• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    一維函數(shù)方法求解原子和分子薛定諤方程

    2021-07-11 16:30:16SARWONOYanoarPribadiURRAHMANFaiz趙潤東張瑞勤
    關(guān)鍵詞:香港城市大學(xué)物理系薛定諤

    SARWONO Yanoar Pribadi,UR RAHMAN Faiz,趙潤東,張瑞勤

    (1.香港城市大學(xué)物理系,香港;2.北京計算科學(xué)研究中心,北京100193;3.北京航空航天大學(xué)物理學(xué)院,北京100191;4.深圳京魯計算科學(xué)應(yīng)用研究院,深圳518131)

    1 Introduction

    Birkhoff[1]and Young[2]suggested that to obtain approximate solutions to the Schr?dinger differential equation with a rigorously numerical technique would be both interesting and desirable.This paper presents the theory and procedure for the application of the one-dimensional functions to the atomic and molecular calculations.One-dimensional hydrogen atom(1D H function),for example,has found many applications in high magnetic fields[3],semiconductor quantum wires[4],carbon nanotubes[5],and polymers[6].Nonetheless,the application of the 1D H functions to atomic and molecular calculations has not yet much investigated.Unlike Gaussian type orbitals(GTO)[7—10]which use the analytical and the recursive integration scheme of the twoelectron integrals calculation or Slater type orbitals(STO)[11,12]which have an extensive analytical energy expressions table[13]as well as basis size efficient,analytical evaluation of energy terms does not exist once the 1D functions such as the 1D H functions are employed.Hence,numerical methods must be applied.The 1D functions are regarded as more general basis functions since the 1D functions can generally repeat the result of the corresponding 3D counterparts but not otherwise,as in the Gaussian orbitals wheree-r2is the product of three 1D Gaussians inx,y,andzcoordinates[14]for the radial distancer= ( )x2+y2+z2.Unlike the standard basis functions,the 1D functions use Cartesian coordinates as a reference frame where common numerical methods of partial-differential equations are formulated so the iteration matrix can be simplified.The normalizability of theNdimensional wavefunction expanded with the 1D functions is satisfied by performing theNproduct of the 1D functions in each dimension like the particle in a box problem.Therefore,the 1D function approach is unique in two aspects.First,the approach is purely numerical methods to solve the Schr?dinger equation.Second,the approach is completely the first application of the 1D functions in atomic and molecular calculations.

    Another compelling reason to develop the 1D function approach is the difficulties of the standard electronic structure methods to treat electronic correlation;hence,an efficient many-body approach is required to deal with the many-body systems with accurate treatment of electron correlations.In the conventional selfconsistent field(SCF)calculations such as density functional theory(DFT)[15,16]and Hartree-Fock(HF)[17,18],the many-electron problems are proceeded with separating them into a set of accessible single-electron selfconsistent eigenvalue equations.Therefore,HF neglects the many-body effects of electron correlation,while DFT includes exchange-correlation functional to account for the effects[19—21].These separations of particles approaches are different in nature from the non-SCF accurate variational calculations of molecular(atomic)energy levels such as the James-Coolidge calculations[22,23](Hylleraas calculations and the hypersherical coordinates methods[24—32])for the H2molecule(the helium atom and its isoelectronic ions).The corresponding accurate variational calculations have tackled the Schr?dinger equations in their full dimensionality and as full many-electron problems.To solve the time-independent Schr?dinger equation of the many-electron atomic and molecular problems which in the present work we limit ourselves to two-electron cases,the employed 1D function approach follows neither the SCF techniques nor the accurate approaches.The 1D function approach assumes the separability ansatz of the many-electron by the Cartesian components,rather than by particle,as stated in recent papers[33—37].

    The primary significance of the 1D function approach over any such particle-separability approaches is to treat exactly the electron correlation.The approach can include many-body effects directly;hence,electron correlation is more accurately treated,as we shall demonstrate when the repulsion energy comparison is performed.Another key advantage is that the basis can be expanded in all Hilbert space directions at once.In the present work,we use real-space grids representation of the Hamiltonian and the wavefunction.The former is represented as sparse matrices and the latter as vectors.This brings to the next benefit which allows iterative methods of residual vector correction[33,34,36,37]to be implemented.Iterations that repeatedly apply the Hamiltonian operatoron a wavefunctionare conveniently carried out since the action of an operator on a wavefunction is implemented numerically as a straightforward matrix-vector product.Unlike the plane wave basis[35]integral evaluations which require additional steps such as the Fourier transform,the numerical integrals are easily performed in the real-space 1D functions basis with the row-vector matrix column-vector product times the power of dimensions of the grid spacing size as the scheme.This feature is particularly useful when dealing with the multicenter integral problem[38,39]present in molecular calculations.Unlike the previous partitioning scheme which divides an integral over the entire molecular volume into a sum of atomic orbitals[40],the 1D function approach does not require such schemes.The residual norm is progressively made small until convergence is reached.Hence,our approach is similar to the previous accurate variational calculations for helium and its isoelectronic ions and hydrogen molecules where SCF is not needed.Furthermore,the behavior of numerical convergence is known.

    In the present work,we present the applications of the 1D function approach to obtain the ground state of the hydrogen atom and the helium atom and their isoelectronic ions and the hydrogen molecule and its ions.The 1D function approach is preferably very general since the only approximations included are the change of a differential equation with a set of finite-difference equations.The resulting error in this approximation can be reduced to suitably small.However,in practice the approach is restricted by computer memory size and computer time cost.Like common perturbation and variational techniques which give accurate results for manyelectron problems provided the number of terms is large so that the correlation effects are properly treated,the present method can describe correlation properly if the grid spacing size is sufficiently small.Hence,any numerical error is understandable from the finite-difference approximation and the results can be straightforwardly compared with the experimental ones.In terms of many-electron problems,if the 1D function approach can reach to four-electrons calculations,large electronic structure problem can be considered as solvedviaa corresponding contracted Schrodinger equation[41].

    2 Theory

    2.1 Vectors and Functions

    In three-dimensional(3D)space,vectors r characterize points and are denoted with the 3-tuple

    On this space,scalar function is usually expressed asφ,

    In a separable solution,the scalar function can be written as the product of its spatial components

    where the functionsX,Y,andZare in 1D.Such problems are not considered in the 1D function approach,rather the nonseparable ones are studied.An example of a nonseparable problem is the Coulombic potential between charges at a distance

    Represented with the 3-tuple Cartesian components,no existence of the product functionφ(x,y,z)equal to the Coulombic potentialV(r).A closer approximation to the Coulombic potential is foundviaa sum of separable-like functions[42,43].

    2.2 Optimally Separable Hamiltonian

    In atomic units,the Hamiltonian of a system ofNelectrons andKnuclei with chargesZwithin Born-Oppenheimer approximations is

    where the first term is the kinetic energy of theith electron,the second term is the electron-electron interaction energy term,the third term is the ionic energy interaction betweenith electron andlth nucleus located at Rl,and the last term is the nuclei repulsion term.The indexiandjrefer to the electrons andl,l′to the nuclei.TheN-electron wavefunctionψ(r1,r2,…,rN)satisfies the Schr?dinger equation

    whereEis the energy of the system.

    Our 1D approach does not necessitate any coordinate transformation from the original Schr?dinger equation formulated in the Cartesian coordinate.Because of the rapid increase of the dimensional size,few-electron problem is suitable for this approach.For a one-electron system found in the hydrogen atom and its isoelectronic series or in the hydrogen molecular ion,a wavefunction constructed by 1D functions has the general form of

    whereφ(x)is an eigenfunction of some 2D single-component one-electron effective HamiltonianSuch basis forms an optimally separable approximate Hamiltonian

    whereφ(x1,x2)is an eigenfunction of some 4D single-component two-electron effective HamiltonianSimilarly,such basis will form an optimally separable approximate Hamiltoniantogether with the 2D single-component two-electron identity operator

    In terms of the 1D single-component one-electron identity operatorand the 2D single-component oneelectron effective Hamiltonianthe optimally separable Hamiltonian for a two-electron system of 6D is written as

    2.3 One-dimensional Functions as the Basis

    The types of 1D functions to be considered generally fall into two categories.First is the one that is the solution of the corresponding 1D equation[44—50],exemplified in the case of the 1D hydrogen(1D H)function which is a symmetric and square-integrable function with the cusp at origin.The expression of the 1D functions in this category is generally very intricate like the composite of the products of more than one simple functions.In practice,the functions are conveniently found through a numerical method.The second category is the one which is the 1D version of the corresponding 3D functions counterpart,such as 1D Gaussian and 1D Slater.Their expressions are simpler than the first category with their properties resembling their 3D functions like the origin-centered 1D Gaussian[51—53]or the cusp possessed in the 1D Slater[54—57].This serves their purposes for convenient integration.In the present work,the 1D H function is implemented to obtain the solution of the hydrogen atom[33].Subsequently,a two-dimensional single-component two-electron of helium and its isoelectronic ions wavefunction is applied to the helium and its isoelectronic ions calculations[33,34,37].Similarly,the ground state of a molecule can be constructed from its single component[36].

    The expression of the 1D H[47]function is given as

    whereAis the normalization constant,nis the principal quantum number,andis the generalized Laguerre polynomial,andkcorresponds to the 1D H energyEasThe trial solutions for the hydrogenic atoms are then constructed from their single component as

    wherecijkare constants andNis the number of the grid points in one-dimension.In this basis,the optimally separable Hamiltonianis represented.As the trial wavefunction Eq.(13)is the eigenstate of thethe eigenvalueE(0)is less superior than the ground state energy of the hydrogenic system.Next,we choose a 2D single-component two-electron of helium and its isoelectronic ions wavefunctionφ(x1,x2)as the basis set for the helium and its isoelectronic ions calculations.The 2D wavefunctionφ(x1,x2)is the solution of the corresponding 2D He Schr?dinger wave equation

    where the first two terms are the kinetic energy terms in each dimensions,the third and the fourth terms are the electron interaction terms with a nucleus at the center,and the last term is the electron-electron interaction term,andζis the corresponding energy eigenvalue of the 2D He.The constantβ=0.00001 has been added to avoid the infinite solution atxi=0,i=1,2[50].In the grid representation(x1i,y1j,z1k,x2l,y2m,z2n),the trial wavefunction is written as

    whereNis the number of grid points of 1D functions in one component andcijklmnis the coefficients of expansion.Thus for two-electron generalization,the basis size grows asN6.

    Like the atomic case,a trial variation function to describe the ground state of a molecule can be constructed from its 1D components.For example,thetrial variation functionψ(0)(r)can be constructed from its 1D componentsφ(x)with coefficients

    whereφ(x)satisfies the corresponding 1D Schrodinger equation of

    for two nuclei atxaandxbwith the 1D energy in one componentεx.For a two-electron molecule such as H2,the trial variation function can be expanded with the product of its 2D two-electron componentsφ(x1,x2)

    The basis functionφ(x1,x2)is obtained from the 2D Schr?dinger equation version of H2

    2.4 Space Discretization and Multidimensional Numerical Integration

    To represent theN-electron Hamiltonian Eq.(6),uniform orthogonal 3N-dimensional grids of equal spacing sizehneed to be generated.Each side of the uniform 3N-D hypercubic grids is of length 2r0.The uniform real-space grids allow for a simple discretization of the Schr?dinger equation so that physical quantities like wavefunctions and potentials are numerically represented by the values at the grid points.The maximum range of each position coordinater0is related with the grid spacing sizehand the number of the grid pointsThe number of the grid pointsNis determined up to computational resources,while the optimum grid spacing sizehis chosen to minimize energy.Derivatives are calculated with finite-differences where in the present work the standard-order is implemented.The Laplacian is the sum of the second-order derivative in each dimension.Therefore,for a two-electron case,six second-order derivatives are constructed.Hence,the accuracy of the discretization is determined by the grid spacing size and the finite-difference order used for the derivatives.The potential energy operators are represented as diagonal matrices as a consequence of a certain quadrature approximation[58].The Coulombic singularity is avoided by setting even number of grid pointsN.The electron repulsion is handled by restricting the two electrons not to occupy the same space.

    Having put all quantities in the real-space grid representations where the operators are represented as matrices and the wavefunctions as vectors,one can perform the convenient multidimensional numerical integration over the Cartesian grids where the accuracy of the scheme largely depends on the number of grid points and the grid spacing size.The action of the operator on the wavefunction is simply a straightforward matrixvector product.In the case of the H2matrix elements calculations,the integrals of the HamiltonianH(r1,r2)over 6D space transform into sum over grid pointsm,nof grid spacing sizehin the grid points positions rm,rn

    The multidimensional integral scheme above clearly provides a convenient alternative route to evaluate any integrals present in the atomic and molecular calculations beside the analytical product theorem formulation of the Gaussian type orbitals or the extensive tables of Slater type orbitals.Unlike the plane wave basis,the integrals evaluation of the 1D functions does not require reciprocal space or Fast Fourier transform library.

    2.5 Residual Vector Correction

    For ground state calculation,the residual vector correction which is based on iterative Krylov subspace methods of orderO(N2)is more suitable than the conventional diagonalization of orderO(N3).The reason is that the Hamiltonian operators are large sparse matrices in the real-space grid representation.Therefore,the computational cost of exact diagonalization scheme becomes very expensive.On the other hand,the sparsity of the matrices causes the iterative diagonalization which involves the matrix-vector multiplication more appealing.The central quantity in the iterative method is the residual function.In the two-electron case atmiteration,the residual functionobtained from the trial wavefunctionand its energyE(m)is in the form of

    The definition of the residual above allows the search to be performed in the direction of the greatest decrease,unlike the preconditioned residual as a consequence of the perturbation theory[59,60].A new and refined wavefunctionψ(m+1)(r1,r2)is given as

    which is the linear combination of the residual functionand the current resultThe expansion coefficientsC0andC1provide the necessary step of the search to minimize the energy withC0approaching unity andC1becoming increasingly small.They can be determined from another smaller 2×2 generalized eigenvalue problem in themstep

    where in the iterative subspace the Hamiltonian matrixHij,the overlap matrixthe optimized energyE,and the trial stepsCj,j=0,m.The iteration is terminated once the difference between the consecutive energiesEin Eq.(23)achieves 0.0003 eV.

    3 Results and Discussion

    3.1 Hydrogen Atom and Its Isoelectronic Ions from 1D H Function

    To test the approach,a single-electron calculation of the hydrogen atom and its isoelectronic ions ground electronic state are performed.The exact ground state energy of the H,He+,Li2+,Be3+are known analytically to be-13.6057,-54.4228,-122.4513 and-217.6912 eV,respectively[61].The results of the 1D function approach can then be straightforwardly assessed in terms of accuracy and convergence.In the hydrogenic cases although only one electron is present and no electron correlation effect,the test is still important,particularly to demonstrate how the Coulomb singularity in the potential energy is to be handled.One can adjust the number of grid points so that none of the grid points lies directly on the nucleus.

    Once the Hamiltonian is represented with the finite-difference method,the residual vector correction Eqs.(21),(22),and(23)is implemented to perform the necessary refinements to obtain more accurate wavefunction.The plot of the coefficientsC0andC1are shown in Fig.S1(see the Supporting Information of this paper).The negative ratio ofC0/C1shows the correct steepest descent search direction where the negative gradient of the Rayleigh quotient is used.TheC0value that is closer to one ensures the correction occurs along the residual vector and the orthonormality with the previous search direction is maintained.The positive coefficientC1means that the gradient of the Rayleigh quotient being in the same direction as the residual unlike the linear system case.The gradient vector trial step is large in the several first iterations.The size ofC1is largely affected by the implemented number of grid pointsNand the grid spacing sizeh,which is understood becauseNdetermines the matrix size of the Hamiltonian whilehaffects the location of the ground state energy through the numerical integration scheme.Correspondingly,the search performed byC1goes to the correct ground state and prevents from moving to the wrong state.Near the convergence pointC1shows a see-saw nature until convergence,showing the stability of the residual vector correction.

    The Rayleigh-quotient minimization performed with the residual vector correction has straightforward effect on the iterated energy and the virial ratio shown in Fig.S2(see the Supporting Information of this paper).In the variational process to obtain a converged energy,the iterated energy is corrected starting from the upper bound until convergence[62].As the energy is refined,the virial ratio shows improvement to be closer to the exact value of two.

    For the shortest grid spacing sizehwhich has the largest number of grid points per dimensionN,the ground state energy of the hydrogen and its isoelectronic ions are converged to a 0.0027 meV or even better(Table 1).This one-electron calculation is a proof of concept that the 0.0027 meV or better accuracy is reachable with the 1D function approach.As seen,the error increases for largerZ.This stems from the fact that in all calculations the number of grid points per Cartesian dimensionNused isN=400 despite heavier nuclei require finer grid spacing size to account for their Coulomb attraction potential and their kinetic energy shown in Table S1(see the Supporting Information of this paper).The kinetic energy and the ionic energy error show the tendency to have increasing error for largerZ,nonetheless the virial ratio remains in the accuracy of 2.0 in the present calculations.The improvement can be proceeded with a technique to evaluate the ionic energy more accurately without requiring too finer grid spacing size known as electron centers displacement[37]which will be demonstrated in the helium atom and its isoelectronic ions calculations part.The Coulombic singularity of the nuclear attraction can also be better treated with the use of various approximations such as“softened”Coulomb potentials[63],reduced dimensionality of physical space[64],approximate 3D discrete variable representation potentials[65,66],or the use of the logarithmic grids[14,67,68]followed with the necessary coordinate transformation of the equation.The kinetic energy may be improved using either higher order second order derivative finite-difference method[69]or analytical form of discrete variable representation kinetic energy matrix[70,71].

    Table 1 Ground state energy of the hydrogen atom and isoelectronic series obtained with the 1D function approach and the exact value a

    If one plots the error relative to the exact energy as a function of the grid spacing sizehas shown in Fig.S3(see the Supporting Information of this paper),a simple quadratic equation can be obtained.This is seen from the value ofR2which is nearly equal to one.The quadratic scaling is anticipated because of the Coulombic singularity,the choice of the Cartesian 1D H basis,and the finite-difference approximation.Such scaling shows that the convergence is fast in the region far from the accurate but slow close to the accurate value.From Fig.S3 the required grid spacing size to reproduce error free relative to exact energy can be calculated from finding the root of the quadratic equation.The roots confirm that finer grid spacing size is required to describe energy more accurately for atoms with an increasingZ.

    In the following,the wavefunction analysis is given.First,the obtained wavefunction can be surfaceplotted against thexandycoordinate,displaying a cusp,a characteristic of 1s,shown in the inset of Fig.S4(A)(see the Supporting Information of this paper).The cusp can be reproduced and the vanishing wavefunction far from nucleus is seen.Second,as seen from Fig.S4(A)the obtained wavefunctions coincide with the exact wavefunctions when plotted against the radial distance.The ion Be3+has the steepest slope and results in the highest cusp than that of other species with lowerZ.The phenomenon is related to the kinetic and the potential energy to conserve the Heisenberg uncertainty principle.Third,F(xiàn)ig.S4(B)shows the radial distribution function of the hydrogen atom and its isoelectronic ions.A single peak of 1scharacteristic is observed.For heavier atoms/ions,the peaks are closer to the nucleus than those of lowerZ.

    3.2 Helium Atom and Its Isoelectronic Ions from 2D He Function

    The developments of the coefficientsC0andC1are shown in Fig.S5(see the Supporting Information of this paper).Overall,the performance of the optimization is the same as that implemented in the hydrogenic cases.The coefficientC0progresses toward one with the opposite sign compared with the coefficientC1,indicating the search occurs in the steepest descent and in the same direction as the negative of the Rayleigh quotient gradient.The magnitude ofC0in the helium optimization is about three times larger than that in the hydrogen cases,corresponding to the larger grid spacing size implemented in the helium case due to the fast-growing scale of the required basis functions.The grid spacing size affects the evaluation of the Rayleigh quotient in the minimization process through the implemented numerical integration scheme which depends onC1.As the Rayleigh quotient minimization is a variational process,the iterative energy of the helium approaches the true energy variationally.Consequently,the virial ratio-V/Tmoves closer to the exact value of 2 which is depicted in Fig.S6(see the Supporting Information of this paper).

    The repulsion energy convergence of the helium atom and its isoelectronic ions where the number of grid points per dimensionN=30 and where the electron centers displacement technique has been implemented[37]are shown in Table 2.The repulsion energy where the non-classical effect is present possesses error within 1.3606 eV or better.In all calculated atomic systems,the calculated repulsion energies are more accurate than the HF results[72]which can reach 2.7211 eV in error.The reason is that the pair-wise electron approach of the 1D function approach treats directly and exactly the electron-electron interaction.If considerably finer grid spacing size and more number of grid points are applied,one will obtain the accurate result for the integration scheme for the repulsion energy.

    Table 2 Electron-electron effect present in the repulsion energy of the total energy component of the helium atom and its isoelectronic ions

    If one plots the error relative to the accurate resultsversusthe grid spacing size,a simple relation will be found from which the required grid spacing size to reproduce error free relative to accurate energy can be obtained[37].In contrast to the hydrogen and its isoelectronic case,the helium case shows a power scaling with the power less than 2.This situation is understandable since the energy of the helium is obtained with larger grid spacing size than that of the hydrogen atom and causes the behavior of the error in the finite-difference method has not reached quadratic scaling.The two-electron wavefunction of the helium atom and its isoelectronic series allow for single-electron functions plot such as single-electron wavefunction,densities,and radial distribution.Fig.1 shows the ground state radial distribution function obtained with the HF,the DFT,and the 1D function approach.Compared with the single-electron methods,the electron correlation effect is seen in the decrease of the probability to find an electron at radial distance between 0.4 and 1.5 bohr[73].

    Fig.1 Radial distribution function obtained with the 1D function approach compared with that obtained with the DFT method and the HF method

    The scaled radial distribution function for the ground state of the helium atom and its isoelectronic ions are given in Fig.S7(see the Supporting Information of this paper).Consistent with the 1s1scharacter of the ground state,a single peak is observed.Fig.S7 also shows that for heavier atoms the peaks are closer to the nucleus.

    3.3 Hydrogen Molecule and Ions from 1D H Function

    The performance and the behavior of the implemented residual vector correction to obtain the ground state and the bond length of molecule are the same as those in the atomic case.The coefficientC0trend toward unity after several number of iterations is observed.CoefficientC1is progressively small and shows“see-saw”pattern.The search occurs in the gradient descent direction whereC0andC1have opposite sign.The magnitude ofC1is influenced by the implemented grid spacing size so thatC1is larger in the H2case than in the H+2case.Inherent in the implemented residual vector correction,the true energy is achieved variationally that is from rigid upper bound to the converged value.

    As seen in Table 3,the present grid specification leads to the energy of the hydrogen molecule and ions which is equal to or more accurate than that of the Hartree-Fock calculation with large basis set(HF/ccpVQZ)[23,72,74─78].The one-electron molecule of H+2allows us to use such finer grid spacing size that an energy accuracy within 0.0027 eV is reached and more improved than that of Ref.[79].The integral evaluations of the kinetic,the ionic,and the repulsion energy of the molecules are presented in Table S2(see the Supporting Information of this paper)with the repulsion energy obtained with the 1D function approach being more improved than that of the HF/cc-pVQZ.

    Table 3 Ground state energy of hydrogen molecule and ions and helium molecule ion obtained with the 1D function approach a

    The equilibrium bond length calculated with the 1D function approach at the present calculation shown in Table 4 is in well agreement with the HF method and the accurate one.

    Table 4 Equilibrium bond length obtained with the 1D function approach,the HF,and the accurate method

    The electron density of the H2obtained with the present method and the HF method differ to each other due to the included electron correlation effect shown in Fig.S8(see the Supporting Information of this paper).The probability to find an electron is enhanced in the left and right anti-binding regions and in the intermediatebinding regions of each of the two nuclei[80].

    4 Conclusions

    In contrast to the conventional electronic structure methods which assume the particle separation,the 1D function approach solves the atomic and molecular Schr?dinger equations by employing their corresponding one-dimensional functions to single out the components.The 1D function scheme is capable of applying onedimensional Cartesian function as the basis,and is specialized for obtaining the ground state of many-electron atomic and molecular systems.In the one-electron(two-electron)system,the corresponding single-component one-electron(two-electron)wavefunction is the choice for the optimal basis for optimally separable Hamiltonian.The accurate Schr?dinger eigenfunction is obtained after successive refinements with a Krylov based iterative method of residual vector correction.The 1D function approach is proven simple,efficient,and applicable to neutral and charged atomic and molecular systems.Accurate treatment of the many-body effects shown in the electron correlation is the key advantage of the 1D function approach which separates the components unlike the Hartree-Fock or density functional method which separate the particles.The results of the molecular calculations indicate that the 1D functions implementation evaluates the multi-center molecular integration without any partitioning of molecular systems into single-center terms and without any Fourier transform required.To develop the approach to calculate larger size atomic and molecular systems with improved accuracy,one can proceed with the use of simply more powerful computing resources or the technical improvements in the underlying theories such as HF or DFT framework.Another potential development includes the calculations of the excited states with modifications in the residual vector correction part.Nonetheless,the 1D function approach completely serves a numerical technique to obtain explicitly the solution of the Schr?dinger equation.

    Acknowledgements

    We acknowledge the Beijing Computational Science Research Center,Beijing,China,for allowing us to use its Tianhe2-JK computing cluster.

    Supporting information of this paper see http://www.cjcu.jlu.edu.cn/CN/10.7503/cjcu20210138.

    This paper is supported by the National Natural Science Foundation of China-China Academy of Engineering Physics(CAEP)Joint Fund NSAF(No.U1930402).

    猜你喜歡
    香港城市大學(xué)物理系薛定諤
    擬相對論薛定諤方程基態(tài)解的存在性與爆破行為
    Chern-Simons-Higgs薛定諤方程組解的存在性
    電子信息與物理系簡介
    香港城市大學(xué)“重探索、求創(chuàng)新”課程教學(xué)改革的路徑探索與啟示
    一類相對非線性薛定諤方程解的存在性
    薛定諤的餡
    幽默大師(2019年6期)2019-01-14 10:38:13
    廣西師大社與香港城市大學(xué)出版社達(dá)成戰(zhàn)略合作
    出版人(2017年8期)2017-08-16 11:05:27
    香港城市大學(xué)今年擬在內(nèi)地招生211名
    高校招生(2017年1期)2017-06-30 08:38:38
    行在科研 育在四方——記清華大學(xué)工程物理系副教授黃善仿
    名校校訓(xùn)
    脱女人内裤的视频| 别揉我奶头~嗯~啊~动态视频| 老汉色∧v一级毛片| 欧美色欧美亚洲另类二区 | 免费观看精品视频网站| 精品久久久久久成人av| 国产成+人综合+亚洲专区| 国产精品 欧美亚洲| 欧美成人免费av一区二区三区| 免费在线观看亚洲国产| 给我免费播放毛片高清在线观看| 国产精品久久视频播放| а√天堂www在线а√下载| 亚洲九九香蕉| 亚洲人成77777在线视频| 午夜亚洲福利在线播放| 亚洲国产精品sss在线观看| 99精品欧美一区二区三区四区| 免费高清在线观看日韩| 久9热在线精品视频| 午夜福利成人在线免费观看| 日日爽夜夜爽网站| 亚洲专区中文字幕在线| 亚洲成人国产一区在线观看| av网站免费在线观看视频| 日本一区二区免费在线视频| 啦啦啦观看免费观看视频高清 | 亚洲九九香蕉| 真人一进一出gif抽搐免费| √禁漫天堂资源中文www| 美女高潮喷水抽搐中文字幕| 欧美日韩亚洲综合一区二区三区_| 日韩大码丰满熟妇| √禁漫天堂资源中文www| 欧美乱码精品一区二区三区| 午夜激情av网站| 亚洲视频免费观看视频| 久久精品91蜜桃| 在线观看免费视频日本深夜| 国产精品98久久久久久宅男小说| 日韩国内少妇激情av| 日韩精品中文字幕看吧| 午夜两性在线视频| av在线播放免费不卡| 国产精品影院久久| 亚洲伊人色综图| 国产精品综合久久久久久久免费 | 在线观看免费日韩欧美大片| 露出奶头的视频| 91麻豆精品激情在线观看国产| 波多野结衣av一区二区av| 午夜两性在线视频| 黑人巨大精品欧美一区二区mp4| 国产视频一区二区在线看| 亚洲五月色婷婷综合| 日日夜夜操网爽| 久久久久国产一级毛片高清牌| 国产国语露脸激情在线看| 国产精品九九99| 午夜福利高清视频| 日日爽夜夜爽网站| 免费在线观看黄色视频的| 国产成人一区二区三区免费视频网站| 色精品久久人妻99蜜桃| 人成视频在线观看免费观看| 国产精品久久久久久人妻精品电影| 亚洲成人久久性| 欧美久久黑人一区二区| 欧美色视频一区免费| 91成年电影在线观看| www.自偷自拍.com| 美国免费a级毛片| 亚洲av电影不卡..在线观看| 青草久久国产| 国产三级黄色录像| 亚洲精品在线美女| 非洲黑人性xxxx精品又粗又长| 日韩欧美免费精品| 男女做爰动态图高潮gif福利片 | 亚洲一区中文字幕在线| 丝袜人妻中文字幕| 天堂√8在线中文| 老熟妇仑乱视频hdxx| 久久青草综合色| 18禁美女被吸乳视频| 亚洲片人在线观看| 精品一区二区三区视频在线观看免费| 人妻丰满熟妇av一区二区三区| 国产单亲对白刺激| 免费不卡黄色视频| 91成人精品电影| 亚洲精品国产区一区二| 天天躁夜夜躁狠狠躁躁| 亚洲美女黄片视频| 国产精品久久久久久亚洲av鲁大| 国产精品亚洲av一区麻豆| av片东京热男人的天堂| 夜夜夜夜夜久久久久| 久久久国产成人精品二区| 成熟少妇高潮喷水视频| 制服人妻中文乱码| 日日爽夜夜爽网站| 色婷婷久久久亚洲欧美| av天堂在线播放| 99国产精品免费福利视频| 亚洲精华国产精华精| 精品午夜福利视频在线观看一区| 少妇熟女aⅴ在线视频| 搡老岳熟女国产| 丝袜美腿诱惑在线| 女人被狂操c到高潮| 在线播放国产精品三级| 免费在线观看日本一区| 亚洲一区二区三区不卡视频| 亚洲av片天天在线观看| 亚洲精品国产区一区二| 久久精品国产亚洲av香蕉五月| 制服丝袜大香蕉在线| 国产麻豆成人av免费视频| 亚洲欧美日韩高清在线视频| 日韩中文字幕欧美一区二区| 久久精品国产亚洲av香蕉五月| 国产伦人伦偷精品视频| 视频在线观看一区二区三区| 成人国语在线视频| 女警被强在线播放| 亚洲一区高清亚洲精品| 国产一区二区在线av高清观看| 日韩精品免费视频一区二区三区| 深夜精品福利| 久久天躁狠狠躁夜夜2o2o| 大码成人一级视频| 曰老女人黄片| 国产一区在线观看成人免费| 99在线人妻在线中文字幕| 亚洲人成伊人成综合网2020| 亚洲中文日韩欧美视频| 人人妻人人澡欧美一区二区 | 国产欧美日韩综合在线一区二区| 亚洲精品av麻豆狂野| 女人爽到高潮嗷嗷叫在线视频| 午夜免费成人在线视频| 免费女性裸体啪啪无遮挡网站| 日韩一卡2卡3卡4卡2021年| 成人精品一区二区免费| 国产高清视频在线播放一区| or卡值多少钱| 黄色丝袜av网址大全| 9热在线视频观看99| 天天躁夜夜躁狠狠躁躁| 人人妻人人澡欧美一区二区 | 18禁美女被吸乳视频| 国产熟女午夜一区二区三区| 可以在线观看毛片的网站| 在线观看66精品国产| 日本 av在线| 大型av网站在线播放| 欧美丝袜亚洲另类 | 欧美+亚洲+日韩+国产| 老汉色av国产亚洲站长工具| www国产在线视频色| 亚洲精品一卡2卡三卡4卡5卡| 久久婷婷人人爽人人干人人爱 | 日日摸夜夜添夜夜添小说| 女人高潮潮喷娇喘18禁视频| 麻豆av在线久日| 久久久国产成人精品二区| 中文字幕av电影在线播放| 亚洲国产日韩欧美精品在线观看 | 狠狠狠狠99中文字幕| 香蕉国产在线看| 婷婷丁香在线五月| 啦啦啦韩国在线观看视频| 12—13女人毛片做爰片一| 亚洲视频免费观看视频| 久久中文字幕人妻熟女| 亚洲精品久久国产高清桃花| 真人一进一出gif抽搐免费| 久久精品国产99精品国产亚洲性色 | 两人在一起打扑克的视频| 精品国内亚洲2022精品成人| av电影中文网址| 久久人人精品亚洲av| 亚洲男人天堂网一区| 欧美乱妇无乱码| 禁无遮挡网站| 老司机午夜福利在线观看视频| 亚洲第一电影网av| 无限看片的www在线观看| 99国产极品粉嫩在线观看| 美女午夜性视频免费| 国产成人精品久久二区二区91| 国产成人精品久久二区二区91| 法律面前人人平等表现在哪些方面| 又黄又爽又免费观看的视频| 午夜福利,免费看| 精品欧美国产一区二区三| 国产在线精品亚洲第一网站| 成年版毛片免费区| 国产一卡二卡三卡精品| 久久精品国产亚洲av高清一级| 亚洲免费av在线视频| 久久精品亚洲精品国产色婷小说| 美女免费视频网站| 一个人观看的视频www高清免费观看 | 夜夜夜夜夜久久久久| 久久国产亚洲av麻豆专区| 亚洲自偷自拍图片 自拍| 国产xxxxx性猛交| 美女免费视频网站| 一级片免费观看大全| 中出人妻视频一区二区| 欧美日本视频| 日本欧美视频一区| 国产精品久久久久久人妻精品电影| 一边摸一边抽搐一进一小说| 99在线视频只有这里精品首页| 超碰成人久久| 国产精品二区激情视频| 国产精品久久电影中文字幕| 亚洲欧美激情在线| 狠狠狠狠99中文字幕| 99精品久久久久人妻精品| 日本三级黄在线观看| 一级毛片精品| 变态另类丝袜制服| 国产不卡一卡二| 日韩精品免费视频一区二区三区| avwww免费| 久久久精品欧美日韩精品| 久久久久九九精品影院| 老司机午夜福利在线观看视频| 亚洲精华国产精华精| 日韩大尺度精品在线看网址 | 人成视频在线观看免费观看| 麻豆国产av国片精品| 久久草成人影院| 一级毛片精品| 一本综合久久免费| 悠悠久久av| 国产精品香港三级国产av潘金莲| 一个人免费在线观看的高清视频| 人成视频在线观看免费观看| 精品熟女少妇八av免费久了| 亚洲成a人片在线一区二区| 50天的宝宝边吃奶边哭怎么回事| 少妇 在线观看| 亚洲国产精品999在线| 日韩欧美免费精品| 老司机午夜福利在线观看视频| 啦啦啦韩国在线观看视频| 97碰自拍视频| 99国产精品一区二区蜜桃av| 看黄色毛片网站| 中文字幕人成人乱码亚洲影| 亚洲人成电影免费在线| 亚洲色图av天堂| 午夜免费观看网址| 热99re8久久精品国产| 亚洲国产日韩欧美精品在线观看 | 精品一区二区三区视频在线观看免费| 国产黄a三级三级三级人| www国产在线视频色| 国产乱人伦免费视频| 国产一区在线观看成人免费| 国产精品自产拍在线观看55亚洲| 黑丝袜美女国产一区| 满18在线观看网站| 国产三级黄色录像| 久久久久久国产a免费观看| 99国产精品一区二区蜜桃av| 欧美日韩福利视频一区二区| 此物有八面人人有两片| 免费无遮挡裸体视频| 纯流量卡能插随身wifi吗| 黄色女人牲交| 长腿黑丝高跟| 亚洲国产精品成人综合色| 97超级碰碰碰精品色视频在线观看| 女警被强在线播放| 一卡2卡三卡四卡精品乱码亚洲| 黄色丝袜av网址大全| 禁无遮挡网站| 别揉我奶头~嗯~啊~动态视频| 性色av乱码一区二区三区2| 国内毛片毛片毛片毛片毛片| 成年人黄色毛片网站| 久久久久国产一级毛片高清牌| 男女床上黄色一级片免费看| 国产精品一区二区三区四区久久 | 人人妻人人澡欧美一区二区 | 久久久久久久精品吃奶| 国产熟女午夜一区二区三区| 在线播放国产精品三级| 精品福利观看| 国产人伦9x9x在线观看| 久久国产精品人妻蜜桃| 又黄又粗又硬又大视频| 日韩有码中文字幕| 久久久久国产一级毛片高清牌| 最新在线观看一区二区三区| 欧美一级毛片孕妇| 中文字幕av电影在线播放| 19禁男女啪啪无遮挡网站| 久久狼人影院| 人人妻人人爽人人添夜夜欢视频| 国产精品 国内视频| 亚洲 欧美一区二区三区| 在线视频色国产色| 视频区欧美日本亚洲| 午夜成年电影在线免费观看| 国产一区在线观看成人免费| 午夜老司机福利片| 国产日韩一区二区三区精品不卡| 午夜精品国产一区二区电影| 午夜福利免费观看在线| 脱女人内裤的视频| 两个人免费观看高清视频| 99久久国产精品久久久| 91九色精品人成在线观看| 日韩成人在线观看一区二区三区| 91av网站免费观看| 伦理电影免费视频| 久久久久国产精品人妻aⅴ院| 亚洲精品粉嫩美女一区| 制服人妻中文乱码| 中文亚洲av片在线观看爽| 日韩欧美三级三区| 无人区码免费观看不卡| 成人国产综合亚洲| 在线观看www视频免费| 久久狼人影院| 国产高清有码在线观看视频 | 91麻豆精品激情在线观看国产| 国产精品自产拍在线观看55亚洲| 桃色一区二区三区在线观看| 在线观看舔阴道视频| 涩涩av久久男人的天堂| 中文字幕另类日韩欧美亚洲嫩草| cao死你这个sao货| 久久精品国产清高在天天线| 好男人在线观看高清免费视频 | 亚洲精华国产精华精| 日本在线视频免费播放| 韩国精品一区二区三区| 黄色视频,在线免费观看| 久久人人爽av亚洲精品天堂| 亚洲少妇的诱惑av| 黄频高清免费视频| 9色porny在线观看| 亚洲欧美日韩无卡精品| 久久性视频一级片| 夜夜爽天天搞| 国产一区二区三区综合在线观看| 亚洲avbb在线观看| 狠狠狠狠99中文字幕| 夜夜躁狠狠躁天天躁| 欧美日韩瑟瑟在线播放| 国产av一区二区精品久久| 日本vs欧美在线观看视频| 老司机靠b影院| 国产成人精品在线电影| 人人澡人人妻人| 欧美亚洲日本最大视频资源| 又大又爽又粗| www日本在线高清视频| 99热只有精品国产| 亚洲avbb在线观看| 国产免费av片在线观看野外av| 免费久久久久久久精品成人欧美视频| 国产蜜桃级精品一区二区三区| 国产极品粉嫩免费观看在线| 国产精品影院久久| 欧美色欧美亚洲另类二区 | 最新在线观看一区二区三区| 久久国产乱子伦精品免费另类| 国产精品 国内视频| 国产高清videossex| 男女午夜视频在线观看| 午夜精品久久久久久毛片777| 黄色成人免费大全| 精品免费久久久久久久清纯| 国产亚洲精品一区二区www| 叶爱在线成人免费视频播放| 精品国内亚洲2022精品成人| 亚洲精品粉嫩美女一区| xxx96com| 1024香蕉在线观看| 精品久久久久久久久久免费视频| 亚洲视频免费观看视频| 中文字幕久久专区| 精品久久久久久久毛片微露脸| 咕卡用的链子| 一个人免费在线观看的高清视频| 久久人人爽av亚洲精品天堂| 日韩欧美国产一区二区入口| 亚洲精品在线观看二区| 国产av在哪里看| 一夜夜www| 国产区一区二久久| 久久国产亚洲av麻豆专区| 此物有八面人人有两片| 香蕉丝袜av| 好看av亚洲va欧美ⅴa在| 日韩中文字幕欧美一区二区| 亚洲性夜色夜夜综合| 亚洲第一电影网av| 午夜视频精品福利| 精品国产乱码久久久久久男人| 男人的好看免费观看在线视频 | 久久伊人香网站| 国产高清视频在线播放一区| 国产伦人伦偷精品视频| 欧美中文日本在线观看视频| 久久精品91蜜桃| 母亲3免费完整高清在线观看| 久久久久久亚洲精品国产蜜桃av| 激情在线观看视频在线高清| 天堂√8在线中文| 成人手机av| 欧美 亚洲 国产 日韩一| 人人妻人人爽人人添夜夜欢视频| 日本撒尿小便嘘嘘汇集6| 97人妻天天添夜夜摸| 一二三四社区在线视频社区8| 日本vs欧美在线观看视频| 侵犯人妻中文字幕一二三四区| 视频区欧美日本亚洲| 亚洲人成伊人成综合网2020| 欧美 亚洲 国产 日韩一| АⅤ资源中文在线天堂| 国产精品久久电影中文字幕| 亚洲一区二区三区色噜噜| 俄罗斯特黄特色一大片| 久久久久国产一级毛片高清牌| 黑人操中国人逼视频| 欧美日韩瑟瑟在线播放| 啦啦啦免费观看视频1| 韩国精品一区二区三区| 亚洲av成人av| 国产成人av教育| 757午夜福利合集在线观看| 黄色女人牲交| 久久久久国产一级毛片高清牌| 岛国在线观看网站| 久久人人精品亚洲av| 好男人在线观看高清免费视频 | 亚洲精品中文字幕在线视频| 99在线人妻在线中文字幕| 日本vs欧美在线观看视频| 欧美乱码精品一区二区三区| 免费av毛片视频| 午夜免费激情av| 50天的宝宝边吃奶边哭怎么回事| 涩涩av久久男人的天堂| 91av网站免费观看| 午夜久久久久精精品| 亚洲国产欧美日韩在线播放| 亚洲人成77777在线视频| 国产成人欧美在线观看| 久久精品亚洲熟妇少妇任你| 亚洲av成人av| 国产精品久久电影中文字幕| 老司机深夜福利视频在线观看| 欧美激情高清一区二区三区| 国产精品亚洲美女久久久| 亚洲国产欧美一区二区综合| 国产成人欧美| 午夜福利视频1000在线观看 | 国产精品免费一区二区三区在线| 亚洲国产中文字幕在线视频| 一二三四社区在线视频社区8| 免费在线观看日本一区| av有码第一页| 精品国产乱子伦一区二区三区| 91麻豆精品激情在线观看国产| 性欧美人与动物交配| 亚洲国产精品sss在线观看| 性欧美人与动物交配| 波多野结衣av一区二区av| 国内精品久久久久久久电影| av中文乱码字幕在线| 69精品国产乱码久久久| 亚洲成人久久性| 搡老妇女老女人老熟妇| 岛国在线观看网站| 日韩欧美免费精品| 亚洲五月色婷婷综合| 免费在线观看黄色视频的| 亚洲成人精品中文字幕电影| 久久久国产成人精品二区| 国产精品久久久久久精品电影 | 国产片内射在线| 好男人电影高清在线观看| 午夜福利成人在线免费观看| 午夜a级毛片| 色尼玛亚洲综合影院| 男女做爰动态图高潮gif福利片 | 激情在线观看视频在线高清| 后天国语完整版免费观看| 纯流量卡能插随身wifi吗| 黄色片一级片一级黄色片| av天堂在线播放| 亚洲第一电影网av| 90打野战视频偷拍视频| 国产精品一区二区精品视频观看| 日韩有码中文字幕| 在线观看免费视频网站a站| 99久久综合精品五月天人人| 天天躁狠狠躁夜夜躁狠狠躁| 国产精品一区二区在线不卡| 在线天堂中文资源库| 看片在线看免费视频| avwww免费| 熟女少妇亚洲综合色aaa.| 在线观看免费视频网站a站| 成人三级做爰电影| 亚洲成人久久性| 亚洲欧美激情综合另类| 丁香欧美五月| 色尼玛亚洲综合影院| 黄色女人牲交| 在线国产一区二区在线| 国产伦人伦偷精品视频| 亚洲色图av天堂| 高清黄色对白视频在线免费看| 看片在线看免费视频| 99国产综合亚洲精品| 又黄又粗又硬又大视频| 国产亚洲精品久久久久5区| 女性被躁到高潮视频| 欧美人与性动交α欧美精品济南到| 搡老岳熟女国产| 怎么达到女性高潮| 大型av网站在线播放| 成人av一区二区三区在线看| 亚洲欧美精品综合久久99| av中文乱码字幕在线| 色尼玛亚洲综合影院| 亚洲av电影在线进入| 黑人巨大精品欧美一区二区mp4| 亚洲欧美激情综合另类| 91麻豆精品激情在线观看国产| netflix在线观看网站| 天天添夜夜摸| 色老头精品视频在线观看| 免费在线观看黄色视频的| 国产高清激情床上av| 免费一级毛片在线播放高清视频 | 91老司机精品| 极品教师在线免费播放| 给我免费播放毛片高清在线观看| 一级a爱片免费观看的视频| 中文字幕久久专区| 男女下面插进去视频免费观看| 老熟妇仑乱视频hdxx| 校园春色视频在线观看| 国产av一区二区精品久久| 精品卡一卡二卡四卡免费| 亚洲一码二码三码区别大吗| 九色亚洲精品在线播放| 国产成人精品久久二区二区91| 久久婷婷成人综合色麻豆| 热99re8久久精品国产| 国产又爽黄色视频| 日日爽夜夜爽网站| 精品国产乱子伦一区二区三区| 女性生殖器流出的白浆| 亚洲va日本ⅴa欧美va伊人久久| 老汉色av国产亚洲站长工具| 国产1区2区3区精品| 亚洲精华国产精华精| 亚洲专区国产一区二区| 给我免费播放毛片高清在线观看| 嫩草影视91久久| 国产精品秋霞免费鲁丝片| 国产精品久久久久久精品电影 | 国产日韩一区二区三区精品不卡| 久久精品影院6| 欧美日韩瑟瑟在线播放| 在线观看免费日韩欧美大片| 亚洲av成人不卡在线观看播放网| 成人国语在线视频| 久久精品影院6| 亚洲自偷自拍图片 自拍| 亚洲精品久久国产高清桃花| 国产乱人伦免费视频| 天天添夜夜摸| 精品国内亚洲2022精品成人| 亚洲熟妇熟女久久| 麻豆一二三区av精品| 91成年电影在线观看| 好男人电影高清在线观看| 免费看十八禁软件| 曰老女人黄片| 视频在线观看一区二区三区| 精品国产超薄肉色丝袜足j| bbb黄色大片| 色在线成人网| 久久久久精品国产欧美久久久| 免费高清视频大片| 精品久久久精品久久久| 国产精品 国内视频| 亚洲国产欧美日韩在线播放| 两人在一起打扑克的视频| 日本 av在线| 9色porny在线观看| 69av精品久久久久久| 此物有八面人人有两片| 精品一区二区三区av网在线观看| av网站免费在线观看视频| 国产三级在线视频| 亚洲精品av麻豆狂野| 一本久久中文字幕| 男女下面进入的视频免费午夜 |