Fock matrix formation

Some convergence problems are due to numerical accuracy problems. Many programs use reduced accuracy integrals at the beginning of the calculation to save CPU time. However, this can cause some convergence problems for difficult systems. A course DFT integration grid can also lead to accuracy problems, as can an incremental Fock matrix formation procedure. 

Ufimtsev and Martinez (UM) have also developed CUDA kernels for the calculation of ERIs and Fock matrix formation involving s- and p-type basis functions on GPUs [20,21]. They opted for the McMurchie-Davidson [22] scheme because it... 

The largest density matrix (or delta-density matrix) elements which will multiply any of the brakets associated with the quartet may be negligibly small [44,97]. This is particularly common in late SCF cycles when incremental Fock matrix formation is being used. 

Superlinear speedups for Fock matrix formation in the iterative part of the Harlree-Fock procedure as a consequence of storing a larger fraction of the integrals on each process as the number of processes increases. Speedups for the entire Hartree-Fock procedure are shown as well. Computations were performed on a Linux cluster for the uracil dimer using the aug-cc-pVTZ basis set (cf. Figure 8.3). A static task distribution of atom quartets was employed (see section 8.3 for details of the algorithm). 

In the remainder of this chapter we will discuss different ways to parallelize the computation of the two-electron integrals with focus on how to achieve load balance. Two-electron integral computation in a quantum chemistry code is often performed in the context of a procedure such as Fock matrix formation or two-electron integral transformation, but we here consider the integral computation separately to specifically concentrate on strategies for distributing the work. Examples of parallel two-electron integral computation in quantum chemical methods are discussed in chapters 8,9, and 10. 

Parallel Fock Matrix Formation with Replicated Data... 

Outline of a parallel algorithm for Fock matrix formation using replicated Fock and density matrices. A, B, C, and D represent atoms M, N, R, and S denote shells of basis functions. The full integral permutational symmetry is utilized. Each process computes the integrals and the associated Fock matrix elements for a subset of the atom quartets, and processes request work (in the form of atom quartets) by caUing the function get quartet. Communication is required only for the final summation of the contributions to F, or, when dynamic task distribution is used, in get quartet. 

Speedups for parallel Fock matrix formation using replicated density and Fock matrices. Speedups were obtained on a Linux duster for the uracil dimer with the aug-cc-pVTZ basis set and were computed relative to single-process tunings using measured wall limes. 

Outline of a parallel algorithm for Fock matrix formation using distributed Fock and density matrices. A, B, C, and D represent atoms, M, N, R, and S denote shells of basis functions, and only unique integrals are computed. 

Fock matrix formation algorithm. Again, we will discuss only the computation of the two-electron part of the Fock matrix. 

To further investigate the scalability of the Fock matrix formation algorithms as well as the effect of running multiple compute threads on a node, a series of runs were performed using two compute threads (and one communication thread) per node, enabling computations to be performed with up to 200 compute threads apart from the number of compute threads per node, the test case was identical to the one used above. The resulting speedups are... 

The Hartree-Fock method is central to quantum chemistry, and an efficient Hartree-Fock program is an essential part of a quanfum chemisfry program package. We oufline fhe Harfree-Fock procedure and presenf and analyze bofh replicafed dafa and disfribufed data Fock matrix formation algorithms. 

In this section we describe parallel versions of selected quantum chemistry algorithms. Electron repulsion integral evaluation and Fock matrix formation are discussed, and the performance of the computation of the HF energy is examined. We conclude with examples from MP perturbation theory. For each of these methods a variety of parallel algorithms exist. We illustrate a few selected, simplified algorithms, and develop performance models that permit comparison of the algorithms. 

Our first task in the parallelization of HF theory is the parallelization of the Fock matrix formation. The computation of the Fock matrix is dominated by the two-electron integral part G = F — A wide variety of approaches to the computation of G has been implemented. Here we will illustrate several possible schemes. Some of these will require that at least one processor is able to store all elements of the matrices involved, whereas in others storage of matrix elements is fully distributed. 

These approaches require that at least one processor of the parallel machine can store all elements of the matrices involved (G and D). This can be a severe limitation, because the sizes of molecules that can be treated could be limited by the memory available on one node rather than by the processing time required. However, this technique is commonly applied. It is the simplest way to sizes of the parallelize existing code and, if memory is abundant or the problems of interest are small or few processors are available, little will be gained by implementing the more complex distributed schemes. So we will consider in more detail two approaches to Fock matrix formation with nondistributed matrices. One uses a master-slave approach, while in the other the processors can work independently to produce a partial Fock matrix on each processor. 

The previously developed performance models now provide us with the tools to model the performance of a complete quantum chemistry calculation - the computation of the HF energy. The HF energy computation involves iterative solution of the Roothaan equations, FC = SCe (see Section 4.2). Depending on the implementation, this entails one Fock matrix formation, one diagonalization, and at least one matrix multiplication in each iteration. 

Fock matrix formation also involves the appreciably more complicated electron repulsion part. A procedure based on the symmetry invariance of the electron density was put forth in 1970. A number of algorithm improvements were made and the presently most-used nomenclature was introduced in 1977. ... 

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