Colloidal silicon quantum dots feature a number of outstanding properties, such as size and termination controlled band gap and photoluminescence, which, in combination with nontoxicity, make them suitable for biomedical applications. Because of the presence of structural disorder and experimental limitations, the atomic structure, especially of the small sub 2.5 nm dots, is not well known. We have developed computational techniques that allow us to model the atomic structures of the small dots and have applied them to heavily B and P codoped Si quantum dots and studied their structural, electronic, and vibrational properties. The study confirms that the structures of the smallest dots are fully amorphous with the onset of formation of a quasi-crystalline core in the larger ones. The models give insights into the dopant distribution in the dot. We find that the core morphology depends strongly on the chemical composition of the dot. Study of the electronic and vibrational properties gives insights into their localization properties and allows validation of the generated models by comparison with experiments. The degree of agreement of the properties of our simulated dots with those from experiments suggests that the fine details in preparation protocols may not critically affect their structure and properties.