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
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 by4, 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 , 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 . 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) , (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)  and the effects of an externally applied electromagnetic field .
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 . The three-level ansatz states that when the hyperpolarizability of a quantum system is at the fundamental limit, only three states contribute ; 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 for a typical Hamiltonian. However, it may be possible that 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 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  and for understanding length-scaling of the second hyperpolarizability . 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.
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 tensor component of the off-resonant dipole-free SOS expression of the second hyperpolarizability is given by 2) holds for a three-dimensional molecule where we have chosen the coordinate system such that 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, , where H is a Hamiltonian of the form3) are given by4) represents an infinite number of equations indexed by the integers . In the text that follows, we refer to a particular sum rule by stating the values . Note that Eq. (4) holds for a broader set of multi-electron Hamiltonians including spin, externally applied electromagnetic fields, and electron correlations . 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 form7, 4) imply that the largest possible transition moment from the ground state is because for .
The three-level ansatz is central to calculating the fundamental limit of the second hyperpolarizability γ . 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 23]. Hence, when γ approaches , approaches unity.
The numerical procedure is the same as in previous studies of the hyperpolarizability β : For a specific number of states s, the energy values are picked arbitrarily and sorted in ascending order so that , where is the energy of the ground state and is the energy of the excited state. For simplicity and with no loss in generality, we shift the energies so that the ground state energy is zero. Since , then and .
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 . 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 given by Eq. (2), we need to only use the sum rules with in Eq. (5) to get all of the required transition moments. Starting with , we get13),5). Since is assumed to be real for all states i and j,2) to calculate . The normalized form is obtained by dividing Eq. (2) by Eq. (9) such that in Eq. (2), , , and . 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, . Thus, the diagonal sum rules give no information about the sign of . 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 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 , 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 ,17) yields18) implies that the largest value of is attained when all other terms of the type (for ) are zero, i.e.,5) results in19) into Eq. (23) gives an upper limit27). 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 with the average energy spacing and transition moments when 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
Figure 1a shows the distribution of the second hyperpolarizability for a 15-state model for weight factors of . Frequency refers to the number of times appears in the histogram. All simulated values lie in the range predicted by Eq. (10). The maximum values of 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 . On the other hand, the smallest 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 , the Monte Carlo method yields values of that approach the limit. We find that the same is true for . The distribution of 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 near the limits, we have plotted the histogram of 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 ) 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 of1]. The three-level ansatz and the fact that when the hyperpolarizability is large are found to be a universal property of all systems studied to date . While the fundamental limit of the off-resonant second hyperpolarizability is calculated using the three-level ansatz , there is no extensive body of literature that supports its validity. As we show below, the Monte Carlo simulations show that when , 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 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 that3 plots as a function of E for a five-state model and iterations. When the intrinsic second hyperpolarizability approaches unity, the energy spacing most often falls in the range , suggesting that is the largest when , 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 , 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 [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
Note that all of the curves appear to be approximately parallel to each other with the exception of the curves for 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 near unity or near −0, 25, the positive and negative limits.
Figure 5 shows 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 approaches the limit, the range of X becomes narrower so that for , X lies in the range .
To better quantify the results, Fig. 6 shows the distribution of X for various ranges of using a ten-level model. The bin size is 0.05 and has been divided into 12 equally spaced ranges except for . is dominated by systems with . For slightly smaller negative values of , the peak in the distribution is less than unity and the rest of the curves peak at .
While any value of is attainable for any arbitrary choice of X, it is clear that the most likely values for the dominant transition moment are when and when . Recall that Fig. 2 implies that the largest values of are for . Interestingly, the three-level ansatz shows that when , yields and yields . 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 can expressed as 17],
Expressing 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 as the fractional contribution of the three states i, j, and k:
The simplest approach for determining the contribution of pairs of states is the missing state analysis , where one calculates in the absence of a particular set of states denoted by . A comparison of with describes the joint contribution of states i and j to the intrinsic second hyperpolarizability. The smaller the value of , the larger the contribution of states i and j to .
Figure 7 shows a logarithmic plot of the absolute value of for a ten-state model with , the largest valued obtained in a Monte Carlo simulation. The most important pair of states are 1 and 2 with . The next most important pair of states are 5 and 1 with . and are also on the order of . Thus, the dominant two states contribute about 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 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
Optimizing the intrinsic hyperpolarizabilities by varying the potential energy function results in a 30% gap between the fundamental limit and the optimized . Experimentally measured second hyperpolarizabilities also fall far short of the limit for a wide range of molecules, but Monte Carlo simulations show that 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 , in agreement with analytical calculations . 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 .
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 resulting when , and for negative γ and 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.
M. G. Kuzyk acknowledges the National Science Foundation (NSF) (ECCS-0756936) and Wright Paterson Air Force Base for generously supporting this work.
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