Vibrational spectroscopic maps are a series of ab initio, semiempirical, or empirical models tailored to specific IR probes to describe vibrational solvatochromic effects on molecular spectra quantitatively. Coherent multidimensional spectroscopy, a nonlinear spectroscopy utilizing multiple time-delayed pulses, is a technique that enables the measurement of solvation-induced frequency shifts and the time-correlations of the fluctuating frequencies. Researchers employ various organic and biochemical methods to introduce small vibrational probes into molecular systems into a variety of chemicals, proteins, nucleic acids, etc. These probes, labeled with infrared (IR) markers, were subject to spectroscopic investigations to obtain quantitative insights into various features of chemical and biological systems. In general, interpreting the experimental multidimensional spectra to get information on the underlying molecular processes requires theoretical modeling. The vibrational frequency shifts observed due to complex intermolecular interactions of small IR probes with surroundings in the condensed phase are minute, often representing fractions of thermal energy. Numerical accuracy associated with advanced quantum mechanical calculations are not sufficient to accurately model these shifts. Consequently, researchers commonly resort to mapping procedures, which correlate certain physical variables calculated for the probe molecule with spectroscopic properties such as vibrational frequencies. These mapping procedures are referred to as vibrational spectroscopic maps within the field. Typically, the physical variables employed in vibrational frequency maps include electric potentials, electric fields, distributed higher multipole moments, and other relevant factors evaluated at specific points surrounding the molecule. As an example, the vibrational frequency associated with a localized vibrational mode is correlated with the electrostatic potential and electric field values at a designated set of points known as distributed sites within the infrared (IR) chromophore.
Theoretical foundation The vibrational frequency shift, denoted as Δ ω j {\displaystyle \Delta \omega _{j}} , for the jth normal mode of a given probe molecule is defined as the difference between the actual vibrational frequency ω j {\displaystyle \omega _{j}} of the mode in a solution and the frequency ω j , 0 {\displaystyle \omega _{j,0}} in the gas phase.
Δ ω ≡ ω j − ω j , 0 {\displaystyle \Delta \omega \equiv \omega _{j}-\omega _{j,0}}
From an effective Hamiltonian for the solute in the presence of molecular environment, one can derive the effective vibrational force constant (or Hessian) matrix approximately as follows:
k j k ≈ M j ω j 2 δ j k + ∂ 2 U ( Q ) ∂ Q j ∂ Q k | 0 − ∑ i q i j k M i ω i 2 ∂ U ( Q ) ∂ Q i | 0 {\displaystyle k_{jk}\approx M_{j}\omega _{j}^{2}\delta _{jk}+{\frac {\partial ^{2}U(\mathbf {Q} )}{\partial Q_{j}\partial Q_{k}}}{\bigg \vert }_{0}-\sum _{i}{\frac {q_{ijk}}{M_{i}\omega _{i}^{2}}}{\frac {\partial U(Q)}{\partial Q_{i}}}{\bigg \vert }_{0}}
where the subscript 0 means the quantity is evaluated at the gas-phase geometry. In the limiting case that the vibrational couplings of the normal mode of interest with other vibrational modes are relatively weak, the vibrational frequency shift under such a weak coupling approximation (WCA) in solution from the gas-phase frequency is given by
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