Figure shows the excited state population decay and the total energy deviation for PSB3. All methods overestimate the initial population transfer to S 0 , and tend to incorrectly plateau around 90 fs for some time before subsequently continuing to transfer. After about 120 fs, the trend in CTMQC-E is close to the exact, giving a small improvement over CTMQC and SH-EDC. In terms of total energy, CTMQC-E reduces the total energy violation of CTMQC with some energy drift after 90 fs due to the numerical implementation as discussed in Sec. III when considering the results for BMA.