Optimizing the second hyperpolarizability with minimally-parametrized potentials
Christopher J. Burke, Joseph Lesnefsky, Rolfe G. Petschek, Timothy J. Atherton
OOptimizing the second hyperpolarizability with minimally-parametrized potentials
C. J. Burke and T. J. Atherton
Department of Physics and Astronomy, Center for Nanoscopic Physics,Tufts University, 4 Colby Street, Medford, MA. 02155
J. Lesnefsky
Department of Physics, University of Illinois at Chicago, 845 W. Taylor St., Chicago, IL 60607-7059
R. G. Petschek
Department of Physics, Case Western Reserve University,10900 Euclid Avenue, Cleveland, Ohio, USA 44106
The dimensionless zero-frequency intrinsic second hyperpolarizability γ int = γ/ E − m − ( e (cid:126) ) was optimized for a single electron in a 1D well by adjusting the shape of the potential. Optimizedpotentials were found to have hyperpolarizabilities in the range − . (cid:47) γ int (cid:47) . ; potentialsoptimizing gamma were arbitrarily close to the lower bound and were within ∼ . of the upperbound. All optimal potentials posses parity symmetry. Analysis of the Hessian of γ int aroundthe maximum reveals that effectively only a single parameter, one of those chosen in the piecewiselinear representation adopted, is important to obtaining an extremum. Prospects for designing newchromophores based on the design principle here elucidated are discussed. I. INTRODUCTION
Developing materials with high electronic nonlinearsusceptibilities is of fundamental importance for a widevariety of applications such as optical solitons, phase con-jugate mirrors and optical self-modulation [1, 2]. Thesesusceptibilities are defined by considering a material inthe presence of an electric field E and expanding the in-duced polarization in a Maclaurin series, P = αE + βEE + γEEE + O ( E ) , (1)where the susceptibilities α , β and γ are generallyfrequency-dependent tensor quantities. Here, we con-sider the zero-frequency limit of these quantities in thenon-resonant regime i..e. where all frequencies are muchless than any resonant frequency of the electrons and mo-tion of the nuclei is neglected. A remarkable result dueto Kuzyk [3] is that quantum mechanics requires thatthe first and second hyperpolarizabilities β and γ arebounded: specifically, γ obeys the inequality, − (cid:18) e (cid:126) √ m (cid:19) N E ≤ γ ≤ (cid:18) e (cid:126) √ m (cid:19) N E ≡ γ max , (2)where N is the number of electrons, E is the energydifference between the ground and first excited states and m is the electron mass. It is natural to define the intrinsichyperpolarizability as a figure of merit to characterize theproximity of a given system to this limit, γ int = γ/γ max (3)and to ask: how to create materials that achieve opti-mal γ int ? The discovery of the bounds (2) has moti-vated a number of experimental studies that have demon-strated that carefully tuning the electronic states and geometry of chromophores can lead to higher secondhyperpolarizabilities[4–7]. Generic design principles mo-tivated by fundamental theory would therefore be desir-able. Unfortunately, the procedure used to derive thebounds (2) cannot directly provide these; they were ob-tained by optimizing γ for a three-level ansatz with re-spect to the dipole matrix elements and energy level spac-ings E = E /E and not by constructing an explicitpotential. Indeed, the assumptions behind the deriva-tion have been questioned[8, 9] and the limits need notbe achievable with a local potential; it has been specu-lated recently these may require exotic Hamiltonians[10].Subsequent work, following approaches developed inearlier studies of β [11, 12], has attempted to addressthis in two ways: First, by identifying universal featuresof Hamiltonians near the fundamental limit by a MonteCarlo search[13] and, secondly, by numerically optimiz-ing γ int with respect to the shape of a local potential[10].This latter work found potentials which have second hy-perpolarizabilities in the range − . ≤ γ int ≤ . ,which represents an apparent bound that is more re-strictive than the bound of (2). Moreover it was demon-strated that the optimized potentials spectra and dipolemoments were broadly consistent with those identified inthe earlier Monte Carlo study.While these strategies provide useful goals for chemistsattempting to design new nonlinear chromophores, theydo not provide insight into which features of the potentialare necessary to optimize γ int , or how many free parame-ters should be necessary to achieve optimal or near opti-mal γ . In a previous paper[14], we developed a techniqueto examine the analogous question for β : by optimizingpotentials described by increasing numbers of free param-eters and examining the eigenvalues of the Hessian matrixat each maximum, we identified the combinations of pa-rameters most important to the optimization. The anal-ysis revealed that effectively only two parameters were a r X i v : . [ phy s i c s . c h e m - ph ] D ec necessary to maximize β , and hence that a surprisinglybroad range of potentials with high β exists around eachmaximum.In this work, we apply the same technique to the prob-lem of optimizing γ . At first sight, the problem appearsto be more difficult than that for β since the expressionfor γ is much more complicated and the bounds for pos-itive and negative γ are different. Remarkably, however,we will show that effectively only one parameter is neces-sary to optimize γ in either direction and, moreover, thatin each case it is one of the parameters utilized in our rep-resentation of the potential. In this sense, we are able tosuggest much more clearly a possible design strategy formaterials with high γ than for β within the limitationsof the model. At least within our representation of thepotentials, we find that the potential that maximizes γ is rapidly varying, while for negative γ , as for β , quitegeneric, slowly varying potentials are adequate. The pa-per is organized as follows: in section II the calculationsperformed are described; the results are presented anddiscussed in section III; conclusions are drawn in sectionIV. II. MODEL
It is first necessary to generalize the method describedour previous paper on optimizing the intrinsic first hy-perpolarizability β int [14]: in the present work, γ int is tobe optimized by adjusting the shape of a one-dimensionalpiecewise-linear potential. Such a potential with N + 1 segments may be represented, V ( x ) = A x + B x < x A n x + B n x n − < x < x n , n ∈ { , ...N − } A N x + B N x > x N − , (4)with the positions x n and slopes A n as the adjustable pa-rameters and where the B n are chosen to enforce conti-nuity. Because γ int is invariant under trivial translationsand rescalings of the potential, some of these parameterswere fixed x = 0 , B = B = 0 , and A = ± . Thesechoices, together with a change of origin and rescalingallow for any potential. Thus maximizing with theseconstraints allows faster optimization. Furthermore, theleft- and right-most slopes are required to be negativeand positive, respectively, ensuring only bound electronstates. Finally, for technical reasons, having to do withthe asymptotic behavior of the Airy functions introducedin eqn 7 below, it is difficult to allow the sign of any slopeto change during an optimization. In consequence, wehave chosen to restrict | A i | > . , and, as appropriateto do separate optimizations for each interesting sign ofeach slope.A second representation for the potential was also con-sidered where parity symmetry was specifically enforced.This was motivated by previous work[15] which identifiesparity as important for optimizing γ int , particularly for the lower bound. The potentials with enforced P sym-metry were constructed on the half line x ≥ with N segments, V ( x ) = (cid:40) A n x + B n x n − < x < x n , n ∈ { , ...N − } A N x + B N x > x N − (5)with x = 0 and requiring V ( − x ) = V ( x ) . Again x n and A n are adjustable parameters and x = 0 , B = B = 0 are fixed. The parameter A was set to either − or +1 to study the consequences of both cases.For such a potential with a uniform applied elec-tric field of strength (cid:15) , the wavefunction obeys theSchrodinger equation in each segment, (cid:20) − d d x + ( A n + (cid:15) ) x + B n (cid:21) ψ n = Eψ n , (6)in units such that e = 1 , (cid:126) = 1 , and m e = 1 . Thesolution in each segment is written in terms of the well-known Airy functions, ψ n ( x ) = C n Ai (cid:34) √ B n − E + x ( A n + (cid:15) ))( A n + (cid:15) ) / (cid:35) ++ D n Bi (cid:34) √ B n − E + x ( A n + (cid:15) ))( A n + (cid:15) ) / (cid:35) . (7)To solve for the coefficients C n and D n the usual bound-ary conditions are imposed, i.e. that the wavefunc-tion ψ ( x ) and its derivative ψ (cid:48) ( x ) are continuous at theboundary between segments. Additionally, in the endsegments, the wavefunction must vanish as x goes to ±∞ fixing D N = 0 . There are a total of N linear equationsin the coefficients for the arbitrary case and N − equa-tions and coefficients for the P -symmetric case, which canbe written in matrix form W · u = 0 (8)where u is a vector comprised of the C n and D n coeffi-cients and W is a matrix which depends on E , (cid:15) and theparameters A n and x n .The allowed energy levels are found by numericallyfinding the roots of det W = 0 (9)with (cid:15) = 0 . It is readily possible, as previously donefor β , to obtain from (8) an expression for the secondhyperpolarizability, γ ≡
16 d E d (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 ; (10)this is achieved by repeatedly differentiating the matrix W using the Jacobi formula, dd (cid:15) det W = Tr (cid:18) adj W · d W d (cid:15) (cid:19) , (11)where adj W is the adjugate of W (since W is singular),and applying the chain rule d W d (cid:15) = ∂W∂(cid:15) + ∂W∂E d E d (cid:15) . (12) Having performed similar calculations to those in [14],we arrive at the expression d E d (cid:15) = − Tr (cid:104)(cid:16) d d (cid:15) adj W (cid:17) · d W d (cid:15) + 3 (cid:16) d d (cid:15) adj W (cid:17) · d W d (cid:15) + 3 (cid:0) dd (cid:15) adj W (cid:1) · d W d (cid:15) + adj W · W (cid:48)(cid:48)(cid:48) (cid:105) Tr (cid:0) adj W · ∂W∂E (cid:1) , (13)where W (cid:48)(cid:48)(cid:48) = d Wd(cid:15) − ∂W∂E d Ed(cid:15) . (14)From this, γ is readily obtained and the intrinsic secondhyperpolarizability γ int = γ/γ max calculated for a givenset of parameters.The quantity γ int was optimized numerically for botharbitrary and P -symmetric potentials with varying num-bers of segments using the FindMaximum function of
Mathematica
8, an implementation of the Interior Pointmethod for constrained optimization. Both maxima andminima of γ int were obtained from a large number ofrandomly generated starting points, and also manuallychosen starting points with large values for γ int . Oncean optimum γ int was found, the extent to which eachof the parameters was important to the extremum wascharacterized by calculating the Hessian matrix, H ij = ∂ ∂P i ∂P j γ int , where the P i are the parameters, and calculating itseigenvalues and eigenvectors. Since the Hessian matrixcharacterizes the local curvature of the objective func-tion in the parameter space around the extremum, thesequantities give the magnitudes and directions of the prin-cipal curvatures. As stressed in previous work[14] thesecurvatures implicitly depend on a measure implied bythis equation that is peculiar to our numerical parame-terization of the problem. More physically relevant mea-sures can also be used to calculate eigenvalues. Whilethese make quantitative changes in the eigenvalues andvectors, they do not make qualitative changes, and wegive results for this “numerically natural” measure below. III. RESULTS AND DISCUSSION
Our optimized potentials, together with the groundand first excited state wavefunctions, are displayed inFig. 1 for both arbitrary (eq. 4) and P -symmetric (eq.5) parametrizations. The associated parameter valuesare listed in Table I. The potentials and wavefunctions are displayed on a transformed position and energy scale, ¯ x = ( x − (cid:104) x (cid:105) ) / ( E − E ) / ¯ V (¯ x, { P } ) = ( V (¯ x, { P } ) − E ) / ( E − E ) (15)such that the ground state energy is E = 0 , the differ-ence between the ground and first excited state energy is E − E = 1 and the position expectation value for theground state is (cid:104) x (cid:105) = 0 . This rescaling does not change γ int and permits convenient comparison of the results ofeach optimization. To identify the relative importance ofeach of the parameters to the optimization, the results ofthe eigenanalysis of the Hessian matrix are also displayedfor selected potentials in Fig. 1; the j -th eigenvalue ofthe Hessian, h j , is listed alongside a plot of the variationin the potential ∆ V j ( x ) in the direction of the associatedeigenvector, ∆ V j ( x ) = ∂ ¯ V (¯ x, { P i + αv ji } ) ∂α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α =0 , (16)where v ji is the i th component of the j -th eigenvector.Note that the values of V and x in the right hand sideof (16) are renormalized as a function of α using (15) sothat the variations presented automatically preserve theproperties (cid:104) x (cid:105) = 0 and E − E = 1 .The first set of results displayed in Fig. 1(a) are op-timized potentials with no enforced symmetry and spec-ified by 3 or 5 free parameters. The optimized γ int forboth the lower and upper bounds of γ int potentials fallsomewhat below the apparent bounds observed in [13].For the upper bound, the best results assuming all slopesexcept the first are positive are what appears to be a lo-cal maximum value of γ int (cid:119) . [Fig. 1(a)(ii)]. Asimilar value was also found in [10] for potentials withno constrained symmetry, but the potentials found heredo not closely resemble those found in that work. Theeigenvalues and eigenvectors displayed below the poten-tial in [Fig. 1(a)(ii)] show that only two of the eigenvaluesare significant in magnitude and are associated with theshape of the potential in the middle while the small eigen-values are associated with the outer slopes. These resultsare reminiscent of those found for the first hyperpolariz-ability, where β int was found to approach its maximum Bound Description Ref. γ int A A A x x Upper 3 param. arb. (a)(i) . . . — . —5 param. arb. (a)(ii) . . . . . . P (b)(i) . — . — . —3 param. P (b)(ii) . — . — . . Lower 3 param. arb. (a)(iii) − . .
471 237 . — . —2 param. P (b)(iii) − . — . — . —Table I: List of parameters of optimized potentials - - - - - - N=3 γ int = γ int = γ int = γ int = - - - - (a)(b) (i) (ii)(i) (ii) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) N=5N=2 N=3 - - - - Figure 1: Optimized potentials for the upper bound of γ int with (a) no enforced symmetry and (b) P -symmetry. Theenergies of the ground and first excited state are indicated byhorizontal lines; the corresponding wavefunctions are also dis-played. The plots have been rescaled to facilitate comparisonby ensuring E − E = 1 and (cid:104) x (cid:105) = 0 while preserving γ int . value for the same class of potentials with a similarlysmall number of parameters and analysis of the Hessianrevealed that only effectively two parameters were im-portant to the maximization. For the lower bound, the3 parameter system [Fig. 2(a)] converges on a shape ap- N=3(a) (b) - - - γ int =- (cid:31) (cid:31) (cid:30) (cid:31) (cid:31) (cid:31) (cid:30) (cid:31) (cid:31) (cid:31) - - - - γ int =- (cid:31) (cid:31) (cid:31) (cid:30) (cid:31) (cid:31) (cid:31) (cid:31) N=2
Figure 2: Optimized potentials for the lower bound of γ int with (a) no enforced symmetry and (b) P -symmetry. (b) - - - - γ int =- γ int = - - - - (a) (cid:31) (cid:30) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:30) (cid:31) (cid:31) (cid:31) (cid:31) (cid:30) (cid:31) (cid:31) (cid:31) (cid:31) Figure 3: Five parameter potentials with no enforced symme-try optimized using two-parameter P -symmetric optima asthe starting point. (a) Upper and (b) lower bounds. proaching a square well. The 5 parameter system alsoconverges on a P -symmetric potential and, because ofthis, further discussion of the lower bound is deferred toa subsequent paragraph.A possible explanation for the fact that the optimizedpositive γ int for arbitrary potentials in Fig. 1 falls shortof the bounds established in [3, 10, 13] is that the hyper-polarizability is sensitive only to some features of thepotential and that many local extrema exist. Ratherthan making extensive runs starting from a variety ofpotentials, we chose to use the more constrained subsetof P -symmetric potentials (5) and found that, indeed,even with only 2-3 parameters, much higher values of γ int could be obtained as shown in Fig. 1(b). This isreminiscent of the observation in [10] that these boundscould only be reached if a P -symmetric starting pointwas used for the search; in this work, we not only enforcethe symmetry of the starting point but at all times in theoptimization.For the upper bound, we found a two-parameter po-tential close to the apparent maximum but below thetheoretical maximum [Fig. 1(b)(i)]. The shape is char-acterized by shallow outer slopes with a divot in the cen-ter; the ground state wave function is localized to thedivot with the highly delocalized first excited state abovethe divot. It was found to be necessary to constrain theslope A > . since the method of calculating γ int pre-sented in section II fails for shallow slopes. To avoid theunphysical feature of delocalized higher excited states, adistant wall was added to construct a three-parameterpotential [Fig. 1(b)(ii)]. Such a change is expected apriori from previous work[16] to make no significant dif-ference to γ int as it is far from the region where ψ and ψ are large. For this well-like potential, we performeda maximization, adjusting A , x , and x while fixingthe outer walls to have a large slope ( A = 100 ). Thebest potential found possesses γ int = 0 . , and has A = 0 . which is the shallowest slope allowed by theconstraint. The eigenvectors of the Hessian, calculatedfor the subspace excluding A , are well aligned with theparameters: the most significant eigenvector correspondsto x , the outer boundary of the divot. The other eigen-vector corresponds to x , the position of the outer walls;this would be expected to have relatively little influenceon γ int since it controls a feature where the ground statewavefunction is small.Fixing the outer slopes at A = 100 as above, we also attempted to optimize γ int with A con-strained to be negative. It was found that γ int increasedas the slope approached zero, until a point was reachedwhere the calculations became numerically unstable dueto the asymptotic properties of the Airy functions. Thehighest value of γ int which was found within a region ofparameter space for which the calculation was still stablewas γ int = 0 . , lower than the current maximum. Wethen performed similar optimizations on potentials withstrict hard wall boundary conditions. The calculationsfor these potentials were numerically stable in all regionsof parameter space which were explored: For A > ,a maximum of γ int = 0 . was found. For A < ahigher maximum of γ int = 0 . was found, though thisis still lower than the current maximum of γ int = 0 . .Since a higher γ int is found in potentials with A < than for potentials with positive A in cases with stricthard wall boundary conditions, we speculate that a value of γ int higher than the current maximum found might befound for the finite A case. Nonetheless, we do not ex-pect to see a significant improvement as the current max-imum is already within ∼ . of the maximum valuefound in previous studies.For the lower bound, a potential with the best valueof γ int = − . was found using only two parame-ters [Fig. 2(b)]. This potential is characterized by steepouter walls and a “bump” in the middle: the ground stateand first excited state wavefunctions cover the same spa-tial extent, but the bump causes the ground state to be-come spread out and relatively flat. Eigenanalysis of theHessian of γ int about this solution shows that one eigen-value is significantly larger than the other, indicating thatonly one of the parameters is physically relevant. More-over, the eigenvectors of the Hessian for this potentialare aligned with the parameter space chosen to representthe potential. The higher eigenvalue is associated withan eigenvector along the x direction, which determinesthe position of the outer walls; the smaller eigenvalue isassociated with the parameter that controls the slope ofthe outer walls. Because γ int is invariant under rescalingsof the form (15), a potential with identical γ int can beconstructed for a well of arbitrary width by tuning theslope of the bump.The two parameter P -symmetric potentials identifiedas the satisfying the apparent lower bound can be equiva-lently represented by a five parameter arbitrary potential.Optimization of the five parameter potential indeed findsthis potential as the apparent global maximum. Thereare therefore no directions in this new parameter spacewhich lead to a higher γ int , despite relaxing the require-ment that x − n = − x n and A − n = − A n . While theexistence of asymmetric potentials with more negative γ int cannot be ruled out, our analysis confirms that a P -symmetric potential satisfies the apparent lower bound.We repeated this procedure for the upper bound usingthe two parameter P -symmetric potential displayed inFig. 2(b) as the starting point for optimization. It wasfound that despite relaxing the symmetry constraint nofurther improvement could be made so that the potentialof Fig. 2(b) is also a local optimum with respect to theexpanded parameter space. Analysis of the hessian wasalso performed on both of these five parameter optima.The eigenvalues and associated eigenvectors, shown inFig. 3, reveal that for both upper and lower boundsthe variation in the potential associated with the largesteigenvalue is indeed asymmetric.The fact that we are able to achieve optimized γ int within 1% of previous limits with far fewer parametersthan in other representations of the potential[10], andalso fewer parameters than required to numerically opti-mize β int , is surprising since the calculated expressionsfor γ int are far more complicated than those for β int .Since the dimensionality of this parameter space is sosmall, it is instructive to visualize it directly: For ourtwo-parameter optimizations, the value of γ int is plot-ted over a portion of the parameter space [Fig. 4(a) and (a) (b)(c) (d) Figure 4: Visualization of the variation of γ int as a functionof the parameters x and A of a two-parameter P -symmetricpotential for the (a) upper bound and (b) lower bound. Theregion of parameter space where γ int is within 2% of the max-imum value is highlighted in (a) and indicated by an arrow. γ int is also shown in the ( E, X ) parameter space for the (c)upper bound and (d) lower bound. (b)]. The plot for the lower bound merely illustrates theresults obtained from analysis of the Hessian, i.e. thatthe minimum is strongly curved about the optimal valueof x but shallow with respect to A . The plot for theupper bound is more interesting: the region of parameterspace for which γ int is within 2% of the maximum valueobtained is highlighted showing a clear ridge.By calculating values of X = x /x max and E = E /E , the natural parameters of the three-levelansatz[3], as a function of ( A , x ) , we are able to dis-play γ int re-parametrized in ( E, X ) space [Fig. 4(c) and(d)]. Here, x max = 1 / √ E in our units. Notice thatthe entire region explored in the numerical parametriza-tion ( A , x ) collapses onto a narrow, elongated regionin ( E, X ) space for the upper bound [Fig. 4(c)] and acomplicated curved line for the lower bound[Fig. 4(d)].These plots confirm the results of eigenanalysis of theHessian: that essentially only a single parameter (or 2for the upper bound) characterizes optimal γ int .The results of the three-parameter optimization andthe plot in fig. (4) both suggests that the truly optimal P -symmetric potential for the upper bound has shallowouter slopes A → . Such a potential can be trans-formed, using (15), to a potential of equivalent γ int butwhere the outer slope is unity and the central well is farnarrower and sharper. Since the central divot for thetransformed potential resembles a Dirac delta function,we studied the second hyperpolarizability of the familyof potentials V ( x ) = | x | − αδ ( x ) (17)where α is the single adjustable parameter. Values of γ int as a function of α are displayed in fig. 5 and the max-imum value is found to be γ int = 0 . which occurs Γ int EX (cid:45) (cid:45) ∆(cid:45) function Α Figure 5: Plot of γ int , E = E /E and X = x /x max as afunction of α for the for the potential V = | x | − αδ ( x ) . when α = 1 . . Despite the simplicity of this one-parameter potential, the result is only smaller thanthe best reported so far and only fractionally smaller thanthose found with the two and three parameter potentialsabove.We now turn to the question of whether the simple, butoptimal or nearly so, potentials that were identified aboveresemble either those previously found [10] or possess theuniversal features identified in [13]. Comparing our bestoptimized potentials and wavefunctions to those in [10],some qualitative similarities are apparent. For the upperbound, the potentials in Watkins et al. are roughly sym-metric near their lowest point. They feature a centraldivot within a wider well where the ground state wavefunction is localized within the central divot and the firstexcited wave function is relatively delocalized comparedto the ground state. For the lower bound, the potentialsfeature a steep well within which both the ground stateand first excited state are localized, and a central bumpwhich causes the ground state to be spread out within thewell. These qualitative features are shared by the poten-tials obtained in this work, but many extraneous detailsare removed by the highly constrained, judiciously chosenrepresentation.In table II we display values of X and E , together withthe values of these parameters that extremize γ int for thethree-level ansatz[3] and those possessed by the best pre-viously found potentials[10]. Values are also shown fortwo elementary potentials, the triangle well and infinitesquare well. The results for the upper bound are quiteconsistent with those of Watkins et al. if the breadth inthe range of these values identified by the Monte Carlostudy[13] is taken into account. The values of E and X are also displayed in Fig. 5 as a function of α , thestrength of the δ function, in the potential (17); E is amonotonically increasing function of α while X is a mono-tonically decreasing function. Crudely, these explain theexistence of a maximum γ int as representing the trade-offbetween increasing the motion of the electron (associatedwith high X ) versus enhancing transitions to other states(associated with low E ). Bound Potential γ int E X
Upper 3-level ansatz[3]
Best from [10] . . . Arb. 5 param. . . . P . . . | x | − αδ . . . Triangle well | x | . . . Lower 3-level ansatz[3] − .
25 0 ± Best from[10] − . . . P − . . . | x | − αδ − . . . Infinite square well − . . . Table II: Second hyperpolarizabilities and physical parame-ters X = x /x max and E = E /E for the optimized po-tentials obtained in this work. IV. CONCLUSION
We have optimized the intrinsic second hyperpo-larizability γ int of a piecewise linear potential wellwith respect to parameters that control the shape ofthe potential. We found solutions that lie withinthe range − . ≤ γ int (cid:47) . in agreement withthe apparent bounds established in previous numericaloptimizations[10]; these both fall short of the Kuzyklimits[3]. By using two types of potential, one whereall slopes were allowed to vary and another with ex-plicitly enforced symmetry, we demonstrated that P -symmetric potentials satisfy the apparent lower boundfor γ int and come within ∼ . of the apparent upperbound. The parametrization used constrains the poten-tial to be smooth, preventing the occurrence of rapidoscillations which do not affect γ int [16]. Because ofthis and the strong symmetry constraint, the optimal P -symmetric potentials found were characterized by only − parameters. Of these, a posteriori analysis of theHessian revealed that effectively only one or two, for thelower and upper bound respectively, were importantThese results are reminiscent of those obtained ear-lier for β int [14], yet the number of parameters requiredto optimize γ int appears to be smaller even though it isa more complex object, containing more terms and in-volving higher derivatives. At least part of the reasonfor this is that for γ int there exists a “compatible” sym-metry operator, P , which can be used to constrain the shape of the potentials; this was not possible for β int where P -symmetric potentials automatically have β = 0 .However, even though we have shown that P -symmetricpotentials can have optimal or near-optimal second hy-perpolarizabilities, it is not clear whether the apparentupper bound γ int ∼ . achieved by Watkins and Kuzykcan be achieved with a P -symmetric potential or whethera small amount of asymmetry is necessary. Moreover,the reason why local potentials fall short of the Kuzykbounds remains opaque.The small parameter space allows us to propose a cleardesign paradigm for new chromophores, within the limi-tations of the model one-electron 1D system studied. Ne-glected here, for example, are multi-electron interactions,molecular ordering and inter-molecular electron hopping.Nonetheless, the potentials obtained could be realized,for example, by a centrosymmetric molecule with a cen-tral attractive or repulsive group—for positive or nega-tive γ int respectively. The strength/electronegativity ofthe central group and the ratio of the length of the cen-tral and peripheral groups can then be tuned to give high γ int . Unfortunately, most practical chromophores are π conjugated systems in which there are approximately asmany electrons as there are sites on the molecule, thusthe consequences for them from this single electron cal-culation are clearly very speculative. Nevertheless, ho-mologous sequences already studied for high γ as a func-tion of chain length, e.g. [6] could likely be enhanced byincluding such a central group with different electroneg-ativity. Any other means of achieving a potential well orinducing a significant phase shift in wavefunctions pass-ing through the center of the molecule is also likely, evenin multi-electron systems, to offer a route for achievinglarger γ int . The present analysis provides other impor-tant insights: first, that since the “true” parameter spacefor γ int is so small, only rough tuning of the moleculardesign ought to be necessary. Secondly, this work againconfirms that there are a large set of modifications to op-timized potentials, e.g. rapid oscillations, that will notchange γ int and need not be considered in planning whatmolecules to synthesize. As has been previously noted,materials with high γ int could also be realized more di-rectly in other ways, such as through composite materi-als. [1] R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press,2008).[2] Y. R. Shen,
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