Modified WKB theory for calculating tunneling splittings in symmetric multi-well systems in full dimensionality is re-derived using Cartesian coordinates. It is explicitly shown that the theory rests on the wavefunction that is exact for harmonic potentials. The theory was applied to calculate tunneling splittings in vinyl radical and some of its deuterated isotopologues in their vibrational ground states and the low-lying vibrationally excited states and compared to exact variational results. The exact results are reproduced within a factor of 2 in most states. Remarkably, all large enhancements of tunneling splittings relative to the ground state, up to three orders in magnitude in some excited mode combinations, are well reproduced. It is also shown that in the asymmetrically deuterated vinyl radical, the theory correctly predicts the states that are localized in a single well and the delocalized tunneling states. Modified WKB theory on the minimum action path is computationally inexpensive and can also be applied without modification to much larger systems in full dimensionality; the results of this test case serve to give insight into the expected accuracy of the method.

中文翻译：

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在笛卡尔坐标系中使用改进的 WKB 方法进行隧道分裂：乙烯基自由基的测试用例

使用笛卡尔坐标重新导出了用于计算全维对称多井系统中隧道分裂的修正 WKB 理论。明确表明该理论依赖于谐波势的精确波函数。该理论被应用于计算乙烯基自由基及其一些氘化同位素同位素体在振动基态和低位振动激发态下的隧道分裂，并与精确的变分结果进行比较。在大多数州，精确结果的再现精度为 2 倍。值得注意的是，隧道分裂相对于基态的所有大幅增强（在某些激发模式组合中高达三个数量级）都得到了很好的再现。还表明，在不对称氘代乙烯基自由基中，该理论正确预测了单井中的局域态和离域隧道态。最小作用路径的修正 WKB 理论计算成本低廉，并且无需修改也可以应用于更大的全维系统；该测试用例的结果有助于深入了解该方法的预期准确性。