def eigen(energy, rot, mx, V, R, mu):
""" determine eigen energy solution based.
Parameters
----------
energy : float
energy (eV) of the attempted solution
rot : int
rotational quantum number
mx : int
matching point index, for inward and outward solutions
V : numpy 3d array
potential energy curve and coupling matrix
R : numpy 1d array
internuclear distance grid
mu : float
reduced mass in kg
Returns
-------
eigenvalue : float
energy of the solution
"""
WI = WImat(energy, rot, V, R, mu)
RI = RImat(WI, mx)
# | R_mx - R^-1_mx+1 |
return linalg.det(linalg.inv(RI[mx])-RI[mx+1])
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