# II.B.i Brownian Oscillator Model

**II.B.i Brownian Oscillator Model**

Here the Multi-mode Brownian Oscillator (MBO) theory is briefly outlined with the response function formalism for calculating the nonlinear optical signals.^{3,6,7} We assume the chromophore-solvent system can be modeled as a two-level electronic system (chromophore) linearly coupled to a near-continuous set of harmonic oscillators (solvent). This model, often referred to as the spin-boson or multi-mode Brownian oscillator (MBO) model, is a common approach for dealing with system-bath interactions and describes many of the salient features observed in nonlinear experiments.

Dynamical information about solvation dynamics can be expressed as a correlation function of the energy gap (transition frequency) fluctuations of the chromophore dissolved in condensed phase. In solutions, the optical transition frequency, w, of the *i*-th chromophore is not static, but fluctuates as the solvent moved around the solute. This fluctuation can be expressed as

(II.2)

where is the average transition frequency, is a static non-fluctuating spectral shift due to the chromophore’s surrounding solvent environment, and represents the fluctuating component. The average transition frequency is also the solvation energy, or stabilization energy, of the chromophore. The two additional terms in Equation II.2 represent the static and dynamics deviations from this average solvation energy. The static term is responsible for the distribution of chromophores energies around the mean. Within the linear response approximation, dynamics of solvation are captured in the fluctuating term. The absorption spectra of an ensemble of chromophores are time-averaged; consequentially both static and dynamic components will lead to broadening of the absorption spectrum. The static and dynamic contributions to the energy gap are often referred to as inhomogeneous and homogenous broadening respectively. The widths of absorption band in solution are then a result of a combination of a static environmental distribution with the dynamic fluctuating effects of the solvent. It is this convolution that obscures the direct observation of the solvent dynamics effects on electronic transitions with linear spectra.

The dynamics solvation are characterized by the transition frequency correlation function:

. (II.3)

Which under certain limits can be equated to the Stokes Shift Solvation function Equation I.3.^{6,8,9} The M(t) function is the central relaxation function used to describe solvent relaxation within the MBO model. The angled brackets represent an ensemble average.

It was shown previously that 3PEPS profiles directly reflect the transition frequency correlation function for times longer than the bath correlation time.^{9} Based on the transition frequency correlation function, we can calculate the linear absorption, fluorescence spectrum and third-order nonlinear optical signals such as TA, TG, and 3PEPS via the response function formalism developed by Mukamel and coworkers.^{3} The application of the MBO model allow both linear and non-linear signals to be calculated from the line broadening function g(t) for the system which can then be expressed in terms of the M(t) function:^{3}

. (II.4)

In previous applications of the MBO model, g(t) was defined as^{6}

. (II.5)

Where in both equations, l is the total reorganization energy and is the total system-bath coupling. Within this formalism, the Stokes shift is then given by 2l.

In the case of Equation II.5, the transition energy, w_{eg}, corresponds to the 0-0 transition of the system, where the average of the maxima between the absorption and fluorescence spectrum. The simulations presented here we use Equation II.4 for calculating the g(t) function. For this definition of g(t), the transition energy is expressed as

(II.6)

where w_{0-0} is the 0-0 transition frequency and l is the total reorganization energy including both intramolecular vibrational and solvation contribution. The central difference between Equations II.4 and II.5 is whether the reorganization energy, l, is explicitly included in the definition of w_{eg} or is accounted for in the calculation of g(t). Both formulae will give identical results.