A new dipole-free sum-over-states expression for the second hyperpolarizability

Abstract

A new dipole-free sum-over-states expression for the second hyperpolarizability The generalized Thomas-Kuhn sum rules are used to eliminate the explicit dependence on dipolar terms in the traditional sum-over-states (SOS) expression for the second hyperpolarizability to derive a new, yet equivalent, SOS expression. This new dipole-free expression may be better suited to study the second hyperpolarizability of non-dipolar systems such as quadrupolar, octupolar, and dodecapolar structures. The two expressions lead to the same fundamental limits of the offresonance second hyperpolarizability; and when applied to a particle in a box and a clipped harmonic oscillator, have the same frequency-dependence. We propose that the new dipole-free equation, when used in conjunction with the standard SOS expression, can be used to develop a three-state model of the dispersion of the third-order susceptibility that can be applied to molecules in cases where normally many more states would have been required. Furthermore, a comparison between the two expressions can be used as a convergence test of molecular orbital calculations when applied to the second hyperpolarizability. I. INTRODUCTIONThe sum-over-states (SOS) expressions have been used for more than three decades in the study of nonlinear optical phenomena, and are perhaps the most universally used equations in molecular nonlinear optics. The sum-over-states expression is obtained from quantum perturbation theory and is usually expressed in terms of the matrix elements of the dipole operator, −ex nm , and the zero-field energy eigenvalues, E n . [1,2,3] The SOS expressions for the first and second hyperpolarizability derived by Orr and Ward using the method of averages [2] are often used because they explicitly eliminate the unphysical secular terms that are present in other derivations.[1] These secular-free expressions contain summations over all excited states.Finite-state approximations are used to apply the theory to experimental results. Oudar and Chemla studied the first hyperpolarizability of nitroanilines by considering only two states, the ground and the dominant excited state. [4] Although the general validity of this "two-level" model has been questioned, especially in its use for extrapolating measurement results to zero frequency, the approximation is still widely used in experimental studies of the nonlinear properties of organic molecules.Several approaches have been used to develop approximate expressions for the second-hyperpolarizability in the off-resonance regime. [5,6,7] While such approximations are helpful, they systematically ignore some of the contributions to the SOS expression. As our goal is to derive a general expression that is equivalent to the traditional SOS one, we choose not to make any assumptions a priori about what type of contributions dominate the response. Furthermore, including all the possible contribution is necessary to properly describe the on-resonance behavior, even * Electronic address: Javier.PerezMoreno@fys.kuleuven.be ...

Experimental verification of a self-consistent theory of the first-, second-, and third-order (non)linear optical response

“…We use the dipole-free expression to calculate γ xxxx because it does not require knowledge of the ground and excited state dipole moments. [15] We do not present the dipolefree-expression here, but rather refer the reader to the literature.…”

“…The resulting algebraic expressions are too complex to present here, but their form can be found in the literature. [15] B. The Dipole Basis…”

“…[11,12,13] Indeed, the dipole-free SOS expression is derived by using the sum rules to eliminate all terms with dipole moment differences. Dipole-free expressions for both the hyperpolarizability, β, [14] and the second hyperpolarizability, γ, [15] have been derived and shown to be mathematically equivalent to the standard SOS results. These dipole-free expressions are key to significantly reducing the number of parameters required to model the nonlinear response.…”

Modeling off-resonant nonlinear-optical cascading in mesoscopic thin films and guest-host molecular systems

“…[29][30][31] The values for the oscillator strengths and their corresponding transition energies were previously reported by Leach, et al [32,33] Truncating the (hyper)polarizabilities to only three states gives k (1) = 1.85 × 10 −23 cm 3 and k (3) = 3.41 × 10…”

Classical model of the upper bounds of the cascading contribution to the second hyperpolarizability

“…[29][30][31] Here, E i0 = E i − E 0 is the energy difference between the i th excited state and the ground state, x ij is the transition moment between state i and state j, and e is the electron charge. The primed sums in Equations 22-24 exclude the ground state from the summation.…”