Department of Inorganic and Physical Chemistry, Ghent University, Krijgslaan 281 S3, 9000 Ghent, Belgium.
J Chem Phys. 2012 Jan 7;136(1):014110. doi: 10.1063/1.3672087.
Despite the importance of non-singlet molecules in chemistry, most variational second order density matrix calculations have focused on singlet states. Ensuring that a second order density matrix is derivable from a proper N-electron spin state is a difficult problem because the second order density matrix only describes one- and two-particle interactions. In pursuit of a consistent description of spin in second order density matrix theory, we propose and evaluate two main approaches: we consider constraints derived from a pure spin state and from an ensemble of spin states. This paper makes a comparative assessment of the different approaches by applying them to potential energy surfaces for different spin states of the oxygen and carbon dimer. We observe two major shortcomings of the applied spin constraints: they are not size consistent and they do not reproduce the degeneracy of the different states in a spin multiplet. First of all, the spin constraints are less strong when applied to a dissociated molecule than when they are applied to the dissociation products separately. Although they impose correct spin expectation values on the dissociated molecule, the dissociation products do not have correct spin expectation values. Secondly, both under "pure spin state conditions" and under "ensemble spin state" conditions is the energy a convex function of the spin projection. Potential energy surfaces for different spin projections of the same spin state may give a completely different picture of the molecule's bonding. The maximal spin projection always gives the most strongly constrained energy, but is also significantly more expensive to compute than a spin-averaged ensemble. In the dissociation limit, both the problem of nondegeneracy of equivalent spin projections, size-inconsistency and unphysical dissociation can be corrected by means of subspace energy constraints.
尽管非单重态分子在化学中很重要，但大多数变分二阶密度矩阵计算都集中在单重态上。确保二阶密度矩阵可以从适当的 N 电子自旋态导出是一个难题，因为二阶密度矩阵仅描述单粒子和双粒子相互作用。为了在二阶密度矩阵理论中对自旋进行一致描述，我们提出并评估了两种主要方法：我们考虑了源自纯自旋态和自旋态集合的约束。本文通过将这些方法应用于氧和碳二聚体不同自旋态的势能表面，对不同方法进行了比较评估。我们观察到应用的自旋约束有两个主要缺点：它们不是大小一致的，并且它们不能再现自旋多重态中不同态的简并性。首先，与应用于分离分子相比，当将自旋约束应用于分离产物时，其强度较小。尽管它们对分离分子施加了正确的自旋期望值，但分离产物没有正确的自旋期望值。其次，无论是在“纯自旋态条件”下还是在“自旋态集合”条件下，能量都是自旋投影的凸函数。同一自旋态的不同自旋投影的势能表面可能会给出分子键合的完全不同的图像。最大自旋投影始终给出约束最强的能量，但与自旋平均集合相比，计算成本也明显更高。在离解极限下，等效自旋投影的非简并性、大小不一致性和非物理离解问题都可以通过子空间能量约束来纠正。
