Analytic energy gradients of constrained nuclear–electronic orbital second-order Møller–Plesset perturbation theory with electron–electron correlation

Nuclear quantum effects are essential for accurately describing hydrogen-containing systems, yet their incorporation into correlated electronic structure methods remains challenging. In this work, we develop a practical constrained nuclear–electronic orbital second-order Møller–Plesset perturbation theory with electron–electron correlation, denoted CNEO-MP2(ee), together with its analytic energy gradients. This method combines the CNEO description of quantum nuclei with an MP2-level treatment of electron correlation, enabling efficient geometry optimization and vibrational analysis on correlated CNEO energy surfaces. Benchmark calculations show that CNEO-MP2(ee) substantially improves conventional MP2 for isolated X–H stretching frequencies and shared-proton transfer modes, demonstrating the importance of nuclear quantum effects in these systems. For bound X–H stretches in hydrogen-bonded complexes, however, the CNEO-induced red shift can be excessive, indicating that a balanced treatment of electron correlation, electron–nuclear correlation, and anharmonicity remains important. Overall, CNEO-MP2(ee) provides an efficient first step toward correlated CNEO methods for molecular systems with significant nuclear quantum effects.