Realizing Cubic-scaling Algorithm And Application Of GW And Bethe-Salpeter Equation With Plane-wave Basis Set

The first-principles calculations based on density functional theory(DFT)have been widely used to predict and study various physical and chemical properties of materials,and have become the third research area independent of experiments and theory.The success of density functional theory lies in the creative use of density as the functional of energy to convert the 3N-dimensional Schrodinger equation into a three-dimensional problem,which greatly reduces the computational complexity.However,density functional theory is a ground state theory based on the variational principle,which makes it have many problems in the application of excited states,such as wrongly predict the spectra and excited state energy levels of molecule and solid systems.At the same time,since there still has no accurate exchange-correlation functional,the bandgaps of semiconductor materials calculated by density functional theory are very dependent on the exchange-correlation functional,especially LDA and GGA will seriously underestimate the bandgap.Meanwhile,based on the wave function,Hartree-Fock theory is also based on the variational principle to obtain the groundstate energy and only considers the single-electron approximation,which makes it unable to accurately describe the properties of the excited state of systems and overstimate the bandgaps of semiconductors.Based on Green function,the many-body perturbation theory can solve the above difficulties to a certain extent,especially the GW approximation based on the single Green function can well describe the quasiparticle properties of semiconductor systems,and obtain bandgaps that are well consistent with experiments;The Bethe-Salpeter equation(BSE)based on two-body Green function can describe the motion of excitons,and obtain the excited properties of systems.This doctoral dissertation mainly studies the periodic system,so we start from density functional theory with plane-wave basis set.We study the implementation of the cubic-scaling algorithm of GW/BSE and its simple application in two-dimensional materials.It is roughly divided into the following six chapters.The first chapter is mainly to review the history of quantum many-body systems and density functional theory.The quantum many-body systems and the difficulty of solving the quantum many-body systems are briefly introduced.The BornOppenheimer(BO)approximation,which separates the motions of nucleus and electrons to simplify the calculation,is further introduced.Hartree-Fock theory is developed under the BO approximation which can reduce the computational complexity.However,Hartree-Fock theory is still a 3N-dimensional probelm.Using density as the functional of energy,the Schr(?)dinger equation can be reduce to three-dimensional problem,which greatly ruduces the computational complexity.The main problem of density functional theory is to find the exact exchange-correlation functional.In this chapter,we only introduce some commonly used exchange-correlation functional.Finally,we also introduce the common DFT software.The second chapter is about the many-body perturvation theory(MBPT)based on Green function.In this chapter,we first introduce the electronic structures of excited states and classification of excitons,and give a brief description of the development of GW/BSE.The concepts of Green function and quasi-particle are introduced by Feynman propagator,and the specific forms of one-body and two-body Green functions are given.In the GW section,the equation of motion of Green function,Dyson equation and Hedin’s equation are mainly described.Based on the equation of motion of Green function,the quasiparticle can be obtained by the self-energy term,which can be solved self-consistently.But starting from a given G0 and solving the iterate equations(scGW)are very expensive and does not give good results.Alternative way to solving GW approximation is by non-self-consistent calculations,which means stop at first order and proceed to calculate self-energy ∑,referred as G0W0,and we give the three common approximations of G0W0.Then we also describe in detail of Bethe-Salpeter equation(BSE)based on the two-body Green function and its difference from Time-dependent density functional theory(TDDFT).Finally,the commonly used software implementation GW and BSE methods are given.The third chapter mainly introduces the realizing cubic-scaling G0W0 calculations with plane wave basis set in periodic and molecular systems using low-rank decomposition algorithm and Cauchy’s integral.A perceived drawback of the G0W0 approximation is its high computational cost and large memory usage which is usually thought(1 or)2 orders of the magnitude more than a typical DFT calculation for thr same systems.The computational complexity of the canolical G0W0 algorith1 is O(Ne4～5),which is mainly because the G0W0 calculation contains multiple two-electron four-center integrals,so how to reduce the two-electron four-center integral is the main focus in this chapter.Firstly,we introduce the inteipolative separable density fitting(ISDF)algorithm.The implementation of ISDF algorithm requires two steps,first use QR factorization with columnpivoting(QRCP)or machine learning K-means algorithm(Kmeans)to obtain interpolation points(IPs),and then obtain interpolation vectors(Ⅳs)by least squares method.To realize the cubic-scaling of self-energy term,we use the ISDF algorithm to obtain the formulas of the low-rank decomposition of Khatri-Rao product,and then use the constracting ordering.However,the dielectric function contains the coupling term of occupied states and the empty states in the deniminator,here we use the Cauchy integral to decouple,and realize the cubic-scaling calculation of the static COHSEX.In the results section,we analyze the computational accuracy and efficiency of the ISDF low-rank decomposition algorithm in static COHSEX,which can achieve an～10× speedup for ISDF-based COHSEX calculations with a plane wave basis set.Using ISDF algorithm in static COHSEX,we find the inversion of dielectric function by LU decomposition become the main time-consuming term.In order to reduce memory and time consumption of dielectric function inversion,the inversion of dielectric function is inverted to the form of product of multiple small matrices using the Sherman-Morrison-Woodburg(SMW)formula.Finally,we demonstrate the computational accuracy of the high-precision GOW0 approximation-generalized plasmon-pole approximation(GPP)under the low-rank decomposition algorithm.The fourth chapter mainly introduces the realizing cubic-scaling BSE calculations with plane wave basis set in periodic and molecular systems using low-rank decomposition algorithm and iterative diagonalization algorithm.Firstly,we introduce the calculational process of BSE,and analyze the bottleneck of BSE calculation.it can be found that the BSE calculation requires two expensive steps,namely constructing the Bethe-Salpeter Hamiltonian(BSH)and diagonalizing the BSH to obtain the eigenvalues and eigenvectors.Firstly,we can reduce the calculational complexity of two-electron four-center integration by using the ISDF low-rank decomposition algorithm.Due to the special term of BSH,named screened exchange,which cannot directly use the iterative algorithm.Here we creatively propose to rearrange the wave vector of subspace and then use implicit iterative diagonalization to realize the cubic-scaling BSE calculations.In the results section,we analyze the computational accuracy and efficiency by using ISDF and itreative algorithms,and find the combination of ISDF low-rank decomposition algorithm and implicit iteration can achieve nearly 25 times speedup in plane wave basis set.The fifth chapter gives a simple application of GW+BSE in two-demensional ferromagnetic semiconductors.By analyzing the magnetic properties of various graphene quantum dots(GQDs),we select triangular zigzag graphene quantum dots(TZGDs),which are inherently ferromagnetic,as the basic unit to constructe the two-dimensional hexagonal lattice with periodic structure through the four-membered ring link of carbon atoms.By first-principles calculations,it is comfirmed that it is a ferromegnetic semiconductor and also a special bipolar magnetic semiconductor(BMS).Then we calculate its quasiparticle band structures and excited electronic properties using GW+BSE,and analyze the optical spectra and exciton wavefunctions.At the end of this chapter,we use the phonon spectra and AIMD to confirm the stability.Finally,the Curie temperature of the magnetic trasition is obtained through empirical formulas and classical Monte Carlo calculations based on the Heisenberg model and it is comfirmed that it is a room-temperature ferromagnetic semiconductor.In sixth chaper,we summarizes the main content of the doctoral dissertation,and look forward to the future development of excited state electronic structures.The appendix mainly introduces the mechanical properties of 2D materials.The main text is about the study of the excited state electronic structures,meanwhile the ground state electronic structures of materials are equally important.Using the firstprinciples calculations,monolayer β-Te has high carrier mobility,suitable bandgap and stable ubder ambient conditions.The maximum electron mobility of β-Te can be changed along armchair direction to along zigzag direction under strain-enginerring,which can realize the rotation of electrical conductance.In order to verify that β-Te can sustain large strain,we studied its stability and mechanical properties,which confirms that it will be useful in the design of stretchable electrical devices.The provious studies have confirmed that monolayer phosphorene or phosphorene-like materials have negative out-of-plane Poisson’s ratios.Our calculations find phosphorene-like materials also have in-plane Poisson’s ratios,which confirms that phosphorene-like materials are double auxetic materials.