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...     

势谐函数展开方法下的快速收敛研究

势谐函数展开方法下的快速收敛研究王沂轩,邓从豪（山东大学化学院,济南,２５０１００）关键词超球坐标,势谐函数,广义Ｌａｇｕｅｒｒｅ函数（ＧＬＦ）超球坐标表象中的势谐（ＰＨ）展开方法是减少超球径耦合微分方程的数目和广义本征矩阵的维数,以解决高维超球谐（...     

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.     

PHGLF calculations of heliumlike three-body systems

By expanding the atomic wave function in potential harmonics (PH ) and generalized Laguerre functions (GLF ), we derived the recurrence relation of the expansion coefficients and then performed calculations for four He-like three-body systems. In comparison with the complete set calculation (HHGLF ), we find that the errors in eigenenergies introduced by the PHGLF scheme are approximately 0.00033, 0.00025, 0.00022, and 0.00021 au for the ground-state H⁻, He, Li⁺, and Be²⁺ systems, respectively, and that for excited states the error is slightly higher and decreases with the grand angular momentum. The present results indicate that the PHGLF scheme is such an efficient means to reduce the degeneracy of the hyperspherical harmonics (HH ) for three-body problems that it can give more satisfactory results than can the HHGLF method with much smaller eigenmatrices. © 1995 John Wiley & Sons, Inc.     

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.     

Performance of Hyperspherical Harmonic Expansionon the Low-lying Pand D States of Helium Atom

IntroductionThe hyperspherical coordinate and corresponding hyperspherical harmonics (HH) havebeen applied to various areas of physics, such as three--body scattering, nuclear and atomicphysics as well as quantum chemistry['J, since the 1950's. The HH--generalized Laguerrefunction method (referred to as HHGLF) [2'3] and several modified versions developed by Dengand his coworkers, such as the potential harmonic--generalized Laguerre function(PHGLF )L'], and the correlation fun ct ion --…     

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。     

Exact Radial Solution of the Non-relativistic Schr?dinger Equation for the Helium Atom with the Potential Harmonic Method

We proposed a simple potential harmonic(PH) scheme for calculating the non-relativistic radial correlation energies of atomic systems.The scheme was applied to the low-lying n1S(n=1,2) and n3S(n=2,3) states of the helium atom.The results exhibit a very stable convergence characterization in both the angular and radial directions with PH and generalized Laguerre functions(GLF) respectively,even though the method is non-variational one.The ninth significant figure of the non-relativistic radial energy(NRE) calculated for the ground state exactly agrees with that of the most accurate literature data from the modified configuration interaction method.The convergent NRE′s for the excited states 21S,23S and 33S with the similar accuracy were also obtained.     

最弱受约束电子势模型理论下使用双广义拉盖尔多项式计算氦原子基态能量

计算了在最弱受约束电子势模型理论下使用双广义拉盖尔多项式的氦原子基态能量. 给出了氦原子基态能量期望值的表达式. 通过搜索它的极小, 找到了氦原子基态能量的极小值. 将我们的结果与双灼(zeta)函数的Hartree-Fock(HF)方法的结果进行比较后发现, 我们的结果略优于双灼(zeta)函数的Hartree-Fock(HF)方法的结果. 如此表明线性组合技术可以应用在最弱受约束电子势模型理论中.     

Improved scheme to solve the atomic Schrödinger equation in hyperspherical coordinates

An improved scheme to accelerate the convergence in the calculations of N-electron atoms, which is based on the exact method we proposed before in hyperspherical coordinates, is presented. The factors influencing the rate of convergence in both parts of expansions in wave function with the hyperspherical harmonics (HHs) of hyperangles and the generalized Laguerre polynomials (GLPs) of hyperradius were investigated. A reselected asymptotic term was introduced by including more structural features in it to accelerate the convergence in the expansion part with the HHs, and a transformation of the hyperradius was used to keep the convergence going properly in the expansion part with the GLPs. Calculations with this scheme for the helium atom were given and compared with some other ones. More accurate results were obtained by considering a simple cusp parameter. © 1996 John Wiley & Sons, Inc.     

Analytical treatment of the oscillating Yukawa potential

Using a suitable Laguerre basis set that ensures a tridiagonal matrix representation of the reference Hamiltonian, we were able to evaluate in closed form the matrix elements of the generalized Yukawa potential with a complex screening parameter. This enabled us to treat analytically both the cosine and sine-like Yukawa potentials on equal footing and compute their bound states spectrum as the eigenvalues of the associated analytical matrix representing their Hamiltonians. Finally we used a carefully designed complex scaling method to evaluate the resonance energies and compared our results satisfactorily with those obtained in the literature for the cosine-like Yukawa potential.     

Orbital relaxation in the Rydberg series of the lithium atom. An excited state SCF calculation

Self-consistent-field (SCF ) calculations for a series of Rydberg states (1s²ns)²S of the Li atom are performed using the generalized Brillouin theorem (GBT) method. The calculated energy is a proper upper bound to the excited state energy. The SCF term values of the Rydberg states are almost the same as those of the frozen-core approximation ones. The orbital behavior shows that the core is slightly expanded by the penetration of the Rydberg orbitals, and the higher Rydberg orbitals can be very well represented by the modified hydrogen-like orbitals.     

Rapidly converging threshold value calculations in screened coulomb potential systems: Critical values of the screening parameter for the Yukawa case

In this work, the threshold values of the screening parameter for Yukawa potential systems are obtained by means of a basis set constructed from Laguerre type functions. The Laguerre basis set is modified by an appropriately chosen extra function in order to imitate the true behaviour of the solutions at the boundary points. The method used is a variational scheme and the numerical results are accurate to thirty decimal points. A scaling parameter is also inserted into the structure of the basis functions, the optimized values of which accelerate the convergence. The main goal of this paper is to develop a method which enables us to calculate the threshold values of the screening parameter for low-lying states. The method is quite general and can be extended to all systems whose potentials decay exponentially when the radial variable goes to infinity.     

Theorie des Orbitals Moleculaires: Recherche d'une Base STO Homogene pour les Atomes Alcalins et Leur Serie Isoelectronique

A method of direct treatment of valence electrons is proposed for the various ²S, ²P,²D, ²F states of the lithium, sodium, and potassium atoms and the corresponding isoelectronic ions. The function describing the outer electron, which is orthogonal to the wave functions of the electrons in the core, is given as a linear combination of generalized Laguerre functions, with effective charges Q = 1, 2 …? equal to the charges of the core. A simpler analytical STO function, with non-integer principal quantum number n is then proposed.     

Study of CO adsorption on perfect and defective pyrite(100)surfaces by density functional theory

First-principles calculations based on density functional theory (DFT) and the generalized gradient approximation (GGA) have been used to study the adsorption of CO molecule on the perfect and defective FeS 2 (100) surfaces. The defective Fe 2 S(100) surfaces are caused by sulfur deficiencies. Slab geometry and periodic boundary conditions are employed with partial relaxations of atom positions in calculations. Two molecular orientations, Cand O-down, at various distinct sites have been considered. Total energy calculations indicated that no matter on perfect or deficient surfaces, the Fe position is relatively more favored than the S site with the predicted binding energies of 120.8 kJ/mol and 140.8 kJ/mol, respectively. Moreover, CO was found to be bound to Fe atom in vertical configuration. The analysis of density of states and vibrational frequencies before and after adsorption showed clear changes of the C–O bond.     

Complex self-consistent field and multireference single- and double-excitation configuration interaction calculations for the 2Pi(g) resonance state of N2-

Self-consistent field and multireference single- and double-excitation configuration interaction calculations employing the complex basis function technique are carried out for the (2)Pi(g) resonance state of the N(2) (-) molecule as well as several other anionic resonance states in the neighboring energy region. The results of calculations employing the same method for the (1)S (2s(2)) state of the He atom and the (1)Sigma(g) (+) (sigma(u) (2)) state of the H(2) molecule are found to be in good agreement with those of earlier work. The present theoretical treatment has succeeded for the first time in satisfying the rigorous criterion of the complex variational principle in computing the N(2) (-) resonance states, namely, a cusp in the plots of real versus imaginary components of the corresponding complex energies has been located at each internuclear distance. On this basis, it is found that the open-shell orbital in the lowest-energy adiabatic N(2) (-) resonance state of (2)Pi(g) symmetry changes its character from quite compact at large internuclear distance to relatively diffuse for r<2.3a(0). This is in contrast to all previous theoretical treatments of this system that have not rigorously satisfied the complex variational principle in their determination of this wave function.     

Solutions of the Schr?dinger equation in the tridiagonal representation with the noncentral electric dipole plus a novel angle-dependent component

A noncentral ring-shaped potential is proposed in which the noncentral electric dipole and a novel angle-dependent component are included, the radial part is selected as the Coulomb potential or the harmonic oscillator potential. The exact solution of the Schr?dinger equation with this potential is investigated by working in a complete square integrable basis that supports a tridiagonal matrix representation of the wave operator. The resulting three-term recursion relation for the expansion coefficients of the wavefunctions (both angular and radial) are presented. The angular/radial wavefunction is written in terms of the Jacobi/Laguerre polynomials. The discrete spectrum of the bound states is obtained by diagonalization of the radial recursion relation.     

Solving the Schrödinger and Dirac equations of hydrogen molecular ion accurately by the free iterative complement interaction method

The nonrelativistic Schr?dinger equation and the relativistic four-component Dirac equation of H(2) (+) were solved accurately in an analytical expansion form by the free iterative complement interaction (ICI) method combined with the variational principle. In the nonrelativistic case, we compared the free ICI wave function with the so-called "exact" wave function as two different expansions converging to the unique exact wave function and found that the free ICI method is much more efficient than the exact method. In the relativistic case, we first used the inverse Hamiltonian to guarantee Ritz-type variational principle and obtained accurate result. We also showed that the ordinary variational calculation also gives a nice convergence when the g function is appropriately chosen, since then the free ICI calculation guarantees a correct relationship between the large and small components of each adjacent order, which we call ICI balance. This is the first application of the relativistic free ICI method to molecule. We calculated both ground and excited states in good convergence, and not only the upper bound but also the lower bound of the ground-state energy. The error bound analysis has assured that the present result is highly accurate.     

Communication: Second-order multireference perturbation theory with explicit correlation: CASPT2-F12

An explicitly correlated complete active space second-order perturbation (CASPT2-F12) method is presented which strongly accelerates the convergence of CASPT2 energies and properties with respect to the basis set size. A Slater-type geminal function is employed as a correlation factor to represent the electron-electron cusp of the wave function. The explicitly correlated terms in the wave function are internally contracted. The required density matrix elements and coupling coefficients are the same as in conventional CASPT2, and the additional computational effort for the F12 correction is small. The CASPT2-F12 method is applied to the singlet-triplet splitting of methylene, the dissociation energy of ozone, and low-lying excited states of pyrrole.     

Decay rates of inner-valence excitations in noble gas atoms

A Fano - algebraic diagrammatic construction - Stieltjes method has been recently developed for ab initio calculations of nonradiative decay rates [V. Averbukh and L. S. Cederbaum, J. Chem. Phys. 123, 204107 (2005)] of singly ionized states. In the present work this method is generalized for the case of electronic decay of excited states. The decay widths of autoionizing inner-valence-excited states of Ne, Ar, and Kr are calculated. Apart from the lowest excitation of Kr, they are found to be in good to excellent agreement with the experimental values. Comparison with the other theoretical studies shows that in many cases the new method performs better than the previously available techniques.