4.6.2 Continuous Fast Multipole Method (CFMM)

The quantum chemical Coulomb problem, perhaps better known as the DFT bottleneck, has been at the forefront of many research efforts throughout the 1990s. The quadratic computational scaling behavior conventionally seen in the construction of the Coulomb matrix in DFT or HF calculations has prevented the application of ab initio methods to molecules containing many hundreds of atoms. Q-Chem Inc., in collaboration with White and Head-Gordon at the University of California at Berkeley, and Gill now at the Australian National University, were the first to develop the generalization of Greengard’s Fast Multipole Method³²⁵ (FMM) to continuous charged matter distributions in the form of the CFMM, which is the first linear scaling algorithm for DFT calculations. This initial breakthrough has since lead to an increasing number of linear scaling alternatives and analogies, but for Coulomb interactions, the CFMM remains state of the art. There are two computationally intensive contributions to the Coulomb interactions which we discuss in turn:

• •

  Long-range interactions, which are treated by the CFMM

• •

  Short-range interactions, corresponding to overlapping charge distributions, which are treated by a specialized “J-matrix engine” together with Q-Chem’s state-of-the art two-electron integral methods.

The Continuous Fast Multipole Method was the first implemented linear scaling algorithm for the construction of the $𝐉$ matrix. In collaboration with Q-Chem Inc., Dr. Chris White began the development of the CFMM by more efficiently deriving⁹⁸⁷ the original Fast Multipole Method before generalizing it to the CFMM.⁹⁹² The generalization applied by White et al. allowed the principles underlying the success of the FMM to be applied to arbitrary (subject to constraints in evaluating the related integrals) continuous, but localized, matter distributions. White and coworkers further improved the underlying CFMM algorithm,^{988, 989} then implemented it efficiently,⁹⁹³ achieving performance that is an order of magnitude faster than some competing implementations.

The success of the CFMM follows similarly with that of the FMM, in that the charge system is subdivided into a hierarchy of boxes. Local charge distributions are then systematically organized into multipole representations so that each distribution interacts with local expansions of the potential due to all distant charge distributions. Local and distant distributions are distinguished by a well-separated (WS) index, which is the number of boxes that must separate two collections of charges before they may be considered distant and can interact through multipole expansions; near-field interactions must be calculated directly. In the CFMM each distribution is given its own WS index and is sorted on the basis of the WS index, and the position of their space centers. The implementation in Q-Chem has allowed the efficiency gains of contracted basis functions to be maintained.

The CFMM algorithm can be summarized in five steps:

1. 1.

   Form and translate multipoles.

2. 2.

   Convert multipoles to local Taylor expansions.

3. 3.

   Translate Taylor information to the lowest level.

4. 4.

   Evaluate Taylor expansions to obtain the far-field potential.

5. 5.

   Perform direct interactions between overlapping distributions.

Accuracy can be carefully controlled by due consideration of tree depth, truncation of the multipole expansion and the definition of the extent of charge distributions in accordance with a rigorous mathematical error bound. As a rough guide, 10 poles are adequate for single point energy calculations, while 25 poles yield sufficient accuracy for gradient calculations. Subdivision of boxes to yield a one-dimensional length of about 8 boxes works quite well for systems of up to about one hundred atoms. Larger molecular systems, or ones which are extended along one dimension, will benefit from an increase in this number. The program automatically selects an appropriate number of boxes by default.

For the evaluation of the remaining short-range interactions, Q-Chem incorporates efficient J-matrix engines, originated by White and Head-Gordon.⁹⁹⁰ These are analytically exact methods that are based on standard two-electron integral methods, but with an interesting twist. If one knows that the two-electron integrals are going to be summed into a Coulomb matrix, one can ask whether they are in fact the most efficient intermediates for this specific task. Or, can one instead find a more compact and computationally efficient set of intermediates by folding the density matrix into the recurrence relations for the two-electron integrals. For integrals that are not highly contracted (i.e., are not linear combinations of more than a few Gaussians), the answer is a dramatic yes. This is the basis of the J-matrix approach, and Q-Chem includes the latest algorithm developed by Yihan Shao working with Martin Head-Gordon at Berkeley for this purpose. Shao’s J-engine is employed for both energies⁸³⁷ and forces,⁸³⁸ and gives substantial speedups relative to the use of two-electron integrals without any approximation—roughly a factor of 10 for energies and 30 for forces at the level of an uncontracted $dddd$ shell quartet, and increasing with angular momentum). Its use is automatically selected for integrals with low degrees of contraction, while regular integrals are employed when the degree of contraction is high, following the state of the art PRISM approach of Gill and coworkers.²⁴

The CFMM is controlled by the following input parameters: