6.22.2 Theory

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

where the 1-RDM (${}^1𝐃$) and 2-RDM are represented in a given spin-orbital basis indexed by $p$, $q$, $r$, and $s$. The one-hole RDM (${}^1𝐐$), two-hole RDM (${}^2𝐐$), particle-hole RDM (${}^2𝐆$), partial three-particle RDMs ($\mathrm{𝐓𝟏}$ and $\mathrm{𝐓𝟐}$), and full three-particle RDMs (${}^3𝐃$, ${}^3𝐐$, ${}^3𝐄$, ${}^3𝐅$) are linear functions of ${}^1𝐃$ and ${}^2𝐃$.³²⁹ Minimizing the electronic energy with respect to ${}^2𝐃$ while enforcing the linear relations among these RDMs, the contraction and spin constraints placed on ${}^2𝐃$, 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.^{922, 727, 769}

The primal formulation of the SDP is

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

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

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

The BPSDP algorithm involves an iterative two-step procedure:

1. 1.

   Solve ${\mathrm{𝐀𝐀}}^T𝐲=𝐀(𝐜-𝐳)+\tau \mu (𝐛-\mathrm{𝐀𝐱})$ for $𝐲$ by conjugate gradient methods.

2. 2.

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

Here, $\tau$ is a step-length parameter that lies in the interval [1.0,1.6] ⁷⁶⁹. 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. 769. 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 $||\mathrm{𝐀𝐱}-𝐛||$, the dual error $||{𝐀}^T𝐲-𝐜+𝐳||$, and the primal/dual energy gap $|E_{\mathrm{primal}}-E_{\mathrm{dual}}|$ 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 molecular orbitals are optimized after a chosen number of v2RDM iterations (Steps 1. and 2. above) indicated by the $rem keyword RDM_ORBOPT_FREQUENCY.