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.
^{
342
}
J. Math. Phys.
(1964),
5,
pp. 1756.
Link
^{,}
^{
341
}
J. Math. Phys.
(1975),
16,
pp. 868.
Link
^{,}
^{
759
}
Nucl. Phys. A
(1975),
237,
pp. 221.
Link
^{,}
^{
946
}
J. Comput. Phys.
(1975),
18,
pp. 300.
Link
^{,}
^{
299
}
J. Math. Phys.
(1979),
20,
pp. 1366.
Link
^{,}
^{
301
}
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 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 ${\widehat{M}}_{S}$ 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 $\u27e8{S}^{2}\u27e9=S(S+1)$, 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 (${}^{2}\mathbf{Q}$) 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.
^{
341
}
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,
^{
300
}
Int. J. Quantum Chem.
(1978),
13,
pp. 697.
Link
^{,}
^{
1253
}
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 (${}^{3}\mathbf{D}$), the three-hole
reduced-density matrix (${}^{3}\mathbf{Q}$), the two-particle-one-hole
reduced-density matrix (${}^{3}\mathbf{E}$), and the one-particle-two-hole
reduced-density matrix (${}^{3}\mathbf{F}$). The full 3-positivity conditions
guarantee partial conditions automatically.
^{
746
}
Phys. Rev. A
(2006),
74,
pp. 032501.
Link
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
procedure
^{
351
}
J. Chem. Phys.
(2008),
129,
pp. 134108.
Link
^{,}
^{
321
}
J. Chem. Theory Comput.
(2016),
12,
pp. 2260.
Link
^{,}
^{
721
}
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).
The current v2RDM and v2RDM-CASSCF 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.