Two-particle reduced Hamiltonian matrices

After the energy is expressed as a functional of the 2-RDM, a systematic hierarchy of V-representabihty constraints, known as p-positivity conditions, is derived [17]. We develop the details of the 2-positivity, 3-positivity, and partial 3-positivity conditions [21, 27, 34, 33]. In Section II.E the formal solution of V-representability for the 2-RDM is presented through a convex set of two-particle reduced Hamiltonian matrices [7, 21]. It is shown that the positivity conditions correspond to certain classes of reduced Hamiltonian matrices, and consequently, they are exact for certain classes of Hamiltonian operators at any interaction strength. In Section II.F the size of the 2-RDM is reduced through the use of spin and spatial symmetries [32, 34], and in Section II.G the variational 2-RDM method is extended to open-shell molecules [35]. 

The formal solution of Al-representability for the 2-RDM is developed in terms of a convex set of two-particle reduced Hamiltonian matrices. To complement the derivation of the positivity conditions from the metric matrices, we derive them from classes of these two-particle reduced Hamiltonian matrices. This interpretation allows us to demonstrate that the 2-positivity conditions are exact for certain classes of Hamiltonian operators for any interaction strength. In this section all of the ROMs are normalized to unity. Much of this discussion appeared originally in Refs. [21, 29]. 

The unitary decomposition may be applied to any Hermitian, antisymmetric two-particle matrix including the 2-RDM, the two-hole RDM, and the two-particle reduced Hamiltonian. The decomposition is also readily generalized to treat p-particle matrices [80-82]. The trial 2-RDM to be purified may be written... 

Here Hd, is the Dirac Hamiltonian for a single particle, given by Eq. [30]. Recall from above that the Coulomb interaction shown is not strictly Lorentz invariant therefore, Eq. [59] is only approximate. The right-hand side of the equation gives the relativistic interactions between two electrons, and is called the Breit interaction. Here a, and a, denote Dirac matrices (Eq. [31]) for electrons i and /. Equation [59] can be cast into equations similar to Eq. [36] for the Foldy-Wouthuysen transformation. After a sequence of unitary transformations on the Hamiltonian (similar to Eqs. [37]-[58]) is applied to reduce the off-diagonal contributions, one obtains the Hamiltonian in terms of commutators, similar to Eq. [58]. When each term of the commutators are expanded explicitly, one arrives at the Breit-Pauli Hamiltonian, for a many-electron system " ... 

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