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Although quantum computation (QC) is regarded as a promising numerical method for computational quantum chemistry, current applications of quantum-chemistry calculations on quantum computers are limited to small molecules. This limitation can be ascribed to technical problems in building and manipulating more qubits and the associated complicated operations of quantum gates in a quantum circuit when the size of the molecular system becomes large. As a result, reducing the number of required qubits is necessary to make QC practical. Currently, the minimal STO-3G basis set is commonly used in benchmark studies because it requires the minimum number of spin orbitals. Nonetheless, the accuracy of using STO-3G is generally low and thus cannot provide useful predictions. We propose to adopt Daubechies wavelet functions as an accurate and efficient method for QCs of molecular electronic properties. We demonstrate that a minimal basis set constructed from Daubechies wavelet basis can yield accurate results through a better description of the molecular Hamiltonian, while keeping the number of spin orbitals minimal. With the improved Hamiltonian through Daubechies wavelets, we calculate vibrational frequencies for H$_2$ and LiH using quantum-computing algorithm to show that the results are in excellent agreement with experimental data. As a result, we achieve quantum calculations in which accuracy is comparable with that of the full configuration interaction calculation using the cc-pVDZ basis set, whereas the computational cost is the same as that of a STO-3G calculation. Thus, our work provides a more efficient and accurate representation of the molecular Hamiltonian for efficient QCs of molecular systems, and for the first time demonstrates that predictions in agreement with experimental measurements are possible to be achieved with quantum resources available in near-term quantum computers.