Analyzing heterogeneous cellular networks (HCNs) has become increasingly complex, particularly due to irregular base station locations, massive deployment of small cells, and flexible resource allocation. The latter is usually not captured by existing stochastic models for analytical tractability. In this work, we develop a more realistic stochastic model for a generic $K$-tier HCN, where users are served in the downlink under frequency diversity. We derive the rate coverage probability for this case, taking into account the interference correlation across different parts of the allocated spectrum. Our results indicate that ignoring this type of correlation may considerably overestimate the offered date rate. Furthermore, the gain of spectrum diversification is significant, particularly at low to moderate target data rates.