That is how the existing system is supposed to work, but the property valuations and bandings have become divorced from the current state of the property market.

Note that, in theory, a secondary function of the council tax should be to prevent habitable properties being left unoccupied for extended periods of time.

Unfortunately when the tax amounts to a rounding error on the ROI for a property, it is completely ineffectual. Hence the large number of empty 'investment' properties in London, further inflating an already overheated property market.