Chem. Mater. 1993,5, 661-671
661
Theoretical Study of the Nonlinear Optical Properties of KTiOP04: Cooperative Effects in Extended --Ti--0--Ti--0-Chains M. Munowitz' and R. H. Jarman Amoco Technology Company, P.O. Box 3011, Naperville, Illinois 60566-7011
J. F. Harrison Department of Chemistry and Center for Fundamental Materials Research, Michigan State University, East Lansing, Michigan 48824-1322 Received November 19, 1992. Revised Manuscript Received February 18,1993
The sum-over-states perturbative formalism, combined with extended Huckel wave functions, is used in a study of second harmonic generation in model systems related to KTiOP04 (KTP). Clusters of one, three, and five interconnected Ti06 groups are constructed to simulate the extended --Ti--0-Ti--0- chains present in the crystal. T o facilitate comparison and illuminate differences among systems of different sizes, we compute a specifically local nonlinear response. Ground and excited states are determined for the full structure, but only those portions of the molecular orbitals relevant to a designated subsystem (typically the central Ti06 group) are used to evaluate dipole transition moments in the s u m over states. Results show, first, that the nonresonant hyperpolarizability is enhanced locally as more Ti06 units are added. An intact and terminated structure, whether an isolated Ti06 monomer or trimer, generally exhibits a smaller nonlinearity, with opposite sign, than a fully integrated structure with the same number of Ti06 groups. A second observation is that, despite differences in magnitude, the changes in hyperpolarizability accompanying an axial distortion of each octahedron follow a nearly uniform pattern in chains of different lengths. These various effects are interpreted by comparing relative changes in total electronic density and by a detailed analysis of the matrix elements and molecular orbitals contributing to the perturbative expression. Indications are that simple molecular orbital pictures developed for isolated fragments may be insufficient to account for the cooperative nonlinear response of highly delocalized systems such as KTP.
Introduction An understanding of the relationship between microscopic structure and macroscopic properties has long been recognized as an important aid in the design of nonlinear optical materials. Computations of molecular and bulk hyperpolarizabilities,for example, have been of interest since the first laser measurements in the 1960s, the motivation for such modeling being as much a need to elucidate qualitative structurefunction relationships as to begin developing a quantitative predictive capabi1ity.l The approaches taken to date encompass a variety of theoreticalmethods, but common to most is an assumption that the aggregate nonlinear response originates from a localized microscopic entity such as a bond, anion, or molec~le.~-~5 Bulk susceptibilities then are derived by (1) For general reviews, see: (a) Robinson, F. N. H. Bell Syst. Tech. J. 1967,46,913. (b) Flytzanis, C. In Quantum Electronics: A Treatise;
Rabin, H.;Tang, C. L.;Eds.; Academic: New York, 1975;Vol. 1, pp 9-201. (c) Chen, C.; Liu, G. Annu. Reo. Mater. Sci. 1986,16,203. (d) Chemla, D. S., Zyss, J., Eds.Nonlinear Optical Properties of Organic Molecules and Crystals; Academic Press: Orlando, FL, 1987. (e) Stucky, G. D.; Phillips, M. L. F., Gier, T. E. Chem. Mater. 1989, 1, 492. (2) Phillips, J. C.; van Vechten, J. A. Phys. Rev. 1969, 183, 709. (3) (a) Levine, B. F. Phys. Rev. Lett. 1969,22,787. (b) Levine, B. F. Phys. Reo. Lett. 1970, 25, 440. (c) Levine, B. F. Phys. Reu. B 1973, 7, 2591. (d) Levine, B. F. Phys. Reu. B 1973,7,2600. (e) Levine, B. F. Phys. Reu. B 1974, 10, 1655. (4) (a) Bergman, J. G.; Crane, G. R. J. Chem. Phys. 1974,60,2470. (b) Tofield, B. C.; Crane, G. R.; Bergman, J. G. J.Chem. SOC.,Trans. Faraday SOC. 1974,1488. (c) Bergman, J. G.; Crane, G. R. J. Solid State Chem. 1976, 12, 172.
geometric superposition of the hyperpolarizabilities attributable to the presumed irreducibleunits, perhaps with some attempt to correct for intermolecular interactions or local-field effects. According to this view, it is the behavior of the microscopic group, supplemented by an appreciation of how the local units combine, that is central to understanding nonlinear optical phenomena in solids. The group approach, in particular Chen's anionic group model, has been applied to a number of inorganic crystals with some s u c c e s ~ .Among ~ the systems investigated have been various titanates, niobates, iodates, phosphates, (5) (a) Wemple, S. H.; DiDomenico, M. Jr.; Camlibel, I. Appl. Phys. Lett. 1968,12,209. (b) DiDomenico, M. Jr.; Wemple, S. H. J.AppLPhys. 1969,40,720. (c) Wemple, S. H.; DiDomenico, M. Jr. J.Appl. Phys. 1969, 40, 735. (6) Zumsteg, F. C.; Bierlein, J. D.; Gier, T. E. J. Appl. Phys. 1976,47, 4980. (7) Morrell, J. A.; Albrecht, A. C. Chem. Phys. Lett. 1979, 64, 46. (8) (a) Lalama, S. J.; Garito, A. F. Phys. Rev. A 1979,20, 1179. (b)
Lalama, S. J.; Singer, K. D.; Garito, A. F.; Desai, K. N. Appl. Phys. Lett. 1981,39,940. (c) Teng, C. C.; Garito, A. F. Phys. Reu. Lett. 1983,50,350. (d) Teng, C. C.; Garito, A. F. Phys. Rev. B 1983,28,6766. (9) (a) Chen, C. T. Acta Phys. Sin. 1976,25,146. (b) Chen, C. T. Acta Phys. Sin. 1977,26, 124. (c) Chen, C. T. Acta Phys. Sin. 1977,26,486. (d) Chen, C. T. Sci. Sin. (Engl. Ed.) 1979,22,756. (e) Chen, C. T.; Chen, X. S. Commun. Fujian Inst. Struct. Matter 1979,2,51. (0Chen, C. T.; Chen, X. S. Acta Phys. Sin. 1980,29,1000. (8) Chen, C. T.; Liu, 2.; Shen, H. ActaPhys. Sin. 1981,30,115. (h) Chen, C. T.; Wu, B.; Jiang,A.;You, G. Sci. Sin. Ser. E (Engl. Ed.) 1986,28, 235. (i) Li, R. K.; Chen, C. T. Acta Phys. Sin. 1986,34,823. (j) Wu, Y. C.; Chen, C. T. Acta Phys. Sin. 1986, 35, 1. (k) Chen, C. T.; Wu, Y. C.; Jiang, A.; Wu, B.; You, G.; Li, R. K.; Lin, S. J. Opt. Am. SOC. Am. B 1989,6,616. (10) Docherty, V. J.; Pugh, D.; Morley, J. 0. J. Chem. SOC., Faraday Tram. 2 1986,81,1179.
Q897-4756/93/2SQ5-Q66l~Q4.QQJQ 0 1993 American Chemical Society
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662 Chem. Mater., Vol. 5, No. 5,1993 (a) h
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(b)
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Figure 1. Crystal structure of KTP. (a) A projection along [OOl], showing alternating Ti06 octahedra and Po4 tetrahedra. The potassium sites are represented by open circles. (b) A view along [Oll], illustrating two intertwined chains of octahedra. Each pair of octahedra in a chain shares one bridging oxygen.
nitrites, and borates, there emerging from these latter studies the new nonlinear crystals b-BaBz04 and LiB305. In many of the cases reported, the anionic groups are quasimolecular and well separated from each other (as with IO3-, for instance), which lends support to the basic assumption of a localized response. In other structures, however, notably the distorted MO6 octahedra characteristic of perovskite and tungsten-bronze materials, identification of a unique anionic group is complicated by the sharing of vertices between adjacent octahedra. KTiOP04 (KTP), a preferred material for second harmonic generation at 1064nm,6J619 is perhaps typical of such a crystal with its chains of Ti06 octahedra and Po4 tetrahedra running parallel to the a and b axes as shown in Figure la.20 The Ti06 groups themselves form helices
Figure 2. Idealized TiOs clusters, used to model the KTP structure. (a) The basic structural unit, a Ti06 octahedron distorted along a C4axis, defines a common coordinate system. (b) Monomer, trimer, and pentamer. The numbers at the far right (2, 1, 0, -1, -2) mark the approximate positions of the individual Ti06 groups in each chain. Groups 1and -1 are linked to the original unit, group 0, via axial oxygens 01 and 02. Ti-0--Ti bond angles are 135'. Note that the six osygens in the central Ti06 of each cluster, shown shaded, retain the labels 01 through 0 6 in subsequent analysis.
1171. (13) Phillips, M. L. F.; Harrison, W. T. A.; Gier, T. E.; Stucky, G. D.; Kulkarni, G. V.; Burdett, J. K. Znorg. Chem. 1990,29, 2158. (14) Lines, M. E. Phys. Reu. B 1990,41,3383. (15) Dovgii, Y. 0.; Kityk, I. V.; D'yakov, V. A. Fiz. Tuerd. Tela tl.pnin"il ~ - - .... --, 1989. _ _ _31. _. _, ,9 -.. (16) Eimerl, D. Proc. SPIE-Int. SOC.Opt. Eng. 1986, 681, 5. Eckhardt, R. C.; Fan, Y. X.; (17) Fan, T. Y.; Huang, C. E.; Hu, B. Q.; Byer, R. L.; Feigelson, R. S.Appl. Opt. 1987, 26, 2391. (18) Vanherzeele, H.; Bierlein, J. D.; Zumsteg, F. C. A .. p p l . Opt. 1988, . 27, 3314. (19) (a) Baumert, J. C.; Schellenberg, F. M.; Lenth, W.; Risk, W. P.; Bjorklund,G. C. Appl.Phys.Lett. 1987,51,2192. (b) Risk, W. P.;Baumert,
in which the octahedra are linked alternately cis and trans such that the Ti--0 bond lengths in the --Ti--0--Ti--0backbones cycle through values of approximately 2.1 A (longer than average), 1.74 A (shorter than average), and 1.96 A (average). Each Ti06 group is therefore distorted along a fourfold axis, and the resulting deviation from strict Oh symmetry is thought to be at least partly responsible for KTP's hyperpolarizability. The extended chains of octahedra, within which each pair has one oxygen in common, are highlighted in Figure lb. In recent work we have studied the nonlinear response of axially distorted TiO&, taking this anionic group as the basic polarizable entity existingin the extended chains of the crystal structure.21 The next stage in the analysis, undertaken here, is to build up such chains computationally and observe the consequences on the hyperpolarizability tensor. We ask, Given the response of a single Ti06 group, how does the hyperpolarizability change when the original unit is placed at the center of an extended chain? How is the electron density at the original site perturbed by the bonding of additional groups? How reasonable is it even to model the nonlinear response of
1988, 52, 85. (20) Tordjman, I.; Masse, R.; Guitel, J. C. Z. Kristallogr. 1974, 139, 103.
(21) Munowitz, M.; Jarman, R. H.;Harrison, J. F. Chem.Mater. 1992, 4, 1296.
(11) (a) Li, D.; Ratner, M. A. Chem. Phys. Lett. 1986, 131, 370. (b) Li, D.; Ratner, M. A.; Marks, T. J. J. Am. Chem. SOC.1988, 110, 1707. (c) Kanis, D. R.; Ratner, M. A.; Marks, T. J. J.Am. Chem. SOC.1990,112, 8203. (d) Kanis, D. R.; Ratner, M. A.; Marks, T. J.; Zerner, M. C. Chem. Mater. 1991, 3, 19. (12) (a) Hurst, G. J. B.; Dupuis, M.;Clementi, E. J.Chem.Phys. 1988, 89,385. (b) Perrin, E.; Prasad, P. N.; Mougenot, P.; Dupuis, M. J.Chem. Phys. 1989, 91, 4728. (c) Karna, S.P.; Dupuis, M. Chem. Phys. Lett. 1990,171,201. (d) Daniel, C.; Dupuis, M. Chem. Phys. Lett. 1990,171, 209. (e) Karna, S. P.; Prasad, P. N.; Dupuis, M. J. Chem. Phys. 1991,94,
_O.
J. C.; Bjorklund, G. C.; Schellenberg, F. M.; Length, W. Appl. Phys. Lett.
Chem. Mater., Vol. 5, No. 5, 1993 663
Nonlinear Optical Properties of KTiOPOr a solid-state system as arising fundamentally from a localized cluster?
As in our previous study, we will be interested entirely in relative effects rather than absolute accuracy. Our concern is only with how a particular property changes in response to a well-defined structural variation-the attachment of an additional group, say, or the lengthening of a specific bond. To this end we again use a perturbative sum-over-states €0rmalism~~-24 based on extended Huckel wave functions.25 The Huckel approximations, though drastic, greatly simplify the calculations while retaining the symmetry and basic structure of the molecular orbitals. Systematic errors introduced by neglect of interelectronic repulsion should affect equally the closely related structures we consider, thus preserving the underlying effects of the geometric variations. S t r u c t u r a l Models We simulate the --Ti--0-Ti--0- chains in KTP by using the Ti06 speciesillustrated in Figure 2a as a basic structural unit. This cluster is an idealized configuration in which the oxygens are arranged to form a distorted octahedron, with the asymmetry along the z axis characterized by two parameters, ro and A, that give the average and difference of the two axial Ti-0 bond lengths:
Each of the remaining four oxygens (03, 04, 05, 06) is placed along an axis in the xy plane at a distance of ro from the central titanium. This arrangement, which remains fixed at the center of the chain as additional octahedra are attached, defines the coordinate system for the calculation of the hyperpolarizability tensors.
It will be convenient to refer to the central Ti06 unit as group 0. The Ti06 cluster sharing 01 then is labeled as group 1,whereas the group attached via 0 2 is labeled as -1. The second Ti06 group in the positive hemisphere (z> 0) is similarly designated as +2, and the second group in the negative hemisphere as -2. Relevant monomeric, trimeric, and pentameric structures with Ti--0--Ti bond angles of 135O are shown in Figure 2b. The chain is formed by appropriate rotations and translations of the central unit. Group 1,for example, is oriented by rotating the coordinates of group 0 according to (2) r(1)= T,(B3)TY(B2) T,(8,) r(O) with B1 = 82 = 83 = -45O and with the rotation matrices (22)Armstrong, J. A.; Bloembergen, N.; Ducuing, J.; Pershan, P. S. Phys. Rev. 1962,127,1918. (23) (a) Ward, J. F. Rev.Mod. Phys. 1965,37,1.(b) Orr, J. B.; Ward, J. F. Mol. Phys. 1971,20,513. (24)Pugh,D.;Morley,J.0.InNonlinear OpticalPropertiesofOrganic Molecules and Crystals; Chemla, D. S.,Zyss, J., Eds.;Academic Press: Orlando, FL, 1987;Vol. 1, pp 193-225. (25) (a) Hoffmann, R.J. Chem. Phys. 1963,39,1397.(b) Janiak, C.; Hoffmann, R. J. Am. Chem. SOC.1990,112,5924.
taken as
(’
T, = 0 cos 8 0sin8
) Ty=(F8p”)
0 -sin8 cos8
sin8 0 cos8
cos8 sin8 0 T, = -sin 8 cos 8 0 (0
O
(3)
1)
The sequence of Euler angles (read right to left) corresponds to a counterclockwise rotation of the atomic coordinates around the axes of the evolving reference frame. Once rotated, group 1 is attached to group 0 by translating 0 2 of group 1 (originally the oxygen at -2) to 01of group 0 (at +z). The original 01 is common to both octahedra. Group -1 is positioned by a rotation TZ(-45O) T,(45’) T,(45O) followed by a displacement of 0 3 (originally at + x ) to 0 2 of group 0. Group 2 is obtained by a 30° rotation of group 0 about z and a subsequent 15O rotation about y, followed by a translation of 0 2 to 0 5 of group 1. For group -2, the rotation is the same but the displacement is from 01 of the rotated group to 0 2 of group -1. In forming the electronic structures we assume oxidation states of Ti4+ and 02-,thereby ensuring closed-shell systems for the species Ti068-, Ti3016~’)-, and Ti502632-. These formal charges are consistent with recent experimental measurements of electron density in KTP, which confirm the presence of strong covalent bonds between Ti4+and 0 2 - ions and which further reveal that the P--0 interactions are primarily electrostatic.26 Implicit in our neglect of both the potassium counterions and the phosphates is the widely held belief that hyperpolarizability in KTP originates within the covalently bonded --Ti--0-Ti--0- system. The effect of the other ions is presumably important, but is more of an indirect influence reflected in the geometric and electronic structure assumed by the --Ti--0-Ti--0- chains in the full crystalline environment. We will touch on some of these points later, in a brief section intended to provide some experimental perspective. Computational Approach The first hyperpolarizability tensor,& is defined through the expression Pi = aij(0)Ej(w)
+ @ijk(-w;wl,w2)Ej(w1) E k ( a 2 ) +
Yijkl(-w;wl’,wi,w;) Ej(O1’)E k ( W i ) El(w,’) + (4) where pi (i = x , y, z ) is a component of the dipole moment induced in the molecule by time-dependent electric fields E(w). Repeated indexes are understood to be summed over the three Cartesian coordinates. For second harmonic generation (SHG), by which an incident field oscillating a t w induces a dipole moment oscillating at 2w, the standard perturbative expression for @ijk depends on excited-state energies and dipole transition moments.22-24 The numerator of every term in the sumover-states expression is a product of three integrals, *.e
(5) (i.ik) = (&In) (nPjIm) (mPB,k) each describing a transition moment between two excited (26) Hansen, N.K.;P r o d , J.; Marnier, G. Acta Crystallogr. 1991, B47, 660.
Munowitz et al.
664 Chem. Mater., Vol. 5, No. 5, 1993 states, Im) and In), and the ground state k) along a particular axis denoted by the N-electron position operator Ri (=A,9, The corresponding denominators involve various combinations of differences among u,20, wg, w,, and wn. Details of the formalism and its implementation in the extended Huckel model, as well as values of the relevant Huckel parameters, may be found in our previous analysisz1of Ti02 and TiOsa. Initially, what we want is to understand how the electronic structure and nonlinear response of a single Ti06 are modified in the presence of a growing chain. For this purpose we will specially compute the microscopic hyperpolarizability relevant to just the central unit of each chain, using only those portions of the molecular orbitals with amplitude at the seven atoms involved. If this grouping (shown shaded in Figure 2b), were electronically isolated from the rest of the system, the values so determined would be identical to those of a single TiOs. With electrons free to delocalize throughout the --Ti--0-Ti--0- chain, however, the partial hyperpolarizability is expected to be affected accordingly. By concentrating on this local response, we are now able to make consistent, direct comparisons between the isolated and integrated Ti06 groups and in so doing clearly discern the influence of the surrounding chain. Thus established, the notion of a local hyperpolarizability can be extended to encompass larger portions of the chain-such as a trimer within a pentamer, a pentamer within a heptamer, and so forth. A model of this sort provides, in each instance, an unperturbed look at a selected piece of the whole system, a view that is undistorted by edge effects from any terminal oxygens. Computation of such a local @ over any region requires, first, that molecular orbitals for the complete chain be expressed as a linear combination of atomic orbitals at all positions, from which a normalized and antisymmetric wavefunction is formed in the usual way. Then, in subsequent evaluations of the matrix elements in (51, the molecular orbital expansions are truncated to include only those atomic orbitals within the desired subset. The overall electronic structure is therefore not disturbed in any way, but attributes such as the population density and dipole transition moments are cleanly identified for the subsystem alone.
a.
Energy Levels and Molecular Orbitals Energy levels for the Ti06 monomer, trimer, and pentamer described above are shown in Figure 3. The axially distorted octahedron used in this illustration is given the values ro = 1.96 A and A = 0.3 A, close to those observed in the KTP crystal.20 The full range of orbital energies is displayed in Figure 3a, and a subset depicting just the highest occupied and lowest unoccupied levels is shown below in Figure 3b. Spanning this latter band of orbitals, which becomes increasingly dense as the chain grows, are most of the virtual transitions associated with the hyperpolarizability. We note that the bandgap so determined is approximately 5.0 eV, as compared to an experimentally measured absorption edge of 3.54 eV.6 Molecular orbitals in the larger systems may, in many instances, be related to those of Ti02 alone. For us the most important mixtures fall into three broad categories involving the titanium and axial oxygens: (1)dp 7r and u bonding orbitals arising from in-phase combinations, (2) the corresponding ?r* and u* antibonding combinations,
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Figure 3. Energy levels for intact monomeric, trimeric, and pentameric structures with ro = 1.96 A and A = 0.3 A, from extended Huckel calculations. In each panel, the monomer is at the left and the pentamer is at the far right. (a) Full structures. The lowermost groups of levels near -32 eV arise from unmixed core atomic orbitals. (b) Expanded view showing the highest occupied and lowest unoccupied levels. The HOMO-LUMO gap is clearly visible. Dense bands of levels begin to form as the chain grows.
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Figure 4. Principal bonding, nonbonding, and antibonding orbitals in a linear arrangement of titanium and two oxygens. Both ?r and u overlaps are illustrated. The diagrams assume equal weight for the oxygens; when the Ti--0 internuclear distances are unequal, the mixing of the oxygen orbitals is altered accordingly.
and (3) various 7r and u nonbonding orbitals in which the titanium does not participate. The basic orbitals as they would appear in symmetric Ti02 are shown schematically in Figure 4; variants of these forms (incorporating the equatorial oxygens from each octahedron) dominate the spectroscopically relevant region shown in Figure 3b, with the nonbonding orbitals generally appearing a t the top of
Nonlinear Optical Properties of KTiOP04 la\
Chem. Mater., Vol. 5, No. 5, 1993 665
I
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Figure 5. Point-to-point plots showing the hyperpolarizability of Ti06 in various monomers, trimers, and pentamers (ro = 1.96 A). Unib are 10-30 cgs, and the coordinate system is as drawn in Figure 2. (a) Curve 1 gives the value of &, in an intact monomer, while curves 2 and 3 display, respectively, the corresponding local values for just the central Ti06 within the trimer and pentamer. Note the differencesin sign and magnitude between &zz for the isolated and embedded monomeric species. (b) The three curves coincide when values for the trimer and pentamer are multiplied by a uniform scaling factor.
the occupied band and the bonding orbitals just below. The relative amplitudes of the oxygen components evolve smoothly with the asymmetry parameter A. Further below the bonding orbitals lie mostly unmixed atomic core orbitals, which appear in each of the three parta of Figure 3a as the lowermost cluster of levels. Above the antibonding orbitals just discussed are additional outof-phase combinations and various other mixtures involving titanium p and d orbitals.
Local Hyperpolarizability Values in Chains of Ti06 We will follow the value of flzZz, generally the largest element in the local SHG tensor, for structures with A I 0.7 A and ro = 1.96 A. The incident wavelength is taken as 1064 nm throughout, and the hyperpolarizabilities are reported in units of cgs. Except in the case of the monomer, where the full set was used, all singly excited states were generated from within the subset of orbitals shown in Figure 3b. This simplification greatly reduces the computational effort required for the chains while having only a small and systematic effect on the value of &, (approximately 1-2 7% at most). Figure 5a provides results for a single Ti06 group, both as an isolated monomer and as a monomer embedded in a trimer and pentamer. The SHG coefficients for the isolated monomer, which have been analyzed in our earlier
work,21 are everywhere negative and show increasing magnitude as the octahedra become more asymmetric. Taking the intact monomer as a reference, we note immediately that the local flzZzvalues Of Ti06 in the trimer are opposite in sign and enhanced in magnitude more than threefold over the range of A considered. The change in local hyperpolarizability effected by going from the trimer to the pentamer, by contrast, is much less pronounced. The sign remains the same, and there is only a 20 % increase in magnitude. Despite these obvious differences between the embedded clusters and the single octahedron, there emerges a striking similarity in the shapes of the j3 vs A curves when the values are scaled to a common reference point. This feature is made clear in Figure 5b, where the data are shown normalized to an arbitrarily selected value observed for the isolated monomer. The procedure, for both trimer and pentamer, is to establish a uniform scaling factor s by taking the ratio of t9,,,(A = 0.6 A) in the isolated monomer to the corresponding local j3zzz(A= 0.6 A) in the particular chain structure. The factor s is different for each chain, but the curves for the monomer, trimer, and pentamer prove to be nearly coincident after the appropriate j3values are renormalized and plotted as s.t9zZZ(A). These same patterns continue to hold as the local unit is extended to encompass a trimer within a pentamer: First, the local hyperpolarizability per unit length over the larger region (the trimer-in-pentamer) is seen to be substantially stronger than for the central octahedron alone (monomer-in-trimer). As an approximate measure of the nonlinear reponse per individual Ti06 group, in Figure 6a we divide by three the local PzZzvalues for the trimer-inpentamer and compare them directly to the local results for the monomer-in-trimer. From these values, whichnow relate to single octahedra in different environments, it is evident that hyperpolarizability at each link of the chain is amplified more than fivefold when the trimer is placed inside the larger structure. This enhancement clearly is not a simple geometric effect brought about by the appropriate transformation and superposition of tensors for groups 1, 0, and -1. It is, instead, a collective effect arising from the change in electronic structure that is brought about when the trimer is allowed to interact with additional groups. Second, the terminating Ti06 groups on the intact trimer (1and -1) determine the intact trimer’s total PZXrin much the same way as do the terminal oxygens (01and 02) for the isolated monomer. The additional nonlinear response originating from the ends of each structure is large and opposed in sign relative to what occurs in the interior. The net effect is illustrated in Figure 6b, where we compare the local j3z,, for the trimer-in-pentamer with the full j3zzz for the terminated trimer. Note that the values plotted here are not divided by 3 but reflect instead the response of the approximately 128electrons shared among the three octahedra. Again, the isolated structure exhibits a smaller hyperpolarizability, with opposite sign, compared to the embedded structure of the same length. Third, the shapes of the various j3 vs A curves are maintained. Any two pairs of curves generated above can be brought into near coincidence by uniform scaling of one set of values.
Analysis It is important to remember that the local B tensor is not necessarily a measure of the full hyperpolarizability
Munowitz et al.
666 Chem. Mater., Vol. 5, No. 5,1993 1
.
.
.
.
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Figure 6. Local hyperpolarizability arising from a 'trimer-inpentamer"-three octahedra situated within a chain of five. (a) Curve 1,obtained by dividingeach local trimer-in-pentamer value by 3, provides a rough estimate of the hyperpolarizability per octahedron in the embedded three-unit structure. Curve 2 shows the corresponding values for a single Ti06 within a trimer. (b) Unscaled local @forthe trimer-in-pentamer (curve l),compared with the full /3 of an isolated trimer (curve 2). The two curves differ in sign and magnitude.
of the larger systems. Rather, the quantity defined here indicates how the nonlinear response in some region of the chain (the central Ti06 unit, for example) changes as additional octahedra are connected via the terminal oxygens. The SHG coefficients thus determined are corrected values that begin to reflect the altered electronic structure at the original site. Bearing in mind this definition, we can now try to account for the principal observations above-namely the immediate local enhancement and sign change of &, as the chain grows, combined with the apparent preservation of shape in the j3 vs A curves for all systems. Electron Densities. For simplicity, we will concentrate on the case of monomer-in-trimer. Here a useful first step is to ascertain where the electrons go upon attachment of groups 1and -1 to the original octahedron. Figure 7 shows, for A = 0.0,0.3,and 0.6 A, the differences in total electron density between the isolated trimer and monomer. The dashed contours delineate regions of depleted density in the trimer relative to the monomer, while the solid contours outline areas of increased density in the trimer. From these diagrams we see that electrons are shifted from both axial oxygens to the titanium atoms of groups 1 and -1, with the shift becoming progressively more skewed as the octahedra are distorted. Whereas 0 2 is the near oxygen in group 0, it finds itself only at an intermediate distance ( 2 A) relative to the titanium in group -1 and is therefore less likely to delocalize into that group. Similarly, the
-
Figure 7. Differences in total electronic density between the trimer and monomer, viewed in the x z plane in the vicinity of the central TiOe. Each panel shows a set of 30 contours, equall spaced between the values -0.2 and 0.6, displayed over an 8 X 8 8,square with the titanium site at the center ( x = 0, z = 0). Contour plots here and in subsequent figures are oriented with z running vertically and x horizontally: (a) A = 0; (b) A = 0.3 A; (c) A = 0.6 A. The dashed contours around the 01 and 0 2 sites correspond to negative values, indicating depleted electronic density in the trimer. The distant oxygen (01)is above, and the near oxygen (02) is below in each plot.
d:
Ti-01 bond goes from being long in group 0 to short in group 1,and consequently electrons flow disproportionately out of 01 with increasing A. Mulliken population^^^^^^ provide a rough picture of the redistribution of ground-state electronic density. At A = 0, where both axial oxygens carry equal populations, the excess charge assigned to each amounts to -1.67 in the monomer and -1.36 in the trimer. Population assigned to all seven atoms in group 0 is 47.44 in the trimer, compared to 48.00 in the monomer. Both sets of figures are consistent with a net outflow of approximately 0.6 electrons to bring about bonding with the 1and -1 groups. As we follow the continuing redistribution with A) we then observe that (27) Mulliken, R. S.J. Chem. Phys. 1955,23,1833,1841,2338,2343.
(28) Noell, J. 0.Inorg. Chem. 1982, 21, 11.
Nonlinear Optical Properties of KTiOP04
Chem. Mater., Vol. 5, No.5, 1993 667
(a)
-W0 n a
0
0.4
0.2
0.6
A (A)
t
it
1
-12
0
0.4
0.2
A
0.6
(A)
Figure 8. Ground-state dipole moment along z for isolated and embedded Ti06 monomers, versus A. The momenta, in atomic units, are calculated in the coordinate system of Figure 2, with the central titanium at the origin. (a) Full values for isolated octahedra (curve l),compared to local values for monomer-intrimer (curve2). (b) Asabove, butwith thelocalvalues uniformly scaled to coincide with those of the isolated structure.
this outflow increases to 0.9 electron (local population = 47.08) in the trimer at A = 0.7 A. In the monomer,the formal charge on the distant oxygen (01)increases from -1.67 to -1.90 as A increases from 0 to 0.7 A;in effect, 01 is physically withdrawing from the group as the species 02-.The charge on the near oxygen, by contrast, decreases from -1.67 to -1.29 over the same range of A, reflecting the enhanced ability of 0 2 to delocalize into the Ti06 as a whole when the T i 4 2 distance is shortened. In the trimer, however, charges on both axial oxygens decrease with increasing A-from -1.36 to -1.20 for 01, and from -1.36 to -1.05 for 02. Electron density associated with the central Ti06 in the trimer thus arises from competing tendencies. For example, 01's tendency to withdraw charge as the Ti-0 separation in group 0 becomes greater is balanced by the increasing demands of the titanium in group 1(to which 01 draws closer). The net result is encapsulated in the z component of the local monomericdipole moment within the trimer, which we show in Figure 8a along with the corresponding quantity for the isolated monomer. The moment for each structure depends approximatelylinearly on A; moreover, as is evident in Figure ab, the curves nearly coincide when the quantities are scaled to a common value in the manner discussed above for Figure 5b. A similar result holds for the full dipole moment of the complete trimer, with its total of 128 electrons. The common dependence of dipole moment on A for monomer and trimer does not by itself explain the common
1
Figure 9. Differences in total electronic density between distorted (A # 0) and symmetric (A = 0) monomers and trimers, viewed in the xz plane as in Figure 7. Thirty equally spaced contours are drawn between values of -1.5 and 1.5,with dashed lines used to indicate negative values. The grid is 8 A X 8 A, and the atoms are positioned as in Figure 7: (a) monomer, A = 0.2 A relative to A = 0; (b) trimer, A = 0.2 A relative to A = 0; (c) monomer, A = 0.4 A; (d) trimer, A = 0.4 A; (e) monomer, A = 0.6 A; (0trimer, A = 0.6 A. Note the similarity between patterns for the monomer and trimer; note also that the titanium remains electronically uninvolved in the distortion.
dependence of &, on A mentioned above, since the former quantity is merely one attribute of the ground-state electronic distribution. Nevertheless the observation suggests immediately that, locally at least, ground-state electronic structure in the monomer and trimer follows a similar pattern in adjusting to the distortion from octahedral symmetry. Furthermore, the pattern is likely to carry over to the excited statesinasmuch as the component orbitals of both ground and excited states are identical in the one-electron extended HUckel model. The similarities become even more pronounced as we inspect the density differences between distorted and symmetric structures. Figure 9 presents difference contours, computed separately for monomer and trimer, showing total density a t various values of A relative to the corresponding density at A = 0. Here it is clear that the change with A is again qualitatively similar for monomer and trimer; apparently whatever the effect is (be it in the context of a chain or an isolated system), it happens in nearly the same way when the octahedra are distorted. Changes in local electronic structure between the trimer and the pentamer are much smaller and not visible a t the contour levels used in the preceding illustrations. A diminution of the effects felt a t the center of the chain by the attachment of increasingly distant groups is, of course, for the pentamer not surprising. That the local , ,@,
668 Chem. Muter., Vol. 5, No. 5, 1993
Munowitz et al.
increases at all, however modestly, demonstrates the extreme sensitivity of the hyperpolarizability, which depends roughly on the third moment of the electron distribution. Sum over States. Having discerned some regularity in the SHG coefficients of the distorted structures, we turn our attention now toward understanding how the net value of &, originates from various transition dipole moments. According to perturbation t h e ~ r y , j~ -~~ 3,,,~ is expressed formally as a s u m of contributions involving singly-excited states In) and Im),given by
(a)
2.5 2
N
c?“ .->
1.5
4-
m -
’
6
1
0
0.5 0
I
1
0
6000
12000
18000
Number of terms
where
(b)
b,,(w)
= f,,(d(ZZZ)
0.4
1
(7)
and (as defined above in eq 5 ) (222)
= (gI&)(nI4m)(mIZb)
(8)
All the frequency dependence is taken up in the energy factors f n m ( w ) . Singly excited states in the extended Huckel model are derived simply by promoting an electron from an occupied orbital la) to an unoccupied orbital lu). Thus each state In) may be represented compactly as la-u), it being understood that the antisymmetrized wavefunction is a single Slater determinant. From the standpoint of the sum over states, there are two immediate consequences attendant upon the lengthening of the chain: an increase in the number of admissible excited states, and a decrease in the average value of each contribution brim. Chain formation is generally accompanied by additional electronic delocalization, and hence the two electrons in each occupied orbital range over a wider area. The atomic orbitals participating in a given molecular orbital contribute proportionately less in the vicinity of the central TiO6, thus reducing the average value of the matrix elements in bn,. Decreases in the individual values are compensated by the larger number of virtual transitions, though, since the larger systems bring with them additional orbitals and excited states. Consider as an example one of the axial oxygens in group 0. Existing in an isolated Ti06 octahedron, either 01 or 0 2 might figure in one of the principal orbitals of either A or u symmetry, namely the bonding, nonbonding, and antibonding combinations discussed previously. As part of a trimer, however, the same oxygen has additional opportunities to mix into orbitals extended over a larger region. One possibility might be for 01to form one of the aforementioned orbitals with the titanium and other axial oxygen of group 1, at the expense of the corresponding atoms in group 0. Another possibility might be for 01 to participate in a combination more evenly delocalized over the 0--Ti--0--Ti--0--Ti--0 chain. And, as we shall elaborate below, adding the 1and -1 groups further lowers the symmetry of the system and thereby makes admissible certain excitations hitherto forbidden. To illustrate how the sum over states develops in the different structures, we arrange the b,, in order of decreasing magnitude (ignoring the sign) and follow the changing sum as the number of terms increases. In Figure 10, the cumulative sum after N terms is plotted as a
.1.2
0
200
400
600
Number of terms
Figure 10. Convergence of the sum over states for.&t Each graph shows the partial s u m for a given number of terms, b,, arranged in order of decreasing magnitude: (a) local of Ti06 in a trimer, A = 0.3 A; (b) j3zzz of isolated TiO6, A = 0.3 A.
function of N for isolated Ti06 and for the central Ti06 in the trimer (A = 0.3 A). Convergence is effectively complete after 200 terms for the isolated monomer, but not so untilnearly 10 000terms for the monomer-in-trimer, Perhaps a more revealing analysis is to subdivide the sum over states according to the orbital la)from which the excited states la-u) are derived. For the dia onal terms, which involve the product (glqn)(nlan) we will allot the full value of b,, to the orbital la) associated with excited state In) = 1u-u). For the off-diagonal terms (glan)(nI&n)(mi.&), an equal division of b,m will be made between orbitals la) and (b)in the states In) = 1u-w) and Im)= 1b-w). The resulting tabulation according to orbital number, presented in Figure 11for both the isolated monomer and for the embedded monomer-in-trimer (A = 0.3 A>,shows how much each occupied orbital contributes to the sum over states. A dashed vertical line separates the bonding from the nonbonding orbitals in each system, and the selective shading in panel (a) identifies certain “new” bonding contributions to the trimer’s local pZ2, (about which see below). The distribution of BILL over orbitals reinforces the conclusion, reached in our earlier study of isolated TiO6, that it is unrealistic to ascribe the hyperpolarizability to just one or a few select transitions. The net value emerging from the sum is determined by numerous contributions similar in magnitude and often opposite in sign. Many of the trimer orbitals referred to in Figure 11 trace back directly to similar forms in the monomer. Among these are two of the A combinations involvinggroup
(nib),
0:
Chem. Mater., Vol. 5, No. 5,1993 669
Nonlinear Optical Properties of KTiOP04 (a)
1.5
-1.5 10
20
30
40
50
60
Orbital number
0.5
-1
I
1
t
I.
4
c
4
t
i
I
-0’5 -1.5I
5
10
15
20
I 25
Orbital number
Figure 11. Breakdown of j3222into specific contributions from the occupied orbitals. Each bar represents the amount attributable to all virtual transitions originating from the designated level, with shading employed to highlight *newnorbital combinations in the trimer. A broken vertical line is used to separate bonding and nonbonding orbitals in the diagram: (a) monomerin-trimer, A = 0.3 A; (b) isolated monomer, A = 0.3 A.
),TI
-
alTi,,)
+ b101,) + ~102,)
which, as orbitals 7 and 8 in the monomer and 17 and 18 in the trimer, retain much the same local character. Neither of these orbitals is in any strict sense completely local to the central Ti06 in the trimer, but that part which is associated with group 0 clearly carries over from its original configuration in the monomer. The precise form of the function is modified in each instance, especially in the vicinity of the bridging oxygens, and it is just such changes that are sufficient to alter the magnitude of the relevant matrix elements. Contours of electronic density orbital (number 171, integrated over in the modified ),TI the dimension y, are displayed in Figure 12a for the trimer with A = 0.3 A. Another example of a modified “old” orbital is the combination
),TI
-
alTi,,) + bl03,) + cJ(O4,) + d105,) + el(06,) which, as orbital 9 in the monomer, is manifested as an in-phase T overlap in the equatorial plane. Being orthogonal to z, this particular orbital does not contribute to &,,. In the trimer, though, the equivalent combination (number 19) draws amplitude from the titanium xy and oxygen p functions of groups 1and -1 as well. The strict orthogonality is eliminated, and now excitations originating from this orbital contribute nearly 0.2 units (approximately 8 % ) to the local value of &,in the trimer.
Figure 12. Electron density for selected trimer orbitals (A = 0.3
A), integrated overy y and projected in the x z plane. Each grid combination. is a 10A x 10A square. (a)Orbital 17,a modified rXz (b) Orbital 19, consisting of the xy function from the central titanium plus additional contributions from the two other octahedra. (c) Orbital 21,in which the central titanium is largely absent.
A view of the integrated orbital density in the x z plane, provided in Figure 12b for A = 0.3 A, shows the extent to which the symmetry is broken in the trimer. Still another type of orbital in the trimer might be classifiedas “new”,at least in the sense that no recognizable connection to the original Ti06 unit is apparent. An example is number 21 (Figure 12c, A = 0.3 A), which is composedpredominantly of titanium and oxygen functions from groups 1and -1. There is almost no amplitude a t the titanium of group 0, and the new orbital enters into the local only through the involvement of the bridging oxygens01and 0 2 from group 0. Clearly, any contribution made here is unique to the trimer. T o demonstrate how matrix elements originating from so-called new orbitals can alter local hyperpolarizability in the trimer, we use shading to distinguish those bars in Figure 11 which relate to new combinations. Note that an orbital such as number 19 (the r X ywith broken symmetry) is modified in such an essential way that it is considered new for this purpose. Any breakdown of this
Munowitz et al.
670 Chem. Mater., Vol. 5, No. 5, 1993
sort admittedly is not without ambiguity and can become particularly confusing for the nonbonding functions. Nevertheless, in addition to the aforementioned bonding orbitals, we will classify as new to the trimer any nonbonding orbitals where less than four of the original equatorial oxygens (03, 0 4 , 05, 06) make substantial contribution^.^^ With these definitions we acquire a readily apprehensible picture of the changes brought about in the trimer, orbital by orbital. It is especially interesting to see that many of the new local contributions in the trimer are opposite in sign to those that prevail in the isolated monomer; this observation helps to account for the overall reversal in sign previously noted. Figure 11 shows, as well, that the highest occupied molecular orbitals make a significant net contribution to the overall 8 but certainly not the only one. Orbitals throughout the entire range influence the s u m over states, and as the Ti06 structures become further extended it grows increasingly unrealistic to attribute the hyperpolarizability to just one or a few low-lying excitations. The same is true for any simplified analyses based on changes in dipole moment for certain favored excitations. Such arguments are often adduced in two-state models, where diagonalterms (&In) (npjln)(n@&) are especially important. Here the integral (npjln) is directly related to the change in dipole moment between the ground state k) and excited state In),so that in simple systems a large change in dipole moment upon excitation often can be associated with an increase in 8. In the Ti06 structures, though, formation of chains tends to reduce charge separation in the interior, and consequently many of the excited states do not exhibit particularly large dipole moments. The local enhancement of 8 does not come about simply from a few favorable transitions but rather is made possible by the large number of new contributions in the bigger systems.
instance, causes a massive reduction in 8, but no concomitant change in the geometry of the individual Ti06 o ~ t a h e d r a .There ~ ~ ~ is, ~ ~however, a correlation between the magnitude of the hyperpolarizability and the Ti--0-Ti angle in the chains. It is not unreasonable to expect that the mixture of titanium and oxygen orbitals in the relevant molecular orbitals will depend on the overall chain geometry. The variations need not be large to exact significant changes in 8, since the latter is very sensitive to small perturbations in the structure. Further evidence for the existence of a cooperative mechanism in the --Ti-0-Ti--0- chain is found in the response of 8 to various substitutions. For example, in the system K(Til.,Sb,)O(P&3ix)04, 8 is reduced by nearly an order of magnitude on going from x = 0 to x = 0.2,34implying that the overall 8 of pure KTP is considerably greater than the s u m of the contributions from isolated octahedra. Delocalization and conjugation have often been cited as factors in determining the nonlinear optical properties of organic molecules.3u3 It is interesting, in the present context, to recall an earlier study in which the Hiickel equations describing the ?r electrons in a conjugated chain were solved analytically in the presence of an electricfield?’ Exact analytical expressions for the energy perturbations up to fourth order were then used to derive values for a, 8, and y along the axis of an N-carbon chain. This procedure demonstrated that the molar linear susceptibility, a,goes roughly as W.8while the second hyperpolarizability, y, is roughly proportional to IP3in such molecules. The results, which are consistent also with calculations based on a free electron model,4l show that nonlinear response per unit length is enhanced as the conjugated chain grows; the effect is clearly similar to what we observe for the local hyperpolarizability in chains of TiO6.
Experimental and Theoretical Perspectives
The notion of a local nonlinear response has been used to compute the hyperpolarizability over selected portions of --Ti--0-Ti--0- chains in model structures related to KTP. Within the constraints of extended HClckel theory, we find that second harmonic generation is enhanced locally as more Ti06 octahedra are added to the chain. A terminated structure, whether an isolated Ti06 monomer or trimer, generally exhibits a smaller hyperpolarizability,
Certain results from experiment and calculation point to the existence of electron delocalization along the --Ti-0-Ti--0- chains in the KTP structure. It has been concluded from fluorescence measurements that KTP has a “delocalized charge-transfer excited state” which is semiconductor-like in nature.30 Results from both theoretical and experimental SHG studies are not inconsistent with this picture. For example, two different approaches to computing the hyperpolarizability using a local Ti06 unit underestimate the experimental value, suggestingthat a cooperative effect involving the chain Of Ti06 octahedra is i m p ~ r t a n t . ~ ” ~ The relationship between the crystal structure and hyperpolarizability of KTP isomorphs also lends support to this picture, given that the nonlinear response in these related systems is strongly dependent on the nature of the exchange cation. Replacement of K by Ag or Na, for (29) Naturally some of the “new” nonbonding orbitals are more new than others. To avoid confusion in Figure 11, we do not specify whether zero, one, or two of the original equatorial p functions are present in the various nonbonding combinations. It is worth noting, though, that many of the larger contributions retain only two or fewer of the monomeric p orbitals. (30) Blasse, G.; Dirksen, G. J.; Brixner, L. H. Mater. Res. Bull. 1986, 20. 989. (31) Hansen, N. K.; Protas, J.; Marnier, G. C. R. Acad. Sci. Paris. Ser. XI 1988, 307, 475.
Summary
(32) Crennell, S. J.; Morris, R. E.; Cheetham, A. K.; Jarman, R. H. Chem. Mater. 1992,4,82. (33) Phillips, M. L. F.; Harrison, W. T. A.; Stucky, G. D.; McCarron, E. M., 111; Calabrese, J. C.; Gier, T. E. Chem. Mater. 1992,4, 222. (34) Ravez, J.; Simon, A,; Boulanger, B.; Crosnier, M. P.; Piffard, Y. Ferroelectrics 1991, 124, 379. (35) Andr6, J.-M.; Barbier, C.; Bodart, V.; Delhalle, J. In Nonlinear Optical Properties of Organic Molecules and Crystals; Chemla, D. S., Zyss, J., Eds.; Academic Press: Orlando, FL, 1987; Vol. 2, pp 137-158. (36) Flytzanis, C. In Nonlinear Optical Properties of Organic Molecules and Crystals; Chemla, D. S., Zyss, J., Eds.; Academic Press: Orlando, FL, 1987; Vol. 2, pp 121-135. (37) Hameka, H.F. J. Chem. Phys. 1977,67,2935. (38) (a) Dirk, C. W.; Twieg, R. J.; Wagnihre, J. Am. Chem. SOC. 1986, 108, 5387. (b) Wagnihre, G. H.; Hutter, J. B. J. Opt. SOC.Am. E 1989, 6, 693. (39) Soos, Z. G.; Ramasesha, S. J. Chem. Phys. 1989, 90,1067. (40) Wu, J. W.; Heflin, J. R.; Norwood, R. A,; Wong, K. Y.; ZamaniKhamiri, 0.;Garito, A. F.; Kalyanaraman, P.; Sounik, J. J.Opt. Am. SOC. B 1989, 6, 707. (41) (a) Rustagi, K. C.; Ducuing, J. Opt. Commun. 1974,10,268. (b) Hermann, J. P.; Ducuing, J. J.Appl. Phys. 1974,45, 5100. (42) Ramasesha, S.; Albert, I. D. L. Chem. Phys. Lett. 1989,154,501. (43) Shuai, A.; Beljonne, D.; Brbdas, J. L. J. Chem. Phys. 1992, 97, 1132.
Nonlinear Optical Properties of KTiOP04
with opposite sign, than a fully integrated structure with the same number of Ti06 groups. Hyperpolarizability per unit length is consistently larger for longer chains. These effects are accounted for in an analysis of the contributions made by individual orbitals in the sum-over-states expression for @. Another observation is that, despite differences in magnitude, the changes in hyperpolarizability brought about by axial distortion of the octahedra follow a nearly uniform pattern in chains of various lengths. Similar relative changes in total electron density accompany the distortions, and the nonlinear response apparently follows suit. These results suggest, for KTP at least, that an approach involving isolated Ti06 groups is likely to underestimate the magnitude of @,as well as produce perhaps the wrong sign. Enough similarity remains, however, for relative
Chem. Mater., Vol. 5, No. 5, 1993 671
effects stemming from geometric distortions to be preserved. Still, a detailed analysis of the composition of the sum over statesshows that the magnitude and distribution of the dipole matrix elements change considerably upon enlargement of the chain. There are more transitions and thus more matrix elements, albeit with smaller values, and there is no single dominant contribution. Our approach has been microscopic and molecular, beginning with a single Ti06 (or, in earlier work, a single TiOd and continuing to add octahedra one at a time. Proceeding in this way we see that changes in local @ become progressively smaller as the surrounding chain is lengthened, with much of the effect associated with larger chains captured simply by embedding one Ti06 group in a trimer. Eventually, of course, any description in terms of molecular orbitals must give way to a model involving bands of states.
