Symmetric Top State

Origin
+ BN(N+1) + (C-B)K²
+ (-2Cζ + η_J*N² + η_K*K²) lK
- D_JN²(N+1)² - D_JKJ(J+1)K² - D_KK⁴
+ H_JN³(N+1)³ + H_JKN²(N+1)²K² + H_KJN(N+1)K⁴ + H_KK⁶
+ L_JN⁴(N+1)⁴ + L_JJKN³(N+1)³K² + L_JKN²(N+1)²K⁴ + L_KKJN(N+1)K⁶ + L_KK⁸
+ ½ε_aa(N₊S_-+N_-S₊) +  ε_ccN_zS_z
+ D^s_NN²N.S + ½D^s_NK(N²N_zS_z+N_zS_zN²) + D^s_KNN.SN_z²
+ α(3S_z²-S²) + ½β(S₊²-S_-²)
+ a_effl_zS_z

<J, K+2, l+1|H|J, K, l-1> = ½ (q₊ + D_qJJ(J+1) + D_qK(K²+(K+2)²)) [ (J(J+1) - K(K-1)) (J(J+1) - K(K+1)) ]^½
<J, K+2, l-1|H|J, K, l+1> = ½ q_- [ (J(J+1) - K(K-1)) (J(J+1) - K(K+1)) ]^½

<J, K+1, l-1|H|J, K, l+1> = (r + D_rJJ(J+1) + D_rK(K²+(K+1)²)) (2K+1) (J(J+1) - K(K+1))^½

A note on the sign convention for these constants is appropriate. This is discussed in a G. J.Cartwright and I. M. Mills, J. Molec. Spectrosc.34, 415 (1970), though that they define their constants, q⁽⁺⁾ and q^(-),  in a way to factor out the expected vibrational dependence, specifically:

The constant ρ defines the sign convention, and Cartwright and Mills suggest a value of -1 should be used for this. The sign convention can also be defined by looking at the spitting of the l = K = ±1 levels. Note that equation (5) of this paper is slightly unclear; the magnitude of the spitting between these otherwise degenerate levels is q₊J(J+1) with the PGOPHER definition of q rather then the Cartwright and Mills definition. The PGOPHER definition means a positive value of q₊ puts the even J, K=1 A₁ rotational level below the A₂ level, whereas a positive value of the Cartwright and Mills constant, q⁽⁺⁾, gives the opposite order.

l doubling is also observed in degenerate electronic states of symmetric top molecules; while the physical origin is different, the J and K dependence of the matrix elements is the same so PGOPHER can be used for such cases also. The classic example is the B state of NH₃ (see for example M. N. R. Ashfold, R. N. Dixon, N. Little, R. J. Stickland and C. M. Western, J. Chem. Phys., 89, 1754 (1988)) with the published q values being simply -q₊ as used by PGOPHER.