Proper and improper zero energy modes in Hartree-Fock theory and their relevance for symmetry breaking and restoration.
Proper and improper zero energy modes in Hartree-Fock theory and their relevance for symmetry breaking and restoration.
复制标题
Hartree-Fock 理论中正确和不正确的零能量模式及其与对称性破缺和恢复的相关性。
DOI：
10.1063/1.4824905
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
G. Scuseria
中科院分区：
文献类型：
--
作者：
Yao Cui;Ireneusz W. Bulik;C. Jiménez;T. M. Henderson;G. Scuseria
我们研究了对称性破缺的Hartree-Fock解的分子轨道Hessian（稳定性矩阵）和随机相位近似（RPA）哈密顿量的谱，重点是零本征值模式。在所有负特征值通过跟随其特征向量下坡而从Hessian中移除之后，仅留下正特征值和零特征值。零模式对应于没有恢复力的轨道旋转。这些旋转决定了戈德斯通流形中的状态，这源于波函数中的自发破缺连续对称性。根据零模的不同数学物理性质，零模可以分为非正常模和正常模。不正确的模式是由对称性破缺引起的，它们的恢复总是会降低能量。另一方面，固有模对应于波函数的简并，它们的对称性恢复不一定会降低能量。我们讨论如何RPA哈密顿区分适当的和不适当的模式，通过加倍与后者相关联的零本征值的数量。Hessian中的固有模式总是成对出现，而在RPA中不会加倍。我们提出了几个教学案例，举例说明上述声明。这些结果的相关性预计Hartree-Fock方法也得到了解决。
We study the spectra of the molecular orbital Hessian (stability matrix) and random-phase approximation (RPA) Hamiltonian of broken-symmetry Hartree-Fock solutions, focusing on zero eigenvalue modes. After all negative eigenvalues are removed from the Hessian by following their eigenvectors downhill, one is left with only positive and zero eigenvalues. Zero modes correspond to orbital rotations with no restoring force. These rotations determine states in the Goldstone manifold, which originates from a spontaneously broken continuous symmetry in the wave function. Zero modes can be classified as improper or proper according to their different mathematical and physical properties. Improper modes arise from symmetry breaking and their restoration always lowers the energy. Proper modes, on the other hand, correspond to degeneracies of the wave function, and their symmetry restoration does not necessarily lower the energy. We discuss how the RPA Hamiltonian distinguishes between proper and improper modes by doubling the number of zero eigenvalues associated with the latter. Proper modes in the Hessian always appear in pairs which do not double in RPA. We present several pedagogical cases exemplifying the above statements. The relevance of these results for projected Hartree-Fock methods is also addressed.