Post-training model quantization is widely used to reduce memory and computational costs of large language models.A novel framework for allocating quantization bitwidths based on sensitivity metrics derived from a Hessian proxy is proposed.The proposed BAQ algorithm achieves a good trade-off between loss minimization and complexity for large language models.Experimental results show that BAQ outperforms GPTQ, achieving up to 56 times lower perplexity at the same bitwidth.