4.5.10 Augmented Roothaan Hall Energy DIIS (ADIIS)

Similar to RCA/EDIIS,^{153, 156, 609} the ADIIS algorithm proposed by Hu and Yang ⁴⁹¹ is also supposed to accelerate SCF convergence in cases where DIIS performs poorly in the initial iterations. This algorithm also involves a Fock matrix extrapolation scheme:

where ${\widetilde{𝐅}}_{n+1}$ is the extrapolated Fock matrix to be diagonalized to generate the updated MOs and electron density, ${𝐅}_i=𝐅[{𝐏}_i]$ is the Fock matrix constructed from the density matrix of the $i$-th iteration, and $\{c_i\}$ are the extrapolation coefficients, which are obtained by minimizing the augmented Roothaan-Hall (ARH) energy function of an extrapolated density ${\widetilde{𝐏}}_{i+1}={\sum }_{i=1}^n{𝐏}_i$

while subjected to the constraint ${\sum }_{i=1}^n{c}_i=1,c_i\ge 0$ for all $i$. As suggested in the original literature, ⁴⁹¹ variable substitutions are conducted ($c_i=t_i^2/{\sum }_i{t}_i^2$) to convert the constrained optimization to a standard, unconstrained optimization problem (optimizers such as L-BFGS can be used to solve the latter). Note that while the sums in Eqs. 4.42 and 4.43 runs from 1 to $n$, in practice it is unnecessary to extrapolate using all ${𝐏}_i$ and ${𝐅}_i$ obtained in the previous SCF cycles. In the Q-Chem implementation of ADIIS, the number of ${𝐏}_i$’s and ${𝐅}_i$’s used in the extrapolation has a maximum of 6.

The ADIIS algorithm is known to become less efficient in the region close to SCF convergence. Therefore, it is desirable to use the “ADIIS+DIIS” algorithm, which carries out ADIIS when the SCF error is below a threshold or the number of ADIIS iterations reaches a certain value. This hybrid algorithm, which can be invoked by setting “SCF_ALGORITHM = ADIIS_DIIS” in Q-Chem, was shown to afford accelerated convergence for cases where DIIS alone was unable or took much longer to converge the SCF problem. ⁴⁹¹