6.22 Variational Two-Electron Reduced-Density-Matrix Methods 6.22.1 Introduction 6.22.3 v2RDM Job Control

6.22.2 Theory

(September 1, 2024)

The electronic energy is an exact functional of the 1-RDM and 2-RDM

E=\frac{1}{2}\sum_{pqrs}{}^{2}D^{pq}_{rs}(pr|qs)+\sum_{pq}{}^{1}D^{p}_{q}h_{pq},

(6.86)

where the 1-RDM ( ${}^{1}\mathbf{D}$ ) and 2-RDM are represented in a given spin-orbital basis indexed by $p$ , $q$ , $r$ , and $s$ . The one-hole RDM ( ${}^{1}\mathbf{Q}$ ), two-hole RDM ( ${}^{2}\mathbf{Q}$ ), particle-hole RDM ( ${}^{2}\mathbf{G}$ ), partial three-particle RDMs ( $\mathbf{T1}$ and $\mathbf{T2}$ ), and full three-particle RDMs ( ${}^{3}\mathbf{D}$ , ${}^{3}\mathbf{Q}$ , ${}^{3}\mathbf{E}$ , ${}^{3}\mathbf{F}$ ) are linear functions of ${}^{1}\mathbf{D}$ and ${}^{2}\mathbf{D}$ . ³⁷¹ Fosso-Tande J. et al.
J. Chem. Theory Comput.
(2016), 12, pp. 2260. Link Minimizing the electronic energy with respect to ${}^{2}\mathbf{D}$ while enforcing the linear relations among these RDMs, the contraction and spin constraints placed on ${}^{2}\mathbf{D}$ , and the positive semidefinite property of all RDMs constitutes a semidefinite program (SDP). The current v2RDM implementation uses a boundary-point SDP (BPSDP) algorithm to solve the SDP. ¹⁰⁴⁵ Povh J., Rendl F., Wiegele A.
Computing
(2006), 78, pp. 277. Link ^, ⁸²⁴ Malick J. et al.
SIAM J. Optim.
(2009), 20, pp. 336. Link ^, ⁸⁷¹ Mazziotti D. A.
Phys. Rev. Lett.
(2011), 106, pp. 083001. Link

The primal formulation of the SDP is

minimize	$\displaystyle E_{\rm primal}$	$\displaystyle={\bf c}^{T}\cdot{\bf x}$	(6.87)
such that	$\displaystyle{\bf A}{\bf x}$	$\displaystyle={\bf b}$
and	$\displaystyle M({\bf x})$	$\displaystyle\succeq 0.$

Here, $\mathbf{x}$ represents the primal solution vector, the vector $\mathbf{c}$ contains all information defining the quantum system (the one- and two-electron integrals), and the mapping $M(\mathbf{x})$ maps the primal solution onto the set of positive semidefinite RDMs:

M({\bf x})=\left(\begin{array}[]{cccc}{}^{1}{\mathbf{D}}&0&0&0\\ 0&{}^{1}{\mathbf{Q}}&0&0\\ 0&0&{}^{2}{\mathbf{D}}&0\\ 0&0&0&{}\ddots\\ \end{array}\right)\succeq 0.

(6.88)

Additional RDMs can be included in $M(\mathbf{x})$ , depending on the choice of N-representability conditions applied. The action of the constraint matrix, ${\mathbf{A}}$ , on ${\mathbf{x}}$ is a compact representation of the $N$ -representability conditions. ${\mathbf{A}}$ maintains the appropriate mappings between each block of $M({\mathbf{x}})$ and enforces the appropriate spin and contraction conditions. Alternatively, one could consider the dual formulation of the semidefinite problem, expressed as

maximize	$\displaystyle E_{\rm dual}$	$\displaystyle=\mathbf{b}^{T}\cdot\mathbf{y}$	(6.89)
such that	$\displaystyle\mathbf{z}$	$\displaystyle=\mathbf{c}-\mathbf{A}^{T}\mathbf{y}$
and	$\displaystyle M(\mathbf{z})$	$\displaystyle\succeq 0$

where ${\mathbf{y}}$ and ${\mathbf{z}}$ are the dual solutions, and $M({\mathbf{z}})$ is constrained to be positive semidefinite.

The BPSDP algorithm involves an iterative two-step procedure:

1.

Solve $\mathbf{AA}^{T}\mathbf{y}=\mathbf{A}(\mathbf{c}-\mathbf{z})+\tau\mu(\mathbf{b}% -\mathbf{A}\mathbf{x})$ for $\mathbf{y}$ by conjugate gradient methods.
2.

Update $\mathbf{x}$ and $\mathbf{z}$ by separating $\mathbf{U}=M(\mu\mathbf{x}+\mathbf{A}^{T}\mathbf{y}-\mathbf{c})$ into its positive and negative components (by diagonalization). The updated primal and dual solutions $\mathbf{x}$ and $\mathbf{z}$ are given by $M(\mathbf{x})=\mathbf{U}(+)/\mu$ and $M(\mathbf{z})=-\mathbf{U}(-)$ .

Here, $\tau$ is a step-length parameter that lies in the interval [1.0,1.6] ⁸⁷¹ Mazziotti D. A.
Phys. Rev. Lett.
(2011), 106, pp. 083001. Link . The penalty parameter $\mu$ controls how strictly the primal or dual constraints are enforced and is updated dynamically according to the protocol outlined in Ref. 871 Mazziotti D. A.
Phys. Rev. Lett.
(2011), 106, pp. 083001. Link . The frequency with which $\mu$ is updated is controlled by the $rem keyword RDM_MU_UPDATE_FREQUENCY. The algorithm is considered converged when the primal error $||\mathbf{A}\mathbf{x}-\mathbf{b}||$ , the dual error $||\mathbf{A}^{T}\mathbf{y}-\mathbf{c}+\mathbf{z}||$ , and the primal/dual energy gap $|E_{\rm primal}-E_{\rm dual}|$ are sufficiently small. The convergence in the primal/dual errors and the primal/dual energy gap are controlled by the $rem keywords RDM_EPS_CONVERGENCE and RDM_E_CONVERGENCE, respectively. The BPSDP algorithm scales $n^{6}$ for the DQG conditions and $n^{9}$ for the T1, T2, and 3POS conditions where $n$ is the number of active orbitals in the v2RDM computation.

In v2RDM-CASSCF, the BPSDP algorithm is carried out to determine the 1- and 2RDM for a subset of active molecular orbitals. These orbitals are optimized with respect to restricted doubly occupied / active and active / external rotations after a chosen number of v2RDM iterations (Steps 1. and 2. above). The frequency of this orbital optimization is controlled by the $rem keyword RDM_ORBOPT_FREQUENCY.

Given converged 1- and 2-RDMs from a v2RDM-CASSCF calculation, an estimate of the remaining correlation effects can be obtained through the formalism of MC-PDFT. In MC-PDFT, the total energy for the system is expressed as

E_{\rm MC-PDFT}=\sum_{i}h^{i}_{i}+\sum_{tu}h^{t}_{u}{}^{1}D^{t}_{u}+E_{\rm H}+% E_{\rm XC}\left[\rho({\bf r}),\Pi({\bf r}),|\nabla\rho({\bf r})|,|\nabla\Pi({% \bf r})|\right]

(6.90)

where the Hartree energy, $E_{\rm H}$ , is the classical Coulomb repulsion

E_{\rm H}=\frac{1}{2}\sum_{ij}(ii|jj)+\sum_{itu}(tu|ii){}^{1}D^{t}_{u}+\frac{1% }{2}\sum_{tuvw}(tu|vw){}^{1}D^{t}_{u}{}^{1}D^{v}_{w}

(6.91)

and $E_{\rm xc}$ represents an on-top pair density functional. In Eq. 6.91, the labels $i$ and $j$ represent doubly occupied spin orbitals, and the labels $t$ , $u$ , $v$ , and $w$ represent active spin orbitals. The symbols $\rho({\bf r})$ and $\Pi({\bf r})$ represent the density and on-top pair density, respectively, which are defined in terms of the molecular orbitals $\{\phi\}$ as

\rho({\bf r})=\sum_{pq}\phi_{p}({\bf r})\phi_{q}({\bf r}){}^{1}D^{p}_{q}

(6.92)

and

\Pi({\bf r})=\sum_{pqrs}\phi_{p}({\bf r})\phi_{q}({\bf r})\phi_{r}({\bf r})% \phi_{s}({\bf r}){}^{2}D^{pq}_{rs}

(6.93)

and $\nabla\rho({\bf r})$ and $\nabla\Pi({\bf r})$ represent the gradients of these quantities. In Q-Chem, $E_{\rm xc}$ is chosen to be a translated ⁸²⁸ Manni G. L. et al.
J. Chem. Theory Comput.
(2014), 9, pp. 3669. Link on-top pair density functional, which is essentially the same as a functional of the density and spin density (and their gradients), with the spin-density (and its gradient) re-expressed as a function of the on-top pair density (and its gradient). The specific choice of on-top pair density functional is controlled through the $rem variables PDFT_EXCHANGE and PDFT_CORRELATION.

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6.22.2 Theory