page_title: Symmetry of diatomic molecules
section_title: Effect of symmetry on the matrix elements of the Hamiltonian
hierarchical_section_title: Symmetry of diatomic molecules / von Neumann-Wigner non-crossing rule / Effect of symmetry on the matrix elements of the Hamiltonian
caption_attribution_description: English: Pictorial representation of action by creation and annihilation operators on Bosonic Fock state.
context_page_description: Molecular symmetry in physics and chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in the application of Quantum Mechanics in physics and chemistry, for example it can be used to predict or explain many of a molecule's properties, such as its dipole moment and its allowed spectroscopic transitions, without doing the exact rigorous calculations. To do this it is necessary to classify the states of the molecule using the irreducible representations from the character table of the symmetry group of the molecule. Among all the molecular symmetries, diatomic molecules show some distinct features and they are relatively easier to analyze.
context_section_description: The electronic terms or potential curves  of a diatomic molecule depend only on the internuclear distance , and it is important to investigate the behaviour of these potential curves as R varies. It is of considerable interest to examine the intersection of the curves representing the different terms.
  Let  and  two different electronic potential curves. If they intersect at some point, then the functions  and  will have neighbouring values near this point. To decide whether such an intersection can occur, it is convenient to put the problem as follows. Suppose at some internuclear distance  the values  and  are close, but distinct (as shown in the figure). Then it is to be examined whether or  and  can be made to intersect by the modification . The energies  and  are eigenvalues of the Hamiltonian . The corresponding orthonormal electronic eigenstates will be denoted by  and  and are assumed to be real.
  The Hamiltonian now becomes , where  is the small perturbation operator (though it is a degenerate case, so ordinary method of perturbation won't work). setting , it can be deduced that in order for  and  to be equal at the point  the following two conditions are required to be fulfilled:
  However, we have at our disposal only one arbitrary parameter  giving the perturbation . Hence the
  two conditions involving more than one parameter cannot in general be simultaneously satisfied (the initial assumption that  and  real, implies that  is also real). So, two case can arise: 
  The matrix element  vanishes identically. It is then possible to satisfy the first condition independently. Therefore, it is possible for the crossing to occur if, for a certain value of  (i.e., for a certain value of ) the first equation is satisfied. As the perturbation operator  (or ) commutes with the symmetry operators of the molecule, this case will happen if the two electronic states  and   have different point group symmetries (for example if they correspond to two electronic terms having different values of , different electronic parities g and u, different multiplicities, or for example are the two terms  and ) as it can be shown that, for a scalar quantity whose operator commutes with the angular momentum and inversion operators, only the matrix elements for transitions between states of the same angular momentum and parity are non-zero and the proof remains valid, in essentially the same form, for the general case of an arbitrary symmetry operator.  
  If  the electronic states  and  have the same point group symmetry, then  can be, and will in general be, non-zero. Except for accidental crossing which would occur if, by coincidence, the two equations were satisfied at the same value of , it is in general impossible to find a single value of  (i.e., a single value of )  for which the two conditions are satisfied simultaneously.  
  Thus, in a diatomic molecule, only terms of different symmetry can intersect, while the intersection of terms of like symmetry is forbidden. This is, in general, true for any case in quantum mechanics where the Hamiltonian contains some parameter and its eigenvalues are consequently functions of that parameter. This general rule is known as von Neumann - Wigner non-crossing rule. 
  This general symmetry principle has important consequences is molecular spectra.
  In fact, in the applications of valence bond method in case of diatomic molecules, three main correspondence between the atomic and the molecular orbitals are taken care of:
  Molecular orbitals having a given value of  (the component of the orbital angular momentum along the internuclear axis) must connect with atomic orbitals having the same value of  (i.e. the same value of ).
  The electronic parity of the wave function (g or u) must be preserved as  varies from  to .
  The von Neumann-Wigner non-crossing rule must be obeyed, so that energy curves corresponding to orbitals having the same symmetry do not cross as  varies from  to .
  Thus, von Neumann-Wigner non-crossing rule also acts as a starting point for valence bond theory.
caption: The non-crossing rule of von Neumann and Wigner. Two potential curves 
    
      
        
          
            
              
                E
              
              
                1
              
            
          
          (
          R
          )
        
      
      {\displaystyle {{E}_{1}}(R)}
    
   and 
    
      
        
          
            
              
                E
              
              
                2
              
            
          
          (
          R
          )
        
      
      {\displaystyle {{E}_{2}}(R)}
    
   cannot cross if the states 1 and 2 have the same point group symmetry
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