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Figure 1 shows the WMEL diagrams for TRIVE-FWM with two lasers of frequency *ω _{1}* and

*ω*. Note that the subscripts in this section label the frequencies of the excitation beams, not their time ordering. Three beams are created by splitting the

_{2}*ω*beam into two beams,

_{2}*ω*and

_{2}*ω*The three beams arecrossed at angles that provide phase matching, $\vec{k}_4 = \vec{k}_1 - \vec{k}_2 + \vec{k}_{2'}$. Scanning

_{2’}.*ω*and

_{1}*ω*while monitoring the output at $\omega_4 = \omega_1 - \omega_2 + \omega_{2'}$ with a monochromator creates a two-dimensional spectrum.

_{2}There are twelve TRIVE pathways that differ in the time ordering of the three infrared excitation pulses (the six columns) and the resonant states (the two rows). The pathways in the top row are parametric processes where the final output coherence emission returns the system to the original state before the nonlinear process so no energy remains in the system from the interactions. The pathways in the bottom row are non-parametric processes where the final output coherence emission returns the system to a vibrationally excited state so energy is delivered to the system. Unlike DOVE-FWM, all of the transitions can be allowed in the harmonic approximation. Two tunable infrared excitation pulses (frequencies labeled as *ω _{1}* and

*ω*) excite the vibrational coherences and then a third excitation (labeled

_{2}*ω*and typically in the uv/visible) excites the Raman transition between the two vibrational states. The three beams are crossed at angles that provide phase matching. Scanning

_{3}*ω*and

_{1}*ω*while monitoring the output at $\omega_4$ with a monochromator creates a two-dimensional spectrum.

_{2}Figure 4 shows an example spectrum for a mixture of a bis-(triphenylphosphine) dicarbonyl nickel and a (triphenylphosphine) tricarbonyl nickel complex in tetrahydrofuran. The sides of the figure show the infrared spectrum of the mixture. Diagonal peaks correspond to both *ω _{1}* and

*ω*exciting the same vibrational state while the cross-peaks correspond to

_{2}*ω*and

_{1}*ω*exciting two different vibrational states that are coupled. The boxes indicate the diagonal and cross-peaks associated with the Ni(CO)

_{2}_{2}(PPh

_{3})

_{2 }(left-most box) and Ni(CO)

_{3}(PPh

_{3})(right-most boxes). The peaks in the inset show that some peaks split because of the frequency domain manifestation of quantum beating that depends on the delay times between pulses.

One can also change the delay times between pulses in order to measure the dynamics of the coherences and populations. Changing the delay times can also change the time ordering of the pulses and thereby change the coherence pathway. Figure 5 shows a diagram of the relationship between the coherence pathways in figure 3 and the nonlinear processes that occur for different delay times $D_{2' 1} \equiv \tau_{2'} - \tau_1$ between the *ω _{1}* and

*ω*and $D_{21} \equiv \tau_2 - \tau_1$ between the

_{2’}*ω*and

_{1}*ω*excitation pulses. The figure also shows the relationship of these pathways to the other FWM spectroscopies. For example, the stimulated photon echo experiment corresponds to region V while the normal photon echo experiment corresponds to boundary between regions III and V.

_{2}An example of the experimental implementation of the delay scans is shown in figure 6. Here, the lasers and monochromator are fixed at positions appropriate for the lower right cross-peak in figure 4 and the $D_{2' 1}$ and $D_{21}$ delay times are scanned. The monochromator monitors a frequency that corresponds to the top row in figure 4. Since *ω _{2}* is detuned from the anharmonically shifted combination band transition, pathways II and IV are not important and one observes the dynamics from the other pathways. As an example, a diagonal cross-section of figure 6 in region I shows the dephasing rate of the

*v’g*coherence in pathway I (see figure 4) while a horizontal cross-section shows the dephasing rate of the

*v’v*coherence. Similarly, a diagonal cross-section in region V shows the population relaxation rate of the

*gg*and

*vv*populations. Similar dynamical data can be obtained by fixing the excitation frequencies and monochromator to emphasize other pathways.

The cross-peaks in these spectra also require coupling, just as they did for DOVE FWM. In the DOVE-FWM case, the coupling appeared as the intensity of the combination bands. For TRIVE FWM, the transitions are all allowed. The cross-peaks would vanish without couplingbecause the parametric and nonparametric pathways in figure 4 have opposite signs and cancel. Coupling creates anharmonic shifts in the frequency or changes in the transition moments or relaxation rates so the two interfering pathways are not equivalent and do not cancel.