A , B , C
Considering the comparable magnitude of contributions of cubic anharmonicity in second order and quartic anharmonicity in first order.
χ r r = 1 16 ϕ r r r r − 1 16 <munderover> ∑ s = 1 3 N − 6 </munderover> ϕ r r s 2 8 ω r 2 − 3 ω s 2 ω s ( 4 ω r 2 − ω s 2 ) χ r s = 1 4 ϕ r r s s − 1 4 <munderover> ∑ t = 1 3 N − 6 </munderover> ϕ r r t ϕ t s s ω t − 1 2 <munderover> ∑ t = 1 3 N − 6 </munderover> 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 ] [ ω r ω s + 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 )
Anharmonicity Constant (Perturbation Theory)
anharmonicity constants, derived by using second order (non-degenerate) perturbation theory