Adding first-order relativistic corrections to the momentum naturally creates the fine-structure version of this operator, since it is equal to one over the twice the system mass times the inner product of the momentum operator with itself, plus the potential operator.  
The diagonal elements of the coupling matrix in an orthonormal n-energy-state system are fully perturbed matrix elements of this object.  Adding first-order relativistic corrections to the momentum naturally creates the fine-structure version of this operator, since it is equal to one over the twice the system mass times the inner product of the momentum operator with itself, plus the potential operator.  FTP, identify this operator, the operation of which on the wave function is equal to operation of the energy operator on the wave function according to the Schr?dinger equation, that is symbolized "h-hat" and that is named for a 19th Century Irish mathematician.
The diagonal elements of the coupling matrix in an orthonormal n-energy-state system are fully perturbed matrix elements of this object.  
Hamiltonian operator or matrix (prompt on energy operator)