论文标题
国家特异性电子结构理论的能源景观
Energy Landscape of State-Specific Electronic Structure Theory
论文作者
论文摘要
特定于状态的近似值可以通过使电子密度放松来提供充满挑战的电子激发的准确表示。尽管已知状态特异性波函数是近似能量的局部最小值或鞍点，但精确电子能的全局结构仍未得到探索。在此贡献中，引入了有关精确电子能量景观的几何观点。在确切的能量景观上，地面和激发状态形成固定点，固定点约束至超晶体的表面，相应的Hessian指数在每个激发水平下增加。研究了精确的固定点之间的连通性，并显示确切能量梯度的平方标志与汉密尔顿方差成正比。然后，使用最小的基础Hartree-fock和兴奋状态均值h $ _2 $（STO-3G）的平均景色表示，用于探索确切的能量景观如何控制状态特异性近似的存在和特性。特别是，近似激发态对应于精确能量景观上的约束固定点，其Hessian指数也增加了更高的能量。最后，使用确切能量的特性来得出方差优化景观的结构，并阐明了方差优化算法所面临的挑战，包括存在非物理鞍点或方差的最大值。
State-specific approximations can provide an accurate representation of challenging electronic excitations by enabling relaxation of the electron density. While state-specific wave functions are known to be local minima or saddle points of the approximate energy, the global structure of the exact electronic energy remains largely unexplored. In this contribution, a geometric perspective on the exact electronic energy landscape is introduced. On the exact energy landscape, ground and excited states form stationary points constrained to the surface of a hypersphere and the corresponding Hessian index increases at each excitation level. The connectivity between exact stationary points is investigated and the square-magnitude of the exact energy gradient is shown to be directly proportional to the Hamiltonian variance. The minimal basis Hartree-Fock and Excited-State Mean-Field representations of singlet H$_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.
