特定态电子结构理论的能量图景
Energy Landscape of State-Specific Electronic Structure Theory
摘要：
状态特定的近似可以通过使电子密度松弛来提供具有挑战性的电子激发的准确表示。虽然已知特定态的波函数是近似能量的局部极小值或鞍点，但精确电子能量的全局结构在很大程度上仍未被探索。在这方面的贡献，几何角度的确切电子能量景观介绍。在精确的能量图上，基态和激发态形成被约束到超球表面的稳定点，并且相应的Hessian指数在每个激发能级处增加。精确的稳定点之间的连通性进行了研究和精确的能量梯度的平方幅度示出成正比的哈密顿方差。最小基Hartree-Fock和激发态平均场表示的单重态H $_2$（STO-3G），然后探索如何精确的能量景观控制的存在和属性的状态特定的近似。特别地，近似的激发态对应于精确能量图上的受约束的稳定点，并且它们的Hessian指数也随着更高的能量而增加。最后，准确的能量的属性被用来推导方差优化景观的结构，并阐明方差优化算法所面临的挑战，包括非物理鞍点或方差的最大值的存在。
英文摘要：
State-specific approximations can provide an accurate representation of challenging electronic excitations by enabling relaxation of the electron density. While state-specific wave functions are known to be local minima or saddle points of the approximate energy, the global structure of the exact electronic energy remains largely unexplored. In this contribution, a geometric perspective on the exact electronic energy landscape is introduced. On the exact energy landscape, ground and excited states form stationary points constrained to the surface of a hypersphere and the corresponding Hessian index increases at each excitation level. The connectivity between exact stationary points is investigated and the square-magnitude of the exact energy gradient is shown to be directly proportional to the Hamiltonian variance. The minimal basis Hartree-Fock and Excited-State Mean-Field representations of singlet H$_2$ (STO-3G) are then used to explore how the exact energy landscape controls the existence and properties of state-specific approximations. In particular, approximate excited states correspond to constrained stationary points on the exact energy landscape and their Hessian index also increases for higher energies. Finally, the properties of the exact energy are used to derive the structure of the variance optimisation landscape and elucidate the challenges faced by variance optimisation algorithms, including the presence of unphysical saddle points or maxima of the variance.
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