Quantum mechanical model of the upper bounds of the cascading contribution to the second hyperpolarizability

Microscopic cascading of second-order nonlinearities between two molecules has been proposed to yield an enhanced third-order molecular nonlinear-optical response. In this contribution, we investigate the two-molecule cascaded second hyperpolarizability and show that it will never exceed the fundamental limit of a single molecule with the same number of electrons as the two-molecule system. We show the apparent divergence behavior of the cascading contribution to the second hyperpolarizability vanishes when properly taking into account the intermolecular interactions. Although cascading can never lead to a larger nonlinear-optical response than a single molecule, it provides alternative molecular design configurations for creating materials with large third-order susceptibilities that may be difficult to design into a single molecule.

Figure 1

The configuration of the molecules studied in this work. The two short black arrows represent the molecules while the long black arrows represent the applied electric field. The red (long dashed) and blue (short dashed) curves represent the electric field lines from each induced dipole.

Figure 2

The four-level quantum model of the second hyperpolarizability of two single molecules. When $r^{\prime }<\sqrt[3]{2}$, the two molecules overlap and the calculations break down.

Figure 3

A diagram illustrating the theoretical approach to cascading. (a) The relationship between ${\gamma }_{\mathrm{eff}}^{\left(0\right)}$ and the individual contributions. (b) The second hyperpolarizability of two molecules each with a set number of electrons with respect to the second hyperpolarizability of a single molecule with the same number of electrons. (c) A comparison between the interacting maximum, ${\gamma }_{\max }$, and the noninteracting case, ${\gamma }_{\max }^{\prime }$.

Figure 4

The cascading contribution of ${\beta }_{\mathrm{pert}}^2/r^{\prime 3}$. The perturbed Hamiltonian keeps the function from diverging as $r^{\prime }\to 0$.

Figure 5

The intrinsic effective second hyperpolarizability of two interacting molecules. The thick black line is the lower limit of the center-of-mass displacement given by the condition $r^{\prime }>\sqrt[3]{2}$. The thick red line is the experimental limit of closest approach for spherical atoms and molecules with large polarizabilities, $r^{\prime }>2$ [23]. The values that exceed the semiclassical limit have been removed from the diagram and are shown as the red plateau.

Figure 6

The top two graphs are the intrinsic hyperpolarizability and second hyperpolarizability as a function of $X$ and $E$ for a single-molecule system. The bottom four graphs show the intrinsic effective second hyperpolarizability of the aligned molecules with no energy corrections as a function of $X$ and $E$ for several $r^{\prime }$ values. The $\beta$ contribution decreases as the distance of separation increases.

Figure 7

The semiclassical intrinsic effective second hyperpolarizability of two end-to-end molecules. The thick black line is the minimum separation distance given by the electromagnetic size of the molecule, or $r^{\prime }>\sqrt[3]{2}$. The thick red line is the experimental limit of closest approach for spherical atoms which gives $r^{\prime }>2$ [23]. The red plateau is the region where ${\gamma }_{\mathrm{eff}}^{\left(0\right)}/{\gamma }_{\max }^{\prime }\le 1$.

Figure 8

The effective second hyperpolarizability of two interacting $p$NA molecules at a temperature of $297$K, where the separation vector between the two molecules is along the direction of the applied field. The molecules are freely rotating, where the alignment is a result of the applied electric field in units of StatV/cm The solid curves include interactions between polarized molecules and the dotted curves include only the effects of the applied electric field.

Figure 9

The contribution to the effective second hyperpolarizability of one freely rotating molecule that is coupled to a second molecule. The second molecule is held in both the direction perpendicular and the direction parallel to the applied field while the other is free to move. The temperature is $297$K and the poling field is ${10}^6$StatV/cm.