A new form of the redfield equation for dissipative systems related to the matrix of correlation functions

In the Redfield theory framework, we consider the problem of the vibrational dynamics in dissipative systems. We decompose the Hamiltonian of interaction of the observed system with a thermal bath into terms that are products of system transition operators and bath transition operators. Using the decomposition, we construct a correlation function matrix containing all the information about the interaction of the system with the bath and obtain a new operator form of the Redfield equation. We consider the procedure for factoring the interaction operator and constructing the correlation function matrix. Using the diagonalization of the obtained matrix, we give correlated dissipation operators, whose introduction simplifies the form of the Redfield equations. We show that for several problems in which fundamental transition frequencies can be chosen, this procedure significantly reduces the dimensionality of the dissipative dynamics problem.