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Correlation-function expressions are derived for the coherent nonlinearresponse of molecules to three resonant ultrafast pulses in the x-ray regime.The ability to create two-core-hole states with controlled attosecond timing infour-wave-mixing and pump probe techniques should open up new windows into theresponse of valence electrons, which are not available from incoherent x-rayRaman and fluorescence techniques. Closed expressions for the necessaryfour-point correlation functions are derived for the electron-boson model byusing the second order cumulant expansion to describe the fluctuatingpotentials. The information obtained from multidimensional nonlinear techniquescould be used to test and refine this model, and establish an anharmonicoscillator picture for electronic excitations

Multiple core-hole coherence in x-ray four-wave-mixing spectroscopies