# How to obtain spectroscopic constants Omega_e, Alpha_e, B_e, D_e and D_0 from centrifugal constants?

I need to compute a spectroscopic constants of $$N_2^+$$ states and I would like to do it using LEVEL16 software, due to its high interpolation flexibility.

But it seems, that LEVEL16 can only compute centrifugal distortion constants directly, while I need even the others like $$B_e, \omega_e, \alpha_e$$ etc.

So, my current LEVEL16 output looks like this (full output is attached in the very bottom of this question):

Matrix element arguments are powers of the distance  r (in Angstroms)
Coefficients of expansion for radial matrix element/expectation value argument:
1.000000D+00 -2.000000D-01  3.000000D-02 -4.000000D-03
Find  4  potential minima:   Vmin=************************************************
at mesh points   R =    1.07800    3.58200    3.72200    4.72500
Find  4  potential maxima:   Vmax=************************************************
at mesh points   R =    3.54100    3.62900    3.75900   10.00000
E(v=  0, J=  0)=**********   Bv=  2.0436791  -Dv= -5.7294D-06   Hv= -2.9963D-11
Lv= -5.2792D-16   Mv=  2.2663D-20   Nv=  4.3228D-25   Ov= -5.1173D-30
E(v=  0, J=  0)=***********   <M(r)>=  0.8129565017       <KE>=    613.747
<X** 1>=   1.08679787   <X** 2>=   1.18212033   <X** 3>=   1.28688369


Now, I know, that

$$G(v) \approx \omega_e\left(v + \frac{1}{2} \right) + \omega_e x_e\left(v + \frac{1}{2} \right)^2 + \omega_e y_e\left(v + \frac{1}{2}\right)^3 + \omega_e z_e\left(v + \frac{1}{2} \right)^4$$

$$B_v \approx B_e - \alpha_e\left(v + \frac{1}{2}\right) + \gamma_e\left(v + \frac{1}{2}\right)^2$$

$$D_v \approx D_e + \beta_e\left(v + \frac{1}{2}\right)$$

but I don't see, how can I achieve the active constants here? Let's take $$D_e = D_0 + G(v)$$ as an example - if I had an energy of a lowest vibrational state, I'd be able to at least determine the dissociation energy, but I don't see $$G(0)$$ anywhere...

So, is there any parameter in LEVEL16 I've overlooked? Or did I just forget about some principle?

### Complete LEVEL16 output

  Reduced masses below are based on atoms 1 & 2 with charges (+0/2) and (+2/2),
respectively, with subtraction/addition of 0 and 2 half-electron masses.

Computation of N2+ (B2 SigmaU+) spectroscopic constants
================================================================================
Generate   ZMU=  7.00126270151(u)   &   BZ= 4.153171609D-01((1/cm-1)(1/Ang**2))
from atomic masses:  14.00307400443  &   14.00197684461(u)
Since the molecule is an ion with charge +1
use Watson's charge-adjusted reduced mass   mu = M1*M2/[M1 + M2 - (+1)*me]
Integrate from  RMIN=  0.800  to  RMAX=  10.00  with mesh  RH= 0.001000(Angst)

Potential-1 for  N( 14)- N( 14)
================================
State has  OMEGA= 0   and energy asymptote:   Y(lim)=************(cm-1)
Perform cubic spline interpolation over the   84 input points
To make input points Y(i) consistent with  Y(lim),  add  Y(shift)=      0.0000
Scale input points:  (distance)* 1.000000000D+00  &  (energy)* 8.065600000D+03
to get required internal units  [Angstroms & cm-1 for potentials]
r(i)         Y(i)        r(i)         Y(i)        r(i)         Y(i)
----------------------   ----------------------   ----------------------
0.800000*************    2.200000*************    3.650000*************
0.850000*************    2.250000*************    3.700000*************
0.900000*************    2.300000*************    3.750000*************
0.950000*************    2.400000*************    3.800000*************
1.000000*************    2.450000*************    3.850000*************
1.050000*************    2.500000*************    3.900000*************
1.100000*************    2.550000*************    3.950000*************
1.150000*************    2.600000*************    4.000000*************
1.200000*************    2.650000*************    4.050000*************
1.250000*************    2.700000*************    4.100000*************
1.300000*************    2.750000*************    4.150000*************
1.350000*************    2.800000*************    4.200000*************
1.400000*************    2.850000*************    4.250000*************
1.450000*************    2.900000*************    4.300000*************
1.500000*************    2.950000*************    4.350000*************
1.550000*************    3.000000*************    4.400000*************
1.600000*************    3.050000*************    4.450000*************
1.650000*************    3.100000*************    4.500000*************
1.700000*************    3.150000*************    4.550000*************
1.750000*************    3.200000*************    4.600000*************
1.800000*************    3.250000*************    4.650000*************
1.850000*************    3.300000*************    4.700000*************
1.900000*************    3.350000*************    4.750000*************
1.950000*************    3.400000*************    4.800000*************
2.000000*************    3.450000*************    4.850000*************
2.050000*************    3.500000*************    4.900000*************
2.100000*************    3.550000*************    4.950000*************
2.150000*************    3.600000*************    5.000000*************
----------------------------------------------------------------------------
Extrapolate to   X .le. 0.8500  with
Y=-23872936.002  +1.108309D+08 * exp(-8.496962D+00*X)
Extrapolate to  X .GE.  4.9500  using
Y=************ - [ 7.145852D+02/X**1 -3.206362D+03/X**3]
----------------------------------------------------------------------------

Calculate properties of the single potential described above
Potential-1 uses inner boundary condition of  zero value  at  RMIN

Eigenvalue convergence criterion is   EPS= 1.0D-06(cm-1)
Airy function at 3-rd turning point is quasibound outer boundary condition

Since state-1 has (projected) electronic angular momentum  OMEGA= 0
eigenvalue calculations use centrifugal potential  [J*(J+1) - 0]/r**2

For  J=  0, seek the first   1 levels of Potential-1   with   VLIM=***********

Matrix element arguments are powers of the distance  r (in Angstroms)
Coefficients of expansion for radial matrix element/expectation value argument:
1.000000D+00 -2.000000D-01  3.000000D-02 -4.000000D-03
Find  4  potential minima:   Vmin=************************************************
at mesh points   R =    1.07800    3.58200    3.72200    4.72500
Find  4  potential maxima:   Vmax=************************************************
at mesh points   R =    3.54100    3.62900    3.75900   10.00000
E(v=  0, J=  0)=**********   Bv=  2.0436791  -Dv= -5.7294D-06   Hv= -2.9963D-11
Lv= -5.2792D-16   Mv=  2.2663D-20   Nv=  4.3228D-25   Ov= -5.1173D-30
E(v=  0, J=  0)=***********   <M(r)>=  0.8129565017       <KE>=    613.747
<X** 1>=   1.08679787   <X** 2>=   1.18212033   <X** 3>=   1.28688369
-------------------------------------------------------------------------------
===============================================================================