p：q共振的n自由度规范化振动哈密顿量的构造与计算
Construction and counting of the number of operators included in a normalized vibrational Hamiltonian with n degrees of freedom with a p:q resonance
摘要：
本文提出了一种构造具有$n$自由度的高激发分子系统在$p:q$共振情况下的归一化振动哈密顿量的方法。我们还提出了所有的独立算子的计数和所有的参数包括在哈密顿量（计数定理1至8）的计数。该方法以系统的方式引入了所有的算子，特别是耦合算子，这些算子可以从李代数的生成元的幂的乘积形成的多项式中构建：在经典力学中从伴随算子$ad_{\mathcal{H}_{0}}$的核$Ker \,ad_{\mathcal{H}_{0}}$中构建的不变多项式的代数（参见[6]或[4]，[5]）。应用到非线性三原子分子ClOH，考虑到费米共振之间的O-Cl伸缩振子和弯曲运动。在[2]中已经实现了在高激发振动态（直到几乎解离极限）下对该分子系统的研究，拟合了725个能级。在我们计算的86个系数（其中31个耦合系数）上，与[2]完全兼容，最小均方根值导致仅保留28个非零系数。在附录中，我们解释了词汇和策略，以证明定理的耦合运营商包括在哈密顿量。
英文摘要：
We propose a method of construction of a normalized vibrational Hamiltonian of a highly excited molecular system with $n$ degrees of freedom in the case of a a $p:q$ resonance. We present also the counting of all the independent operators and the counting of all the parameters included in the Hamiltonian (Counting theorems 1 to 8). The method introduces, on a systematic way, all the operators, in particular the coupling operators, that can be built from the polynomials formed by products of powers of the generators of a Lie algebra: the algebra of the invariant polynomials built in classical mechanics from the the kernel $Ker \,ad_{\mathcal{H}_{0}}$ of the adjoint operator $ad_{\mathcal{H}_{0}}$ (see [6] or [4],[5]). Application to the non-linear triatomic molecule ClOH is then given, taking into account the Fermi resonance between the O-Cl stretching oscillators and the bending motion. The study of this molecular system in highly excited vibrational states (until almost the dissociation limit) has been realized in [2], with a fit of 725 levels of energy. On the 86 coefficients (among which 31 coupling coefficients) that we count, and completely compatible with [2], the smallest rms value leads to keep only 28 non-zero coefficients. In the appendix, we explain the vocabulary and the strategy employed in order to demonstrate the theorems of coupling operators included in the Hamiltonian.
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