A time correlation function theory of two-dimensional infrared spectroscopy with applications to liquid water

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Abstract
A theory describing the third-order response function R (3) (t 1 ,t 2 ,t 3 ), which is associated with two-dimensional infrared (2DIR) spectroscopy, has been developed. R (3) can be written as sums and differences of four distinct quantum mechanical dipole (multi)time correlation functions (TCF’s), each with the same classical limit; the combination of TCF’s has a leading contribution of order ℏ 3 and thus there is no obvious classical limit that can be written in terms of a TCF. In order to calculate the response function in a form amenable to classical mechanical simulation techniques, it is rewritten approximately in terms of a single classical TCF, B R (t 1 ,t 2 ,t 3 )=〈μ j (t 2 +t 1 )μ i (t 3 +t 2 +t 1 )μ k (t 1 )μ l (0)〉, where the subscripts denote the Cartesian dipole directions. The response function is then given, in the frequency domain, as the Fourier transform of a classical TCF multiplied by frequency factors. This classical expression can then further be quantum corrected to approximate the true response function, although for low frequency spectroscopy no correction is needed. In the classical limit, R (3) becomes the sum of multidimensional time derivatives of B R (t 1 ,t 2 ,t 3 ). To construct the theory, the response function’s four TCF’s are rewritten in terms of a single TCF: first, two TCF’s are eliminated from R (3) using frequency domain detailed balance relationships, and next, two more are removed by relating the remaining TCF’s to each other within a harmonic oscillator approximation; the theory invokes a harmonic approximation only in relating the TCF’s and applications of theory involve fully anharmonic, atomistically detailed molecular dynamics (MD). Writing the response function as a single TCF thus yields a form amenable to calculation using classical MD methods along with a suitable spectroscopic model. To demonstrate the theory, the response function is obtained for liquid water with emphasis on the OH stretching portion of the spectrum. This approach to evaluating R (3) can easily be applied to chemically interesting systems currently being explored experimentally by 2DIR and to help understand the information content of the emerging multidimensional spectroscopy.