包括电子间相关势谐函数的CFPHGLF方法

包括电子间相关势谐函数的ＣＦＰＨＧＬＦ方法王沂轩,邓从豪（山东大学化学院,济南,２５０１００）关键词超球坐标,势谐函数,快速收敛前文［１,２］吸收相关函数（ＣＦ）的思想,与势谐函数－广义Ｌａｇｕｅｒｒｅ函数（ＧＬＦ）方法（ＰＨＧＬＦ）［３,４］相结合...     

超球谐展开的收敛行为He原子的^3S态

利用相关函数－超球谐－广义Ｌａｇｕｅｒｒｅ函数方法，研究Ｈｅ原子＾３Ｓ态波函数向超球谐函数展开的收敛行为。截止于ｌ＝０，１，２的超球谐函数给出的本征能分别与组态相互作用的ｓ－，ｓｐ－，ｓｐｄ－极限能一致。仅用４４个超球谐函数，便得到了与精确的ＨｙｌｌｅｒａａｓＣＩ变分能量小数点后第５位的２＾３Ｓ，３＾３Ｓ的４＾３Ｓ态本征能吻合。     

三电子原子的超球坐标分析和基态能态计算

用超球坐标表示多电子原子的薛定谔方程,从符合Pauli原理的全反称波函数出发,推导出相应的超球径耦合微分方程,其形式与双电子原子的相似。以九维超球谐为投影函数构造S₃群不可约表示的对称基,将超球谐-广义Laguerre展开法(HHGL)推广到Li原子,首次对其Schrdinger方程的精确求解进行了研究,基态能量的计算值与实验值接近。最后,还从波函数出发对提高计算的精度进行了讨论。     

超球坐标下相关函数对锂原子基态本征能收敛行为的影响

在超球坐标下,由3个对称相关函数exp[-α(r₁+r₂+r₃)](α分别为,√-2E/3,2.76和核电荷数z),利用相关函数-超球谐-广义Laguerre函数方法(CFHHGLF)直接求解Li原子的Schrdinger方程,计算基态本征能。从数值来看,α=z结果最好,其次是α=2.76,变参数α较差;而从径向的收敛速度来看,变参数α收敛最快,α=2.76收敛较慢,α=z时本征能只有当对称基超过100时才呈现收敛趋势。结果表明,在超球谐展开方案下只有α=z的相关函数才能得到高精度的本征能量。     

类氦离子Schrdinger方程直接求解的精确势谐方法

给出了一种直接求解三体原子Schrdinger方程的完整势谐展开方法,包括联立的超球径耦合微分方程和广义本征能方程;将其应用于类氦离子,得到了非常精确的基态和低躺激发态的本征能.     

原子Schrǒdinger方程直接求解的相关函数-势谐-广义Laguerre函数方法

引入相关函数,将势谐-广义Laguerre函数(PHGLF)方法发展为相关函数-势谐-广义Laguerre函数(CFPHGLF)方法,应用于He,H~-,Li~+和Be~(2+)等三体体系的n~1S(n=1～4)态.得到了比由PHGLF方法更精确的激发态n~1S(n=2～4)本征能量,且从n=2至4,与Hylleraas组态相互作用(HCI)能量愈来愈近,如He原子的3个激发态的本征能与最精确的HCI之差分别为0.001 827,0.000 666,0.000 376(a. u.);但基态的能量本征值逊于PHGLF方法.结果表明,非势谐函数对基态波函数尚有较显著的贡献,对激发态波函数的贡献较小;在势谐方案下,对基态(1~1S)应考虑电子间的相关,对于激发态重要的是消除电子与核间的歧点.     

类氦离子1,3P激发态Schrodinger方程的直接解

利用相关函数（CF）-超球谐（HH）-广义Laguerre(GLF)方法直接求解类氦离子n1,3P（n=1,2,3）低躺激发态的Schrodinger方程,得氦原子的本征能量分别为-2.13317Eh (13P),-2.12383Eh(11P),-2.05810Eh(23P),-2.05516Eh(21P),-2.03235Eh(33P)和-2.03109Eh(31P),它们与文献值在........第6位上很好地吻合。这说明CFHHGLF方法也适用于类氦离子1,3P激发态。还给出了总角动量L=1的对称超球谐基函数和有关矩阵元的解析式。     

SPK势能面上Cl和H2反应的量子动力学计算

给出了在ＳＰＫ势能面上应用广义牛顿变分法（ＧＮＶＰ）和振幅密度衡量法（ＭＭＡＤ）对于反应系统Ｃｌ＋Ｈ２→ＨＣｌ＋Ｈ态－态之间反应几率的三维量子力学计算。总反应能量从９ｋｃａｌ．ｍｏｌ＾－１至１６ｋｃａｌ．ｍｏｌ＾－１。对于两种方法的计算结果进行了比较。为了同超球谐密耦合方法结果进行比较，还对总反应能量１９ｋｃａｌ．ｍｏｌ＾－１的情况进行了计算和分析。     

地球磁场的离散高斯分析理论

本文从地磁场的高斯(球谐)分析是“三维问题”出发,研究了高斯分析由于测点离散化而产生的一系列理论问题。包括:1.离散球谐函数的正交性;2.高斯分析在测点离散条件下的误差分析,给出了估计高斯系数误差的基本关系式,并由此分析了影响待求高斯系数误差的各种因素;3.测点离散条件下高斯系数的独立性,指出测点的分布和权重选取不满足离散球谐函数正交关系将影响高斯分析的“三维意义”。     

超球坐标下原子、分子体系的变分计算：—氦原子和氢负离子基态

应用超球会标表示氦原子和氢负离子的薛定谔方程，将二电子原子在三维空间中的运动转化为单电子原子在六维空间中受广义库仑力作用的运动，我们给出了六维空间广义角动量算符的本征值与本征函数，并以此本征函数微基构造超球波函数，得到超球径微分方程，以广义Laguerre 多项式表示超球径波函数，运用密度矩阵和线性变分法得到非正交基下超球径波函数满足的久期方程，最后求得能量和波函数，计算结果与精确的计算符合良好。     

The eigenenergies for ~3S excited states of the helium atom with CFPHGLF method

The matrix elements of the correlation function between symmetric potential harmonics were first simplified into the analytical summation of the grand angular momentum. The correlation-function potential-harmonic and generalized Laguerre function method (CFPHGLF) proposed by us recently was then applied to directly solve the Schrodinger equation for n3S(n=2-5) excited states of the helium atom. With only 12 PHs, the convergent eigenenergies of 23S, 33S, 43S and 53S states were 2.17427, 2.06849, 2.03644, 2.02257 Eh, respectively. The errors only were 0.00096, 0.00020, 0.00007, 0.00005 Eh, when compared with the exact Hylleraas variational results respectively.     

Correlation-function potential-harmonic and generalized Laguerre function method for directly solving Schrodinger equations of atoms

The potential-harmonic and generalized Laguerre function method (PHGLF) was modified into the correlation-function potential-harmonic and generalized Laguerre function method (CFPHGLF). The eigenenergies for 21S, 31S and 41S states of helium-like systems from the CFPHGLF are much more accurate than those from the previous PHGLF, but the eigenenergy for the 11S is not as good as that from the PHGLF method. The results indicate that the electron-nucleus cusp plays more important role than the electron-electron cusp and the cluster structure for the loosely bound excited states, and that the electron-electron cusp is absolutely essential for the tightly bound ground state.     

On the accuracy of the potential harmonic functions

The matrix elements of the correlation function between symmetric potential harmonics are simplified into the analytical summations of the grand angular momenta by smartly using the recurrence and coupling relations of the potential harmonics. The correlation-function potential-harmonic and generalized-Laguerre-function method (CFPHGLF) , recently developed by us, is applied to the S states of the helium-like systems for Z = 2 to 9. The results exhibit good convergence with the bases in terms of both the angular and radial directions. The final eigen-energies agree excellently with the best s-limits of the variational configuration interaction (CI) method for the involved low-lying S states. The accuracy of the potential harmonic (PH) expansion scheme is discussed relative to the exact Hylleraas CI results (HCI), and Hartree-Fock results. Moreover, suggestion is given for the future improvement of the PH scheme.     

Excited states of boron isoelectronic series from explicitly correlated wave functions

The ground state and some low-lying excited states arising from the 1s2 2s2p2 configuration of the boron isoelectronic series are studied starting from explicitly correlated multideterminant wave functions. One- and two-body densities in position space have been calculated and different expectation values such as , , , , , and , where r, r12, and R stand for the electron-nucleus, interelectronic, and two electron center of mass coordinates, respectively, have been obtained. The energetic ordering of the excited states and the fulfillment of the Hund's rules is analyzed systematically along the isoelectronic series in terms of the electron-electron and electron-nucleus potential energies. The effects of electronic correlations have been systematically studied by comparing the correlated results with the corresponding noncorrelated ones. All the calculations have been done by using the variational Monte Carlo method.     

Molecular structure calculations: a unified quantum mechanical description of electrons and nuclei using explicitly correlated Gaussian functions and the global vector representation

We elaborate on the theory for the variational solution of the Schro?dinger equation of small atomic and molecular systems without relying on the Born-Oppenheimer paradigm. The all-particle Schro?dinger equation is solved in a numerical procedure using the variational principle, Cartesian coordinates, parameterized explicitly correlated Gaussian functions with polynomial prefactors, and the global vector representation. As a result, non-relativistic energy levels and wave functions of few-particle systems can be obtained for various angular momentum, parity, and spin quantum numbers. A stochastic variational optimization of the basis function parameters facilitates the calculation of accurate energies and wave functions for the ground and some excited rotational-(vibrational-)electronic states of H(2) (+) and H(2), three bound states of the positronium molecule, Ps(2), and the ground and two excited states of the (7)Li atom.     

Non-Born-Oppenheimer variational calculation of the ground-state vibrational spectrum of LiH+

Very accurate, rigorous, variational, non-Born-Oppenheimer (non-BO) calculations have been performed for the fully symmetric, bound states of the LiH(+) ion. These states correspond to the ground and excited vibrational states of LiH(+) in the ground (2)Sigma(+) electronic state. The non-BO wave functions of the states have been expanded in terms of spherical N-particle explicitly correlated Gaussian functions multiplied by even powers of the internuclear distance and 5600 Gaussians were used for each state. The calculations that, to our knowledge, are the most accurate ever performed for a diatomic system with three electrons have yielded six bound states. Average interparticle distances and nucleus-nucleus correlation function plots are presented.     

量子Monte Carlo处理激发态

提出了用于电子激发态的剩余函数变分量子ＭｏｎｔｅＣａｒｌｏ（ＳＦＶＭＣ）方法,已经证明：若激发态的初始波函数与基态的初始波函数属于对称性不同的不可约表示时,该激发态的ＳＦＶＭＣ方法与基态的ＳＦＶＭＣ方法完全相同;若激发态的初始波函数与基态的初始波函数有相同的对称性时,只要对激发态的初始波函数作正交性修正,则其态的ＳＦＶＭＣ方法亦可推到该激发态的情况。文章导出了这第二类激发态的ＳＦＶＭＣ方法的详细计     

Configuration interaction study of the ground and excited states of TiO2 ring structures

Theoretical studies of the ground and lowest excited singlet and triplet states of a series of titanium dioxide ring structures, (TiO(2))(2n), n = 3-9, are reported. Calculations are based on many-electron configuration theory, where energies of states and geometrical structures are determined by variational energy minimization. The lowest energy excited states correspond to excitations from oxygen 2p levels to unoccupied 3d orbitals on titanium. For each ring system, two types of excited state solutions are investigated: those that maintain periodic symmetry for individual orbitals and solutions that allow the symmetry to be broken. The latter solutions which correspond to localized states or excitons are found to be significantly lower in energy than the symmetric solutions. We compare the vertical excitation energy of these well-defined geometrical structures with size effects reported in experimental studies.     

分子离子NO+激发态的势能函数

100公里以上的大气分子离子主要为NO~+。它的辐射特点及它与电子、原子或分子的相互作用,对于理解大气的化学过程具有特别重要的意义。为了研究这些过程,确定NO~+分子离子基态及其各个激发态的分子势能函数是非常重要的。精确的X~1Σ~+,A~1Ⅱ和a~3∑~+势能曲线已发表;基于光电子谱的研究发现了NO~+的其它激发态,但对于这些激发态的研究尤其是势能函数的研究不多。本文研究并导出NO~+的基态和10个激发态的势能函数。