# 6.18.1 Introduction

The methods described in this section involve the direct variational optimization of the two-electron reduced-density matrix (2-RDM, ${}^{2}\mathbf{D}$), subject to necessary ensemble $N$-representability conditions.286, 285, 635, 792, 251, 253 Such conditions place restrictions on the 2-RDM in order to ensure that it is derivable from an antisymmetrized $N$-electron wavefunction. In the limit that the $N$-representability of the 2-RDM is exactly enforced, the variational 2-RDM (v2RDM) approach is equivalent to full configuration interaction (CI). Such computations are, in general, computationally infeasible, so the v2RDM optimization is typically carried out under a subset of two- or three-particle conditions. When only partially enforcing $N$-representability, the v2RDM approach yields a lower bound to the full CI energy.

In Q-Chem, all v2RDM optimizations are carried out under the following conditions:

• the 2-RDM is positive semidefinite

• the one-electron reduced-density matrix (1-RDM) is positive semidefinite

• the trace of the 2-RDM is equal to the number of pairs of electrons, $N(N-1)$

• each spin block of the 2-RDM properly contracts to the appropriate spin block of the 1-RDM

• the expectation value of $\hat{M}_{S}$ is $\frac{1}{2}(N_{\alpha}-N_{\beta})$ (the maximal spin projection)

Additionally, an optional spin constraint can be placed on the 2-RDM such that $\langle S^{2}\rangle=S(S+1)$, where the $S$ is the spin quantum number. Note that this constraint on the expectation value of $\hat{S}^{2}$ does not strictly guarantee that the 2-RDM corresponds to an eigenfunction of $\hat{S}^{2}$. Without additional constraints, a v2RDM optimization would yield poor-quality 2-RDMs with energies far below those of full CI. Reasonable results require, at a minimum, that one enforce the positivity of additional pair-probability density matrices, including the two-hole reduced-density matrix (${}^{2}\mathbf{Q}$) and the particle-hole reduced-density matrix (${}^{2}\mathbf{G}$). The positivity of ${}^{2}\mathbf{D}$, ${}^{2}\mathbf{Q}$, and ${}^{2}\mathbf{G}$ constitute the DQG constraints of Garrod and Percus.285 For many systems, the DQG constraints yield a reasonable description of the electronic structure. However, if high accuracy is desired, it is sometimes necessary to consider constraints on higher-order reduced-density matrices (e.g. the three-electron reduced-density matrix [3-RDM]). In Q-Chem, v2RDM optimizations can be performed under the T1 and T2 partial three-particle conditions,252, 1054 which do not explicitly depend upon the 3-RDM. The positivity conditions imposed in v2RDM computations are controlled through the $rem keyword RDM_POSITIVITY. The main utility of the v2RDM approach is in the context of active-space-based descriptions of strong or nondynamical correlation. The most common active-space-based approach for strong correlation is the compete active space self-consistent field (CASSCF) method. By performing a v2RDM computation within an active space and coupling v2RDM to an orbital optimization procedure, one can achieve a v2RDM-driven CASSCF procedure294, 269, 605 that provides a lower bound the conventional CI-based CASSCF energy. Because the v2RDM-CASSCF method scales polynomially with respect to the number of active orbitals, v2RDM-CASSCF can handle much larger active spaces (e.g., 50 electrons in 50 orbitals) compared to CI-CASSCF (e.g., 18 electrons in 18 orbitals). The current v2RDM and v2RDM-CASSCF implementations can make use of the density fitting (DF) approximation to the two-electron integrals. The use of DF integrals is particularly advantageous for v2RDM-CASSCF computations with large active spaces because of the increased efficiency in the orbital optimization/integral transformation step. The use of DF integrals is triggered by using the$rem keyword AUX_BASIS. Analytic gradients are only available for DF integrals and are not available when frozen molecular orbitals are requested. Specification of the active space is demonstrated in the examples below. Additionally, a GPU-accelerated implementation of v2RDM and v2RDM-CASSCF employing the DQG conditions is available.