Clustering of Electrons and the Filling Factor in Quantum Hall Effect

The relative angular momentum of two particles, L-2 is used to define a projection operator P-2(p) which corresponds to the relative angular momentum L-2+p the filling factor is defined by v = 1 /(L-2(min) + p). The number of particles is g. If one more particle comes near it for the purpose of forming a cluster, the filling factor becomes v = [(1/2)g(g + 1) + p](-1). The g=2, p=4 is called Haffnian and g=2, p=3 is called Gaffnian. For g=2, p=3, v=1/6. This state has a ground state of a special pseudopotential type Hamiltonian. Given a fraction, we find the ground state energy of a special Hamiltonian. The state wave function, the ground state energy and the special Hamiltonian are linked together. In this methodology, the flux quanta can not be attached to the electrons and the projection operator is not linked to the Coulomb Hamiltonian. Therefore, composite fermions (CF) with the flux quanta attached to the electrons, suggested by Jain cannot satisfy the wave functions, the Hamiltonian and the ground state requirements. The calculated ground state does not attach flux quanta.