We check the ab initio $GW$ approximation and Bethe-Salpeter equation (BSE) many-body methodology against the exact solution benchmark of the hydrogen molecule ${\mathrm{H}}_2$ ground state and excitation spectrum, and in comparison with the configuration interaction (CI) and time-dependent Hartree-Fock methods. The comparison is made on all the states we could unambiguously identify from the excitonic wave functions' symmetry. At the equilibrium distance $R=1.4a_0$, the $GW+\mathrm{BSE}$ energy levels are in good agreement with the exact results, with an accuracy of $\sim 0.1\text{–}0.2\mathrm{eV}$. $GW+\mathrm{BSE}$ potential-energy curves are also in good agreement with the CI and the exact result up to $2.3a_0$. The solution no longer exists beyond $3.0a_0$ for triplets ($4.3a_0$ for singlets) due to instability of the ground state. We tried to improve the $GW$ reference ground state by a renormalized random-phase approximation (r-RPA), but this did not solve the problem.

中文翻译：

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通过多体GW逼近和Bethe-Salpeter方程计算氢分子光谱

我们检查从头算起 $G\mathrm{w\; ^}$ 逼近和Bethe-Salpeter方程（BSE）多体方法针对氢分子的精确溶液基准 ${\mathrm{H}}_2$基态和激发光谱，并与配置相互作用（CI）和与时间有关的Hartree-Fock方法进行比较。对所有我们可以从激子波函数的对称性中明确确定的状态进行比较。在平衡距离$\mathrm{[R}=1.4{\mathrm{一种}}_0$， $G\mathrm{w\; ^}+\mathrm{疯牛病}$ 能级与精确结果高度吻合，精确度为 $〜0.1\text{–}0.2\mathrm{电子伏特}$。 $G\mathrm{w\; ^}+\mathrm{疯牛病}$ 势能曲线也与CI一致，直到 $2.3{\mathrm{一种}}_0$。解决方案不再存在$3.0{\mathrm{一种}}_0$ 三胞胎$4.3{\mathrm{一种}}_0$对于单重态），因为基态不稳定。我们试图改善$G\mathrm{w\; ^}$ 通过重新规范化的随机相位近似（r-RPA）来确定参考基态，但这并不能解决问题。