Efficient Hybrid Density Functional Calculations For Large Periodic Systems Using Numerical Atomic Orbitals

First principles electronic structure calculation plays an important role in sev-eral research fields,including physics,chemistry,and materials science.The wide-spread application of first-principles methods are particularly due to the steady increase of the achieved computational accuracy and efficiency,and the availability of high-performance computing resources.Hybrid density functional(HDF)approximations are a type of first principles methods that can yield accurate results for a large range of chemical and physical properties.However,the computation cost of HDFs is much higher than the local and semi-local density functional approximations in typical imple-mentations,preventing their applications to large-scale systems.The primary objective of the present work is to develop numerical algorithms and computer codes within the framework of numerical atomic orbitals(NAO)that allow for efficient and massively parallel HDF calculations.Our implementation is based on a localized variant of the resolution-of-the-identity(LRI)technique,which enables a linear-scaling build of the Hartree-Fock exchange matrix-the computational bottleneck of HDF calculations.One of the key points of the LRI technique is the auxiliary basis functions(ABFs),the quality of which significantly affects the computation accuracy.The construction of high-quality ABF set is one of the emphases in this thesis.For one thing an ad-vanced scheme based on traditional on-site ABFs is proposed,reducing the number of ABFs without compromising the accuracy.For another we propose a new algorithm for constructing an optimized ABF(opt ABF)set.Different from the on-site ABFs which depend only on a single atom,opt ABFs take the chemical environment of a given system into account,and thus represent the system in a better way.Furthermore,the quality of the opt ABF set can be systematically improved by increasing the size of the bases,correcting the problematic cases where only on-site ABFs are not adequate.With these developments,better accuracy of the HDF calculations can be achieved at reduced computational cost.To develop an efficient and scalable HDF code,we employed several numeri-cal techniques in our implementation,including sparse matrices pre-screening,per-spective conversion of the computing framework,memoization of two-centre integral,acceleration of periodic calculation,a proper choice of the order of matrix multipli-cations,Cauchy-Schwarz inequality matrix screening and Cauchy-Schwarz inequality ERI screening.These numerical techniques,added together,lead to an efficient HDF implementation in the ABACUS code package.In addition,to achieve massively par-allel computation,we employ "Machine-scheduling distribution" and "K-means distri-bution" for parallel tasks distribution,and several parallel communication algorithms about density matrices and Hamiltonian matrices,which guarantee the extendibility of the program.In summary,a highly efficient and massively parallel implementation for HDF cal-culations within the NAO framework is presented.The wall time of HDF calculations increases perfectly linearly with the number of atoms,and decreases linearly with the in-verse number of CPU cores.HDF calculations for systems with thousands of atoms can now be routinely calculated at a relatively fast speed with relatively few computational resources.