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Article pubs.acs.org/JPCA

Quantum Mechanical Study of Pre-Dissociation Enhancement of Linear and Nonlinear Polarizabilities of (TeO2)n Oligomers as a Key to Understanding the Remarkable Dielectric Properties of TeO2 Glasses Mikhail Smirnov,*,† Andrei Mirgorodsky,‡ Olivier Masson,‡ and Philippe Thomas‡ †

Physical Department, Saint-Petersburg State University, 198504 Petrodvorets, Russia Laboratoire Science des Procedes Ceramiques et de Traitements de Surface, UMR 7315 CNRS, Centre Europeen de la Ceramique, Université de Limoges, 12 rue Atlantis, 87068 Limoges Cedex, France

‡

ABSTRACT: The effects of intermolecular interactions of TeO2 molecules in the (TeO2)n oligomers on the polarizability (α) and second hyperpolarizability (γ) are investigated by the use of a density functional method. A significant intermolecular distance dependence of both quantities is observed. The huge dissociation-induced polarizability enhancement is analyzed in terms of the molecular orbital evolution. It is shown that the obtained results can provide a new look at the microscopic origin of the extraordinary dielectric properties of TeO2 glass.

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INTRODUCTION During the last two decades the computational quantum chemistry studies of the interaction induced polarizability and hyperpolarizability surfaces came to the attention of material science chemists.1−8 It was found that such an approach may provide a valuable information on polarization properties of polymerized3,4 or condensed-state systems.5 This approach was applied mostly to two-atomic6,7 or three-atomic8 systems or to organic compounds.5,9 It seems rather promising to use it for studying the inorganic clusters which can be considered as elementary structural units of the glasses. Ones of the greatest electronic linear susceptibilities χ(1) and hyper-susceptibilities χ(3) of oxide glasses presently known are those of pure TeO2 glass (2 and 50 times higher than their homologues of pure SiO2 glass, respectively10). Since establishing this fact, it was widely suggested that such extraordinary optical properties of a p-element dioxide framework like TeO2 should be related to the electron lone pairs (LP) of TeIV atom. Actually, these LPs seemed to be a single particularity of electron structure of TeO2 glass, capable of being so extraordinarily linearly and nonlinearly polarizable. So, as far as they are always present in the TeO2-based structures, it could be expected that the optic characteristic of M2O−TeO2 glasses (M = alkali metal elements), in particular their dielectric hyper-susceptibilities, would linearly decrease upon adding the M2O modifier, causing the Te atom concentration to decrease. However, the modifierinduced degradation of the nonlinear optic constants of those glasses has a rate much higher than linear. Moreover, the ab initio estimation11 of the polarizability characteristics of TeO4 units (considered as structural elementary units of condensed TeO2) has revealed that neither the LP of TeIV atoms nor the isolated TeO4 units in whole can be the source of so high χ(3) value in TeO2 glass. Thus experimental and theoretical evidence set one thinking that the extraordinary nonlinear © 2012 American Chemical Society

dielectric properties of TeO2 glass should be associated with its structural fragments other than those considered in paper,11 in particular, with the Te−O−Te bridges. Consequently, the ab initio study12 of a large variety of (TeO2)p molecular oligomers containing such bridges and having stable equilibrium structures of various shapes and dimensions were performed. Its objective was to establish which of them possess values of linear polarizability α and third-order hyperpolarizability γ capable of reproducing the gigantic χ(1) and χ(3) values of TeO2 glass. Thus it was found that this capability is inherent exclusively in the molecules having the form of the OTe⟨OO⟩Te⟨OO...⟩TeO linear chains framed from p polymerized TeO2 species, and that the chain length is a factor determining the occurrence of such properties. Actually, the ab initio calculations have revealed that specific (i.e., per one TeO2 unit) polarizability αs and hyperpolarizability γs of those chains augment nonlinearly as p augments up to 12−14, thus reaching exceptionally high values that remain practically invariable during the further chain length increase. Such an effect, called the chain length ef fect (CLE), is known for the organic polymers in which it is associated with the delocalized conjugated π-bonding inherent in the CCC− chains. Driving mainly by intuition, the idea of a strong delocalization of a dielectric response along the (TeO2)p chains was proposed in ref 12 and the relevant theoretical formalism was developed to describe the αs(p)- and γs(p)-dependencies obtained from the ab intio calculations. The chemical particularity of the OTe⟨OO⟩Te⟨OO...⟩TeO chains is reflected in the structure of TeO4 coordination polyhedrons that have a rather particular form called bisphenoid or Received: March 29, 2012 Revised: July 30, 2012 Published: August 28, 2012 9361

dx.doi.org/10.1021/jp303014k | J. Phys. Chem. A 2012, 116, 9361−9369

The Journal of Physical Chemistry A

Article

Table 1. Properties of TeO2 Molecule Calculated within Different Approximations B3LYP

MP2

property

3-21G*

SBKJC(p,2d)

TZVPPD

3-21G*

SBKJC(p,2d)

exp

R, Å Ech, kJ/mol α0, au γ0, au

1.8437 321.5 34.6 1208

1.8036 206.1 39.9 3544

1.7950 220.5 42.2 4545

1.8301 221.1 33.7 −972

1.8302 156.16 43.9 1973

1.84 238.5 45.8 ∼105

tensors were calculated. Linear polarizability α was computed analytically by using standard coupled-perturbed Hartree−Fock method,20 and then the third-order hyperpolarizability γ was evaluated numerically by double differentiation of α with respect to the external electric field. In this study we focus our attention on: • cohesive energy Ech = E(pTeO2) − pE(TeO2)

trigonal bipyramid. The two shortest TeO bonds (about 1.9 Å in length) form a OTeO angle of about 100°, and together with the Te atom LP form an equatorial plane of the polyhedrons. Consequently, these bonds are termed equatorial bonds, TeOeq. Two longer TeO bonds (about 2.15 Å in length) are almost perpendicular to the equatorial plane and are called axial bonds, TeOax. The TeOeq bonds have mainly covalent nature with bond order equal to 1.7.13 Thus, the OeqTeOeq units resemble the TeO2 molecules. The TeOax bonds are much weaker and, having bond order about 0.3,13 could be presumably considered as intermolecular contacts mainly of electrostatic origin. However, in light of the results of the ab initio study14 aimed at understanding the roles of various electronic structure fragments of the polarization properties of (TeO2)p chains, the question of the chemical nature of the TeOax bonds is a highly challenging and open question. Actually, during that study, the total molecular dipole moment (and hence polarizability and hyperpolarizability) of the chains was decomposed into the partial contributions of so-called localized molecular orbitals (LMO),15 which can be assigned either to lone-pair LMO localized on the particular atoms, or to valence bond LMO localized between two atoms. Such a decomposition for the specific polarizability and specific hyperpolarizability has offered extremely intriguing results;14 namely, the contributions of the weak TeOax bonds in both the αs and γs values were surprisingly high (providing 30% and 52% of total values, respectively), thus sending us in search of new ab initio information clarifying the nature of those bonds and thus explaining their importance in the polarization mechanism of polymerized TeO2. To make such information as objective as possible, it should be derived from ab initio calculations in a direct way, excluding any intermediate transformation of their initial results. In realizing such a program, the α and γ values of the two simplest (TeO2)p oligomers with p = 2 and 3 were studied as functions of the intermolecular separation jointly with their optimized structures and the relevant formation energies, and the results obtained are presented in this paper.

• structure variations characterized by the Te−O bond lengths • longitudinal components of polarizability and hyperpolarizability tensors, i.e, on αxx and γxxxx, with axis x directed along Te−Te line It is well-known that the calculation of (hyper)polarizabilities requires large basis sets and extraordinary accuracy in the wave function,6,8 whereas replacing the core electrons with an effective core potential (ECP) does not reduce the accuracy of (hyper)polarizabilities.21 To test the effect of the basis set extension, we used the ECP22 with extended basis sets SBKJC(p,2d)23 and Def2TZVPPD24 both containing polarization and diffuse functions. To reveal role of the electron correlation effect, we performed the calculations for the single TeO2 molecule and the (TeO2)2 dimer within MP2 routine.25 Experimental information related to the questions studied in this paper is rather scanty. It is limited by structure of TeO2 molecule26 and the energy of the dimerization,27 i.e., the cohesive energy for p = 2. No direct experimental information is available for the main object of the study: polarization properties of the molecules. One can only use values of refractive index and the nonlinear susceptibility χ(3) measured on the TeO2 glass to estimate the isotropic polarizability α0 and hyperpolarizability γ0 of the TeO2 molecule. Such estimations give α0 = 45.8 au.10 Similar estimations for the third-order hyperpolarizability derived from experimental value of nonlinear optic susceptibility χ(3) lead to value γ0 ∼ 105 au.11 Thus, we estimate quality of different computational methods judging from their ability to reproduce experimental data on geometry of TeO2 molecule, energy of dimerization, average α0 and γ0 values. Computational results are shown in Table 1. One can see that basis set extension leads to the bond length shortening, the cohesive energy decrease and augments both the polarizability and hyperpolarizability values. Use of a larger basis set (TZVPPD) gives a too short Te−O bond and does not lead to any drastic changes for other properties. So, we have decided do not use it in further analysis. Use of electron correlation correction within the MP2 method does not change markedly the Te−O bond length and reduces the cohesive energy by about 30%. The MP2 correction affects mostly the hyperpolarizability values diminishing them markedly. In total, we can conclude that extension of the basis set and refinement of the exchange−correlation correction produce considerable and opposite impacts on the calculated polarizability and hyperpolarizability values. The tested approximations do not meet the convergence criteria. Nevertheless, the

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METHODS As a computational method, the density functional theory realized in Beck’s three-parameter hybrid method using the Lee−Yang− Parr correlation functional16 was chosen. This method, being run within the 3-21G* basis set by the GAUSSIAN program17 was found to be capable of reproducing satisfactorily the energies, geometries, and polarization properties of many-electron systems such as molecules containing several tellurium and oxygen atoms.12,18,19 The oligomer dissociation process was treated as chemical reaction (TeO2)p → pTeO2. The coordinate of this reaction was Te−Te distance R. Thus, geometry of the (TeO2)p systems was optimized with respect to all degrees of freedom but R. After geometry optimization for every fixed R value, the polarizability and hyperpolarizability 9362



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results presented in Table 1 allow suggesting that the B3LYP/ 3-21G* approximation provides us quite reasonable values for all analyzed properties. It is pertinent to note that the main aim of the study is not an accurate theoretical estimation of the properties under study but an analysis of the rather strong effect: the dissociation-induced enhancement of polarizability and hyperpolarizability values. Below we show that neither further basis set extension nor refinement of the correlation correction removes this phenomenon, even if they may change its quantitative characteristics. Inasmuch as we deal with the dissociation process, it is pertinent to estimate the basis set superposition error (BSSE),28 which implies an artificial energy lowering due to inconsistent compensation of basis set incompleteness. This problem is not important for the 2(TeO2) and 3(TeO2) systems in the vicinity of their equilibrium states (because they are real covalently bonded polyatomic system) but may induce considerable errors in vicinity of the dissociation points. To test this issue, we have used the counterpoise method 29 incorporated in the GAUSSIAN-03 program. Our calculations showed that for both 2(TeO2) and 3(TeO2) systems the BSSE energy is monotonous with respect to the intermolecular separations and does not exceed 25% of the cohesive energy. Thus, the BSSE correction would lead to some flattening of the calculated Ech(R) curves but does not change considerably location of the dissociation points. It is important that this correction does not affect the polarization properties of the molecules.

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RESULTS AND ANALYSIS (TeO2)2 Dimer. The (TeO2)2 fully optimized structure (Figure 1a) corresponds to a Te−Te separation R0 equal to

Figure 2. Cohesive energy (a), lengths of Te−O bonds within Te− O−Te bridges (b), and polarizability and hyperpolarizabilty (c) of the 2(TeO2) system as functions of the Te−Te separation.

into two chemically different Te−O bonds. The shorter of them (Te1−O6 and Te2−O4) becomes intramolecular Te−O bonds at further R increase. Other ones (Te1−O4 and Te2−O6) sharply evolve becoming long intermolecular contacts which linearly lengthens with increasing R. Thus the dimer system evolves into a couple of individual TeO2 species. As a result, the α and γ values undergo the following evolutions. (iii) The α(R) curve, achieving its maximum at R = Rc, goes down and tends to an asymptote,

Figure 1. Structure of the (TeO2)2 dimer (a) and the 2(TeO2) system (b) obtained by geometry optimization at fixed Te−Te separation: R = R0 (a), R > Rc(b).

α(R ) ∼ 2α0/(1 − α0/R3)

which is the isotropic polarizability of an isolated TeO2 molecule. Such a course corresponds to the electrostatic mechanism of dipole−dipole induction polarization (a molecular analogue of the Lorentz field polarization). (iv) Being almost constant at R < Rc, the γ value vigorously increases and at R = 4.2 Å reaches maximal value, which is about 20 times as large as γ(R0) = 1.75 × 104 au. At longer intermolecular separations, the γ value monotonically decreases approaching the value 2γ0 (here γ0 is the isotropic hyperpolarizability of isolated TeO2 molecule). Figure 2c illustrates the essence of the dissociation-induced enhancement effect. To reveal how it depends on the computational scheme, we have simulated the α(R)- and γ(R)dependencies with extended basis set and within MP2 routine. These results are shown in Figure 3. One can see that shapes of all α(R) curves are similar and their maxima always occur in the points of dissociation. The three calculated γ(R) curves are also similar: they are almost flat up to the point of dissociation, at larger R values they exhibit a sharp increase and reach their

3.106 Å and possesses the symmetry of C2h group with C2 axis coinciding with the O−O line inside a double bridge Te⟨OO⟩Te. All the bridging Te−O bonds are equivalent, and their length is 2.015 Å. The calculated R-dependencies of the structural, energetic and polartizability characteristics of the dimer are show in Figure 2a−c. When R changes from R0 to a critical value Rc = 3.65 Å, the C2h symmetry of the dimer is kept, and the above-mentioned characteristics vary monotonously. At R = Rc, all the properties under consideration dramatically alter their behaviors, so that at R > Rc the following effects occur: (i) The Ech(R) curve changes sign of curvature and assumes the form close to Van-der-Waals curve −C/R6. (ii) The bridging Te−O bond length curve bifurcates into two branches, and the symmetry of the system undergoes a C2h → Ci variation. The Te−O−Te simple bridges become essentially asymmetric (Figure 1b) and divide 9363



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Figure 4. Structure of the (TeO2)3 trimer (a) and the 3(TeO2) system (b) obtained by geometry optimization at fixed Te−Te separation: R = R0 (a), R > Rc (b).

value is very closed to that found for the (TeO2)2 structure. As a model object, the trimer offers a fundamental advantage over the dimer owing to occurrence of a “classic” TeO4 bisphenoid inside it. This allows us to discriminate the roles of different structural fragments, namely, Te−Oeq (Te3−O8 and Te3−O9 in Figure 4a) and Te−Oax (Te3−O6 and Te3− O7) bonds. Indeed, neither single curve for the Te−O bond lengths (similar to that in Figure 2b for R < Rc) can be found in Figure 5b, and no bifurcation point Rc can be objectively indicated for the trimer.

Figure 3. α(R)- and γ(R)-dependencies for 2(TeO2) system calculated within B3LYP/3-21G* (squares), MP2/3-21G* (triangles), and B3LYP/SBKJC(p,2d) (circles) methods.

maxima at R ∼ 4−5 Å. The basis set extension and the electron correlation correction produce opposite effects on the dissociation enhancement effect: the former amplifies it, whereas the latter reduces. One can see that the B3LYP/3-21G* provides us a reasonable compromise. The results presented in Figure 2 unequivocally imply that the α value behavior at R > R0 is mainly (if not totally) determined by the Te−O bond length behavior. Actually, the α value increase for R < Rc is attributable exclusively to the bond lengthening, and the α decrease for R > Rc apparently results from the fact the half bridging covalent bonds occur no more, becoming the intermolecular contacts, and the remaining molecular bonds shorten. The chemical origin of the γ-hyperpolarizability variation seems to differ drastically from that mentioned above for the linear polarizability α. Actually, the γ(R) curve for R < Rc in Figure 2c unequivocally shows that as long as the system is built up exclusively of the short covalent Te−O bonds, its γ-hyperpolarizability is kept very small. Therefore, the increase of the γ value at R > Rc should be attributed exclusively to the appearance of the Te−O bonds related to the upper branch in Figure 2b, and indicated in Figure 1b by dotted lines. Our calculations showed that the relevant overlap population, although being small for such bonds, remains nonzero up to R = 4.2 Å, whereas the form of the relevant Van-der-Waals potential valley becomes more and more flat. The overlap population evidently drops for R > 4.2 Å, because all the electron density forming the broken bonds “return” into the covalent bonds of the isolated TeO2 molecules. (TeO2)3 Trimer. The optimized (TeO2)3 structure is schematically presented in Figure 4a, and the relevant R0

Figure 5. Cohesive energy (a), Te−O bond lengths (b), polarizability and hyperpolarizabilty of the 3(TeO2) system in dependence of the Te−Te separation.

At the same time, a well pronounced peak of the α curve is perfectly seen in Figure 5c at R = 3.63 Ǻ which is the inflection point of the curves in Figure 5a,b. So, conventionally, this R value (practically coinciding with the Rc value for the dimer) is 9364



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accepted as Rc in Figure 5. It corresponds to the bond lengths equal to 1.953 Å (equatorial) and 2.433 Å (axial), which are practically the same as in the dimer for Rc. At any R value, the trimer possesses C2 symmetry, which assures the equivalence of two double bridges forming its central block (Figure 4a). Figure 5b shows that, at varying R, the behaviors of the axial (Te3−O7 and Te3−O6) and equatorial (Te3−O8 and Te3−O9) bonds of the central polyhedron are practically the same as the behaviors of the long (Te1−O8 and Te2−O9) and short (Te1−O7 and Te2−O6) bonds of the terminal polyhedrons, respectively. Look now at the α(R) and γ(R) curves at Figure 5c. The shape of the first of them manifests, as itself, nothing new in comparison with that of its homologue in Figure 2c. However, particularities of its R-dependence between 3 and 4.5 Å, including the peak near 3.63 Å, must be now attributed to the evolution of the long Te−O bonds, because the lengths of the short (covalent) bonds are practically kept constant. In contrast to Figure 2c, the γ(R) curve in Figure 5c is never plane but monotonously increases with increasing R up to its huge value at R = 4.2 Ǻ , manifesting no peculiarity at the Rc point. In light of the evidence presented in Figure 2c for the dimer, such a γ value augmentation in Figure 5c cannot be related to the trimer short bonds of the Te3−O8 and Te1−O7 type, which are chemically close to the dimer short (covalent) bonds. Actually, Figure 2c clearly shows that the variation of the short bonds should keep the γ values small and constant until the “broken” long bonds appear at the Rc point. In line with this, the most vigorous γ value increase in Figure 5c takes place for the interval 3.7 Å < R < 4.0 Å, i.e., for that in which the short bond lengths are minimal and practically constant, whereas the long bonds markedly lengthen. Therefore, the role of the short bonds like Te3−O8 and Te1−O7 in the γ-hyperpolarizability mechanism can be considered as negligible, and the source of the γ value variations can be limited only by the contributions from the long bonds. A detailed comparative analysis of the curves in Figures 2 and 4 allows concluding that as long as the axial bonds (Te3−O6 and Te3−O7) exist as chemical reality (see below Discussion), their contributions dominate in the γ values of the trimer structures. In particular, for R = 4.2 Å they are 5−7 times as high as those of the long nonaxial bonds (Te1−O8 and Te2−O9).

γ=

∑ n

μn0 2 μnn 2 En0 3

−

∑ n,m

μn0 2 μm0 2 En0Em0 2

⎛ μ μ μ μ + ⎜⎜ ∑ n0 nn nm m0 + ⎝ n ≠ m En0Em0

∑ l≠n≠m

μn0 μnm μml μl 0 ⎞ ⎟ En0Em0El 0 ⎟⎠

(2)

In these two series, indexes 0 and n label the ground and the nth excited electronic states, respectively, and En0 = En − E0 is the difference between their energies. The term μn0 = ⟨n|x|0⟩

(3)

is the 0 → n transition moment. The excited states are numbered in ascending order of energy scale. Thus, the term with n = 1 gives maximal contribution to the perturbation theory series (1−2). Within the framework of molecular orbital (MO) theory using the one-electron wave functions,30 E01 value is equal to difference between orbital energies of highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO):31 E01 = Eg = εHOMO − εLUMO

(4)

Therefore, to discuss the main contributions in the series (1−2), we should focus our attention on the orbital energies of HOMO and LUMO. The orbital energies εn of the 3(TeO2) system calculated for different R are given in Figure 6 jointly with the valence shell AO energies of isolated Te and O atoms. The energy interval presented in Figure 6 corresponds to the

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DISCUSSION The central point of our discussion is a remarkable influence of the Te−Oax bond length on the polarizability properties presented in Figure 5. Actually, a huge contribution of Te−Oax bond to the γ value at R = 4.2 Ǻ (which makes it one order higher than the γ(R0) value) drastically drops for the groundstate structure in which the length of the axial bonds is only about 10% superior that of the equatorial bonds. In such a case, that contribution remains still dominating but assumes the same order of magnitude as all others.13 This fact is a challenge for the theory, and below we attempt to understand its origin on the microscopic level in analyzing the MO characteristics of the relevant molecular species. Molecular Orbital Analysis. The quantum mechanics within the framework of a static perturbation approximation yields the following expressions for the α and γ values:1 α=2∑ n>0

Figure 6. Orbital energies for Te and O atoms and for the 3(TeO2) system at different R values: A, R = ∞; B, R = 4.5 Å; C, R = Rc; D, R = R0.

valence-shell MOs and covers the interval between −1 and 0 au. It should be noted that the εn energies of the MOs related to the 4d(Te) atomic orbitals (AO) lie markedly deeper (near −1.825 au), and their positions shift by less than 0.01 au during the R variation. Their total contribution to cohesive energy is about 0.1 au Consequently, their role may be neglected in our

μn0 2 En0

(1) 9365



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Figure 7. Orbital energies of A-type (black solid lines) and B-type (red dashed lines) TMOs. Orbital energies of π-type MMOs are shown by arrows on the right vertical axis. Structure of these MMOs and the related B-type TMOs are sketched on the right.

study. Data in Figure 6 show that, throughout the whole R interval, the MO energies below −0.5 au are mainly grouped around the two levels close to those of 5s(Te) and 2s(O) AOs, whereas positions of the higher εn levels indicates a considerable 2p(O) and 5p(Te) AO mixing. The energy gap between HOMO and LUMO energies remains always well detectable. It is shown in Figure 6 by vertical arrows. It is seen that that R-dependence of Eg value is not monotonous and exhibits sharp minimum in vicinity of Rc (column C in Figure 6). It is remarkable that just at this point two factors coincide: LUMO energy is minimal and HOMO energy is maximal. More detailed presentation of εn(R)-dependencies in the vicinity of HOMO and LUMO levels is given in Figure 7 on the left. Solid and dotted lines in this figure correspond to the A-type and B-type MOs, which are symmetric and antisymmetric with respect to C2 rotation, respectively. It is remarkable that the LUMO and HOMO states belong to different symmetry types. Hence, the transition moment integral (3) between them is nonzero. Thus, the sharp maximum of the polarizability variation during the molecular association process can be directly related with the R-induced variations the LUMO and HOMO energies. Origin of the nonmonotonous R-dependencies of HOMO and LUMO energies can be revealed from a detailed analysis of corresponding molecular orbitals. For this purpose, it is appropriate to represent MOs of the trimer system (below referred as TMO) as combinations of MOs of individual TeO2 molecules (below referred as MMO). The HOMO and LUMO of an isolated TeO2 molecule are shown in Figure 8a,b. The HOMO and LUMO of the 3(TeO2) system calculated at different R values are shown in Figure 8c−h. It is seen that at large R values the trimer’s HOMO and LUMO (Figure 8c,d) are superpositions of HOMO and LUMO of the constituent molecules.

Figure 8. HOMO and LUMO of the 3(TeO2) system at different intermolecular distances.

The LUMO of a TeO2 molecule corresponds to the antibonding π-state. Figure 8d shows that a specific spatial arrangements of the trimer structure (almost parallel alignment of the constituent TeO2 molecules) gives rise to a noticeable positive intermolecular overlapping between LUMOs of neighboring molecules even at large R. This effect endows this TMO with partly bonding character, thus resulting in a progressive LUMO energy lowering. This finding allows a suggestion that all other similar TMOs constructed from molecular π-state also may be strongly Rdependent. The π-type MMOs (asymmetric with respect to molecular plane) consist of p-AOs perpendicular to this plane. There are three such MMOs in a TeO2 molecule. They are 9366



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shown in the middle part of Figure 7. The first one, symbolized by b, is the bonding MMO, which ensures the Te−O bond order higher than 1. The second one, symbolized by n, is nonbonding MMO, which corresponds to the lone pairs of oxygen atoms. The third one, symbolized by a, is the antibonding π-MMO. The b and n-MMOs are occupied and the a-MMO is LUMO. In the 3(TeO2) system, each of these πMMOs gives rise to three TMOs which are triply degenerated at large R values. Among these nine TMOs there are five B-type TMOs. As the molecules approach each other and the intermolecular overlapping occurs, the degenerated TMOs split. Character of the splitting is schematically shown in the right part of Figure 7. The TMOs constructed from a-MMO and b-MMO as MMO1+MMO2+MMO3 are labeled as a+ and b+; those constructed as MMO1−MMO2+MMO3 are labeled as a− and b− (here MMO2 corresponds to contribution of the central TeO2 molecule). It is seen that the a+ and b− TMOs give rise to positive intermolecular overlap and the a− and b+ TMOs give rise to negative intermolecular overlap. According to general criterion of covalent bonding, the former should induce energy lowering, and the latter will cause the energy increase. Such splitting is well represented in our computations: two pairs of εn(R) curves with opposite slopes are well recognized in the left graph of Figure 7. The B-type TMO constructed from n-MMO does not include any contribution from central TeO2 molecule (because of symmetry constraint); thus it does not induce any intermolecular overlap, and corresponding εn(R) curve is flat. So, at decreasing R the εn value of the a+-TMO lowers, and that of b−-TMO rises, and both approach the εn value of n-TMO. It is not possible to trace evolution of these three TMOs against a background of all other B-type TMOs because of their considerable mixing. We can only reconstruct this process reasoning from the two terminal states which are schematically depicted in Figure 9.

shown on the right. The B1-TMOs involve bonding overlap within the intermolecular axial bonds and the π-bonding within the terminal Te−O bonds. Both factors ensure lowering of the corresponding εn value. The B2-TMO does not involve any overlap. Hence, it possess the εn value close to that of nonbonding n-TMO. The B3-TMO exhibits weak negative intermolecular overlap and strong antibonding overlap within the terminal Te−O bonds. Hence, its εn value is somewhat higher than that of the antibonding a+-TMO. It is precisely this TMO which plays role of LUMO at R < Rc. The above analysis allows concluding that origin of the sharp minimum of LUMO energy at R = Rc is due to anticrossing behavior of the two TMOs originated from π-MMOs. As a result of mixing between these MMOs, the Te−Oeq bonds loss the π-bonding contribution, and the Te−Oax bonds are formed. We now turn attention to the R-dependence of the HOMO energy. The HOMO MMO is the σ-state shown in Figure 8a. Such electronic states, localized primarily within molecular planes, are weakly sensitive to the trimer association process. Hence, the A-TMOs constructed from such MMOs (Figure 8c) should give rise to a rather flat εn(R) curve. This is just what we see in Figure 7 at R > 4 Ǻ . However, at lower R values the HOMO energy suddenly increases and exhibits a sharp maximum in the vicinity of R = Rc. This is accompanied by a change of LUMO structure. Figure 8e evidence that at R = Rc the LUMO state is TMO originated from nonbonding π-MMOs. At the same time, Figure 8g shows that at R < Rc role of LUMO returns to the σ-TMO. This finding allows the following interpretation (Figure 10). At large R values, the

Figure 10. Scheme of R-induced variation of the A-type TMOs in vicinity of HOMO level.

trimer’s LUMO is σ-TMO shown in Figure 8c. This state exhibits a stable εn value. Just below this level there is another A-type TMO originated from nonbonding π-MMO. It is shown in Figure 10 on the right. Negative intermolecular overlap inherent in this π-TMO causes the corresponding εn level to rise at decreasing R. The sudden increase of the LUMO energy at R < 4 Ǻ appears when the orbital energy of the π-TMO overpasses that of the σ-TMO. At R < Rc, the π-TMO commixes with vacant A-states lying above energy gap, and role of HOMO returns to the σ-TMO (Figure 8g). To summarize, we can conclude that, along with decreasing R, the TMOs originated from π-MMOs are strongly influenced by intermolecular overlap. As a result, the low lying (in energy scale) TMOs with negative intermolecular overlap rise and the high lying TMOs with positive intermolecular overlap lower.

Figure 9. Schematic representation of R-induced evolution of B-TMOs originated from π-MMOs.

The TMOs presented in Figure 9 were determined from analysis of electronic states calculated at two extreme R-values. TMOs shown on the right have been discussed above. The arrows show directions of the εn(R) evolution. The three TMOs shown on the left can be viewed as resulting from mixing of those 9367



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Such εn variations affect the LOMO and HOMO energies, thus producing the Eg minimum, which in turn gives rise to the α(R) maximum, as suggested by eq 1. In contrast to linear polaizability α, whose general formula 1 quite transparently predicts a particularity of the α(R) curve in question, the theory of the hyperpolarizability, yielding the eq 2, hides all the peculiarities of the γ(R) curve. So, in our case, the occurrence of its maximum of γ(R) in the vicinity of R = 4.2 Ǻ cannot be explained analytically. Notwithstanding, the logic obliges us to say that the formation of the axial bonds is the only source of the high γ for so large R, and the vigorous evolution of their “embryos” results in the maximum near R = 4.2 Ǻ . Recently, absolutely the same character of the γ(R) behavior was found intrinsic in the H2 molecule in the moment when the H−H bond length is maximal just before dissociation.1,2 The purely σ-character of the H−H bond means that the similar effects observed in our work, most likely, have no relation to the intramolecular π-bonding. So, this is the long length of the weak Te−Oax bonds determining the quasimolecular nature of the solid TeO2 is one of the sources of its huge hyperpolarizability. To clarify the latter point, it can be repeated that, as this study shows, the Te−Oax bonds are the links between the delocalized π-bonds of the TeO2 species forming the TeO2 framework, thus representing the canals for intermolecular electron density migrations when the external field is applied. So, there is a good reason to think that, in being the source of the high γ value of the TeO2 olygomers, the Te−Oax bonds, at the same time, ensure the strong delocalization of the dielectric charge response. As a result, the CLE arises, and the γ hyperpolarizability of TeO2 achieves its huge values measured for TeO2 glass and calculated for TeO2 crystalline polymorphs. These aspects will be considered in a forthcoming publication. Overlook of Experimental Data. According to the results presented in Figures 2c and 5c, the highest linear susceptibility values of the TeO2-based compounds should be inherent in those in which all the long Te−O bonds have the lengths of about 2.4 Å, thus implying that those values would increase with the increasing relative content of such bonds. In this connection, it is rather interesting to compare the experimental linear susceptibilities of the two forms of the solid TeO2, namely, the paratellurite crystalline lattice and TeO2 glass. According to precise X-rays diffraction structural studies,32 the Te−Oeq and Te−Oax bond lengths in paratellurite are equal to 1.87 and 2.12 Ǻ .. The experimental information on the TeO2 glass structure is restricted to radial distribution function (RDF). According to ref 33, the RDF for pure TeO2 glass exhibits three Gaussian-like peaks attributed to Te−O distances. The first peak (the highest and narrowest one) has maximum position very close the length of the Te−Oeq bonds in paratellurite. The second peak, with a maximum near 2.13 Å, is twice broader and is much lower. Indeed, it can be attributed to the Te−Oax bonds. The presence of such two peaks in the RDF means that the “averaged” TeO4 bisphenoid in TeO2 glass is close to that in paratellurite. The special attention must be paid to the third peak centered about 2.39 Ǻ . This peak indicates the presence of the bonds which are essentially longer than the axial bonds of the “averaged” bisphenoid. In light of the idea of the chain-like constitution of TeO2 glass,34 those extraordinarily long bonds can be considered as interchain contacts which shorten (down to 2.22 Å) when the glass evolves into the metastable γ-TeO2 lattice, which is the intermediate step of the crystallization process, finally resulting in

the formation of the highest-density ground-state structure of condensed TeO2, the paratellurite lattice. The results of the present work allow us to suggest that, owing to those interchain Te−O contacts, the average specific polarizability αs of the TeO2 units in TeO2 glass (αsg) must be greater than that in paratellurite (αsp). To verify this suggestion, the linear susceptibilities of the TeO2 glass and paratellurite can be presented as χ(1)(TeO2 glass) = αsgNg and χ(1)(α-TeO2) = αspNp, in which Ng and Np are the concentrations of the TeO2 units in the glass and paratellurite, respectively. Consequently, we obtain the relation αsg:αsp = χ (1) (TeO2 glass): χ (1) (α ‐TeO2 )

Np Ng

In estimating the ratio of isotropic linear dielectric susceptibilities (reduced to zero light frequency as described in11) χ(1)(TeO2 glass):χ(1)(α-TeO2) as 3.46:3.92 = 0.88,9,35 and the Np/Ng value as the ratio of corresponding mass densities 6.04:5.11 = 1.12,32,36 we obtain the αsg:αsp ratio equal to 1.05. This fact can be considered as an experimental confirmation of the high sensibility of polarization characteristics of the condensed forms of the tellurium dioxide to their structural organization, and first of all, to the nature of the chemical bonding between TeO2 units, which is theoretically predicted by the results of the present work.

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SUMMARY AND CONCLUDING REMARKS The results of this study offer the clearer understanding of the chain length effect inherent in (TeO2)p chain-like oligomers and can be summarized as follows: (a) The HOMO of the (TeO2)3 oligomer originates from the HOMOs of isolated molecules TeO2. Corresponding electron density distribution is a sum of the LPs of the tellurium atoms and oxygen atoms. Because such a form of HOMO provides energy minimum in the course of association with the TeO2 molecules, there is a good reason to think that it is inherent to the HOMOs of all TeO2 chain-like oligomers. Consequently, the important contribution of the LPs to the total α value can be considered as the common property of the ground-state structures of (TeO2)p oligomers. (b) When the (TeO2)p oligomer evolves from equilibrium state to the predissociation state, the α-polarizability doubles its value, thus achieving its maximal value for the longest Te−Oax bond length equal to 2.4 Å, which corresponds to its breaking point. Such bond lengths are absent in the paratellurite crystal lattice, but its minor amount in the framework of TeO2 glass explains why the specific polarizability αs of the glass is 5% above that of the crystal despite the decrease of the TeO2 unit concentration. (c) An exceptional role of long Te−Oax bonds in mechanisms of a huge γ-hyperpolarizability polymerized or condensed forms of TeO2 (first revealed in ref 13) is confirmed in this study. The MO analysis shows that, during the TeO2 molecule association, the germs of these bonds appear as a consequence of the electron density overlapping between the π-electron states of neighboring molecules when the relevant Te−O separation is about 3 Ǻ. For such a nonstationary oligomer geometry, the fieldinduced charge redistribution is highly and nonlinearly 9368



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(21) Cundari, T. R; Kurtz, H. A.; Zhou, T. J. Phys. Chem. A 1998, 102, 2962. (22) Stevens, W. J.; Krauss, M.; Basch, H.; Jasien, P. G. Can. J. Chem. 1992, 70, 612. (23) Labello, N. P.; Ferreira, A. M.; Kurtz, H. A. Int. J. Quantum Chem. 2006, 106, 3140−3148. (24) Rappoport, D.; Furche, F. J. Chem. Phys. 2010, 133, 134105. (25) Head-Gordon, M.; Pople, J. A.; Frisch, M. J. Chem. Phys. Lett. 1988, 153, 503. (26) Konings, R. J. M.; Booij, A. S.; Kovacs, A. Chem. Phys. Lett. 1998, 292, 447. (27) Muenow, D. W.; Hastie, J. W.; Hauge, R.; Bautista, R.; Margrave, J. L. Trans. Faraday Soc. 1969, 65, 3210. (28) Boys, S. F.; Bernardi, F. Mol. Phys. 1970, 19, 553. (29) Simon, S.; Duran, M.; Dannenberg, J. J. J. Chem. Phys. 1996, 105, 11024. (30) Roothaan, C. C. J. Rev. Mod. Phys. 1951, 23, 69. (31) Pouchan, C.; Begue, D.; Zhang, D. Y. J. Chem. Phys. 2004, 121, 4628. (32) Thomas, P. A. J. Phys. C 1988, 21, 4611. (33) Niida, H.; Uchino, T.; Jin, J.; Kim, S.; Fukunaga, T.; Yoko, T. J. Chem. Phys. 2001, 114, 459. (34) Noguera, O.; Merle-Mejean, T.; Mirgorodsky, A. P.; Smirnov, M. B.; Thomas, P.; Champarnaud-Mesjard, J.-C. J. Non-Cryst. Solids 2003, 330, 50. (35) Uchida, N. Phys. Rev. B 1971, 4, 3736. (36) El-Mallawany, R. A. H. Tellurite Glasses Handbook; CRC Press: Boca Raton, FL, 2002.

dependent on the overlapping strength, which is manifested by a giant variation of the γ-hyperpolarizability values. It can be thought that, in the case of the equilibrium structures, the Te−Oax bonds endow the charge redistribution process with a delocalized character, thus assuring the chain length effect (i.e., an increase of specific α and γ values along with length of −Oax−Te−Oeq− chains). (d) This work, clarifying the nature of the Te−Oax bonds and evaluating their critical length, provides important contribution for the fundamental crystal chemistry of tellurites, as well as for applied material science. At the same time, the data on the Te−O bond length dependences of the α and γ values can represent a considerable interest for the linear and nonlinear optic device engineering.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the University of Limoges and the Conseil Régional du Limousin for financial support. REFERENCES

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