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The behavior of polyatomic molecules around their equilibrium positions can be regarded as quantum coupled anharmonic oscillators. Solving the corresponding Schr\"odinger equations can interpret or predict experimental spectra of molecules. In this study, we develop a novel approach to solve excited states of anharmonic vibrational systems. The normal coordinates of molecules are transformed into new coordinates through a normalizing flow parameterized by a neural network, facilitating the construction of a set of orthogonal many-body variational wavefunctions. Our methodology has been validated on an exactly solvable $64$-dimensional coupled harmonic oscillator, yielding numerical results with a relative error on the order of $10^{-6}$. Furthermore, the neural canonical transformations are also applied to calculate the energy levels of two specific molecules, acetonitrile ($\text{C}\text{H}_3\text{CN}$) and ethylene oxide ($\text{C}_2\text{H}_4\text{O}$) involving $12$ and $15$ vibrational modes respectively, which are in agreement with experimental findings. One of the key advantages of our approach is its flexibility concerning the potential energy surface, requiring no specific form for its application. Moreover, this method can be easily implemented on large-scale distributed computing platforms, making it particularly suitable for investigating more complex vibrational structures. |
