Anharmonicity Constant (Perturbation Theory)

Available identifiers

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

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

Defining Formula: $\begin{matrix} χ_{r r} & = & \frac{1}{16} ϕ_{r r r r} - \frac{1}{16} \sum_{s = 1}^{3 N - 6} ϕ_{r r s}^{2} \frac{8 ω_{r}^{2} - 3 ω_{s}^{2}}{ω_{s} (4 ω_{r}^{2} - ω_{s}^{2})} \\ χ_{r s} & = & \frac{1}{4} ϕ_{r r s s} - \frac{1}{4} \sum_{t = 1}^{3 N - 6} \frac{ϕ_{r r t} ϕ_{t s s}}{ω_{t}} - \frac{1}{2} \sum_{t = 1}^{3 N - 6} \frac{ϕ_{r s t}^{2} ω_{t} (ω_{t}^{2} - ω_{r}^{2} - ω_{s}^{2})}{Δ_{r s t}} \\ + & [A (ζ_{r, s}^{(a)})^{2} + B (ζ_{r, s}^{(b)})^{2} + C (ζ_{r, s}^{(c)})^{2}] [\frac{ω_{r}}{ω_{s}} + \frac{ω_{s}}{ω_{r}}] \\ Δ_{r s t} & = & (ω_{r} + ω_{s} + ω_{t}) (ω_{r} - ω_{s} - ω_{t}) (- ω_{r} + ω_{s} - ω_{t}) (- ω_{r} - ω_{s} + ω_{t}) \end{matrix}$
$A, B, C$ symbol represents Rotational Constant
$N$ symbol represents Number of Particles
$χ$ symbol represents Anharmonicity Constant
$ω$ symbol represents Vibration Frequency (Harmonic)
$ϕ$ symbol represents Force Constant (Anharmonic)
$ζ$ symbol represents Coriolis Coupling Constant