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The electron correlations near the half-filling of the lowest and excitedLandau levels (LL's) are studied using numerical diagonalization. It is shownthat in the low lying states electrons avoid pair states with relative angularmomenta ${\cal R}$ corresponding to positive anharmonicity of the interactionpseudopotential $V({\cal R})$. In the lowest LL, the super-harmonic behavior of$V({\cal R})$ causes Laughlin correlations (avoiding pairs with ${\cal R}=1$)and the Laughlin-Jain series of incompressible ground states. In the firstexcited LL, $V({\cal R})$ is harmonic at short range and a different series ofincompressible states results. Similar correlations occur in the pairedMoore-Read $\nu={5\over2}$ state and in the $\nu={7\over3}$ and ${8\over3}$states, all having small total parentage from ${\cal R}=1$ and 3 and largeparentage from ${\cal R}=5$. The $\nu={7\over3}$ and ${8\over3}$ states aredifferent from Laughlin $\nu={1\over3}$ and ${2\over3}$ states and, in finitesystems, occur at a different LL degeneracy (flux). The series of Laughlincorrelated states of electron pairs at $\nu=2+2/(q_2+2)={8\over3}$,${5\over2}$, ${12\over5}$, and ${7\over3}$ is proposed, although only in the$\nu={5\over2}$ state pairing has been confirmed numerically. In the secondexcited LL, $V({\cal R})$ is sub-harmonic at short range and (near thehalf-filling) the electrons group into spatially separated larger $\nu=1$droplets to minimize the number of strongly repulsive pair states at ${\calR}=3$ and 5.Comment: 10 pages, 8 figures, submitted to PR

Electron correlations in partially filled lowest and excited Landau levels