4.5.1 Introduction

As for any numerical optimization procedure, the rate of convergence of the SCF procedure is dependent on the initial guess and on the algorithm used to step towards the stationary point. Q-Chem features a number of SCF optimization algorithms which can be selected via the $rem variable SCF_ALGORITHM, including:

Methods that are based on extrapolation or interpolation:

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  The highly successful DIIS procedures. These are the default (except for restricted open-shell SCF calculations) and are available for all orbital types (see Section 4.5.3). The damping and level-shifting technique ⁵⁴⁰ Hillier I. H., Saunders V. R.
  Proc. Roy. Soc. Ser. A
  (1970), 320, pp. 161. Link ^{, 1121} Saunders V. R., Hillier I. H.
  Int. J. Quantum Chem.
  (1973), 7, pp. 699. Link ^{, 465} Guest M. F., Saunders V. R.
  Mol. Phys.
  (1974), 28, pp. 819. Link can also be invoked together with DIIS (R, U only).

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  ADIIS: the combination of augmented Roothaan-Hall (ARH) energy function and DIIS developed by Hu and Yang, ⁵⁶⁴ Hu X., Yang W.
  J. Chem. Phys.
  (2010), 132, pp. 054109. Link which isavailable for R and U only.

Methods that make use of orbital gradient:

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  Direct Minimization (DM), which has been re-implemented as simple steepest descent with line search, and is available for all orbital types. DM can be invoked after a few DIIS iterations.

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  Limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm (L_BFGS), which is a quasi-Newton method using the gradients and steps taken in the previous iterations to construct the approximate Hessian.

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  Geometric Direct Minimization (GDM) which is an improved and highly robust version of DM and is the recommended fall-back when DIIS fails. Like DM, It can also be invoked after a few iterations with DIIS to improve the initial guess. GDM is the default algorithm for restricted open-shell SCF calculations and is available for all orbital types (see Section 4.5.7). GDM comes in three variants:

  • –

    GDM: the default implementation.

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    GDM_LS: Uses a line search at every iteration.

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    GDM_QLS: Employs a quadratic energy model to perform the line search at every iteration.

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  Square Gradient Minimization (SGM) family: SGM is a GDM-inspired method for excited state orbital optimization. Currently, three variants of this approach are supported: SGM (for RO and and OS_RO), SGM_LS (for R, U, RO and OS_RO) and SGM_QLS (for R and U), which are based upon the corresponding GDM approaches. For further details, see Section 4.5.14.

Methods that require orbital Hessian:

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  NEWTON_CG/NEWTON_MINRES (solve $\mathrm{𝐇𝐝}=-𝐠$ for the update direction with CG/MINRES solvers).

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  SF_NEWTON_CG (the “saddle-free" version of NEWTON_CG).

The analytical orbital Hessian is available for R/U/RO/G/CR unless special density functionals (e.g., those nonlocal functionals except for VV10 ¹³²⁰ Vydrov O. A., Van Voorhis T.
J. Chem. Phys.
(2010), 133, pp. 244103. Link ) are used, while the use of finite-difference Hessian is available for all orbital types by setting FD_MAT_VEC_PROD = TRUE.

In addition to these algorithms, there is also the maximum overlap method (MOM) which ensures that DIIS always occupies a continuous set of orbitals and does not oscillate between different occupancies. MOM can also be used to obtain higher-energy solutions of the SCF equations (see Section 7.6). The relaxed constraint algorithm (RCA), which guarantees that the energy goes down at every step, is also available via the old SCF code (set GEN_SCFMAN = FALSE). Nevertheless, the performance of the ADIIS ⁵⁶⁴ Hu X., Yang W.
J. Chem. Phys.
(2010), 132, pp. 054109. Link algorithm should be similar to it.

Since the code in GEN_SCFMAN is highly modular, the availability of different SCF algorithms to different SCF (orbital) types is largely extended in general. For example, the old ROSCF implementation requires the use of the GWH guess and the GDM algorithm exclusively. Such a limitation has been eliminated in GEN_SCFMAN based RO calculations.