Table Of ContentThe effect of electron interactions on the universal properties of systems with
optimized off-resonant intrinsic hyperpolarizability
David S. Watkins* and Mark G. Kuzyk∗
Department of Physics and Astronomy, Washington State University,
Pullman, Washington 99164-2814; and *Department of Mathematics
Becauseofthepotentiallylargenumberofimportantapplicationsofnonlinearoptics,researchers
have expended a great deal of effort to optimize the second-order molecular nonlinear-optical re-
sponse,calledthehyperpolarizability. Thefocusofourpresentstudiesistheintrinsichyperpolariz-
ability,whichis ascale-invariant quantitythatremovestheeffectsof simplescaling, thusbeingthe
1 relevant quantity for comparing molecules of varying sizes. Past theoretical studies have focused
1 on structural properties that optimize the intrinsic hyperpolarizability, which have characterized
0 the structure of the quantum system based on the potential energy function, placement of nuclei,
2 geometry, and the effects of external electric and magnetic fields. Those previous studies focused
on single-electron models under the influence of an average potential. In the present studies, we
n
generalize our calculations to two-electron systems and include electron interactions. As with the
a
J single-electron studies, universal properties are found that are common to all systems – be they
molecules, nanoparticles, or quantumgases – when thehyperpolarizability isnear thefundamental
6
limit.
1
]
s I. INTRODUCTION focus on (1) the non-resonant regime, were all frequen-
c cies aresmallcomparedwith the Bohrfrequencies ofthe
i
t Materials with large nonlinear-optical suscep- molecule;and,(2)thexxxtensorcomponentofβ,where
p
o tibilities are central for optical applications such x is the axis along which the hyperpolarizability is the
. as telecommunications,[1] three-dimensional nano- largest.
cs photolithography,[2, 3] and making new materials[4] We consider only the electronic hyperpolarizability in
i for novel cancer therapies.[5] Semi-empirical theoretical the off-resonant limit - where the nucleiareassumednot
s
y modeling of complex molecules, with feedback from to contribute. Chernyak and coworkers showed that in
h experiments, has resulted in a better understanding of secondandthirdharmonicgenerationexperiments,when
p how to make materials with larger nonlinear-optical the wavelengths are tuned below the excited electronic
[ response.[6, 7] state energy and above the nuclear excitation energies,
1 Accurate theoretical modeling of a variety of spe- thepurelyelectronichyperpolarizabilitiesaccountfor90-
v cific molecules is a critical part in the development of 95% of the total.[14] Our theoretical analysis and ex-
3 structure-property relationships.[8–11] In contrast, our perimental results come to the same conclusion.[15, 16]
4 work seeks to build a broad theoretical understanding of Bishop and coworkers point out that while this may
0 the nonlinear-optical response that applies to all quan- be true for harmonic generation hyperpolarizabilties, vi-
3 tum systems,[12] of which molecules are a small subset. bronic contributions can be dominant when any of the
.
1 The uniqueness of this approachis that it seeks to iden- frequencies vanish or in self action effects where nega-
0 tify universal properties that are common to all systems tive frequencies cancel positive ones.[17] The literature
1 sharing a nonlinear response that is close to the funda- is extensive on the topic of conditions when the vibronic
1
mental limit.[13] In the present work, we show that in- contributions are dominant.[18–21]
:
v teractionsbetweenelectronsdo nothaveaneffectonthe In the theoretical studies presented here, we focus
i universalproperties,andthatscalingduetotheaddition specifically on the zero-frequency limit of the electronic
X
ofanelectronis aspredictedfromquantumlimittheory. response. Our results can be compared with second-
r
a For electric fields that are small compared with in- harmonic generation measurements of the off-resonant
ternal molecular fields, the induced dipole moment of a hyperpolarizability,whichisdominatedbytheelectronic
molecule, p, can be expressed as a series in the applied response,upon extrapolationto zero frequency[22] using
electric field, E, a two-level dispersion model.[23] All future references to
hyperpolarizabilitiesrefertothefirstelectronichyperpo-
p=αE+βE2+γE3+..., (1)
larizability at zero-frequency. We stress that our goal
is to understand the zero-frequency electronic response,
where α is the polarizability, β, the hyperpolarizability,
independent of the subtleties associated with measuring
γ the second hyperpolarizability, and so on. In general,
it.
thesecoefficientsaretensorsthatdependonthefrequen-
The hyperpolarizability can be calculated for any
cies of the applied electric field. However, our studies
quantum system with the dipole energy due to the ap-
plied electric field as a perturbation. This results in a
sum-over-states(SOS) expansion,[24] which is expressed
∗ [email protected]
in terms of the position matrix elements and energy
2
eigenvalues of the unperturbed molecule. there are many one-dimensional potentials - each of dis-
Thefundamentallimitofthehyperpolarizabilityisde- tinct shape, but different from each other – whose op-
fined as the upper bound determined from fundamental timized intrinsic hyperpolarizability is 0.709. For these
quantum principles, without approximation. Thus, the systems, the ratio of second to first excited state energy
fundamental limit of the hyperpolarizability is an abso- is E E /E 0.48. Furthermore, typically over
10 20
≡ ≈
lute quantity that applies to any quantum system that 90% of the optimized hyperpolarizability is due to two
is described by the Schr¨odinger Equation. Rather than excited states. Thus, all optimized materials can be well
being a single number, the limit is a function of the en- approximated by a three-level model.
ergy difference between ground and first excited state,
E =E E , and the number of electrons.
10 1− 0 When maximizing the off-resonance hyperpolarizabil-
This upper bound is calculated using the Thomas-
ityofaone-dimensionalmolecule,planarmoleculesmade
Reiche-Kuhn sum rules,[25] which are exact and follow
of point nuclei, or systems with externally applied elec-
directly from the Schr¨odinger equation. They relate the
tricandmagneticfields,one-electronmodelswereunable
energy eigenvalues and position matrix elements to each
to get intrinsic hyperpolarizabilities that exceed about
other,andareusedtosimplify the SOSexpansionofthe
0.708. However,Monte Carlosimulations that randomly
hyperpolarizability.[24]Thisapproachwasappliedtothe
assign energies and position matrix elements under the
calculationthefundamentallimitoftheoff-resonancehy-
constraint that the sum rules are obeyed result in β
int
perpolarizability, which yields[26]
that approaches unity.[38] Since such large values have
neverbeen observedunder any Hamiltonian-basedsimu-
e¯h 3 N3/2
β = √43 , (2) lation, this suggests that one may need to invoke exotic
MAX (cid:18)√m(cid:19) · E170/2 Hamiltonians to reach the fundamental limit.
whereN isthenumberofelectrons,E theenergydiffer-
10
encebetweenthefirstexcitedstateandthegroundstate, Thevaluesofβint 0.708thatwehaveobtainedinour
≈
E = E E , and m is the electron mass. The fun- computersimulationsaremuchhigherthananythathave
10 1 0
damentalli−mitofthe secondhyperpolarizabilityhas also been measured for real molecules. For example, it was
been calculated.[27] These calculations show that there reported that a new molecule with asymmetric modula-
is a large gap between the hyperpolarizabilities and the tion of conjugationhas a measured value of βint =0.048
fundamental limit.[26–31] The existence of such a limit [34], but this is still more than a factor of ten lower
is both scientifically interesting and of practical utility than what our simulations have produced. It is natu-
because it provides a target for making optimized mate- ral to ask whether our simulations give unrealistically
rials. largeβintvaluessimplybecauseweareusingone-electron
One approach to the development of structure- models.[39]
property relationships is to correlate the nonlinear-
optical response with variations in the structure. How- Forasystemoftwonon-interactingelectrons,thevalue
ever,it is difficult to decouple structuralproperties from of β is twice that of a system with one electron. But
simple scaling - i.e. the change in the effective size of a β scales as 23/2 due to the factor N3/2 in Equation
MAX
molecule with a change in its structure. It canbe shown 2. Thus the highest value of β that we will be able
int
that the intrinsic hyperpolarizability βint, which is ob- to get with the intrinsic hyperpolarizability of the two
tained by dividing the actual hyperpolarizability β by electron system is about .708/√2 0.5. As the num-
the fundamental limit βMAX: berofnon-interactingelectronsisin≈creased,theintrinsic
hyperpolarizability decreases. For a 10-electron system,
β =β/β , (3)
int MAX this would reduce the intrinsic hyperpolarizability by a
factorof30,thegapobservedbetweenthebestmolecules
is invariant to simple scaling.[13] A comparison of the
and the fundamental limit.
intrinsic hyperpolarizabiltiesofmolecules canbe used to
determine the propertiesthatarerelevantfor optimizing
the nonlinear-optical response. In the present work we consider two-electron models
Past studies have considered the optimization of the and ask whether it is possible to produce values of β
int
intrinsic hyperpolarizability by varying the shape of that are substantially greater than 0.5 when including
the potential energy function,[32, 33] which lead to the electron-electron interactions. We will show that β
int
new paradigm of large-β molecules using modulation of the two-electron system is as large as about 0.708.
of conjugation for which there is some experimental Furthermore, we find that the same universal properties
evidence;[34,35]studyingtheeffectofgeometrybyvary- hold as in the one-electron case. Thus, the high β
int
ing the chargesandpositions ofpoint nuclei;[36]andthe values that we obtained in previous work are not simply
effect of applying an arbitrary electromagnetic field.[37] an artifact of one-electron models. More importantly,
All of these studies find that when the hyperpolariz- thissuggeststhatonecanusesimpleone-electronmodels
ability of a quantum system is near the fundamental whenpaintingourunderstandingofthenonlinear-optical
limit, certain universal properties result. For example, response with the broadest of brushes.
3
II. THEORY tion 4, and the energy levels are sums of energy levels
of Equation 4. The largest β values that we can obtain
Weuseatwo-particleHamiltonianthatincludesaone- fromEquation5areexactlytwicethosethatwegetfrom
dimensionalpotentialenergyfunction,V(x),thatactson Equation4,butthelargestvalueofβint islowerbyafac-
bothelectrons,andaninteractiontermbetweentheelec- torofexactly√2becauseofthefactorN3/2 thatappears
trons. Notethatpaststudiesintwodimensionsshowthe in Equation 2 for βMAX.
same universal behavior as in one dimension, so we find To include interactions between the electrons, we con-
no particular utility in studying these other classes. If sider a model of the form
electroninteractionsdonotchangethe universalproper- ¯h2
ties in 1D, it is unlikely that they will have any effect in 2Ψ+(V(x1)+V(x2)
− 2m∇
2D.
+ V (x x ))Ψ=EΨ. (6)
Thepotentialfunctionismodeledbyapiecewiselinear 2 | 1− 2|
function that has 19 degrees of freedom in the compu- Here V(x ) is the external potential that acts on both
i
tational region. With 39 degrees of freedom, the results electrons equivalently. It can be due to the nuclei or
arenearlythesame;but,thecomputationsaremoretime external influences such as an applied electromagnetic
consuming. Wethususe19degreesoffreedombecauseit field. V (x x ) is the interaction potential between
2 1 2
providesagoodcomprmisebetweenaccuracyandspeed. electrons,|wh−ich is|a sum of two components
Starting from a given potential function, we use the
Nelder-Mead simplex algorithm [40] to vary the shape V (x x )=V (x x )+V (x x ),
2 1 2 21 1 2 22 1 2
| − | | − | | − |
of the potential to maximize β . Since there are
int
19 degrees of freedom, we are maximizing the hyper- whereV21 andV22 denoteelectrostaticrepulsionbetween
polarizability over a 19-dimensional space. We have the electrons and spin interaction terms, respectively.
three waysof computing β (SOS,[24]dipole free,[41] and These terms are piecewise linear, which is appropriate
non-perturbative[42]), all of which require solving the for a one-dimensional model.
Schr¨odinger eigenvalue problem for the given potential Defining d= x1 x2 , weassume thatthe interaction
| − |
(and in some cases also for neighboring potentials). We potentials take the form
solvethe eigenvalueproblemnumericallyusing the finite
L d
element method.[43] V (d)=¯h2 max C − ,0 , (7)
21 1
· L
Once we have solved the eigenvalue problem, we can (cid:26) (cid:27)
compute transition moments and then obtain β by the
where L is the length of the molecule (we use L=25 for
standard Orr and Ward SOS expression β [24], the
SOS all calculations); and,
dipole free expression β [41], or a non-perturbative
DF
finitedifferenceapproximationβ .[42]Intheoptimiza-
NP Lˆ d
βtion =coβde w/eβuse β.NOPn.ceTthhaetoips,tiwmeizaseteiokntios cmoamxpimleitzee, V22(d)=S·max(C2 L−ˆ ,0), (8)
int NP MAX
we use a comparison of β with β and β as a
NP SOS DF
test of convergence. where Lˆ = C L with C 1 to ensure that the spin
3 3
≪
Theexactcomputationofβ andβ requiressums interaction acts only at close range. Note that this is a
SOS DF
over infinitely many states. We approximate them by one-dimensionalcoulombpotentialthatsatisfiesthegen-
summing over the 30 lowest energy levels. This is found eralcondition 2V(x x ) δ(x x ),whereδ(x x )
1 2 1 2 1 2
togiveanaccuratevalue forthehyperpolarizabilitythat is the Dirac de∇lta func−tion. ∝Inone-−dimension, 2 =−∂2 .
is almost identical with the result we get when using up S is given by ∇ ∂x2
to 100 states.[33]
To illustrate the difference between the one- and two- h¯2 (triplet)
S = 4 (9)
electron cases, we take a closer look at the Schr¨odinger ( 3h¯2 (singlet).
eigenvalue problems. For the one-electron problem in − 4
one-dimension, we solve
Note that the parameters that we obtain during numer-
¯h2 ∂2Ψ ical optimization may be unphysical in the sense the in-
+V(x)Ψ=EΨ, (4) teraction energies that result can be much larger than
− 2m ∂x2
is typically observed between electrons; but, we stress
where the wave function Ψ is a function of x. For two that our goal is to understand the extremes of what is
electrons without interactions, the Schr¨odinger equation consistent with the Schr¨odinger equation. As discussed
is later, the resulting universal properties are unchanged if
¯h2 ∂2Ψ ∂2Ψ theparametersareconstrainedtoaphysically-reasonable
− 2m ∂x2 + ∂x2 +(V(x1)+V(x2))Ψ=EΨ, (5) range.
(cid:18) 1 2(cid:19) The piece-wise potential energy function is the most
where Ψ is a function of x and x . The solutions of general one-dimensional potential energy function inso-
1 2
Equation 5 are tensor products of solutions of Equa- far as the pieces are small compared with the size of the
4
smallest features of the system. The form of the inter- integrand is antisymmetric and the integral x is zero.
0m
action potential, on the other hand, is precisely the one- Thus this term contributes nothing to β.
dimensional coulomb interaction. Thus, our model ac- In addition to the computations of β , we compute
int
counts for any system with interacting point charges, of the matrix[32, 36]
which a two-electron one-dimensional molecule is a sub-
set. 1 N
τ(N) =δ (E +E )x x , (10)
For most of the experiments reported here we use mp m,p− 2 nm np mn np
C1 = 0.1, C2 =0.5, C3 =0.1. These are ad hoc choices. nX=0
We found that our results are not too sensitive to the
where N is the number of states used. Each matrix ele-
choice of constants. However, if V21 is made too large ment of τ(N), indexed by m and p, is a measure of how
relative to V , the electrons are separated into two sep-
22 well the (m,p) sum rule is obeyed when truncated to N
arate wells and we get small values of β . In some
int states. If the sum rules are exactly obeyed, τ(∞) =0 for
experiments reportedbelow,we allowedthe constants to mp
all m andp. We note that if the sum rulesaretruncated
vary. We found that the spin interaction term V is
22 to an N-state model, the sum rules indexed by a large
much more important for making the intrinsic hyperpo-
value of m or p (i.e. m,p N) may be disobeyed even
larizabilitylargerthanis theelectrostaticrepulsionterm ∼
when the position matrix elements and energies are ex-
V .
21 act. We have found that the values of τ(N) are small for
Subspaces of symmetric and antisymmetric func- mp
exact wavefunctions when m < N/2 and p < N/2. So,
tions are invariant under any Hamiltonian of the form
whenevaluatingtheτ matrixtotestourcalculations,we
Given by Equation 6. Therefore we can consider
the symmetric and antisymmetric eigenvalue problems consider only the components τ(N) .
m≤N/2,p≤N/2
separately. For the symmetric spatial wavefunctions Since the hyperpolarizability depends critically on the
(Ψ(x ,x ) = Ψ(x ,x )) the spin part is a singlet, while transition dipole moment from the ground state to the
2 1 1 2
fortheantisymmetricspatialwavefunctions(Ψ(x2,x1)= excited states, we use the value of τ0(030) as one impor-
Ψ(x1,x2)) the spin part is a triplet. tanttestoftheaccuracyofthecalculatedwavefunctions.
−
We approximate Equation 6 on a 20 20 grid of Additionally, we use the standard deviation of τ(N),
×
quadratic serendipity finite elements [43]. On the sym-
metric subspace the number of degrees of freedom is cut 2
N/2 N/2 τ(N)
nearly in half by the restriction Ψ(x2,x1) = Ψ(x1,x2). ∆τ(N) = r m=0 p=0 mp , (11)
On the antisymmetric subspace it is more than halved. P PN/2 (cid:16) (cid:17)
On each of the subspaces we use the implicitly restarted
Arnoldi method [44] to compute the wave functions and which quantifies, on average,how well the sum rules are
energy levels. obeyedinaggregate,making∆τ(N) a broadertestofthe
In all of our computations, the ground state is a sym- accuracy of a large set of wavefunctions.
metric function (spin singlet), so the value of β depends Wementionedearlierthatwedidourcomputationson
only on the states that are symmetric in space. To see a 20 20 grid of finite elements. As a check on whether
×
this,weconsiderthedefinitionofthetransitionmoments this mesh is fine enough, we repeatedour final computa-
x , which appear as factors in every term of the SOS tionsona100 100gridandfoundthatwegotthesame
0m
×
or dipole-free expression for β, results.
Our code is written in MATLAB.
L L
x = (t +t )Ψ (t ,t )Ψ (t ,t )dt dt .
0m 1 2 0 1 2 m 1 2 1 2
Z0 Z0 III. RESULTS AND DISCUSSIONS
If Ψ is symmetric and Ψ is antisymmetric, then the
0 m
We use various analytical functions as starting poten- pending on the choice of the starting potential. Table I
tials, and then apply the numerical optimization tech- summarizes the result of optimizing β for two inter-
int
nique as described above to determine the optimal po- acting electrons with C = 0.1, C = 0.5, and C = 0.1
1 2 3
tential function for each. The tables that follow give the whenstartingfromsixdifferentpotentials,whicharethe
hyperpolarizability of the starting potential, β followed same as in previous work[33] except that they have been
S
by the hyperpolarizabilities of the optimized potentials. re-scaledforconvenience. Eachstartingpotential(except
So, for example, when V = 0, we see that β = 0; but, V(x)=0) was scaled so that V(L)=¯h2/10. Also, when
S
the optimized potentials are all nonzero. we state that our starting potential is V(x)=x2, for ex-
ample,wemeanthatitisthepiecewiselinearinterpolant
Becausethe Nelder-Meadalgorithmdeliversonlya lo-
of x2.
calmaximum,the finaloptimizedpotentialcanvary,de-
5
TABLE II. Summary of results for the hyperpolarizability computed from the subspace of antisymmetric functions (triplet
states). βs istheintrinsichyperpolarizabilityofthestartingpotentialwhileβint isthevalueafteroptimization. Thetransition
moments and energies are in thesame dimensionless unitsas in Table I.
Function βS βint τ0(030) ∆τ(30) E10 E20 E10/E20 x00 x10 xxm101a0x
V(x)
0 0 0.6692 0.0014 0.0546 0.0528 0.1153 0.4582 5.7865 3.4477 0.7925
x 0.0636 0.6690 0.0056 0.1472 0.0813 0.1817 0.4472 1.5841 2.7723 0.7902
x2 0.0057 0.6670 0.0025 0.1181 0.0650 0.1426 0.4556 2.4667 3.0848 0.7863
√x 0.1047 0.6682 0.0001 0.0999 0.0713 0.1582 0.4507 2.0296 2.9681 0.7926
x+cos(x) 0.2113 0.6676 0.0023 0.0614 0.0782 0.1664 0.4698 4.4549 2.8294 0.7912
tanh(x) 0 0.6691 0.0032 0.0570 0.0538 0.1174 0.4586 7.4199 3.4047 0.7900
TABLE I. Summary of results for two interacting electrons – with C =0.1, C =0.5, and C =0.1 – using different starting
1 2 3
potentials. βs is the intrinsic hyperpolarizability of the starting potential while βint is the value after optimization. The
transition moments and energies are in dimensionless units. To convert to specific units, consider an example where xnm is
interpreted to be in units of angstroms. The energies would then be determined by multiplying all values of En0 by ¯h2/ma2,
with a=10−10m (1˚A).In this case, theenergy is in unitsof 1.2 10−18J or about 7.6eV.
×
Function βS βint τ0(080) ∆τ(80) E10 E20 E10/E20 x00 x10 xxm101a0x
V(x)
0 0 0.7068 0.0027 0.0704 0.0235 0.0488 0.4816 11.2267 -5.1493 0.7888
x 0.6319 0.7063 0.0044 0.0574 0.0253 0.0528 0.4787 -11.8570 -4.9505 0.7872
x2 0.5523 0.7064 0.0035 0.0589 0.0249 0.0520 0.4799 -11.7233 -4.9913 0.7883
√x 0.6648 0.7063 0.0042 0.0599 0.0259 0.0540 0.4808 -12.0313 -4.8971 0.7888
x+cos(x) 0.0311 0.7021 0.0044 0.0631 0.0381 0.0792 0.4808 -11.0082 -4.0336 0.7869
tanh(x) 0.0018 0.7068 0.0024 0.0601 0.0234 0.0488 0.4800 -11.2016 -5.1530 0.7886
random 0.1066 0.7022 0.0080 0.0770 0.0395 0.0832 0.4749 14.6972 3.9642 0.7882
It is importantto stressthat ourapproachis basedon nitude of the fundamental limit of the position matrix
invariance under simple scaling,[13] so that the magni- element x for a two-electron system, and is given by,
10
tudesoftheenergies,dipolematrixelements,andhyper-
polarizabilities are unimportant to the analysis. Rather, ¯h
ratiosoftheobservables,suchasE10/E20,x10/xm10ax and xm10ax = √mE . (12)
10
β/β arethekeyintensivequantities. Assuch,anyrow
max
of Table I can be transformed by simple scaling to arbi-
trarilyadjustallthe quantities,but leavingthe intensive Aswasfoundfortheone-electroncase,[13,33]aquan-
parameters unaffected. It is this property of simple scal- tum system with a hyperpolarizability near the funda-
ing that makes possible a comparison of a broad range mental limit shares certain universal properties. For ex-
of molecular shapes and sizes using the intrinsic hyper- ample, all optimized values of βint are just over 0.7, for
polarizability as the metric. whichthe energyratioE10/E20 isjust under0.5andthe
Thevaluesofβint reportedinthetablewerecomputed intrinsictransitionmoment xxm101a0x isabout0.79. Notethat
using β = β /β . When we checked the com- ifelectroninteractionas characterizedby the C parame-
int NP MAX
putation using β in place of β , we got the same ters is decreased smoothly to zero, the optimized hyper-
SOS NP
values to at least four decimal places, thus, we choose to polarizabilityapproaches0.5,asis predictedanalytically
use β in all of our optimization calculations. When forthe non-interactinglimit. When electroninteractions
NP
weusedβ ,wehadagreementtowithinafractionofa are made very large, the optimized intrinsic hyperpolar-
DF
percent. In the last column of Table I, xmax is the mag- izability does not increase beyond β =0.708.
10 int
Whilewehavefocusedmainlyonwell-definedpotential thefirstsimulation,wepickedtherandomnumberstobe
energyfunctions,itisalsopossibletoapplyMonteCarlo normally distributed with mean zero and standard devi-
methods to randomly pick the shape of the potential. In ation 1. The optimization algorithm got trapped in the
6
parameterspacewhereβ wassmall. However,foraset trixelementsareatleast10timessmaller. Thus,thenon-
int
of randomnumbers with the standarddeviationreduced
to 0.1, the simulation gave β 0.7. The resulting
int
≈
parameters are shown in the last row of Table I. Once
again,the same universalparametersareobservedas for
0.4
the other optimized potentials.
0.2
It is not likely that the universal properties that
0
we observe are coincidental or artifacts of the calcu-
−0.2
lation given independent calculations using widely dif-
−0.4
ferent approaches. In addition to the large number of
−0.6
one-dimensional potentials for one and two electrons,
for which these same universal properties result,[32, −0.8
33] the same universal properties are found when al- −1
20
lowing point nuclei to move in a plane to form a 15 20
molecule,[45] applying electromagnetic fields to two- 10 15
10
dimensional molecules,[37] and using constrained po- 5 5
tentials with properties more similar to real molecules 0 0
and including continuum states.[46] Monte Carlo calcu-
lations,whichprobeamuchlargerparameterspacethan
theformerexamplesarealsoobservedtosharesomeuni- FIG. 1. Surface plot of βnm that is representative of all the
optimized hyperpolarizabilities in Table I.
versalproperties,thoughthey areless restrictivethanin
the cases where a potential energy function and vector
potential is used.
Inallsimulations,theenergiesofthetripletstateswere interacting electron potentials that optimize β lead to
int
trackedandfoundtobeofhigherenergythantheground hyperpolarizabilitiesthatobeythethree-levelansatz,[32]
statesingletstate. Sincethetripletstatesareorthogonal thatis,asystemwithβ nearthefundamentallimitisde-
to the space of singletstates, no triplet states contribute scribed by a three-level model.
to the results. Toinvestigatethe hyperpolarizabilitydue
The three-level ansatz has been shown to be univer-
to the triplet states, we used the lowest energy triplet
sally obeyed in systems described by a one-dimensional
state to calculate the dipole moment from which the hy-
potential,[33] point nuclei in a plane,[45] and for one-
perpolarizability is determined using the nonperturba-
dimensional monte-carlo simulations.[38] In the case of
tive method. We alsoused the SOSexpressionswith the
arbitrarily applied electric fields to point nuclei in two-
lowest energy triplet state as the ground state to cal-
dimensions,theoptimizeddiagonalcomponentofthehy-
culate the hyperpolarizability. Thus, these calculations
perpolarizability obeys the three-level ansatz. However,
only consider transitions between triplet states. Table II optimized β yields β = β /βmax 0.9 and the
summarizes the results. xxy int xxy xxx ≈
intrinsic transition moment is 0.51. In this case, the hy-
Inallcases,the optimizedintrinsichyperpolarizability
perpolarizabilityisdominatedbytwoexcitedstates;but,
forthetripletsubspaceisjustshortof0.67,whichislower
anadditionalpairofexcitedstatescontributeabout25%.
thanthevalueof0.71foundinthesingletsubspace. Sim-
Infurther experiments,starting with the potentialde-
ilarly,theotherparameters,suchastheenergyratioand
picted in Figure 2, we attempted to increase β even
normalizedtransitionmoment,areclosebutnotidentical int
further by varying the constants C , C , and C appear-
tothesingletvalues. Thus,weconcludethatthequalities 1 2 3
ing in the interaction terms (Equation 7 and Equation
of the optimized system using only triplet states is simi-
8). These constants are all constrained to be positive.
lartothoseofthesingletsubspace. However,whenusing
We achieved only a modest increase from β = 0.7068
the true ground state of the system, the same universal int
to β = 0.7080, still no better than the largest intrin-
propertiesareobservedfortwointeractingelectronsasis int
sic hyperpolarizability attained by varying the potential
observed for a single electron.
energy function;[33] but the largestfound in our simula-
We define β as the contribution from states n and
nm
tions. The final values of C , C , and C were 0.0034,
m to the SOS sum,[33] 1 2 3
0.8787, and 0.4942. The strength of the electrostatic re-
x0nxnmxm0 pulsion term is governed by C1, which decreased from
β = , (13)
nm 0.1 to 0.0034 during the optimization. Similar results
E E
n0 m0
were seen in other experiments of this type. Thus the
where x x x δ . The hyperpolarizability is electrostatic repulsion term seems not to be important
nm nm 00 n,m
≡ −
calculatedbysumming β overallexcitedstates ofthe for maximizing β . The strength of the spin interac-
nm int
system. Figure 1 shows a plot of β that is representa- tiontermis governedby C , whichincreasedfrom0.5to
nm 2
tiveforallthepotentialsplottedinFigureI. Typicalval- 0.8787 during optimization. This suggests that the spin
ues are β = 0.9, β = β = 0.09, and all other ma- interaction term is important for maximizing β .
11 21 12 int
−
7
TABLE III. Parameters resulting from optimizing the intrinsic hyperpolarizability starting with the potentials shown and
varying the strengths of the electron interaction terms. For the V(x) = 0 starting potential, both the electron interaction
parameters and potentialfunction were varied. βs is theintrinsichyperpolarizability ofthestarting potentialwhile βint isthe
valueafter optimization.
Function βS βint τ0(080) ∆τ(80) E10 E20 E10/E20 x10 xxm101a0x
V(x)
Figure 2 0.7068 0.7080 0.0018 0.0627 0.0235 0.0490 0.4803 5.1265 0.7865
x 0.6319 0.6663 0.0069 0.0149 0.0442 0.0804 0.5494 -4.1252 0.8670
0 0 0.7074 0.0010 0.0549 0.0249 0.0516 0.4827 -4.9887 0.7870
optimized potentials are found to be dominated by only
TABLE IV. Electron interaction parameters resulting from
one pairofexcitedstates. Thus,the three-levelansatzis
optimizing the intrinsic hyperpolarizability starting with the
again obeyed when electron interactions are important.
potentialsshownandvaryingthestrengthsoftheelectronin-
teractionterms. FortheV(x)=0startingpotential,boththe Based on these observations, we conclude that when
electron interaction parameters and potential function were the diagonal component of the hyperpolarizability is op-
varied. βs is the intrinsic hyperpolarizability of the starting timized,allsystems-includingoneswithtwointeracting
potential while βint is the valueafter optimization. electrons - share the same universal properties; and, the
three-level ansatz holds. This observation suggests that
Function βS βint C1 C2 C3
when investigating the properties of a quantum system,
V(x)
thenumericaloptimizationofthesimplestmodelscanbe
Figure 2 0.7068 0.7080 0.0034 0.8787 0.4942
used to identify universal properties. This paradigm is
x 0.6319 0.6663 0.000087 14.75 0.0612 in stark contrast to studies that seek to understand the
0 0 0.7074 0.0933 0.5283 0.1033 propertiesofaspecific system,[8,47]andismoreakinto
the theoretical studies by Meyers et al, who studied the
effectofanelectricfieldonpolyenes[48]whichconfirmed
Inanotherexperimentweusedthe potentialV(x)=x the concept of bond length alternation.[49, 50] However,
and started with C = 0.1, C = 0.5, and C = 0.1. our approach of optimizing the nonlinear response using
1 2 3
Holding V(x) fixed, we tried to increase β by varying the potential energy function is even more general. The
int
C , C , and C . With this potential we already have a onlyconstraintweapplyisthatthesumrulesareobeyed,
1 2 3
rather goodβ =0.6319to begin with. By varying the whichdemandsconsistencywithquantummechanicsbut
int
constants C , we were only able to realize a modest in- is otherwise general.
i
crease in βint to about 0.6663, which was obtained with Arguably,themostimportantresultofuniversalprop-
C1 =0.000087,C2 =14.75,andC3 =0.0612. With such erty studies is that there appear to be a very large num-
a small C1, the electrostatic repulsion term is essentially ber of ways to make a system with a nonlinear-optical
zero. The large value of C2 means that the spin interac- response near the fundamental limit with βint =0.7.[13]
tiontermisverystrong,andthesmallvalueofC3 means Szafruga and coworkers performed numerical optimiza-
that the spin interaction operates only over a very short tion studies under the constraint that the potential en-
range. We concludethatvaryingtheelectroninteraction ergy varies by an amount that is typical in a molecule,
potentialalonedoes nothavea largeeffect onimproving aswellasextendingthecomputationalspacebeyondthe
the intrinsic hyperpolarizability. potential well to approximate continuum states.[46] It
In another experiment we started with the potential wasfoundthatwiththoserestrictionsthesameuniversal
V(x) = 0 and the initial constants C = 0.1, C = 0.5, propertieswereobserved,andβ 0.708. Therefore,it
1 2 int
≈
and C =0.1. When allowing both the potential energy appears that it should be possible to make many classes
3
function and electron interaction parameters to vary, we ofyet-to-be-foundmoleculesthatareatthefundamental
obtained β = 0.7074. The final values of the electron limit.
int
interaction parameters were C1 = 0.0933, C2 = 0.5283, Some of the parameters shown in Table IV yield elec-
andC3 =0.1033,sothey changedlittle fromtheir initial tron interaction energies that are larger than typically
values. observed. However, even in the strong interaction limit,
TableIIIsummarizessomeoftheparametersfoundfor the effect on the optimized hyperpolarizability is not
the optimized intrinsic hyperpolarizabilities when elec- large. Thus, we conclude that when electron interac-
tron interactions are present. In the two cases where tions are included, the universal properties remain the
the intrinsic hyperpolarizability is greater than 0.7, the same and that electron interactions may often be unim-
sameuniversalvaluesareobserved;i.e. the energyratios portant when studying universal properties of optimized
andtheintrinsictransitionmomentsarealmostidentical systems. This is notto saythatelectroninteractionsare
to the ones in Table I. Furthermore, β for all three unimportantwhencalculatingthepropertiesofaspecific
nm
8
0.05 V(x)
ρ (x) 0.12
0.04 n
0.03 0.1
0.02
x) x) 0.08
( (
ρn 0.01 ρn
x), x), 0.06
( 0 (
V V
−0.01 0.04
−0.02
0.02 V(x)
−0.03 ρ (x)
n
0
−10 −5 0 5 10 −10 −5 0 5 10
x x
FIG. 2. Optimized potential energy function and electron FIG. 3. Optimized potential energy function and electron
density for the lowest-energy 15 states of a 30-state model. density for 15 states. Starting potential is V(x)=tanh(x).
Starting potential is V(x)=0.
10 10 10
5 5 5
molecule. When we limited the parameter space in the
0 0 0
present studies in the way described in Szafruga[46], the
−5 −5 −5
observed universal properties with electron interactions
−10 −10 −10
remained the same. −10 0 10 −10 0 10 −10 0 10
Finally,wecomparethechargedensitiesthatresultfor
the optimized potential energy function with two elec- 10 10 10
5 5 5
trons with prior results with one electron. Figure 2 and
0 0 0
Figure 3 show the optimized potential energy functions
−5 −5 −5
for starting potentials V(x) = 0 and V(x) = tanh(x).
−10 −10 −10
Also shown is the electron density for the lowest-energy
−10 0 10 −10 0 10 −10 0 10
fifteen states of a thirty -state model. Note that the nu-
merical labels on the y axes refer only to the potential 10 10 10
energy functions and all electron densities are positive. 5 5 5
0 0 0
This is a common observation that very different poten-
−5 −5 −5
tialenergyfunctionsthatleadtoverydifferentwavefunc-
−10 −10 −10
tions always share the same universal properties. Thus,
−10 0 10 −10 0 10 −10 0 10
there is a unifying underlying structure that is shared
bytheseoptimizedsystemsthatarenotapparentsimply FIG. 4. Contour plots of electron density functions for the
from the electron densities. ninelowestenergylevels(lefttoright,toptobottom)forthe
Inpaststudies,[33,42]itwasfoundthattwoclassesof optimized potential with starting potential V =tanhx.
wavefunction lead to optimized hyperpolarizability: (1)
Localized charge density resulting from randomly oscil-
lating potentials, reminiscent of Anderson Localization the four lowest energy eigenstates. Many of the higher
for the tanh(x) starting potential, and (2) delocalized states show that the two electrons are localized in dif-
wavefunctions. In both cases, two excited states domi- ferent spatial regions. This is reminiscent of the one-
nated intheir contributionto the hyperpolarizability. In dimensionalone-electronpotential,wherethewavefunc-
the present studies, there is a hint of localization in the tions do not overlap. Note that the electron densities
groundandfirstexcitedstates,buttheothersaredelocal- appear similar in the excited states of all optimized sys-
ized. However,thisresultmaybeduetothefactthatthe tems.
present calculation used piecewise continuous potentials The transition dipole moment amplitude from the
rather than splines, which were used in previous studies. groundstate,ψ (x ,x )(x +x )ψ (x ,x ),isshownin
n 1 2 1 2 0 1 2
Tostudylocalization,itismoreinstructivetoconsider Figure 5. The only transition dipole moment densities
the full two-electron wavefunctions. Figure 4 shows the thatarelargearefortransitionsfromthegroundstateto
electron density ψ (x ,x )2 as a function of x and x thefirstandsecondexcitedstates. Forthehigherstates,
n 1 2 1 2
| |
for an optimized quantum system using the starting po- the transition moment amplitudes are either small; or,
tential V(x) = tanh(x). The electrons are correlated in large positive regions are balanced by small negative
9
10 10 10 teractionbetweenelectronsthatleadstotheN3/2scaling
5 5 5 law. The fact that the fundamental limit of the hyper-
0 0 0 polarizability scales as N3/2 follows from the sum rules
−5 −5 −5 andthe limits they setforthe oscillatorstrengths. Ifthe
−10 −10 −10
full oscillator strength resides in one state, the modulus
−10 0 10 −10 0 10 −10 0 10
squared of the transition moment to that state from the
10 10 10 ground state is proportional to the number of electrons,
5 5 5 thus, the transition moments scale as the square root
0 0 0 of the number of electrons. Since the hyperpolarizabil-
−5 −5 −5 ity scales as the third power of the transition moment,
−10 −10 −10 it scales as N3/2. Note that the sum over the dipole
−10 0 10 −10 0 10 −10 0 10 terms of the form ∆x x 2 can be re-expressed using
n n0
| |
the sum rules in terms of sums of the products of the
10 10 10
5 5 5 form x0nxnmxm0.[41] Thus, while counterintuitive, the
0 0 0 dipole terms acting together will also scale as N3/2.
−5 −5 −5 In the absence of interactions, the optical properties
−10 −10 −10 are obviously additive over the molecular units. When
−10 0 10 −10 0 10 −10 0 10 allowedto interact, the largestpower law allowedby the
fundamental limits is N3/2, and the optimization algo-
FIG. 5. Contour plots of transition dipole moment ampli-
tude ψn(x1,x2)(x1+x2)ψ0(x1,x2) with n = 1 to 9 (left to rithm finds this local maximum. The fact that the pa-
right and top to bottom) for the optimized potential energy rameters that characterize the interaction show no uni-
function with starting potential V =tanhx. versal behavior (see Table IV) implies that there is no
unique interaction that leads to the optimal response.
Thus, the question about the nature of the interactions
regions. Thus, the three-level ansatz is obeyed either that leads to optimized scaling does not have a unique
throughminimaloverlapbetweenthedifferenteigenfunc- answer.
tions,orcancelationbetweenpositiveandnegativeprob- Inclosing,wenotethatthefocusofourpresentworkis
ability amplitude. ontheoff-resonant hyperpolarizability. Sincethefunda-
One might ask how the potential energy function can mental limit of the dispersion of the hyperpolarizability
be manipulated to make a real system with an opti- has also been calculated,[56] it would be a straightfor-
mized nonlinear optical response. There are many ar- ward matter to calculate the potential energy function
tificial structures, such as nanoparticles, quantum wires, thatisoptimizedatanysetofwavelengths. Suchcalcula-
and multiple quantum wells in which there is some con- tionswouldbe usefulindeterminingtheopticalmaterial
troloverthestructureandtheresultingpotentialprofile. properties required for a particular device. Since there
Forexample,nanowirescanbeusedasbuildingblocksfor are an infinite number of combinations of wavelengths,
nanoscale electronic and optoelectronic devices,[51] and we have limited our present studies to the off-resonance
multiplequantumwellscanbemade,forexample,witha regime.
IV. CONCLUSIONS
bandedgethatcanbeusedtomakeasystemwithalarge
quadraticelectroabsorptioncoefficient.[52]Recently,Hall
andcoworkersusedgradedbarrierstocontroltheoptical Ratherthancalculatingthenonlinear-opticalresponse
properties of ZnO/ZnMgO quantum wells with an in- of large numbers of complex molecules and searching for
trinsic internal electric field.[53] Clearly, there are many correlations between structure and nonlinear response,
alternativestomoleculesthatcanbeusedtodeliberately our work focuses on developing general principles that
engineer a potential profile. broadly apply to all quantum systems; and, endeavors
Wang and coworkers have shown that the potential to identify universal properties that can be applied to
energy profile of a molecule can be approximated by us- buildanunderstandingofwhatleadstoalargenonlinear-
ing a linear combination of atomic potentials.[54] Thus, optical susceptibility.
the results of our work may be able to be connected The present work seeks to determine whether or not
to real molecules using such an approach. We stress electron interactions and Fermi correlations change the
that optimization of the potential energy function is universalbehaviorof anoptimized systemas is observed
a more general approach than other numerical opti- foroneelectronsystems. Wehavefoundthattheuniver-
mization techniques, with the exception of the Monte sal behavior is indeed the same. The intrinsic transition
Carlo techniques,[38, 55] because potentials apply to the moment, energy ratio of the first two dominant excited
broader set of systems of which molecules are a small states,andthe intrinsic hyperpolarizabilitiesfor anopti-
subset. Our present work shows that the observed uni- mizedsystemareallthe samefora broadrangeofstart-
versal properties are not restricted to just one electron ingpotentials. Also,thethree-levelansatzisagainfound
systems. to be obeyed. Since electron interactions appear to have
Itisnosimplemattertodeterminethenatureofthein- a small effect on the conclusions derived from numerical
10
optimization,simpleone-electronmodelsmaybeenough wholeisrepresentativeofthepropertiesofthetrueglobal
to study the broadprinciples underlying light-matterin- maxima. Similarly, with only a handful of exceptions,
teractions. experimental work that focuses on the development of
newmoleculesfindsthatthe hyperpolarizabilitiesarefar
Optimizationstudiesareinherentlytrickybecauseone shortofthefundamentallimit. Thebroadprinciplesthat
can never be certain whether or not a global maximum we seek to develop aims to bridge the gap between the
has been found. However, as we continue to gener- limits and real molecular systems. Clearly, altogether
ate more numerical data from a broader range of start- new paradigms will need to be developed to attain what
ing potentials, molecular symmetries, externally-applied is possible.
electromagneticfields, andelectroninteractions,we con- Acknowledgements: MGK thanks the National
tinue to observe the same universal properties - leading Science Foundation (ECCS-0756936) and Wright Pater-
to the tentative conclusion that perhaps the data as a son Air Force Base for generously supporting this work.
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