DIIS: A method to accelerate SCF convergence

Introduction

Direct Inversion of the Iterative Subspace (DIIS) is a method first proposed by Pulay¹ to accelerate the convergence of a standard self-consistent field (SCF) procedure.

Method

Iterative procedures generate a set of trial vectors $\mathbf{p}^i$. We can define an error term between successive iterations:

\[\mathbf{e}^i = \mathbf{p}^{i+1} - \mathbf{p}^i\]

The DIIS method assumes that the final solution can be extrapolated as a linear combination of the current iterative subspace:

\[\mathbf{p} = \sum_{i=1}^{m} c_i \mathbf{p}^i\]

The coefficients are obtained by minimization of the error vector

\[\mathbf{e} = \sum_{i=1}^{m} c_i \mathbf{e}^i\]

subject to the constraint

\[\sum_{i=1}^{m} c_i = 1\]

This leads to the standard Pulay linear system involving the B matrix, where

\[\mathbf{B}_{ij} = \mathbf{e}_i \cdot \mathbf{e}_j\]

Error vector in C-DIIS

The commutator-DIIS (C-DIIS) error vector is built from the requirement that the Fock matrix and density matrix commute in the orthonormal MO basis:

\[[\mathbf{F'}, \mathbf{D'}] = 0\]

This can be written in the AO basis as

\[\mathbf{e}_i = \mathbf{F}_i \mathbf{D}_i \mathbf{S} - \mathbf{S} \mathbf{D}_i \mathbf{F}_i\]

Algorithm

• Compute the error matrix in each SCF iteration.
• Build the B matrix from previous error vectors.
• Solve the Pulay linear system for the DIIS coefficients.
• Form the extrapolated Fock matrix and continue the SCF iterations.