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This paper numerically analyzes the influence of second harmonics on the characteristics of Josephson junction quantum bits (qubits). It is shown that the splitting of energy states of phase qubits is sensitive to energy scale and unharmonicity parameters. On the other hand, for charge qubits, the expectation value of the Cooper-pair number is not affected by the inclusion of unharmonicity parameters, whereas the expectation value of the superconducting current is strongly influenced by energy scale and unharmonicity parameters. The numerical results seem to present the importance of second harmonics in the realization of superconducting qubits.