Electron Repulsion Integral (ERI) and its derivative evaluation is the limiting factor for self-consistent-field (SCF) and Density Functional Theory (DFT) calculations. Therefore, calculation of these quantities on graphical processing Units (GPUs) can significantly accelerate quantum chemical calculations. Recurrence relations, one of the fastest ERI evaluation algorithms currently available, are used to compute ERIs. A direct-SCF scheme to assemble the Fock matrix and gradient efficiently is presented, wherein ERIs are evaluated on-the-fly to avoid CPU-GPU data transfer, a well known architectural bottleneck in GPU specific computation. A machine-generated code is utilized to calculate different ERI types efficiently. However, only s, p and d ERIs and s, p derivatives can be executed on GPUs using the current version of CUDA and NVidia GPUs. Hence, we developed an algorithm to compute f type ERIs and d type ERI derivatives on GPUs. Our benchmarks shows the performance GPU enable ERI and ERI derivative computation yielded speedups of 10~100 times relative to traditional CPU execution. An accuracy analysis using double-precision calculations demonstrates the accuracy is satisfactory for most applications. Besides ab inito quantum chemistry methods, GPU programming can be applied to a number of computational chemistry applications, for example, The Weighted Histogram Analysis Method (WHAM), a technique to compute potentials of mean force. We present an implementation of multidimensional WHAM on Graphical Processing Units (GPUs), which significantly accelerates its computational performance. Our test cases, that simulate two-dimensional free energy surfaces, yielded speedups up to 1000 times in double precision. Moreover, speedups of 2100 times can be achieved when single precision is used whose use introduces errors of less than 0.2 kcal/mol. These applications of GPU computing in computational chemistry can significantly benefit the whole computational chemistry community.
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