6.22.1 Introduction

The methods described in this section involve the direct variational optimization of the two-electron reduced-density matrix (2-RDM, ${}^2𝐃$), subject to necessary ensemble $N$-representability conditions. ⁴⁰⁰ Garrod C., Percus J. K.
J. Math. Phys.
(1964), 5, pp. 1756. Link ^{, 399} Garrod C., Mihailović M. V., Rosina M.
J. Math. Phys.
(1975), 16, pp. 868. Link ^{, 886} Mihailović M. V., Rosina M.
Nucl. Phys. A
(1975), 237, pp. 221. Link ^{, 1105} Rosina M., Garrod C.
J. Comput. Phys.
(1975), 18, pp. 300. Link ^{, 345} Erdahl R. M. et al.
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(1979), 20, pp. 1366. Link ^{, 347} Erdahl R. M.
Rep. Math. Phys.
(1979), 15, pp. 147. Link Such conditions place restrictions on the 2-RDM in order to ensure that it is derivable from an ensemble of $N$-electron density matrices. 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 ${\widehat{M}}_S$ 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 $⟨{\widehat{S}}^2⟩=S(S+1)$ (in units of ${\mathrm{\hslash }}^2$), where the $S$ is the spin quantum number. Note that this constraint on the expectation value of ${\widehat{S}}^2$ does not strictly guarantee that the 2-RDM corresponds to an eigenfunction of ${\widehat{S}}^2$. 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 (${}^2𝐐$) and the particle-hole reduced-density matrix (${}^2𝐆$). The positivity of ${}^2𝐃$, ${}^2𝐐$, and ${}^2𝐆$ constitute the DQG constraints of Garrod and Percus. ³⁹⁹ Garrod C., Mihailović M. V., Rosina M.
J. Math. Phys.
(1975), 16, pp. 868. Link 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, ³⁴⁶ Erdahl R. M.
Int. J. Quantum Chem.
(1978), 13, pp. 697. Link ^{, 1453} Zhao Z. et al.
J. Chem. Phys.
(2004), 120, pp. 2095. Link which do not explicitly depend upon the 3-RDM; or the full 3-positivity conditions, which include the three-particle reduced-density matrix (${}^3𝐃$), the three-hole reduced-density matrix (${}^3𝐐$), the two-particle-one-hole reduced-density matrix (${}^3𝐄$), and the one-particle-two-hole reduced-density matrix (${}^3𝐅$). The full 3-positivity conditions guarantee partial conditions automatically. ⁸⁷⁰ Mazziotti D. A.
Phys. Rev. A
(2006), 74, pp. 032501. Link The positivity conditions imposed in v2RDM computations are controlled through the $rem variable 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 procedure ⁴¹⁰ Gidofalvi G., Mazziotti D. A.
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(2008), 129, pp. 134108. Link ^{, 371} Fosso-Tande J. et al.
J. Chem. Theory Comput.
(2016), 12, pp. 2260. Link ^{, 842} Maradzike E. et al.
J. Chem. Theory Comput.
(2017), 13, pp. 4113. Link 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).

Note that v2RDM-CASSCF only describes electron correlations among the active orbitas. A computationally inexpensive estimate of the remaining correlation effects can be achieved with the multiconfiguration pair-density functional theory (MC-PDFT) approach, ⁸²⁸ Manni G. L. et al.
J. Chem. Theory Comput.
(2014), 9, pp. 3669. Link ^{, 395} Gagliardi L. et al.

(2017), 50, pp. 66. Link which evaluates the energy as a functional of the on-top pair density (OTPD). In the MC-PDFT implementation in Q-Chem, the OTPD is derived from a variationally optimized 2-RDM as described in Ref.  903 Mostafanejad M., DePrince III E. A.
J. Chem. Theory Comput.
(2019), 15, pp. 290. Link .

The current v2RDM, v2RDM-CASSCF, and MC-PDFT implementations must 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 v2RDM computation will fail without the $rem keyword AUX_BASIS. Analytic gradients are not available when frozen molecular orbitals are requested. Specification of the active space is demonstrated in the examples below. When the formatted checkpoint file is requested, natural orbitals are saved in it.