Anharmonicity Constant (Perturbation Theory) (Q6674132)

anharmonicity constants, derived by using second order (non-degenerate) perturbation theory

Language	Label	Description	Also known as
English	Anharmonicity Constant (Perturbation Theory)	anharmonicity constants, derived by using second order (non-degenerate) perturbation theory

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A, B, C

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N

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χ

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ω

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ϕ

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ζ

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Considering the comparable magnitude of contributions of cubic anharmonicity in second order and quartic anharmonicity in first order.

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\begin{matrix} χ_{r r} & = & \frac{1}{16} ϕ_{r r r r} - \frac{1}{16} \sum_{s = 1}^{3 N - 6} ϕ_{r r s}^{2} \frac{8 ω_{r}^{2} - 3 ω_{s}^{2}}{ω_{s} (4 ω_{r}^{2} - ω_{s}^{2})} \\ χ_{r s} & = & \frac{1}{4} ϕ_{r r s s} - \frac{1}{4} \sum_{t = 1}^{3 N - 6} \frac{ϕ_{r r t} ϕ_{t s s}}{ω_{t}} - \frac{1}{2} \sum_{t = 1}^{3 N - 6} \frac{ϕ_{r s t}^{2} ω_{t} (ω_{t}^{2} - ω_{r}^{2} - ω_{s}^{2})}{Δ_{r s t}} \\ + & [A (ζ_{r, s}^{(a)})^{2} + B (ζ_{r, s}^{(b)})^{2} + C (ζ_{r, s}^{(c)})^{2}] [\frac{ω_{r}}{ω_{s}} + \frac{ω_{s}}{ω_{r}}] \\ Δ_{r s t} & = & (ω_{r} + ω_{s} + ω_{t}) (ω_{r} - ω_{s} - ω_{t}) (- ω_{r} + ω_{s} - ω_{t}) (- ω_{r} - ω_{s} + ω_{t}) \end{matrix}

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