clusters: F4, Cl4, F6, and Cl6. No deviations from the corresponding symmetries, tetrahedral or octahedral, have been permitted. Therefore, the optimized structures do not represent the lowest-lying isomers for these species and are only of theoretical interest. Our results confirm the multireferential character of Z4 and Z6 compounds (Z = H, F, Cl), in contrast with the single-reference character of the XZ4 (X = C, Si, Ge) and YZ6 (Y = S, Se, U) compounds; therefore, despite the clear patterns within a group, it is not possible to extrapolate the results to the case when the atomic number, X or Y, becomes zero. In the case of SF6, the geometry optimization with fixed octahedral symmetry afforded an equilibrium distance of 2.95 au, in perfect agreement with experiment.

Figure 3. Equilibrium distances (Re), in au, between the center and a vertex of the tetrahedron.

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Figure 4. Equilibrium distances (Re), in au, between the center and a vertex of the octahedron.

AUTHOR INFORMATION

Corresponding Author

*Tel: +34 94301 8328. E-mail: [email protected]. Notes

The obtained Re = 2.95 au for the equilibrium S−F bond length is in perfect agreement with the experimental data.21 We can see immediately that the extrapolated to z = 0 values from the MP2 calculations4 are far from those obtained by PNOF6(3) for low-lying isomers of free-space halogen clusters. For instance, we have obtained a binding energy of 39.7 kcal/ mol with respect to four fluorine atoms and an Re = 2.95 au for the t-F4 cluster, which is larger than the extrapolated ∼2.1 au. (See Figure 2.) These equilibrium distances are closer to those corresponding to the known stable clusters with an heavy atom in the center. As in the t-H4 model, the differences among the ground states that describe the free-space clusters (Zn, Z = F, Cl) of clusters with an X atom in the center (XZn) explain this behavior. In free-space clusters, the obtained occupation numbers are close to one for highest strongly occupied orbitals, specifically, four and six natural orbitals in a case of tetrahedron and octahedron, respectively. These partial occupancies indicate the intrinsic multireferential character of these states. Thus, a large number of configurations is needed to properly account for the nondynamic correlation if a wave function method is employed. On the contrary, for XZ4 and YZ6, the calculated occupation numbers are close to zero or two, so the ground state of these stable species can be easily described by a single-

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge the important contribution of Dr. Eloy Ramos-Cordoba to the full CI calculations of the tetrahedron H4. Financial support comes from Eusko Jaurlaritza (ref. IT588-13) and Ministerio de Economa y Competitividad (ref. CTQ2012-38496-C05-01). The SGI/IZO−SGIker UPV/ EHU is gratefully acknowledged for generous allocation of computational resources. N.H.M. completed his contribution to this article during a stay at DIPC. He thanks Professor P. M. Echenique for much stimulation and very generous hospitality during the visit. We also thank Professors D. Lamoen and C. Van Alsenoy for making possible N.H.M.’s continuing affiliation with the University of Antwerp. Finally, N.H.M. is indebted to Prof. C. Amovilli of the University of Pisa for invaluable discussions on the general area embraced by this article.

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REFERENCES

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