Effect of different subsets on convergence patterns of hyperspherical harmonic expansion for the S states of the helium atom

Effects of different subsets on convergence patterns of hyperspherical harmonic (HH) expansions for the low-lying ¹S and ³S states of the helium atom have been investigated with the correlation-function-hyperspherical-harmonic-generalized-Laguerre-function (CFHHGLF) method by successively introducing HH subsets with the fixed three-dimensional angular momentums (l) into the atomic wave functions. The eigenenergies given by the HH subsets of l=0, 1, 2, and 3 are in good agreement with the best s-, sp-, spd-, and spdf limits of variational configuration interaction (CI) calculations, respectively. The final eigenenergies of the ground state as well as the examined low-lying excited ¹S and ³S states are quite close to the exact Hylleraas CI (HCI) values at the sixth decimal place. Moreover, l=0 and l≠0 expansion results also tell us that it is not necessary to take into account too many HHs at the given l, especially for higher l, and that the more the absolute electron correlation energies the bigger l it takes to obtain precise eigenenergies. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 64 : 661–668, 1997     

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。     

The effect of electron correlation on one-electron properties in the 2 3S and 2 1S excited states of the helium atom

The density difference ΔD(r₁), defined as the difference between the exact and the Hartree-Fock radial density functions is reported for the 2 ³S and the 2 ¹S excited states of the helium atom. The density differences in both cases resemble the inverse of the helium ground-state density difference, but are more long-ranged due to the diffuseness of the excited states. The effect of electron correlation on one-electron distributions is also demonstrated by comparison of (r₁ⁿ) expectation values and several indices of the relative sizes of atoms. Electron correlation leads to a contraction of the Hartree-Fock electron clouds.     

Integral evaluation in the case of a helium atom positioned off‐center in a spherical box

A variational approach to the problem of multielectron atoms placed off‐center in a spherical box leads to the difficult evaluation of electron repulsion integrals in an environment where spherical symmetry is no longer present. A technique for the evaluation of the electron repulsion integrals generated by this situation is developed and tested for the case of a helium atom placed off‐center in a spherical box. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 459–467, 1999     

Solving the electron and electron-nuclear Schrodinger equations for the excited states of helium atom with the free iterative-complement-interaction method

Very accurate variational calculations with the free iterative-complement-interaction (ICI) method for solving the Schrodinger equation were performed for the 1sNs singlet and triplet excited states of helium atom up to N=24. This is the first extensive applications of the free ICI method to the calculations of excited states to very high levels. We performed the calculations with the fixed-nucleus Hamiltonian and moving-nucleus Hamiltonian. The latter case is the Schrodinger equation for the electron-nuclear Hamiltonian and includes the quantum effect of nuclear motion. This solution corresponds to the nonrelativistic limit and reproduced the experimental values up to five decimal figures. The small differences from the experimental values are not at all the theoretical errors but represent the physical effects that are not included in the present calculations, such as relativistic effect, quantum electrodynamic effect, and even the experimental errors. The present calculations constitute a small step toward the accurately predictive quantum chemistry.     

One and two body densities for excited states of the helium confined atom

A study of the first excited states of the helium atom confined under impenetrable spherical walls is carried out. Both single particle and two body, intracule and extracule, densities are constructed. Crossing levels and Hund's rule are analyzed in terms of the contribution to the total energy from kinetic, electron–nucleus, and electron–electron energies. A study about the behavior of the single particle and two body densities is carried out. The Multiconfiguration Parameterized Optimized Effective Potential method is employed with a cut-off factor to account for Dirichlet boundary conditions. Single particle density is analytically constructed whereas the Monte Carlo algorithm is used to calculate two body densities.     

The helium atom in a semi-infinite space limited by a paraboloidal boundary

The ground state of the helium atom in a semi-infinite space limited by a paraboloidal boundary, with the nucleus at the focus, is evaluated variationally. The pressure and electric dipole moment of the semiconfined atom are evaluated as functions of the position of the boundary. Estimates of the boundary positions and pressure variations at the successive ionization thresholds are established. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 269–275, 1997     

Multi-configuration electron-hole potential method for excited states

The recently proposed electron-hole potential (EHP) method for excited states is extended to the multi-configurational case. The variation equation is solved using the quadratic convergence method. The EHP methods are shown to be approximations to the complete singly excited configuration interaction (CSECI) in the variational sense. Extended Brillouin theorems are proved for the EHP methods. The excitation energies and wave functions obtained by one and two configurational EHP methods agree well with those of the CSECI method. The EHP methods have clear advantage in the computer time requirement over the CI method and are especially suited for a calculation of approximate excited states of large molecules. The EHP methods are applicable to excited states which belong to the same irreducible representation as the ground state.     

Calculations of D‐wave bound states and resonance states of the screened helium atom using correlated exponential wave functions

We have investigated the effects of screened Coulomb (Yukawa) potentials on the bound ^1,3D states and the doubly excited ^1,3 D^e resonance states of helium atom using highly correlated exponential basis functions. The Density of resonance states are calculated using stabilization method. Highly correlated exponential basis functions are used to consider the correlation effect between the charged particles. A total of 18 resonances (nine each for ¹ D^e and ³ D^e states) below the n = 2 He ⁺ threshold has been calculated. For each spin states, this includes four members in the 2pnp series, three members in the 2snd series, and two members in 2pnf series. The resonance energies and widths for various screening parameters ranging from infinity to a small value for these ^1,3 D^e resonance states are reported along with the bound‐excited 1s3d ^1,3 D state energies. Overall behavior of the spectral profile of 1s3d ¹D state of helium atom due to electron‐electron and electron‐nucleus screening are also presented. Accurate resonance energies and widths are also reported for He in vacuum. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Power series with rational coefficients for two-electron atom energies

The method of o(4, 2) operator replacements is generalized. As a result, the series whose limiting values when the variable goes to +∞ should correspond to the two-electron atom energies now have rational coefficients. This generalization allows one also to compute the series for the case of singlet S symmetry, a case which could not be considered in the previous original formulation of the method. Series with rational coefficients for the helium singlet and triplet S ground-state energy are calculated up to order 41 and 45, respectively. Moreover, symbolic computations also allow one to give the first few coefficients of these series for arbitrary values of the nuclear charge Z. Finally, a new method for analytic continuation to the limit +∞ is presented for the energies of the helium singlet and triplet ground states. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63 : 1079–1089, 1997     

Convergence properties of Fock's expansion for S-state eigenfunctions of the helium atom

It is proved by functional analytic methods that for S-state solutions of Schrödinger's equation for the helium atom, Fock's expansion in powers of R ^1/2 and R ln R, where R is the hyperspherical radius r ₁ ² +r ₂ ² , converges pointwise for all R, thereby generalising a result of Macek that the expansion converges in the mean for all R<1/2. It is shown that for any value (even complex) of the energy E, Schrödinger's equation, considered as a partial differential equation with no boundary condition at R=, has infinitely many solutions representable by an expansion of the type proposed by Fock. Some of the open problems are discussed in determining whether for E in the point spectrum of the atomic Hamiltonian the physical eigenfunction _E, which has exponential decay as R , is representable by Fock's expansion.     

Evaluation of Coulomb and exchange integrals for higher excited states of helium atom by using spherical harmonics series

In this work we study the higher excited states of Helium Atom. The purpose is to evaluate Coulomb and exchange integral via spherical harmonics series. The Coulomb and exchange integrals energy shift is evaluated up to sixth order. This is the energy when the atom is perturbed by Coulomb potential between electrons. The energy levels obtained from both integrals are in agreement with the experimental data. For highly-excited states, the calculated energy approaches ?54.416?eV, in agreement with the graphical results from the book by Powell and Crasemann [1].     

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.     

A potential harmonic method for the three-body coulomb problem

A potential harmonic method that is suitable for the three-body coulomb systems is presented. This method is applied to solve the three-body Schroedinger equations for He and e ⁺ e ⁻ e ⁺ directly, and the calculations yield very good results for the energy. For example, we obtain a ground-state energy of −0.26181 hartrees for e ⁺ e ⁻ e ⁺, and −2.90300 hartrees for He with finite nuclear mass, in good agreement with the exact values of −0.26200 hartrees and −2.90330 hartrees. Compared with the full-set calculations, the errors in the total energy for ground and excited states of e ⁺ e ⁻ e ⁺ are very small, around −0.0001 hartrees. We conclude that the present method is one of the best PH methods for the three-body coulomb problem. Received: 5 September 1996 / Accepted: 14 July 1997     

Ritz variational method for the high‐lying nonautoionizing doubly excited 1,3Fe states of two‐electron atoms

Energy eigenvalues of nonautoionizing doubly excited states originating from 2pnf ( ) configuration of two‐electron atoms have been calculated by expanding the basis set in explicitly correlated Hylleraas coordinates under the framework of Ritz variational method. A detailed discussion on the evaluation of correlated basis integrals is given. The energy eigenvalues of a number of these doubly excited states are being reported for the first time especially for the high lying states. The effective quantum numbers ( ) for the states mentioned above have been calculated by using the theory of quantum defect.     

Correlated wave functions for the ground and some excited states of the iron atom

We study the states arising from the [Ar]4s(2)3d6 and [Ar]4s(1)3d7 configurations of iron atom with explicitly correlated wave functions. The variational wave function is the product of the Jastrow correlation factor times a model function obtained within the parametrized optimized effective potential framework. A systematic analysis of the dependence of both the effective potential and the correlation factor on the configuration and on the term is carried out. The ground state of both, the cation, Fe+, and anion, Fe-, are calculated with correlated wave functions and the ionization potential and the electron affinity are obtained.