Abstract

The first- and second-order hyperpolarizabilities have been extensively studied to identify universal properties near the fundamental limit. Here, we employ the Monte Carlo method to study the fundamental limit of the second hyperpolarizability. As was found for the first hyperpolarizability, the largest values of the second hyperpolarizability approach the calculated fundamental limit. The character of transition moments and energies of the energy eigenstates are investigated near the second hyperpolarizability’s upper bounds using the missing state analysis, which assesses the role of each pair of states in their contribution. In agreement with the three-level ansatz, our results indicate that only three states (ground and two excited states) dominate when the second hyperpolarizability is near the limit.

© 2010 Optical Society of America

1. INTRODUCTION

In 2000, the fundamental limit of the off-resonant electronic first hyperpolarizability and second hyperpolarizability was calculated in an attempt to understand constraints imposed by quantum mechanics [1, 2, 3]. (Note that for the rest of this paper, we say hyperpolarizability when referring to the first hyperpolarizability.) Using Thomas–Kuhn sum rules and assuming that when the hyperpolarizability β is optimized only three states (including the ground state) contribute, an upper limit of β was found to be given by

βmax=31/4(em)3(N3/2E107/2),
where e is charge of the electron, is Planck’s constant, N denotes the number of electrons in the atom or molecule, and E10 is the energy difference between the first excited state and ground state. In conjugated systems, it is known that only the π-electrons contribute to the nonlinear-optical response, so the effective number of electrons, N=Neff, is used instead of the total number [4, 5].

A comparison of the largest experimentally measured second-order molecular susceptibilities with the fundamental limit reveals a large gap between the two. Measurements have never crossed the limit and are typically well below it. Until about 2005, all molecules ever measured mysteriously fell a factor of about 30 below the limit [6, 7], until a new class of molecules was reported by Kang and co-workers [8, 9].

In early studies [10], several experimental tools such as linear spectroscopy, Raman spectroscopy, hyper-Rayleigh scattering, and Stark spectroscopy were used to investigate the possible reasons for the gap between theory and experiment. There it was explained that the gap is mainly due to the unfavorable arrangement of excited state energies.

In an effort to understand how one might make a real quantum system that has a hyperpolarizability near the limit, Zhou and co-workers applied a numerical approach to determine the shape of the potential energy function that yields the largest value of the hyperpolarizability [11, 12]. The potentials were chosen among a wide range of possibilities including polynomials, power laws, and piecewise continuous functions. The best hyperpolarizabilities found were given by β0.7βmax. Thus, a new but smaller gap appeared between numerical simulations and the fundamental limit. These studies suggested that the concept of conjugation of modulation could be used as a new design paradigm for making better molecules, which was later experimentally verified with one-dimensional molecules that broke above the apparent limit [13, 14].

Those numerical optimization studies showed the first hint of the appearance of universal properties of quantum systems that approach the fundamental limit. In all cases, independent of the starting potential, a large range of optimized systems were found to have (1) β0.7βmax, (2) they are all well approximated by a three-level model, (3) the energy ratio between the first two excited states is always near 1/2, and (4) the normalized transition moment to the first excited state is about 0.75 [11, 12]. A broader set of studies found similar universal properties, for example, when considering the effects of geometry (i.e., the optimal arrangement of point nuclei) [15] and the effects of an externally applied electromagnetic field [16].

Since the 30% gap may be an indication of a flaw in the calculation of the fundamental limits, Kuzyk and Kuzyk (KK) focused on investigating the validity of the three-level ansatz, with the only assumption underlying limit theory [17]. The three-level ansatz states that when the hyperpolarizability of a quantum system is at the fundamental limit, only three states contribute [5]; that is, the contributions from all other states are negligible. Note that the ansatz does not claim the converse. When three states are involved, the hyperpolarizability is not necessarily at the limit. KK used Monte Carlo simulations to numerically generate a set of β values to investigate if any of the data attain the predicted fundamental limit. They found a continuous distribution of values up to the fundamental limit. These studies verified the validity of the theory of limits and therefore indirectly verified the three-level ansatz.

More interestingly, these results imply that a set of wave functions exist that are consistent with the sum rules, yet may not be derivable from typical Hamiltonians; that is, the sum rules may permit state vectors that are not derivable from the Schrödinger equation. This statement may seem contradictory since sum rules are directly derivable from the Schrödinger equation; but the sum rules apply under broader conditions, which we can understand as follows.

The sum rules are derived using the fact that [x,[x,H]]=2/2m for a typical Hamiltonian. However, it may be possible that [x,[x,H+A]]=2/2m for an operator A, where A can take a form that is never observed in molecular systems. As an example, it is straightforward to show that the unusual operator A=f(x,y,z)pxpypz obeys the commutation relationship yet is of a form that is to our best knowledge never found in nature. Other such operators can be invented. Thus, while all standard Hamiltonians obey the sum rules, more exotic Hamiltonians can also obey the sum rules.

Sum rules have been used to determine unknown transition moments when fitting the dispersion data of the second harmonic hyperpolarizability [18] and for understanding length-scaling of the second hyperpolarizability [19]. Fundamental limits have also been used to investigate the essential ingredients required of small molecules to approach the fundamental limit [20, 21]. However, no numerical optimization studies have targeted the second hyperpolarizability. In the present work, we focus on Monte Carlo simulations of γ, the second hyperpolarizability, to test the assumptions used in calculating the fundamental limits, to search for the broadest universal properties, and to determine if this approach is capable of generating γ values close to the limit. Finally, we will analyze the number of states that contribute to γ when it approaches the limit—a first step in establishing the validity of the three-level ansatz for the second hyperpolarizability.

2. APPROACH

The second hyperpolarizability can be expressed using the dipole-free sum-over-states (SOS) equation, which is equivalent to the traditional SOS expression when enough states are included in the sum. The xxxx tensor component of the off-resonant dipole-free SOS expression of the second hyperpolarizability is given by [22]

γDF=18(2nmnln{(2Em0En0)(2El0En0)En05(2El0En0)Em0En03}x0mxmnxnlxl0+2nmnlm{1El0Em0En0(2El0Em0)Em03En0}x0lxlmxmnxn0mn{1Em02En0+1En02Em0}x0m2x0n2),
where a prime indicates that the ground state is excluded in the summation. Eij=EiEj, where Ei and Ej are energies of ith and jth states, respectively. xij is the matrix element of position between states i and j. Note that Eq. (2) holds for a three-dimensional molecule where we have chosen the coordinate system such that xxxx is the largest diagonal component of the second hyperpolarizability.

The Thomas–Kuhn sum rules are the direct consequence of the commutator of the Hamiltonian with the position operator, [[H,x],x], where H is a Hamiltonian of the form

H(p,r)=p22m+V(r).
Here, p is the momentum at position r and V(r) is the potential energy of the particle. The Thomas–Kuhn sum rules for the non-relativistic Hamiltonian (3) are given by
n=0[En12(Em+Ep)]xmnxnp=2N2mδmp,
where δij is the Kronecker delta. Equation (4) represents an infinite number of equations indexed by the integers (m,p). In the text that follows, we refer to a particular sum rule by stating the values (m,p). Note that Eq. (4) holds for a broader set of multi-electron Hamiltonians including spin, externally applied electromagnetic fields, and electron correlations [23]. Most of the cases can be described by the non-relativistic Hamiltonian. For the relativistic case, some modifications of Eq. (4) are required [24, 25, 26].

The normalized form of the sum rules takes the form

n=0[en12(em+ep)]ξmnξnp=δmp,
where ei=Ei0/E10 can take on values from 1 to infinity for i>1 (the ground state is labeled by 0). Finally the normalized transition moments are defined as
ξij=xij|x01max|,
where
|x01max|2=2N2mE10.
Equations (7, 4) imply that the largest possible transition moment from the ground state is x01 because E10<Ei0 for i>1.

The three-level ansatz is central to calculating the fundamental limit of the second hyperpolarizability γ [2]. It states that when γ is at the limit, only three states contribute. Using this ansatz and the sum rules given by Eq. (4), the second hyperpolarizability given by Eq. (2) is found to be constrained to the range [2]

e44m2(N2E105)γ4e44m2(N2E105).
Hence, defining the fundamental limit of the second hyperpolarizability as
γmax=4e44m2(N2E105),
then
14γint1,
where γint=γ/γmax. Instead of studying γ directly, we use the dimensionless intrinsic second hyperpolarizability because it takes into account simple scalings [23]. Hence, when γ approaches e4N24/m2E105, γint approaches unity.

The numerical procedure is the same as in previous studies of the hyperpolarizability β [17]: For a specific number of states s, the energy values are picked arbitrarily and sorted in ascending order so that E0<E1<<Es1, where E0 is the energy of the ground state and Ei is the energy of the ith excited state. For simplicity and with no loss in generality, we shift the energies so that the ground state energy is zero. Since ei=Ei0/E10, then e0=0 and e1=1.

The next step is assigning transition moments that together with the energies are forced to be consistent with sum rules. For a time-invariant Hamiltonian, it can be shown that the wave functions are real provided that there are no degeneracies [27]. Our method is not well suited to situations where degenerate states are present. As such, we treat the case where transition moments are real and intentionally avoid degeneracies. We are thus ignoring the class of problems in which (1) the Hamiltonian is not time invariant and (2) there are degeneracies. These cases will be considered in future studies.

Since there are no dipole moments (the diagonal components of the transition moments) in γDF given by Eq. (2), we need to only use the sum rules with m=p in Eq. (5) to get all of the required transition moments. Starting with (m,p)=(0,0), we get

e10|ξ10|2+e20|ξ20|2+e30|ξ30|2++en0|ξn0|2=1.
Since e10=e1e0=1, ξ01 can be randomly picked from the interval 1<ξ01<1. Then ξ02 is obtained based on the fact that
n=2en0ξn02=1e10ξ102,
which implies
ξ202(1e1ξ102)/e20.
A random number 1r1 is used to get ξ20 from Eq. (13),
ξ20=r(1e1ξ102)/e20.
For an s-state model the same method is used for all other transition moments, ξi0, except for ξs1,0, which is directly determined from the sum rules. For example, for a ten-state model,
ξ10,0=(1e1ξ102e2ξ202e9ξ902)/e10,0.
The remaining transition moments are calculated using the higher-order sum rules in sequence with p=1,2,3, in Eq. (5). Since ξij is assumed to be real for all states i and j,
ξij=ξji.
The energy and transition moment values are inserted in normalized form into Eq. (2) to calculate γint. The normalized form is obtained by dividing Eq. (2) by Eq. (9) such that in Eq. (2), γγint, Eijeij, and xijξij. Note that one set of parameters obtained with the procedure given by Eqs. (12, 13, 14, 15, 16) is called a single trial or single iteration. To identify common properties of the second hyperpolarizability, hundreds or thousands of iterations are performed, and the distributions of the results are analyzed.

Before proceeding, we need to address the issue of the signs of the transition moments. Recall that the diagonal sum rules suffice to fix all the transition moments. However, all the transition moments in the diagonal sum rules appear as a squared modulus, |ξij|2. Thus, the diagonal sum rules give no information about the sign of xij. So one can choose the sign of each transition moment independently from the values of all others. The question arises of whether or not the signs are constrained by the sum rules that were neglected.

To answer this question, we first refer to the non-diagonal sum rules for which mp in Eq. (5), which can be used to determine a condition on the signs by solving the equations for all different pairs of m and p. Considering the number of states, s, this approach demands solving s coupled equations. Since this is time consuming, we confronted the problem using the simpler approach of randomly picking all transition moments to be positive or negative. Neither β nor γ, calculated from the resulting matrix elements and energies, exceeded the fundamental limit. Thus we concluded that the sign of the transition moments can be picked randomly. We have studied the issue of truncated sum rules [28], and in the future we will address the self-consistency between the sum rules when truncated.

The energies and transition moments are related via sum rules that impose a limit on all the transition moments. To express the limit mathematically we start with Eq. (5) with m=p,

n=0enp|ξnp|2=1.
For p=0 Eq. (17) yields
e10|ξ10|2+e20|ξ20|2+e30|ξ30|2++en0|ξn0|2=1.
Equation (18) implies that the largest value of ei0|ξi0|2 is attained when all other terms of the type ej0|ξj0|2 (for ji) are zero, i.e.,
ei0|ξi0|21,0<in,
or
(1ei0)1/2ξi0(1ei0)1/2.
ξi0 is largest when ei0 is minimized, and vice versa. Since based on our definition for energy spacing, when i1, we have ei01 then ξi0 is limited to
1ξi01  for 0<in.
Following the same procedure for p=1 in Eq. (5) results in
e01|ξ01|2+e21|ξ21|2+e31|ξ31|2+=1,
whence
e21|ξ21|2+e31|ξ31|2++en1|ξn1|2=1+e10|ξ01|2.
Here we have used the fact that e01=e10. Substituting Eq. (19) into Eq. (23) gives an upper limit
e21|ξ21|2+e31|ξ31|2++en1|ξn1|22.
This implies that ei1|ξi1|22 (when 1<in) and is obtained when for all ji, ej1|ξj1|2=0. Then
ei1|ξi1|22,1<in,
or
(2ei1)1/2ξi1(2ei1)1/2  for 1<in.
Using the same method for p=3,4,5,,n1, the constraint on the transition moments is found to be
2p/2eip1/2ξi,p2p/2eip1/2  for p<in.
Recall that we have assumed that the transition moments are real so ξij=ξji. The sum rules specify the range of validity of all non-diagonal transition moments according to Eq. (27). The Monte Carlo method typically yields energy spacings that lead to transition moments that lie below unity. Equation (27) is a general result that applies to quantum systems that are described by the Hamiltonian given by Eq. (3).

3. RESULTS AND DISCUSSION

In this section, we use Monte Carlo simulations to study the statistics of the distribution of the intrinsic second hyperpolarizability with the aim of understanding the physical properties of a quantum system that leads to a nonlinear response that approaches the fundamental limit. Correlations of the values of γint with the average energy spacing and transition moments when γint is at the limit will be used to investigate whether such systems behave in a way that suggests universal behavior. Finally, we discuss the effect that pairs of states have on the second hyperpolarizability using a missing state analysis.

When energies are chosen randomly, on average they are equally spaced. Thus the energy spacing is similar to the eigenenergies of a harmonic oscillator. To increase the domain of energy spacing that is probed by the Monte Carlo approach, we define an energy weighing factor f, such that

EiEif.
For f<1 the energies are denser at higher energies and for f>1 the states are denser at lower energies.

Figure 1a shows the distribution of the second hyperpolarizability for a 15-state model for weight factors of f=0.75,1,1.25. Frequency refers to the number of times γint appears in the histogram. All simulated γint values lie in the range predicted by Eq. (10). The maximum values of γint approach unity, which suggests that the set of energies and transition moments that are consistent with sum rules can create an intrinsic second hyperpolarizabilities that can be close to the fundamental limit (γint1). On the other hand, the smallest γint peaks at zero and the largest negative value approaches −0.25, as predicted.

In contrast to studies that determine the shape of the class of potential energy functions that optimize the intrinsic hyperpolarizabilities, which leads to βint<0.708, the Monte Carlo method yields values of βint that approach the limit. We find that the same is true for γint. The distribution of γint for all three weighting factors peaks near zero. However, the distributions that are generated with larger weighting factors have larger tails near the limits as shown in the insets of Fig. 1.

To better illustrate the distribution of γint near the limits, we have plotted the histogram of γint in the range [0.75,1] [Fig. 1b] and in the range [−0.1,−0.25] [Fig. 1c] where the bin size is 0.05. Both of these diagrams suggest that systems whose energy difference between adjacent states gets larger for higher energies (black bars with f=1.25) appear more frequently near the limit.

Recall that the three-level ansatz states that when the hyperpolarizability is near the fundamental limit, the ground and two excited states dominate. When the sum rules are applied to the three-level model of the hyperpolarizability and the model is parameterized in terms of

E=e1e2=E10E20,
X=ξ10=x10x10max,
where
x0nmax=2N2mEn0,
the hyperpolarizability is then found to be at the fundamental limit when X=1/34 and E=0 [1]. The three-level ansatz and the fact that X=1/34 when the hyperpolarizability is large are found to be a universal property of all systems studied to date [23]. While the fundamental limit of the off-resonant second hyperpolarizability is calculated using the three-level ansatz [2], there is no extensive body of literature that supports its validity. As we show below, the Monte Carlo simulations show that when γint1, the system can be modeled with three dominant states.

Figure 2 shows the resulting distributions for Monte Carlo simulations of the intrinsic second hyperpolarizability for 5, 10, and 15 states. The inset [Fig. 2b] focuses on the intrinsic hyperpolarizabilities in the range [0.75,1]. Clearly, when there are a fewer number of states, the frequency of γint near the limit increases significantly: the five-state model yields 1 order of magnitude higher frequency than the 15- and 40-state models. Also, the 15-state systems appear with higher frequencies than the 40-state systems. This is a general trend that suggests that the three-level ansatz holds for the second hyperpolarizability.

Figure 2 shows that the largest second hyperpolarizabilities are associated with greater energy spacing, suggesting that

limE0γint=1.
Figure 3 plots γint as a function of E for a five-state model and 1×106 iterations. When the intrinsic second hyperpolarizability approaches unity, the energy spacing most often falls in the range E<0.1, suggesting that γint is the largest when E20E10, the same result that is found for the first hyperpolarizability.

To provide a more quantitative measure of the distribution of energies for various ranges of γint, we generate histograms from the data shown in Fig. 2. The distribution of E for a ten-state model with 100,000 iterations is shown in Fig. 4 . Included are 12 equally spaced intervals of γint [except for the range of (0.85,1)]. The points represent the Monte Carlo data and the curves are fits to a stretched exponential model of the form

ln(F/F0)=(E/E0)n,
where F0, E0, and n are fit parameters and F is the frequency. The stretched exponential was chosen as a model because it best fits the data with the fewest number of parameters.

Note that all of the curves appear to be approximately parallel to each other with the exception of the curves for γint in the ranges of (0.85,1), (0.75,0.85), and (−0.25,−0.15), which fall off more steeply as a function of E. Thus, larger energy spacing is correlated with γint near unity or near −0, 25, the positive and negative limits.

Figure 5 shows γint as a function of X for a five-state model and 100,000 runs when the second hyperpolarizabilities are larger than 0.95. According to Fig. 5, as γint approaches the limit, the range of X becomes narrower so that for γint>0.98, X lies in the range 0.2<X<0.2.

To better quantify the results, Fig. 6 shows the distribution of X for various ranges of γint using a ten-level model. The bin size is 0.05 and γint has been divided into 12 equally spaced ranges except for γint[0.85,1]. γint[0.25,0.15] is dominated by systems with X=±1. For slightly smaller negative values of γint, the peak in the distribution is less than unity and the rest of the curves peak at X=0.

While any value of γint is attainable for any arbitrary choice of X, it is clear that the most likely values for the dominant transition moment are X=±1 when γint=0.25 and X=0 when γint=1. Recall that Fig. 2 implies that the largest values of γint are for E=0. Interestingly, the three-level ansatz shows that when E=0, X=0 yields γint=1 and |X|=1 yields γint=0.25 [2]. Thus, while the Monte Carlo simulations include systems where many states may be contributing to the second hyperpolarizability, on average, this statistical result gives the same result as the three-level ansatz. What remains to be investigated is the validity of the three-level ansatz.

The three-level ansatz is fundamental to the calculations of the limits of the first- and second-order nonlinear-optical responses. The intrinsic hyperpolarizability βint can expressed as [17]

βint=n,mβintn,m,
where the prime indicates that the ground state is excluded in the summation. βintn,m is the fractional contribution of pairs of individual states n and m [17],
βintn,m=(34)3/4ξ0nξnmξm0(1enem2emenen3).
The total hyperpolarizability β can be calculated as the sum over all the fractional contributions. When the hyperpolarizability is near the fundamental limit, we find that βintn,m is large for only one pair of states. This is in agreement with the three-level ansatz.

Expressing γint in terms of fractional contributions of pairs of states is complicated by the fact that three excited states contribute to each term in Eq. (2). We define γintijk as the fractional contribution of the three states i, j, and k:

γint=ijkγintijk=ijkγintijk+i=jkγintijk+ij=kγintijk+i=kjγintijk+i=j=kγintijk.
In contrast to the hyperpolarizability, where βintnn=0, the second hyperpolarizability does not vanish when i=j=k.

The simplest approach for determining the contribution of pairs of states is the missing state analysis [29], where one calculates γint in the absence of a particular set of states (i,j) denoted by γintmissing(i,j). A comparison of γint with γintmissing(i,j) describes the joint contribution of states i and j to the intrinsic second hyperpolarizability. The smaller the value of γintmissing(i,j), the larger the contribution of states i and j to γint.

Figure 7 shows a logarithmic plot of the absolute value of γintmissing(i,j) for a ten-state model with γint=0.9886, the largest valued obtained in a Monte Carlo simulation. The most important pair of states are 1 and 2 with γintmissing(2,1)=2.76444×1014. The next most important pair of states are 5 and 1 with γintmissing(5,1)=1011. γintmissing(8,1) and γintmissing(9,1) are also on the order of 1011. Thus, the dominant two states contribute about 103 times the next most important state. The same procedure can be followed for other optimized values, and similar results are found. It is worth mentioning that γintmissing(i,j)γintmissing(j,i) because the individual terms in Eq. (2) are not symmetric when two incidences are interchanged. Since each pair of states in Eq. (2) contributes twice, we can define

γintsymmetrized(i,j)=γintmissing(i,j)+γintmissing(j,i),
for ij and restrict the sum to ij. We have applied such a symmetrization to all values obtained with the missing state analysis.

4. CONCLUSION

Optimizing the intrinsic hyperpolarizabilities by varying the potential energy function results in a 30% gap between the fundamental limit and the optimized βint. Experimentally measured second hyperpolarizabilities also fall far short of the limit for a wide range of molecules, but Monte Carlo simulations show that βint can approach unity. In the present work, we find that Monte Carlo simulations lead to values of γ arbitrarily close to the fundamental limit, with 0.25<γint<1, in agreement with analytical calculations [2]. Also, the three-level ansatz appears to be obeyed as shown using the missing state analysis, where the two dominant states account for over 99% of γint.

It is important to stress that the Monte Carlo approach may lead to a broader set of transition moments and energies than are attainable with standard Hamiltonians; that is, Hamiltonians that include kinetic and potential energies of the many electrons in an atom as well as interactions with an external electromagnetic field. As such, universal behavior that may be typical of systems described by standard Hamiltonians may not be observed in our calculations. Nevertheless, we find that the Monte Carlo simulations, on average, are consistent with analytical results, with optimized γint resulting when E0, and |X|=1 for negative γ and X=0 for positive γ.

This is, to the best of our knowledge, the first study to confirm that there may be universal properties associated with the second hyperpolarizability when it is near the fundamental limit. As such, our approach may lead to the design of better materials for third-order nonlinear-optical applications if the universal properties can be re-expressed in terms of parameters that can be varied by a synthetic chemist.

ACKNOWLEDGMENTS

M. G. Kuzyk acknowledges the National Science Foundation (NSF) (ECCS-0756936) and Wright Paterson Air Force Base for generously supporting this work.

 

Fig. 1 (a) Distribution of simulated intrinsic second hyperpolarizabilities γint for a 15-state model and weight factors of 0.75, 1, and 1.25. The inset shows histograms of γint near the limits. The bin size for the main figure is 0.003; and, for Insets (b) and (c), the bin size is 0.05. The error bars are defined as the square root of the frequency.

Download Full Size | PPT Slide | PDF

 

Fig. 2 (a) Histogram of γint for 5-, 15-, and 40-state models. (b) Enlarged histogram for γint[0.75,1]. The bin size is 0.05.

Download Full Size | PPT Slide | PDF

 

Fig. 3 The distribution of γint versus E. For large values of the second hyperpolarizability, the majority of E values lie in the range [0,0.1].

Download Full Size | PPT Slide | PDF

 

Fig. 4 Distribution of E for different ranges of γint for a ten-state model with 100,000 iterations. Each symbol corresponds to the specified range of γint.

Download Full Size | PPT Slide | PDF

 

Fig. 5 γint as a function of X for second hyperpolarizabilities within 5% of the limit.

Download Full Size | PPT Slide | PDF

 

Fig. 6 Distribution of X for different ranges of γint.

Download Full Size | PPT Slide | PDF

 

Fig. 7 A logarithmic plot of |γint(i,j)| using the missing state analysis for a ten-state model and γint=0.9886. We remark that |γint(i,j)|=|γint(j,i)| so that the two smallest values in the diagram are identical and represent only one pair of states.

Download Full Size | PPT Slide | PDF

1. M. G. Kuzyk, “Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 85, 1218–1221 (2000). [CrossRef]   [PubMed]  

2. M. G. Kuzyk, “Fundamental limits on third-order molecular susceptibilities,” Opt. Lett. 25, 1183–1185 (2000). [CrossRef]  

3. M. G. Kuzyk, “Quantum limits of the hyper-Rayleigh scattering susceptibility,” IEEE J. Sel. Top. Quantum Electron. 7, 774–780 (2001). [CrossRef]  

4. J. Zhou and M. G. Kuzyk, “Intrinsic hyperpolarizabilities as a figure of merit for electro-optic molecules,” J. Phys. Chem. C 112, 7978–7982 (2008). [CrossRef]  

5. M. G. Kuzyk, “Using fundamental principles to understand and optimize nonlinear-optical materials,” J. Mater. Chem. 19, 7444–7465 (2009). [CrossRef]  

6. M. G. Kuzyk, “Fundamental limits on third-order molecular susceptibilities: erratum,” Opt. Lett. 28, 135–137 (2003). [CrossRef]  

7. M. G. Kuzyk, “Erratum: Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 90, 039902 (2003). [CrossRef]  

8. H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005). [CrossRef]  

9. H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007). [CrossRef]   [PubMed]  

10. K. Tripathy, J. Pérez Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. M. Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932–7945 (2004). [CrossRef]   [PubMed]  

11. J. Zhou, M. G. Kuzyk, and D. S. Watkins, “Pushing the hyperpolarizability to the limit,” Opt. Lett. 31, 2891–2893 (2006). [CrossRef]   [PubMed]  

12. J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76, 053831 (2007). [CrossRef]  

13. J. Pérez-Moreno, Y. Zhao, K. Clays, and M. G. Kuzyk, “Modulated conjugation as a means for attaining a record high intrinsic hyperpolarizability,” Opt. Lett. 32, 59–61 (2007). [CrossRef]  

14. J. Pérez-Moreno, Y. Zhao, K. Clays, M. G. Kuzyk, Y. Shen, L. Qiu, J. Hao, and K. Guo, “Modulated conjugation as a means of improving the intrinsic hyperpolarizability,” J. Am. Chem. Soc. 131, 5084–5093 (2009). [CrossRef]   [PubMed]  

15. M. G. Kuzyk and D. S. Watkins, “The effects of geometry on the hyperpolarizability,” J. Chem. Phys. 124, 244104 (2006). [CrossRef]   [PubMed]  

16. D. S. Watkins and M. G. Kuzyk, “Optimizing the hyperpolarizability tensor using external electromagnetic fields and nuclear placement,” J. Chem. Phys. 131, 064110 (2009). [CrossRef]   [PubMed]  

17. M. C. Kuzyk and M. G. Kuzyk, “Monte Carlo studies of the fundamental limits of the intrinsic hyperpolarizability,” J. Opt. Soc. Am. B 25, 103–110 (2008). [CrossRef]  

18. X. Hu, D. Xiao, S. Keinan, I. Asselberghs, M. J. Therien, K. Clays, W. Yang, and D. N. Beratan, “Predicting the frequency dispersion of electronic hyperpolarizabilities on the basis of absorption data and Thomas–Kuhn sum rules,” J. Phys. Chem. C 114, 2349–2359 (2010). [CrossRef]  

19. A. D. Slepkov, F. A. Hegmann, S. Eisler, E. Elliot, and R. R. Tykwinski, “The surprising nonlinear optical properties of conjugated polyyne oligomers,” J. Chem. Phys. 120, 6807–6810 (2004). [CrossRef]   [PubMed]  

20. J. C. May, J. H. Lim, I. Biaggio, N. N. P. Moonen, T. Michinobu, and F. Diederich, “Highly efficient third-order optical nonlinearities in donor-substituted cyanoethynylethene molecules,” Opt. Lett. 30, 3057–3059 (2005). [CrossRef]   [PubMed]  

21. J. C. May, I. Biaggio, F. Bures, and F. Diederich, “Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules,” Appl. Phys. Lett. 90, 251106 (2007). [CrossRef]  

22. J. Pérez-Moreno, K. Clays, and M. G. Kuzyk, “A new dipole-free sum-over-states expression for the second hyperpolarizability,” J. Chem. Phys. 128, 084109 (2008). [CrossRef]   [PubMed]  

23. M. G. Kuzyk, “A bird’s-eye view of nonlinear-optical processes: Unification through scale invariance,” Nonlinear Opt., Quantum Opt. 40, 1–13 (2010).

24. S. P. Goldman and G. W. F. Drake, “Relativistic sum rules and integral properties of the Dirac equation,” Phys. Rev. A 25, 2877–2881 (1982). [CrossRef]  

25. P. T. Leung and M. L. Rustgi, “Relativistic corrections to Bethe sum rule,” Phys. Rev. A 33, 2827–2829 (1986). [CrossRef]   [PubMed]  

26. S. M. Cohen, “Aspects of relativistic sum rules,” Adv. Quantum Chem. 46, 241–265 (2004). [CrossRef]  

27. J. J. Sakurai, Modern Quantum Mechanics—Revised Edition (Addison Wesley Longman, 1994).

28. M. G. Kuzyk, “Truncated sum rules and their use in calculating fundamental limits of nonlinear susceptibilities,” J. Nonlinear Opt. Phys. Mater. 15, 77–87 (2006). [CrossRef]  

29. C. W. Dirk and M. G. Kuzyk, “Missing-state analysis: A method for determining the origin of molecular nonlinear optical properties,” Phys. Rev. A 39, 1219–1226 (1989). [CrossRef]   [PubMed]  

References

  • View by:
  • |
  • |
  • |

  1. M. G. Kuzyk, “Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 85, 1218–1221 (2000).
    [Crossref] [PubMed]
  2. M. G. Kuzyk, “Fundamental limits on third-order molecular susceptibilities,” Opt. Lett. 25, 1183–1185 (2000).
    [Crossref]
  3. M. G. Kuzyk, “Quantum limits of the hyper-Rayleigh scattering susceptibility,” IEEE J. Sel. Top. Quantum Electron. 7, 774–780 (2001).
    [Crossref]
  4. J. Zhou and M. G. Kuzyk, “Intrinsic hyperpolarizabilities as a figure of merit for electro-optic molecules,” J. Phys. Chem. C 112, 7978–7982 (2008).
    [Crossref]
  5. M. G. Kuzyk, “Using fundamental principles to understand and optimize nonlinear-optical materials,” J. Mater. Chem. 19, 7444–7465 (2009).
    [Crossref]
  6. M. G. Kuzyk, “Fundamental limits on third-order molecular susceptibilities: erratum,” Opt. Lett. 28, 135–137 (2003).
    [Crossref]
  7. M. G. Kuzyk, “Erratum: Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 90, 039902 (2003).
    [Crossref]
  8. H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
    [Crossref]
  9. H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
    [Crossref] [PubMed]
  10. K. Tripathy, J. Pérez Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. M. Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932–7945 (2004).
    [Crossref] [PubMed]
  11. J. Zhou, M. G. Kuzyk, and D. S. Watkins, “Pushing the hyperpolarizability to the limit,” Opt. Lett. 31, 2891–2893 (2006).
    [Crossref] [PubMed]
  12. J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76, 053831 (2007).
    [Crossref]
  13. J. Pérez-Moreno, Y. Zhao, K. Clays, and M. G. Kuzyk, “Modulated conjugation as a means for attaining a record high intrinsic hyperpolarizability,” Opt. Lett. 32, 59–61 (2007).
    [Crossref]
  14. J. Pérez-Moreno, Y. Zhao, K. Clays, M. G. Kuzyk, Y. Shen, L. Qiu, J. Hao, and K. Guo, “Modulated conjugation as a means of improving the intrinsic hyperpolarizability,” J. Am. Chem. Soc. 131, 5084–5093 (2009).
    [Crossref] [PubMed]
  15. M. G. Kuzyk and D. S. Watkins, “The effects of geometry on the hyperpolarizability,” J. Chem. Phys. 124, 244104 (2006).
    [Crossref] [PubMed]
  16. D. S. Watkins and M. G. Kuzyk, “Optimizing the hyperpolarizability tensor using external electromagnetic fields and nuclear placement,” J. Chem. Phys. 131, 064110 (2009).
    [Crossref] [PubMed]
  17. M. C. Kuzyk and M. G. Kuzyk, “Monte Carlo studies of the fundamental limits of the intrinsic hyperpolarizability,” J. Opt. Soc. Am. B 25, 103–110 (2008).
    [Crossref]
  18. X. Hu, D. Xiao, S. Keinan, I. Asselberghs, M. J. Therien, K. Clays, W. Yang, and D. N. Beratan, “Predicting the frequency dispersion of electronic hyperpolarizabilities on the basis of absorption data and Thomas–Kuhn sum rules,” J. Phys. Chem. C 114, 2349–2359 (2010).
    [Crossref]
  19. A. D. Slepkov, F. A. Hegmann, S. Eisler, E. Elliot, and R. R. Tykwinski, “The surprising nonlinear optical properties of conjugated polyyne oligomers,” J. Chem. Phys. 120, 6807–6810 (2004).
    [Crossref] [PubMed]
  20. J. C. May, J. H. Lim, I. Biaggio, N. N. P. Moonen, T. Michinobu, and F. Diederich, “Highly efficient third-order optical nonlinearities in donor-substituted cyanoethynylethene molecules,” Opt. Lett. 30, 3057–3059 (2005).
    [Crossref] [PubMed]
  21. J. C. May, I. Biaggio, F. Bures, and F. Diederich, “Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules,” Appl. Phys. Lett. 90, 251106 (2007).
    [Crossref]
  22. J. Pérez-Moreno, K. Clays, and M. G. Kuzyk, “A new dipole-free sum-over-states expression for the second hyperpolarizability,” J. Chem. Phys. 128, 084109 (2008).
    [Crossref] [PubMed]
  23. M. G. Kuzyk, “A bird’s-eye view of nonlinear-optical processes: Unification through scale invariance,” Nonlinear Opt., Quantum Opt. 40, 1–13 (2010).
  24. S. P. Goldman and G. W. F. Drake, “Relativistic sum rules and integral properties of the Dirac equation,” Phys. Rev. A 25, 2877–2881 (1982).
    [Crossref]
  25. P. T. Leung and M. L. Rustgi, “Relativistic corrections to Bethe sum rule,” Phys. Rev. A 33, 2827–2829 (1986).
    [Crossref] [PubMed]
  26. S. M. Cohen, “Aspects of relativistic sum rules,” Adv. Quantum Chem. 46, 241–265 (2004).
    [Crossref]
  27. J. J. Sakurai, Modern Quantum Mechanics—Revised Edition (Addison Wesley Longman, 1994).
  28. M. G. Kuzyk, “Truncated sum rules and their use in calculating fundamental limits of nonlinear susceptibilities,” J. Nonlinear Opt. Phys. Mater. 15, 77–87 (2006).
    [Crossref]
  29. C. W. Dirk and M. G. Kuzyk, “Missing-state analysis: A method for determining the origin of molecular nonlinear optical properties,” Phys. Rev. A 39, 1219–1226 (1989).
    [Crossref] [PubMed]

2010 (2)

X. Hu, D. Xiao, S. Keinan, I. Asselberghs, M. J. Therien, K. Clays, W. Yang, and D. N. Beratan, “Predicting the frequency dispersion of electronic hyperpolarizabilities on the basis of absorption data and Thomas–Kuhn sum rules,” J. Phys. Chem. C 114, 2349–2359 (2010).
[Crossref]

M. G. Kuzyk, “A bird’s-eye view of nonlinear-optical processes: Unification through scale invariance,” Nonlinear Opt., Quantum Opt. 40, 1–13 (2010).

2009 (3)

M. G. Kuzyk, “Using fundamental principles to understand and optimize nonlinear-optical materials,” J. Mater. Chem. 19, 7444–7465 (2009).
[Crossref]

J. Pérez-Moreno, Y. Zhao, K. Clays, M. G. Kuzyk, Y. Shen, L. Qiu, J. Hao, and K. Guo, “Modulated conjugation as a means of improving the intrinsic hyperpolarizability,” J. Am. Chem. Soc. 131, 5084–5093 (2009).
[Crossref] [PubMed]

D. S. Watkins and M. G. Kuzyk, “Optimizing the hyperpolarizability tensor using external electromagnetic fields and nuclear placement,” J. Chem. Phys. 131, 064110 (2009).
[Crossref] [PubMed]

2008 (3)

M. C. Kuzyk and M. G. Kuzyk, “Monte Carlo studies of the fundamental limits of the intrinsic hyperpolarizability,” J. Opt. Soc. Am. B 25, 103–110 (2008).
[Crossref]

J. Zhou and M. G. Kuzyk, “Intrinsic hyperpolarizabilities as a figure of merit for electro-optic molecules,” J. Phys. Chem. C 112, 7978–7982 (2008).
[Crossref]

J. Pérez-Moreno, K. Clays, and M. G. Kuzyk, “A new dipole-free sum-over-states expression for the second hyperpolarizability,” J. Chem. Phys. 128, 084109 (2008).
[Crossref] [PubMed]

2007 (4)

J. C. May, I. Biaggio, F. Bures, and F. Diederich, “Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules,” Appl. Phys. Lett. 90, 251106 (2007).
[Crossref]

H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
[Crossref] [PubMed]

J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76, 053831 (2007).
[Crossref]

J. Pérez-Moreno, Y. Zhao, K. Clays, and M. G. Kuzyk, “Modulated conjugation as a means for attaining a record high intrinsic hyperpolarizability,” Opt. Lett. 32, 59–61 (2007).
[Crossref]

2006 (3)

J. Zhou, M. G. Kuzyk, and D. S. Watkins, “Pushing the hyperpolarizability to the limit,” Opt. Lett. 31, 2891–2893 (2006).
[Crossref] [PubMed]

M. G. Kuzyk and D. S. Watkins, “The effects of geometry on the hyperpolarizability,” J. Chem. Phys. 124, 244104 (2006).
[Crossref] [PubMed]

M. G. Kuzyk, “Truncated sum rules and their use in calculating fundamental limits of nonlinear susceptibilities,” J. Nonlinear Opt. Phys. Mater. 15, 77–87 (2006).
[Crossref]

2005 (2)

J. C. May, J. H. Lim, I. Biaggio, N. N. P. Moonen, T. Michinobu, and F. Diederich, “Highly efficient third-order optical nonlinearities in donor-substituted cyanoethynylethene molecules,” Opt. Lett. 30, 3057–3059 (2005).
[Crossref] [PubMed]

H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
[Crossref]

2004 (3)

K. Tripathy, J. Pérez Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. M. Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932–7945 (2004).
[Crossref] [PubMed]

A. D. Slepkov, F. A. Hegmann, S. Eisler, E. Elliot, and R. R. Tykwinski, “The surprising nonlinear optical properties of conjugated polyyne oligomers,” J. Chem. Phys. 120, 6807–6810 (2004).
[Crossref] [PubMed]

S. M. Cohen, “Aspects of relativistic sum rules,” Adv. Quantum Chem. 46, 241–265 (2004).
[Crossref]

2003 (2)

M. G. Kuzyk, “Fundamental limits on third-order molecular susceptibilities: erratum,” Opt. Lett. 28, 135–137 (2003).
[Crossref]

M. G. Kuzyk, “Erratum: Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 90, 039902 (2003).
[Crossref]

2001 (1)

M. G. Kuzyk, “Quantum limits of the hyper-Rayleigh scattering susceptibility,” IEEE J. Sel. Top. Quantum Electron. 7, 774–780 (2001).
[Crossref]

2000 (2)

M. G. Kuzyk, “Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 85, 1218–1221 (2000).
[Crossref] [PubMed]

M. G. Kuzyk, “Fundamental limits on third-order molecular susceptibilities,” Opt. Lett. 25, 1183–1185 (2000).
[Crossref]

1994 (1)

J. J. Sakurai, Modern Quantum Mechanics—Revised Edition (Addison Wesley Longman, 1994).

1989 (1)

C. W. Dirk and M. G. Kuzyk, “Missing-state analysis: A method for determining the origin of molecular nonlinear optical properties,” Phys. Rev. A 39, 1219–1226 (1989).
[Crossref] [PubMed]

1986 (1)

P. T. Leung and M. L. Rustgi, “Relativistic corrections to Bethe sum rule,” Phys. Rev. A 33, 2827–2829 (1986).
[Crossref] [PubMed]

1982 (1)

S. P. Goldman and G. W. F. Drake, “Relativistic sum rules and integral properties of the Dirac equation,” Phys. Rev. A 25, 2877–2881 (1982).
[Crossref]

Asselberghs, I.

X. Hu, D. Xiao, S. Keinan, I. Asselberghs, M. J. Therien, K. Clays, W. Yang, and D. N. Beratan, “Predicting the frequency dispersion of electronic hyperpolarizabilities on the basis of absorption data and Thomas–Kuhn sum rules,” J. Phys. Chem. C 114, 2349–2359 (2010).
[Crossref]

Beratan, D. N.

X. Hu, D. Xiao, S. Keinan, I. Asselberghs, M. J. Therien, K. Clays, W. Yang, and D. N. Beratan, “Predicting the frequency dispersion of electronic hyperpolarizabilities on the basis of absorption data and Thomas–Kuhn sum rules,” J. Phys. Chem. C 114, 2349–2359 (2010).
[Crossref]

Biaggio, I.

J. C. May, I. Biaggio, F. Bures, and F. Diederich, “Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules,” Appl. Phys. Lett. 90, 251106 (2007).
[Crossref]

J. C. May, J. H. Lim, I. Biaggio, N. N. P. Moonen, T. Michinobu, and F. Diederich, “Highly efficient third-order optical nonlinearities in donor-substituted cyanoethynylethene molecules,” Opt. Lett. 30, 3057–3059 (2005).
[Crossref] [PubMed]

Brown, E. C.

H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
[Crossref] [PubMed]

Bures, F.

J. C. May, I. Biaggio, F. Bures, and F. Diederich, “Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules,” Appl. Phys. Lett. 90, 251106 (2007).
[Crossref]

Cariati, E.

H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
[Crossref] [PubMed]

H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
[Crossref]

Clays, K.

X. Hu, D. Xiao, S. Keinan, I. Asselberghs, M. J. Therien, K. Clays, W. Yang, and D. N. Beratan, “Predicting the frequency dispersion of electronic hyperpolarizabilities on the basis of absorption data and Thomas–Kuhn sum rules,” J. Phys. Chem. C 114, 2349–2359 (2010).
[Crossref]

J. Pérez-Moreno, Y. Zhao, K. Clays, M. G. Kuzyk, Y. Shen, L. Qiu, J. Hao, and K. Guo, “Modulated conjugation as a means of improving the intrinsic hyperpolarizability,” J. Am. Chem. Soc. 131, 5084–5093 (2009).
[Crossref] [PubMed]

J. Pérez-Moreno, K. Clays, and M. G. Kuzyk, “A new dipole-free sum-over-states expression for the second hyperpolarizability,” J. Chem. Phys. 128, 084109 (2008).
[Crossref] [PubMed]

J. Pérez-Moreno, Y. Zhao, K. Clays, and M. G. Kuzyk, “Modulated conjugation as a means for attaining a record high intrinsic hyperpolarizability,” Opt. Lett. 32, 59–61 (2007).
[Crossref]

K. Tripathy, J. Pérez Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. M. Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932–7945 (2004).
[Crossref] [PubMed]

Coe, B. J.

K. Tripathy, J. Pérez Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. M. Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932–7945 (2004).
[Crossref] [PubMed]

Cohen, S. M.

S. M. Cohen, “Aspects of relativistic sum rules,” Adv. Quantum Chem. 46, 241–265 (2004).
[Crossref]

Diederich, F.

J. C. May, I. Biaggio, F. Bures, and F. Diederich, “Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules,” Appl. Phys. Lett. 90, 251106 (2007).
[Crossref]

J. C. May, J. H. Lim, I. Biaggio, N. N. P. Moonen, T. Michinobu, and F. Diederich, “Highly efficient third-order optical nonlinearities in donor-substituted cyanoethynylethene molecules,” Opt. Lett. 30, 3057–3059 (2005).
[Crossref] [PubMed]

Dirk, C. W.

C. W. Dirk and M. G. Kuzyk, “Missing-state analysis: A method for determining the origin of molecular nonlinear optical properties,” Phys. Rev. A 39, 1219–1226 (1989).
[Crossref] [PubMed]

Drake, G. W. F.

S. P. Goldman and G. W. F. Drake, “Relativistic sum rules and integral properties of the Dirac equation,” Phys. Rev. A 25, 2877–2881 (1982).
[Crossref]

Eisler, S.

A. D. Slepkov, F. A. Hegmann, S. Eisler, E. Elliot, and R. R. Tykwinski, “The surprising nonlinear optical properties of conjugated polyyne oligomers,” J. Chem. Phys. 120, 6807–6810 (2004).
[Crossref] [PubMed]

Elliot, E.

A. D. Slepkov, F. A. Hegmann, S. Eisler, E. Elliot, and R. R. Tykwinski, “The surprising nonlinear optical properties of conjugated polyyne oligomers,” J. Chem. Phys. 120, 6807–6810 (2004).
[Crossref] [PubMed]

Facchetti, A.

H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
[Crossref] [PubMed]

H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
[Crossref]

Goldman, S. P.

S. P. Goldman and G. W. F. Drake, “Relativistic sum rules and integral properties of the Dirac equation,” Phys. Rev. A 25, 2877–2881 (1982).
[Crossref]

Guo, K.

J. Pérez-Moreno, Y. Zhao, K. Clays, M. G. Kuzyk, Y. Shen, L. Qiu, J. Hao, and K. Guo, “Modulated conjugation as a means of improving the intrinsic hyperpolarizability,” J. Am. Chem. Soc. 131, 5084–5093 (2009).
[Crossref] [PubMed]

Hao, J.

J. Pérez-Moreno, Y. Zhao, K. Clays, M. G. Kuzyk, Y. Shen, L. Qiu, J. Hao, and K. Guo, “Modulated conjugation as a means of improving the intrinsic hyperpolarizability,” J. Am. Chem. Soc. 131, 5084–5093 (2009).
[Crossref] [PubMed]

Hegmann, F. A.

A. D. Slepkov, F. A. Hegmann, S. Eisler, E. Elliot, and R. R. Tykwinski, “The surprising nonlinear optical properties of conjugated polyyne oligomers,” J. Chem. Phys. 120, 6807–6810 (2004).
[Crossref] [PubMed]

Ho, S. T.

H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
[Crossref] [PubMed]

H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
[Crossref]

Hu, X.

X. Hu, D. Xiao, S. Keinan, I. Asselberghs, M. J. Therien, K. Clays, W. Yang, and D. N. Beratan, “Predicting the frequency dispersion of electronic hyperpolarizabilities on the basis of absorption data and Thomas–Kuhn sum rules,” J. Phys. Chem. C 114, 2349–2359 (2010).
[Crossref]

Jiang, H.

H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
[Crossref] [PubMed]

H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
[Crossref]

Kang, H.

H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
[Crossref] [PubMed]

H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
[Crossref]

Keinan, S.

X. Hu, D. Xiao, S. Keinan, I. Asselberghs, M. J. Therien, K. Clays, W. Yang, and D. N. Beratan, “Predicting the frequency dispersion of electronic hyperpolarizabilities on the basis of absorption data and Thomas–Kuhn sum rules,” J. Phys. Chem. C 114, 2349–2359 (2010).
[Crossref]

Kelley, A. M.

K. Tripathy, J. Pérez Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. M. Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932–7945 (2004).
[Crossref] [PubMed]

Kuzyk, M. C.

Kuzyk, M. G.

M. G. Kuzyk, “A bird’s-eye view of nonlinear-optical processes: Unification through scale invariance,” Nonlinear Opt., Quantum Opt. 40, 1–13 (2010).

D. S. Watkins and M. G. Kuzyk, “Optimizing the hyperpolarizability tensor using external electromagnetic fields and nuclear placement,” J. Chem. Phys. 131, 064110 (2009).
[Crossref] [PubMed]

J. Pérez-Moreno, Y. Zhao, K. Clays, M. G. Kuzyk, Y. Shen, L. Qiu, J. Hao, and K. Guo, “Modulated conjugation as a means of improving the intrinsic hyperpolarizability,” J. Am. Chem. Soc. 131, 5084–5093 (2009).
[Crossref] [PubMed]

M. G. Kuzyk, “Using fundamental principles to understand and optimize nonlinear-optical materials,” J. Mater. Chem. 19, 7444–7465 (2009).
[Crossref]

J. Zhou and M. G. Kuzyk, “Intrinsic hyperpolarizabilities as a figure of merit for electro-optic molecules,” J. Phys. Chem. C 112, 7978–7982 (2008).
[Crossref]

M. C. Kuzyk and M. G. Kuzyk, “Monte Carlo studies of the fundamental limits of the intrinsic hyperpolarizability,” J. Opt. Soc. Am. B 25, 103–110 (2008).
[Crossref]

J. Pérez-Moreno, K. Clays, and M. G. Kuzyk, “A new dipole-free sum-over-states expression for the second hyperpolarizability,” J. Chem. Phys. 128, 084109 (2008).
[Crossref] [PubMed]

J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76, 053831 (2007).
[Crossref]

J. Pérez-Moreno, Y. Zhao, K. Clays, and M. G. Kuzyk, “Modulated conjugation as a means for attaining a record high intrinsic hyperpolarizability,” Opt. Lett. 32, 59–61 (2007).
[Crossref]

J. Zhou, M. G. Kuzyk, and D. S. Watkins, “Pushing the hyperpolarizability to the limit,” Opt. Lett. 31, 2891–2893 (2006).
[Crossref] [PubMed]

M. G. Kuzyk and D. S. Watkins, “The effects of geometry on the hyperpolarizability,” J. Chem. Phys. 124, 244104 (2006).
[Crossref] [PubMed]

M. G. Kuzyk, “Truncated sum rules and their use in calculating fundamental limits of nonlinear susceptibilities,” J. Nonlinear Opt. Phys. Mater. 15, 77–87 (2006).
[Crossref]

K. Tripathy, J. Pérez Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. M. Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932–7945 (2004).
[Crossref] [PubMed]

M. G. Kuzyk, “Fundamental limits on third-order molecular susceptibilities: erratum,” Opt. Lett. 28, 135–137 (2003).
[Crossref]

M. G. Kuzyk, “Erratum: Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 90, 039902 (2003).
[Crossref]

M. G. Kuzyk, “Quantum limits of the hyper-Rayleigh scattering susceptibility,” IEEE J. Sel. Top. Quantum Electron. 7, 774–780 (2001).
[Crossref]

M. G. Kuzyk, “Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 85, 1218–1221 (2000).
[Crossref] [PubMed]

M. G. Kuzyk, “Fundamental limits on third-order molecular susceptibilities,” Opt. Lett. 25, 1183–1185 (2000).
[Crossref]

C. W. Dirk and M. G. Kuzyk, “Missing-state analysis: A method for determining the origin of molecular nonlinear optical properties,” Phys. Rev. A 39, 1219–1226 (1989).
[Crossref] [PubMed]

Leung, P. T.

P. T. Leung and M. L. Rustgi, “Relativistic corrections to Bethe sum rule,” Phys. Rev. A 33, 2827–2829 (1986).
[Crossref] [PubMed]

Lim, J. H.

Liu, Z.

H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
[Crossref]

Liu, Z. F.

H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
[Crossref] [PubMed]

Macchioni, A.

H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
[Crossref] [PubMed]

H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
[Crossref]

Marks, T. J.

H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
[Crossref] [PubMed]

H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
[Crossref]

May, J. C.

J. C. May, I. Biaggio, F. Bures, and F. Diederich, “Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules,” Appl. Phys. Lett. 90, 251106 (2007).
[Crossref]

J. C. May, J. H. Lim, I. Biaggio, N. N. P. Moonen, T. Michinobu, and F. Diederich, “Highly efficient third-order optical nonlinearities in donor-substituted cyanoethynylethene molecules,” Opt. Lett. 30, 3057–3059 (2005).
[Crossref] [PubMed]

Michinobu, T.

Moonen, N. N. P.

Pérez Moreno, J.

K. Tripathy, J. Pérez Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. M. Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932–7945 (2004).
[Crossref] [PubMed]

Pérez-Moreno, J.

J. Pérez-Moreno, Y. Zhao, K. Clays, M. G. Kuzyk, Y. Shen, L. Qiu, J. Hao, and K. Guo, “Modulated conjugation as a means of improving the intrinsic hyperpolarizability,” J. Am. Chem. Soc. 131, 5084–5093 (2009).
[Crossref] [PubMed]

J. Pérez-Moreno, K. Clays, and M. G. Kuzyk, “A new dipole-free sum-over-states expression for the second hyperpolarizability,” J. Chem. Phys. 128, 084109 (2008).
[Crossref] [PubMed]

J. Pérez-Moreno, Y. Zhao, K. Clays, and M. G. Kuzyk, “Modulated conjugation as a means for attaining a record high intrinsic hyperpolarizability,” Opt. Lett. 32, 59–61 (2007).
[Crossref]

Qiu, L.

J. Pérez-Moreno, Y. Zhao, K. Clays, M. G. Kuzyk, Y. Shen, L. Qiu, J. Hao, and K. Guo, “Modulated conjugation as a means of improving the intrinsic hyperpolarizability,” J. Am. Chem. Soc. 131, 5084–5093 (2009).
[Crossref] [PubMed]

Ratner, M. A.

H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
[Crossref] [PubMed]

Righetto, S.

H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
[Crossref] [PubMed]

H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
[Crossref]

Rustgi, M. L.

P. T. Leung and M. L. Rustgi, “Relativistic corrections to Bethe sum rule,” Phys. Rev. A 33, 2827–2829 (1986).
[Crossref] [PubMed]

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics—Revised Edition (Addison Wesley Longman, 1994).

Shen, Y.

J. Pérez-Moreno, Y. Zhao, K. Clays, M. G. Kuzyk, Y. Shen, L. Qiu, J. Hao, and K. Guo, “Modulated conjugation as a means of improving the intrinsic hyperpolarizability,” J. Am. Chem. Soc. 131, 5084–5093 (2009).
[Crossref] [PubMed]

Slepkov, A. D.

A. D. Slepkov, F. A. Hegmann, S. Eisler, E. Elliot, and R. R. Tykwinski, “The surprising nonlinear optical properties of conjugated polyyne oligomers,” J. Chem. Phys. 120, 6807–6810 (2004).
[Crossref] [PubMed]

Stern, C. L.

H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
[Crossref] [PubMed]

H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
[Crossref]

Szafruga, U. B.

J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76, 053831 (2007).
[Crossref]

Therien, M. J.

X. Hu, D. Xiao, S. Keinan, I. Asselberghs, M. J. Therien, K. Clays, W. Yang, and D. N. Beratan, “Predicting the frequency dispersion of electronic hyperpolarizabilities on the basis of absorption data and Thomas–Kuhn sum rules,” J. Phys. Chem. C 114, 2349–2359 (2010).
[Crossref]

Tripathy, K.

K. Tripathy, J. Pérez Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. M. Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932–7945 (2004).
[Crossref] [PubMed]

Tykwinski, R. R.

A. D. Slepkov, F. A. Hegmann, S. Eisler, E. Elliot, and R. R. Tykwinski, “The surprising nonlinear optical properties of conjugated polyyne oligomers,” J. Chem. Phys. 120, 6807–6810 (2004).
[Crossref] [PubMed]

Ugo, R.

H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
[Crossref] [PubMed]

H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
[Crossref]

Watkins, D. S.

D. S. Watkins and M. G. Kuzyk, “Optimizing the hyperpolarizability tensor using external electromagnetic fields and nuclear placement,” J. Chem. Phys. 131, 064110 (2009).
[Crossref] [PubMed]

J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76, 053831 (2007).
[Crossref]

J. Zhou, M. G. Kuzyk, and D. S. Watkins, “Pushing the hyperpolarizability to the limit,” Opt. Lett. 31, 2891–2893 (2006).
[Crossref] [PubMed]

M. G. Kuzyk and D. S. Watkins, “The effects of geometry on the hyperpolarizability,” J. Chem. Phys. 124, 244104 (2006).
[Crossref] [PubMed]

Xiao, D.

X. Hu, D. Xiao, S. Keinan, I. Asselberghs, M. J. Therien, K. Clays, W. Yang, and D. N. Beratan, “Predicting the frequency dispersion of electronic hyperpolarizabilities on the basis of absorption data and Thomas–Kuhn sum rules,” J. Phys. Chem. C 114, 2349–2359 (2010).
[Crossref]

Yang, W.

X. Hu, D. Xiao, S. Keinan, I. Asselberghs, M. J. Therien, K. Clays, W. Yang, and D. N. Beratan, “Predicting the frequency dispersion of electronic hyperpolarizabilities on the basis of absorption data and Thomas–Kuhn sum rules,” J. Phys. Chem. C 114, 2349–2359 (2010).
[Crossref]

Yang, Y.

H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
[Crossref]

Zhao, Y.

J. Pérez-Moreno, Y. Zhao, K. Clays, M. G. Kuzyk, Y. Shen, L. Qiu, J. Hao, and K. Guo, “Modulated conjugation as a means of improving the intrinsic hyperpolarizability,” J. Am. Chem. Soc. 131, 5084–5093 (2009).
[Crossref] [PubMed]

J. Pérez-Moreno, Y. Zhao, K. Clays, and M. G. Kuzyk, “Modulated conjugation as a means for attaining a record high intrinsic hyperpolarizability,” Opt. Lett. 32, 59–61 (2007).
[Crossref]

Zhou, J.

J. Zhou and M. G. Kuzyk, “Intrinsic hyperpolarizabilities as a figure of merit for electro-optic molecules,” J. Phys. Chem. C 112, 7978–7982 (2008).
[Crossref]

J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76, 053831 (2007).
[Crossref]

J. Zhou, M. G. Kuzyk, and D. S. Watkins, “Pushing the hyperpolarizability to the limit,” Opt. Lett. 31, 2891–2893 (2006).
[Crossref] [PubMed]

Zhu, P.

H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
[Crossref]

Zuccaccia, C.

H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
[Crossref] [PubMed]

H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
[Crossref]

Adv. Quantum Chem. (1)

S. M. Cohen, “Aspects of relativistic sum rules,” Adv. Quantum Chem. 46, 241–265 (2004).
[Crossref]

Angew. Chem., Int. Ed. (1)

H. Kang, A. Facchetti, P. Zhu, H. Jiang, Y. Yang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, and T. J. Marks, “Exceptional molecular hyperpolarizabilities in twisted π-electron system chromophores,” Angew. Chem., Int. Ed. 44, 7922–7925 (2005).
[Crossref]

Appl. Phys. Lett. (1)

J. C. May, I. Biaggio, F. Bures, and F. Diederich, “Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules,” Appl. Phys. Lett. 90, 251106 (2007).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

M. G. Kuzyk, “Quantum limits of the hyper-Rayleigh scattering susceptibility,” IEEE J. Sel. Top. Quantum Electron. 7, 774–780 (2001).
[Crossref]

J. Am. Chem. Soc. (2)

H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted π-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129, 3267–3286 (2007).
[Crossref] [PubMed]

J. Pérez-Moreno, Y. Zhao, K. Clays, M. G. Kuzyk, Y. Shen, L. Qiu, J. Hao, and K. Guo, “Modulated conjugation as a means of improving the intrinsic hyperpolarizability,” J. Am. Chem. Soc. 131, 5084–5093 (2009).
[Crossref] [PubMed]

J. Chem. Phys. (5)

M. G. Kuzyk and D. S. Watkins, “The effects of geometry on the hyperpolarizability,” J. Chem. Phys. 124, 244104 (2006).
[Crossref] [PubMed]

D. S. Watkins and M. G. Kuzyk, “Optimizing the hyperpolarizability tensor using external electromagnetic fields and nuclear placement,” J. Chem. Phys. 131, 064110 (2009).
[Crossref] [PubMed]

A. D. Slepkov, F. A. Hegmann, S. Eisler, E. Elliot, and R. R. Tykwinski, “The surprising nonlinear optical properties of conjugated polyyne oligomers,” J. Chem. Phys. 120, 6807–6810 (2004).
[Crossref] [PubMed]

K. Tripathy, J. Pérez Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. M. Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932–7945 (2004).
[Crossref] [PubMed]

J. Pérez-Moreno, K. Clays, and M. G. Kuzyk, “A new dipole-free sum-over-states expression for the second hyperpolarizability,” J. Chem. Phys. 128, 084109 (2008).
[Crossref] [PubMed]

J. Mater. Chem. (1)

M. G. Kuzyk, “Using fundamental principles to understand and optimize nonlinear-optical materials,” J. Mater. Chem. 19, 7444–7465 (2009).
[Crossref]

J. Nonlinear Opt. Phys. Mater. (1)

M. G. Kuzyk, “Truncated sum rules and their use in calculating fundamental limits of nonlinear susceptibilities,” J. Nonlinear Opt. Phys. Mater. 15, 77–87 (2006).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. Chem. C (2)

X. Hu, D. Xiao, S. Keinan, I. Asselberghs, M. J. Therien, K. Clays, W. Yang, and D. N. Beratan, “Predicting the frequency dispersion of electronic hyperpolarizabilities on the basis of absorption data and Thomas–Kuhn sum rules,” J. Phys. Chem. C 114, 2349–2359 (2010).
[Crossref]

J. Zhou and M. G. Kuzyk, “Intrinsic hyperpolarizabilities as a figure of merit for electro-optic molecules,” J. Phys. Chem. C 112, 7978–7982 (2008).
[Crossref]

Nonlinear Opt., Quantum Opt. (1)

M. G. Kuzyk, “A bird’s-eye view of nonlinear-optical processes: Unification through scale invariance,” Nonlinear Opt., Quantum Opt. 40, 1–13 (2010).

Opt. Lett. (5)

Phys. Rev. A (4)

J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76, 053831 (2007).
[Crossref]

S. P. Goldman and G. W. F. Drake, “Relativistic sum rules and integral properties of the Dirac equation,” Phys. Rev. A 25, 2877–2881 (1982).
[Crossref]

P. T. Leung and M. L. Rustgi, “Relativistic corrections to Bethe sum rule,” Phys. Rev. A 33, 2827–2829 (1986).
[Crossref] [PubMed]

C. W. Dirk and M. G. Kuzyk, “Missing-state analysis: A method for determining the origin of molecular nonlinear optical properties,” Phys. Rev. A 39, 1219–1226 (1989).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

M. G. Kuzyk, “Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 85, 1218–1221 (2000).
[Crossref] [PubMed]

M. G. Kuzyk, “Erratum: Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 90, 039902 (2003).
[Crossref]

Other (1)

J. J. Sakurai, Modern Quantum Mechanics—Revised Edition (Addison Wesley Longman, 1994).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1 (a) Distribution of simulated intrinsic second hyperpolarizabilities γ int for a 15-state model and weight factors of 0.75, 1, and 1.25. The inset shows histograms of γ int near the limits. The bin size for the main figure is 0.003; and, for Insets (b) and (c), the bin size is 0.05. The error bars are defined as the square root of the frequency.
Fig. 2
Fig. 2 (a) Histogram of γ int for 5-, 15-, and 40-state models. (b) Enlarged histogram for γ int [ 0.75 , 1 ] . The bin size is 0.05.
Fig. 3
Fig. 3 The distribution of γ int versus E. For large values of the second hyperpolarizability, the majority of E values lie in the range [0,0.1].
Fig. 4
Fig. 4 Distribution of E for different ranges of γ int for a ten-state model with 100,000 iterations. Each symbol corresponds to the specified range of γ int .
Fig. 5
Fig. 5 γ int as a function of X for second hyperpolarizabilities within 5% of the limit.
Fig. 6
Fig. 6 Distribution of X for different ranges of γ int .
Fig. 7
Fig. 7 A logarithmic plot of | γ int ( i , j ) | using the missing state analysis for a ten-state model and γ int = 0.9886 . We remark that | γ int ( i , j ) | = | γ int ( j , i ) | so that the two smallest values in the diagram are identical and represent only one pair of states.

Equations (37)

Equations on this page are rendered with MathJax. Learn more.

β max = 3 1 / 4 ( e m ) 3 ( N 3 / 2 E 10 7 / 2 ) ,
γ DF = 1 8 ( 2 n m n l n { ( 2 E m 0 E n 0 ) ( 2 E l 0 E n 0 ) E n 0 5 ( 2 E l 0 E n 0 ) E m 0 E n 0 3 } x 0 m x m n x n l x l 0 + 2 n m n l m { 1 E l 0 E m 0 E n 0 ( 2 E l 0 E m 0 ) E m 0 3 E n 0 } x 0 l x l m x m n x n 0 m n { 1 E m 0 2 E n 0 + 1 E n 0 2 E m 0 } x 0 m 2 x 0 n 2 ) ,
H ( p , r ) = p 2 2 m + V ( r ) .
n = 0 [ E n 1 2 ( E m + E p ) ] x m n x n p = 2 N 2 m δ m p ,
n = 0 [ e n 1 2 ( e m + e p ) ] ξ m n ξ n p = δ m p ,
ξ i j = x i j | x 01 max | ,
| x 01 max | 2 = 2 N 2 m E 10 .
e 4 4 m 2 ( N 2 E 10 5 ) γ 4 e 4 4 m 2 ( N 2 E 10 5 ) .
γ max = 4 e 4 4 m 2 ( N 2 E 10 5 ) ,
1 4 γ int 1 ,
e 10 | ξ 10 | 2 + e 20 | ξ 20 | 2 + e 30 | ξ 30 | 2 + + e n 0 | ξ n 0 | 2 = 1.
n = 2 e n 0 ξ n 0 2 = 1 e 10 ξ 10 2 ,
ξ 20 2 ( 1 e 1 ξ 10 2 ) / e 20 .
ξ 20 = r ( 1 e 1 ξ 10 2 ) / e 20 .
ξ 10 , 0 = ( 1 e 1 ξ 10 2 e 2 ξ 20 2 e 9 ξ 90 2 ) / e 10 , 0 .
ξ i j = ξ j i .
n = 0 e n p | ξ n p | 2 = 1.
e 10 | ξ 10 | 2 + e 20 | ξ 20 | 2 + e 30 | ξ 30 | 2 + + e n 0 | ξ n 0 | 2 = 1.
e i 0 | ξ i 0 | 2 1 , 0 < i n ,
( 1 e i 0 ) 1 / 2 ξ i 0 ( 1 e i 0 ) 1 / 2 .
1 ξ i 0 1   for 0 < i n .
e 01 | ξ 01 | 2 + e 21 | ξ 21 | 2 + e 31 | ξ 31 | 2 + = 1 ,
e 21 | ξ 21 | 2 + e 31 | ξ 31 | 2 + + e n 1 | ξ n 1 | 2 = 1 + e 10 | ξ 01 | 2 .
e 21 | ξ 21 | 2 + e 31 | ξ 31 | 2 + + e n 1 | ξ n 1 | 2 2.
e i 1 | ξ i 1 | 2 2 , 1 < i n ,
( 2 e i 1 ) 1 / 2 ξ i 1 ( 2 e i 1 ) 1 / 2   for 1 < i n .
2 p / 2 e i p 1 / 2 ξ i , p 2 p / 2 e i p 1 / 2   for p < i n .
E i E i f .
E = e 1 e 2 = E 10 E 20 ,
X = ξ 10 = x 10 x 10 max ,
x 0 n max = 2 N 2 m E n 0 ,
lim E 0 γ int = 1.
ln ( F / F 0 ) = ( E / E 0 ) n ,
β int = n , m β int n , m ,
β int n , m = ( 3 4 ) 3 / 4 ξ 0 n ξ n m ξ m 0 ( 1 e n e m 2 e m e n e n 3 ) .
γ int = i j k γ int i j k = i j k γ int i j k + i = j k γ int i j k + i j = k γ int i j k + i = k j γ int i j k + i = j = k γ int i j k .
γ int symmetrized ( i , j ) = γ int missing ( i , j ) + γ int missing ( j , i ) ,

Metrics